Macaulay duality and its geometry

S. L. Kleiman]Steven L. Kleiman
Dept. of Math., 2-172 MIT

77 Mass. Ave.

Cambridge, MA 02139, USA

J. O. Kleppe]Jan O. Kleppe
Faculty of Technology, Art and Design

Oslo Metropolitan University

St. Olavs Plass

NO-0130 Oslo, Norway

Both authors are deeply grateful to Tony Iarrobino for many
discussions in person and in email about the content of this article and
for providing many pertinent references, in particular about smoothable
quotients and elementary components, which are discussed in
The second author also thanks
Arvid Siqveland for discussions about deformations of modules.
Finally, both authors are very grateful for the referee's many
thoughtful suggestions for improving the paper.

[2010]13H10, 13C40, 14M05, 14C05

§ ,

Macaulay Duality, between quotients of a polynomial ring over a field,
annihilated by powers of the variables, and finitely generated
submodules of the ring's graded dual, is generalized over any Noetherian
ring, and used to provide isomorphisms between the subschemes of the
Hilbert scheme parameterizing various sorts of these quotients, and the
corresponding subschemes of the Quot scheme of the dual.  Thus notably
the locus of recursively compressed algebras of permissible socle type
is proved to be covered by open subschemes, each one isomorphic to an
open subscheme of a certain affine space.  Moreover, the polynomial
variables are weighted, the polynomial ring is replaced by a graded
module, and attention is paid to induced filtrations and gradings.
Furthermore, a similar theory is developed for (relatively) maximal
quotients of a graded Gorenstein Artinian algebra.

0pt plus 0pt minus 3pt

§ INTRODUCTION

This introduction surveys previous work by many, explains our approach,
and summarizes our advances.

In 1972, Iarrobino <cit.> initiated the study of reducible
punctual Hilbert schemes.  Since then, many researchers have carried on,
studying various loci of thick points, often with common support.  Thus
certain subtle geometric issues arose, for which the present article
aims to give the first proper treatment in natural generality.

A key tool has been Macaulay duality.  It provides a means to transform
loci in the local punctual Hilbert scheme into more tractable loci in a
related Quot scheme.  However, previous treatments of Macaulay duality
are inadequate to the job, since (with a few limited exceptions) they're
developed over a base field, and cannot properly handle flat families.
Thus in the present article, our first goal is to develop a suitable
version (<ref>) of Macaulay duality over any Noetherian base

Our main goal stems from Iarrobino and Emsalem's 1978 landmark paper
<cit.>.  They fixed a polynomial ring $P$ in $r$ variables over an
algebraically closed field $k$.  Notably, they considered local
Gorenstein quotients $C$ of $P$ with Hilbert vector $(1,r,r,1)$.  Using
Macaulay duality, they found that the homogeneous $C$ fill an open
subset of the projective space on the cubic forms, and that over it, the
adically filtered $C$ fill an affine-space bundle of fiber dimension
$\binom r2$, whose structure map takes $C$ to its associated graded

Our main goal is to treat Iarrobino's remarkable 1984 generalization in
<cit.> of the preceding results.  The original $C$ are generalized
by the compressed $C$; these are the $C$ whose length is maximal
compressed@compressed $C$
among quotients that are annihilated by a power of each variable and
are of fixed “permissible” socle type.  Iarrobino discovered and
Fröberg and Laksov reproved, see (<ref>) below, that,
strikingly, the length of a compressed $C$ can be computed a priori from
its socle type.  This basic result grew out of <cit.>*Thm. 3.31,
p. 173; related work is discussed in the note at the bottom of p. 350
in <cit.>.  This result is assumed in the present paper.  Rather,
we give the geometry a proper treatment.

We fix two flaws in <cit.>, beyond the lack of use of a suitable
Macaulay duality.  The first concerns the locus (in the Hilbert scheme)
of compressed homogeneous $C$.  It is not true, as asserted in
<cit.>*Prop. 3.6, p. 356, that every $C$ remains compressed
under recursion on the diameter of the socle type; see (<ref>).
So to reach our main goal, we work with recursively compressed
recursively compressed
$C$; these are the compressed $C$ that do remain compressed under
recursion.  It is their locus that is covered by open subschemes of
affine space; see (<ref>)(2).  However, we show in
(<ref>) on the basis of the work of Iarrobino and of
Fröberg and Laksov mentioned above that a general $C$ of socle type
bounded by $\mb t$ is, in fact, recursively compressed of socle type
$\mb t$.

The second flaw is the poor justification of the description in
<cit.>*Prop. 3.11, p. 359 of the retraction from the locus of
filtered $C$ to that of homogeneous $C$.  That justification proceeds by
induction on the minimal number of generators of the dual $C^*$.
However, adding new generators, one after the other, leads not to a
single quotient $C$, but to a flag of quotients.  Therefore, we provide
a noninductive proof in (<ref>).

The theory of compressed Artinian local rings $C$ has been extended by
many authors in various directions.  In his 1985 thesis
<cit.>, Miri studied such $C$ that are quotients of direct
sums of shifts $B := \bigoplus_q P(q)^{\oplus \mb t(q)}$.  However, in
his 1993 paper <cit.>, which is based on <cit.>, he
took $B := P^{\oplus t}$ with $t\ge 1$.  He focused on the $C$ with
cyclic duals of maximal length of socle type $\mb t$.  A stronger
version of his existence theorem <cit.>*Prp. 2.8, p. 2846
with $B := \bigoplus_q P(q)^{\oplus \mb t(q)}$ is given in
(<ref>)(1) below with a simpler proof.  He also described the
varieties of graded $C$ and of filtered $C$ in
<cit.>*Thm. 3.1, p. 2853; his descriptions are special
cases of (<ref>)(2) and (<ref>).

In 2000, Boij studied graded level quotients $C$ of $B := P(s)^{\oplus
t}$ for $s,\,t \ge 1$.  A version of of his existence theorem is given
for comparison with Miri's in (<ref>)(2).

In 2001, Cho and Iarrobino <cit.> initiated the study of
(relatively) compressed quotients $C$ of a proper quotient $A$ of $P$;
these are the $C$ whose length is maximal among quotients of fixed
“permissible” socle type.  They gave examples of $A$ where this
maximal length is not the a priori bound computed from the socle type and
the Hilbert function of $A$.  A few years later, they and others gave
additional examples.  There's more discussion of this matter near the
end of the introduction.

In 2014, Rossi and Şega <cit.> studied the Poincaré
series of finitely generated modules over compressed local Gorenstein
$C$; they treated mixed characteristic $C$, a notable advance.  In 2018,
Kustin, Şega, and Vraciu <cit.> extended that work to
compressed local $C$ of arbitrary permissible socle type.  Both papers
recall that such $C$ exist over an infinite field, but otherwise, they
study such a $C$, assuming $C$ exists.  However, since a mixed
characteristic $C$ isn't flat over any base ring $k$, such a $C$ isn't
covered by our theory.

In our work, $P$ is replaced by a finitely generated, graded algebra $A
:= \bigoplus_{n\ge0}A_n$ over an arbitrary Noetherian base ring $k$;
from Section 4 on, $A_0 = k$.  Further, $C$ is now a quotient of a
finitely generated graded $A$-module $B = \bigoplus_{n\ge 0}B_n$ with
$B_n$ a locally free $k$-module of finite rank $\mb b(n)$.  For example,
$k$ could be a field, $A$ a weighted polynomial ring, and $B =
A$; see (<ref>) and (<ref>), which illustrate an
advantage gained from a nonstandard weighting.  However, unless
explicitly stated otherwise, a polynomi ring has the standard grading,
in which all the variables have weight 1.  Furthermore, it may well be
useful to generalize our work to the case where $B$ is filtered,
especially in the linkage results (<ref>) and (<ref>) when
$C$ is filtered, but not graded.

In general, the $C$ of interest are locally free $k$-modules of finite
rank.  Below, such a $C$ is termed $k$-Artinian.  Of course, if
$k$ isn't Artinian, then neither is $C$.  However, $C$ can be viewed as
providing a flat family of Artinian quotients.

To treat the various loci $L$ of Artinian quotients and the maps between
them, we use the rudiments of Grothendieck's theory of representable
functors.  The first job then is to take a $k$-scheme $T$ of finite type
and to identify the $T$-points of $L$, namely, the $k$-maps $\tau\:T\to
L$.  Below, each $L$ of interest turns out to parameterize a universal
family $\mc L$ of quotients.  Then $\tau^*\mc L$ is a family
parameterized by $T$; conversely, every family parameterized by $T$ is
obtained via pullback along a unique $\tau$.  Thus the $T$-points $\tau$
are given by certain quotients $\mc C$ on $T$. The first job is to
identify, a priori, these $\mc C$.  Doing so means generalizing the
definition of the locus $L$ in terms of $T$ and $\mc C$.  Here
$T$-flatness plays a central role.  Further, it suffices to use affine
$T$.  The second job is to give $L$ a scheme structure and to construct
the universal family $\mc L$; in other words, the job is to represent
the functor of these $\mc C$.

To do the second job, we use Grothendieck's theory of flattening
flattening stratification
(as it's named in <cit.>*Lect. 8).  The key
tool is Grothendieck's Lem. 3.6 on p. 221-15 in <cit.>: given a
coherent sheaf $\mc F$ on a locally Noetherian scheme $Q$ and an integer
$r$, there's a (unique) subscheme $Z$ of $Q$ such that a map $\tau\:
T\to Q$ factors through $Z$ iff $\tau^*\mc F$ is locally free of rank
$r$.  (To prove it, let $F_r$ be the closed subscheme of $Q$ defined by
the $(r-1)$st Fitting Ideal of $\mc F$, and set $Z = F_r - F_{r-1}$.)
Historically speaking, flattening stratification was introduced into the
study of loci of Artinian algebras by the second author in <cit.>
and <cit.>.

For example, filter $B$ by $F^nB := \bigoplus_{p\ge n}B_p$, and consider
the locus of filtered, $k$-Artinian quotients of $B$ with given Hilbert
function $\mb h$.  What's the value of the associated functor at $T :=
\Spec(K)$ with $K$ a $k$-algebra?  It's the set of $A\ox_kK$-quotients
$C$ of $B\ox_kK$ with the induced filtration such that $F^nC/F^{n+1}C$
is locally free over $K$ of rank $\mb h(n)$ for all $n$.  In
(<ref>), we prove these sets form a functor.

Let's represent this functor.  Set $\ell := \sum_p\mb h(p)$ and $X :=
\Spec(A)$.  Denote the $\mc O_X$-module associated to $B$ by $\wt{B}$.
Set $\smash{Q := \Quot_{\wt{B}/X/k}^\ell}$, and denote by $\mc Q$ the
quotient of $B_Q$ whose associated $\mc O_{X_Q}$-module is the universal
quotient of $\wt B_Q$.  Given $q$, Grothendieck's lemma yields a
subscheme $Q_q$ of $Q$ such that a map $\tau\:T\to Q$ factors through
$Q_q$ iff $\tau^*(\mc Q/F^{q}\mc Q) := (\mc Q/F^{q}\mc Q)\ox K$ is
locally free of rank $\sum_{p < q}\mb h(p)$.  Then the functor is
represented by $\bb F\Psi_B^{\mb h} := \bigcap_{q} Q_q$ and the pullback
FbbPsi@$\bb F\Psi_B^{\mb h}$
of $\mc Q$; see (<ref>).  Note it's not enough to render all
$\tau^*(F^q\mc Q/F^{q+1}\mc Q)$ flat, as possibly $\tau^*F^{q}\mc Q \neq
F^{q}\tau^*\mc Q$, whereas $\tau^*(\mc Q/F^{q}\mc Q) = \tau^*\mc
Q\big/F^{q}\tau^*\mc Q$; a simple example was given by Granger
<cit.>*p. 3, top.

Next, consider the sublocus of homogeneous quotients.  Its subfunctor
has, as value at $T$, the subset of $C = \bigoplus_p C_p$ with $C_p$
locally free of rank $\mb h(p)$ for all $p$.  To represent it, set $Z :=
\bb F\Psi_{B}^{\mb h}$.  Define $\mc S$ by $\mc Q =: \wt B_Q/\mc S$.
Consider the pullback $\mc Q_Z$ of $\mc Q$ and its associated graded
quotient $G_\bu (\mc Q_Z)$.  Then $\mc Q_Z = \wt B_Z/\mc S_Z$, and
$G_\bu (\mc Q_Z) = \wt B_Z/G_\bu(\mc S_Z)$.  Set $\mc R := \wt B_Z/(\mc
S_Z + G_\bu(\mc S_Z))$.  Grothendieck's lemma yields a subscheme $\bb
H\Psi_{ B}^{\mb h}$ of $Z$ such that a map $T\to Z$ factors through $\bb
H\Psi_{ B}^{\mb h}$ iff $\mc R_T$ is locally free of rank $\ell$; so iff
both $\mc Q_T \onto \mc R_T$ and $G_\bu(\mc Q_T) \onto \mc R_T$ are
isomorphisms, see (<ref>).  Thus the subfunctor is represented by
$\bb H\Psi_{ B}^{\mb h}$ and the pullback of $\mc Q$.
HbbPsi@$\bb H\Psi_B^{\mb h}$

Alternatively, form the maps $\beta_n\: (\wt B_Q)_n \to \mc Q$.
Grothendieck's lemma yields a subscheme $Y_n$ of $Q$ such that a map
$\tau\:T\to Q$ factors through $Y_n$ iff $\tau^*\Cok(\beta_n)$ is
locally free of rank $\ell - \mb h(n)$.  Then the subfunctor is
represented by $\bb H\Psi_{ B}^{\mb h} := \bigcap_{n} Y_n$ and the
pullback of $\mc Q$; see (<ref>) again.  Note it's not enough to
render $\Ker(\beta_n)$ locally free, as forming a kernel does not always
commute with pullback, whereas forming a cokernel does.

The previous four paragraphs fix and generalize the second author's 1998
proof of <cit.>*Thm. 1.1, p. 609 and his 2006 proof of
<cit.>*Prp. 9, p. 3145.  (Note that $\bb H\Psi_B^h$ is
denoted by ${\rm GradAlg}(h)$ in <cit.> and <cit.>.)  The
particular case of homogeneous quotients also results from the 2004
general theory of Haiman and Sturmfels, see <cit.>*6.2, p. 766,
but both of our proofs above are simpler and more direct.

There are two natural maps: a closed embedding $\bb H\Psi_{ B}^{\mb h}
\into \bb F\Psi_{ B}^{\mb h}$ and a retraction $\bb F\Psi_{ B}^{\mb
h}\onto \bb H\Psi_{ B}^{\mb h}$.  They represent maps of functors: the
inclusion of the homogeneous $C$ into the filtered $C$, and the
retraction taking a filtered $C$ to its associated graded quotient
$G_\bu C$.  A priori, $\bb H\Psi_{ B}^{\mb h} \into \bb F\Psi_{ B}^{\mb
h}$ is a monomorphism, but a posteriori, it's a closed embedding, as the
schemes are separated and a retraction exists.  See (<ref>).

Jelisiejew kindly pointed out in email to the authors the following
interesting way to construct $\bb F\Psi_{ B}^{\mb h}$ and $\bb H\Psi_{
B}^{\mb h}$ and to study the retraction map $\bb F\Psi_{ B}^{\mb h}\onto
\bb H\Psi_{ B}^{\mb h}$.  This way involves the Białynicki-Birula
decomposition of the above Quot scheme $Q$ under the natural action of
the multiplicative group.  Namely, $\bb F\Psi_{ B}^{\mb h}$ appears as a
disjoint union of connected components of the decomposition, and $\bb
H\Psi_{ B}^{\mb h}$ is its subscheme of fixed points.  Consequently, the
retraction map is affine; in fact, near a point of $\bb H\Psi_{ B}^{\mb
h}$ where the map is smooth, it's an affine-space fibration.
Please see Prp. 3.1 and Rmk. 3.2 on p. 255 of <cit.>,
and Thm. 1.1 and Prp. 5.3(3) and Sec. 7 of <cit.>, and Sec. 5
of <cit.>; note that in these papers, $\bb F\Psi_{ B}^{\mb h}$
would be denoted by $\bigl(\bb H\Psi_{ B}^{\mb h}\bigr)^+$.  These
papers place various restrictive hypotheses on $A$, $B$, and $S$;
however, Jelisiejew said he believes that, without change, the proofs
cover our setup.  We do not pursue these ideas here, except indirectly
in (<ref>) through references to <cit.>.

To analyze the structure of $\bb H\Psi_{ B}^{\mb h}$ and of the
retraction $\bb F\Psi_{ B}^{\mb h}\onto \bb H\Psi_{ B}^{\mb h}$,
following <cit.> and <cit.>, we generalize Macaulay duality.
Macaulay introduced it in 1916 in Chap. IV of <cit.>.  In essence
(cf. Robert's introduction in <cit.>), Macaulay fixed a power
series ring $\wh P$ over a field $k$, and formed the span, $P^\dg$ say,
of the Laurent monomials $1/M$ for all monomials $M \in \wh P$.  He
viewed $P^\dg$ as an $\wh P$-module: the product of a monomial $L\in \wh
P$ with $1/M$ is $L/M$ if $L$ divides $M$ and is 0 if not.  He called a
$\psi\in \wh P$ and an $f\in P^\dg$ “inverse to one another” if
$\psi\cdot f = 0$, and he <cit.>* p. 68 defined the “inverse
system” of an ideal $I$ containing powers of the variables to be the
set of all $f\in P^\dg$ inverse to all $\psi \in I$.  Thus $P^\dg$
is the graded dual of the polynomial ring $P$, and the inverse system is
graded dual inverse system
just $\Hom_k(P/(I\cap P),\, k)$.  For instance, take $I := (X_1X_2,\,
X_1^2-X_2^3)$ in $\wh P := k[[X_1,X_2]]$; its inverse system in $P^\dg :=
k[X_1^{-1},X_2^{-1}]$ is generated by a single element $f :=
X_1^{-2}+X_2^{-3}$; see (<ref>).  The bijection between $I$, or
equivalently $\wh P/I$, and its inverse system is commonly known as
“Macaulay Duality” (or “apolarity”). Our generalization is

Macaulay Duality is a special case of Matlis Duality, as introduced in
1958 in <cit.>*Sec. 4, pp. 525–28 for an arbitrary complete
Noetherian local ring.  Indeed, $\Hom_k(\wh P/I,\, k) = \Hom_{\wh P}(\wh
P/I,\, P^\dg)$, see the end of (<ref>); hence, $P^\dg$ is the
injective hull of $k$ by the uniqueness of dualizing modules.  In 2018,
in <cit.>*Sec 3, pp. 15–18; Sec. 5, pp. 20–22, Smith
developed a relative version of Matlis duality, for an arbitrary
Noetherian domain $k$ and a Noetherian $k$-algebra $P$ equipped with an
ideal $J$ such that $P/J$ is a finitely generated $k$-module.  For any
$P$-module $M$, set $M^\vee := \varinjlim \Hom_k(M/J^nM,\ k)$.  Smith
proved $M^\vee = \Hom_P(M,\,P^\vee)$, and when $P = k[[X_1,\dotsc,X_n]]$
and $J = \la X_1,\dotsc,X_n \ra$, then $P^\vee = H_J^n(P)$.  Notice
$P^\vee = P^\dg$.

Another version of relative Matlis Duality is given in (<ref>).
In it, $P$ is replaced by a $k$-Artinian $k$-algebra $A$ where $k$ is a
Noetherian ring.  Set $A^* := \Hom_k(A,k)$.  This version asserts that
$A^*$ is a dualizing module for the $k$-Artinian $A$-modules.

In 1978, the first relative version of Macaulay Duality was stated,
without proof, by Emsalem <cit.>*Prp. 18, p. 415 for a power
series ring over a local algebra of finite dimension over an
algebraically closed field $K$ of characteristic 0 in order to study
deformations of local Artinian $K$-algebras.

In 1998, in Rmk. 1.9 on p. 613 of <cit.>, the second author
conjectured there should be a relative Macaulay duality yielding an
isomorphism of schemes $L\risom \Gamma$ where $\Gamma$ is the open
subscheme of $\bb H\Psi_P^{\mb h}$ of Gorenstein quotients of a
polynomial ring $P$ over an algebraically closed field $k$ of
characteristic not $2$ and where $L$ is the appropriate catalecticant
determinantal locus of $s$-forms, with $s := \sup\{\,n\mid\mb
h(n)\neq0\,\}$; see (<ref>) and (<ref>).  As evidence, he
constructed a canonical map of reductions $L_{\rm red}\to \Gamma_{\rm
red}$; it's bijective on $k$-points by Macaulay duality, and
corresponding $k$-points of $L$ and $\Gamma$ have the same tangent
space.  In 2006, he elaborated on this construction in the proof of
Thm. 11 on p. 3146 in <cit.>, and in the following subsection,
Prb. 12, he noted that the arguments there prove that the map is a
topological isomorphism and so the dimensions at corresponding points
are equal.  In 2007, on p. 694 in <cit.>, he extended this proof
to the case of level quotients.  In particular, the equality of the
tangent spaces and of the dimensions yield a smoothness criterion for

In 2018 in <cit.>*Subsec. 2.2, pp. 271–78, Jelisiejew
developed a relative Macaulay Duality for a power series ring with
coefficients in a Noetherian algebra over an algebraically closed field
$K$ in order to study loci of Gorenstein Artinian $K$-algebras.  The
Gorenstein hypothesis is weakened a bit in Sec. 4.4 of his 2017 thesis
<cit.>.  In the recent paper <cit.>*Sec. 4.1, he and
Šivic developed a Macaulay Duality for a free module over a
polynomial ring with coefficients in an algebraically closed field.

Macaulay Duality was also generalized to higher-dimensional Gorenstein
quotients of the polynomial ring and the power series ring over a field
in 2017 by Elias and Rossi <cit.>.  However, here we do not pursue
that direction.

Generalizing Macaulay Duality is our first goal.  Our treatment is
independent of earlier work, and goes further in the way explained next;
yet it's rather simple.  For it, recall we fix a Noetherian ring $k$, a
finitely generated graded $k$-algebra $A := \bigoplus_{n\ge0}A_n$, and a
$k$-flat finitely generated graded $A$-module $B = \bigoplus_{n\ge

Let $C$ be a $k$-module.  Set $C^* := \Hom_k(C,k)$.  If $C$ is an
$A$-module, then so is $C^*$ via the adjoint action: $(af)\psi :=
f(a\psi)$ for all $a\in A$ and $f\in C^*$ and $\psi\in C$.

Assume $C$ is locally free of finite rank.  Then $C^*$ is too, and the
natural map $C\to (C^*)^*$ is an isomorphism.  In other words, $C\mapsto
C^*$ is a dualizing functor on these $C$.  Further, see
(<ref>), it induces a dualizing functor on the filtered $C$,
which restricts to a dualizing functor on the graded $C$, and which
respects the retraction that takes a filtered $C$ to its associated
graded module.

Moreover, see (<ref>) and (<ref>), form the graded dual
$B^\dg := \bigoplus_{n\ge0} (B_n)^*$.  Then the functor $C\mapsto C^*$
yields a bijection from the $k$-Artinian quotients $C$ of $B$ onto the
$k$-Artinian submodules $D$ of $B^\dg$ with $B^\dg/D$ flat.  Further,
this bijection respects the induced filtrations on $C$ and $D$, and it
respects the retractions taking $C$ and $D$ to their associated graded
modules.  To recast this bijection into the more traditional form of
apolarity, note that, if $C = B/I$, then $C^* = (0:_{B^\dg}I)$, where
$(0:_{B^\dg}I)$ is the apolar annihilator, which consists of all $f\in
B^\dg$ with $\psi\cdot f = 0$ for all $\psi \in I$.  Note $B^{\dg\dg} =
B$; so $D^* = B/(0:_BD)$.  Thus the bijection becomes $I\mapsto
(0:_{B^\dg}I)$ with inverse $D\mapsto (0:_BD)$.  That then is our
generalization of Macaulay Duality.

Our Macaulay Duality facilitates the study of the retraction $\bb
F\Psi_{ B}^{\mb h}\onto \bb H\Psi_{ B}^{\mb h}$ as follows.
Recall that, if $T := \Spec(K)$ with $K$ a $k$-algebra, then a $T$-point
of $\bb F\Psi_{ B}^{\mb h}$ is given by a filtered $A\ox_kK$-quotient
$C$ of $B\ox_kK$ with $F^nC/F^{n+1}C$ locally free over $K$ of rank $\mb
h(n)$.  Now, every map $K\to L$ yields a natural isomorphism
$\Hom_K(C,K)\ox_KL \risom \Hom_L(C\ox_KL,L)$.  Thus Macaulay Duality
provides an isomorphism from the functor of points of $\bb F\Psi_{
B}^{\mb h}$ onto the functor whose value at $T$ is the set of filtered
submodules $D$ of $B\ox_kK$ with $F^nD/F^{n+1}D$ locally free over $K$
of rank $\mb h^*(n)$ where $\mb h^*(n) := \mb h(-n)$ and with
$(B^\dg\ox_kK)/D$ flat over $K$.
hmbzst@$\mb h^*(n)$

The latter functor is automatically representable by a copy of $\bb
F\Psi_{ B}^{\mb h}$, say $\bb F\Delta_{ B^\dg}^{\mb h^*}$, and by
a subsheaf of the pullback of $ B^\dg$ to $\bb F\Delta_{
B^\dg}^{\mb h^*}$; this subsheaf is obtained from the universal quotient
on $\bb F\Psi_{ B}^{\mb h}$ by “Hom'ing" it into the structure
sheaf.  Similarly, the subfunctor of homogeneous $D$ is representable by
a copy of $\bb H\Psi_{ B}^{\mb h}$, say $\bb H\Delta_{ B^\dg}^{\mb
h^*}$.  Also, $\bb F\Psi_{ B}^{\mb h}\onto \bb H\Psi_{ B}^{\mb h}$
becomes the retraction $\bb F\Delta_{ B^\dg}^{\mb h^*} \onto \bb
H\Delta_{ B^\dg}^{\mb h^*}$ given by $D \mapsto G_\bu D$.

Thus a locus of quotients with given Hilbert function $\mb h$ can be
transformed into an isomorphic dual locus of submodules.  Now, our main
goal is to treat, via this duality, the geometry of recursively
compressed quotients.  Since they're defined in terms of the socle type
$\mb t$, the next step is to relativize and dualize this notion.

Assume $A/F^1A =k$.  Fix a $k$-Artinian $A$-module $C$, set $\Soc_k(C)
:= \Hom_A(k,\,C)$, and call this $A$-module the $k$-socle of
$C$.  If $k$ is a field, we recover the traditional notion.  In general,
$\Soc_k(C) = (C^*\ox_A k)^*$.  A form of this equation was proved by
Iarrobino <cit.>*Lem. 2.1, p. 346. It's important, because
$C^*\ox_A k$ is more useful than $\Soc_k(C)$, as by Nakayama's Lemma,
$C^*\ox_A k$ locally governs the generators of $C^*$.
The equation holds, as both sides are equal to $\Hom_A(C^*,\,k^*)$;
the left, as $C\mapsto C^*$ is a dualizing functor, and the right, by
Adjoint Associativity; see (<ref>).

For instance, take $P := k[X_1,X_2]$ and $C := P/I$ with $I :=
(X_1X_2,\, X_1^2- X_2^3)$.  Then, as is easy to see, $C^*$ is generated
by $X_2^{-3}+ X_1^{-2}$, and so $X_2^{-3}$ and $X_1^{-1}$ form a minimal
generating set of $G_\bu(C^*)$.  Thus we obtain $\Soc(G_\bu(C)) =
(X_1,\,X_2^3)$ without even finding $G_\bu(C)$; cf. (<ref>).

Assume $C^*\ox_A k$ is $k$-Artinian.  Then so is $\Soc_k(C)$, and their
Hilbert functions, say $\mb t^*$ and $\mb t$, satisfy $\mb t^*(p) = \mb
t(-p)$, as $\Soc_k(C) = (C^*\ox_A k)^*$.  Extending tradition, we call
$\mb t$ the $k$-socle type of $C$, and $\mb t^*$ the local
tmb@$\mb t$ tmbst@$\mb t^*$
ksot@$k$-socle type   local generator type
generator type of $C^*$.  Notice that, given any maximal ideal of
$k$, its complement contains an element $f$ such that the localization
$C^*_f$ is generated by a set with $\mb t^*(p)$ elements of degree $p$
for all $p$.

For every $k$-algebra $K$, the $A\ox_kK$-module $(C\ox_kK)^*$ is also of
local generator type $\mb t^*$ as $(C\ox_kK)^* = C^*\ox_kK$; so
$C\ox_kK$ is of $K$-socle type $\mb t$.  Set $T := \Spec(K)$.  It
follows that the subsets of the $T$-points of $\bb F\Psi_{ B}^{\mb h}$
and $\bb H\Psi_{ B}^{\mb h}$ given by those quotients of socle type $\mb
t$ form functors in $T$, which are compatible with retraction.
Similarly, the subsets of the $T$-points of $\bb F\Delta_{ B^\dg}^{\mb
h^*}$ and $\bb H\Delta_{ B^\dg}^{\mb h^*}$ given by those submodules of
local generator type $\mb t^*$ form functors in $T$, which are
compatible with retraction.

The former functors are represented by subschemes of $\bb F\Psi_{
B}^{\mb h}$ and $\bb H\Psi_{ B}^{\mb h}$, say $\bb F\Psi_{ B}^{\mb h,
\mb t}$ and $\bb H\Psi_{ B}^{\mb h, \mb t}$.  Indeed, flattening
FbbPsit@$\bb F\Psi_{ B}^{\mb h,\mb t}$
HbbPsit@$\bb H\Psi_{ B}^{\mb h,\mb t}$
stratification yields $\bb F\Psi_{ B}^{\mb h, \mb t}$ much as above for
$\bb F\Psi_{ B}^{\mb h}$, and we can take $\bb H\Psi_{ B}^{\mb h, \mb t}
:= \bb F\Psi_{ B}^{\mb h, \mb t} \cap \bb H\Psi_{ B}^{\mb h}$; see
(<ref>).  Now, Macaulay Duality gives isomorphisms from the
former functors to the latter.  So they're represented by copies of $\bb
F\Psi_{ B}^{\mb h, \mb t}$ and $\bb H\Psi_{ B}^{\mb h, \mb t}$, say $\bb
F\Delta_{ B^\dg}^{\mb h^*, \mb t}$ and $\bb H\Delta_{ B^\dg}^{\mb h^*,
\mb t}$,
FbbDeltst@$\bb F\Delta_{ B^\dg}^{\mb h^*, \mb t}$
HbbDeltst@$\bb H\Delta_{ B^\dg}^{\mb h^*, \mb t}$
which are subschemes of $\bb F\Delta_{ B^\dg}^{\mb h^*}$ and
$\bb H\Delta_{ B^\dg}^{\mb h^*}$.  Also, $\bb H\Delta_{ B^\dg}^{\mb h^*,
\mb t} = \bb H\Delta_{ B^\dg}^{\mb h^*} \cap \bb F\Delta_{ B^\dg}^{\mb
h^*, \mb t}$.  Now, for $\mb h$ fixed, as $\mb t$ varies, the $\bb
F\Psi_{ B}^{\mb h, \mb t}$ and $\bb H\Psi_{ B}^{\mb h, \mb t}$ stratify (that is, disjointly cover) $\bb F\Psi_{ B}^{\mb h}$ and
$\bb H\Psi_{ B}^{\mb h}$; see (<ref>). So by Macaulay Duality,
the $\bb F\Delta_{ B^\dg}^{\mb h^*, \mb t}$ and $\bb H\Delta_{
B^\dg}^{\mb h^*, \mb t}$ stratify $\bb F\Delta_{ B^\dg}^{\mb h^*}$ and
$\bb H\Delta_{ B^\dg}^{\mb h^*}$.

From Section 7 on, many arguments proceed via recursion on the diameter $(s-\bar s)$ of $\mb t$, where $s := \sup\{\,p\mid \mb t(p)
> 0\,\}$ and $\bar s := \inf\,\{\, p \mid \mb t(p) \neq 0\,\}$.  Note
that, if $C$ has Hilbert function $\mb h$, plainly $s := \sup\{\,p\mid
\mb h(p) > 0\,\}$; also, $\mb t(s) := \mb h(s)$ by (<ref>)(2).
diameter  s@$s$  sbar@$\bar s$

The recursion exits when $\bar s = s$, or equivalently, $\mb t(p) := 0$
for $p \neq s$.  Then we call $\mb t$ and $C$ level, and set
$\bb F\Lambda_{ B}^{\mb h} := \bb F\Psi_{ B}^{\mb h, \mb t}$ and
$\bb H\Lambda_{ B}^{\mb h} := \bb H\Psi_{ B}^{\mb h, \mb t}$.
FbbLam@$\bb F\Lambda_{ B}^{\mb h}$
HbbLam@$\bb H\Lambda_{ B}^{\mb h}$
In (<ref>), there's an alternative construction of $\bb
F\Lambda_{ B}^{\mb h}$; it shows $\bb F\Lambda_{ B}^{\mb h}$ is an open
subscheme of $\bb F\Psi_{ B}^{\mb h}$.  So $\bb H\Lambda_{ B}^{\mb h}$
is an open subscheme of $\bb H\Psi_{ B}^{\mb h}$.  Sometimes, $ \bb
H\Psi_{ B}^{\mb h} - \bb H\Lambda_{ B}^{\mb h}$ contains $k$-points; an
example is the quotient $D^*$ in (<ref>).  Also, the
retraction $\bb F\Psi_{ B}^{\mb h} \to \bb H\Psi_{ B}^{\mb h}$ needn't
carry $\bb F\Lambda_{ B}^{\mb h}$ into $\bb H\Lambda_{ B}^{\mb h}$;
indeed, (<ref>) discusses a (standard) example of a
($k$-Gorenstein) level filtered $C$ with $G_\bu C$ not level.  However,
by (<ref>), if $G_\bu C$ is level, then so is $C$; thus the
preimage of $\bb H\Lambda_{ B}^{\mb h}$ lies in $\bb F\Lambda_{ B}^{\mb
h}$.  Jelisiejew develops some of the theory of $k$-Gorenstein modules
for arbitrary $k$ in <cit.>, especially in Subsection 2.2.

There's a third construction of $\bb H\Lambda_{ B}^{\mb h}$ studied in
(<ref>)–(<ref>).  This time, $\bb H\Lambda_{ B}^{\mb h}$
arises, via flattening stratification, as a subscheme of the
Grassmannian $\bb G$ of rank-$\mb t(s)$ locally free quotients of $
B_s$. Moreover, for $\mb t$ fixed, as $\mb h$ varies, the $\bb
H\Lambda_{ B}^{\mb h}$ stratify $\bb G$.  Hence, if $k$ is a domain,
then $\bb G$ is reduced and irreducible, and its generic point lies in
some $\bb H\Lambda_{ B}^{\mb h}$.  This $\bb H\Lambda_{ B}^{\mb h}$ is
reduced and irreducible, and it's covered by open subschemes, each one
isomorphic to an open subscheme of the affine space over $k$ of fiber
dimension $t(\mb b(s) - t)$ where $t := \mb h(s) = \mb t(s)$ and $\mb b$
is the Hilbert function of
bbmb@$\mb b$ tmi@$t$
$B$.  Also, by lower semicontinuity of rank, if $\bb H\Lambda_{ B}^{\mb
h^\di} \neq \emptyset$, then $\mb h^\di(p) \le \mb h(p)$ for all $p$; so
$\sum_p\mb h^\di(p) \le \sum_p\mb h(p)$, with equality iff $\mb h^\di =
\mb h$.  This third construction yields the isomorphism $L\risom \Gamma$
conjectured in Rmk. 1.9 on p. 613 of <cit.> and discussed above.

For a moment, suppose $B = A$.  Then $C$ is a $k$-algebra, and $C^*$ is,
by (<ref>), its dualizing module.  Moreover, $C^*$ is invertible
iff $C$ is level and $\mb t(s) = 1$.  If so, we say $C$ is $k$-Gorenstein.  (Likely, Grothendieck coined the term “Gorenstein”,
and Serre in 1960 was the first to publish it, but historically it's
more correct to honor any one of several others, starting with Macaulay;
see <cit.>*pp. 56, 61–62 and <cit.>*Sec. 21.2,
pp. 529–30.)  The condition for $C$ to be $k$-Gorenstein doesn't
depend on its filtration, but that for $G_\bu C$ does; see (<ref>)
for an example.  If $B$ is $k$-Artinian and $k$-Gorenstein, but $C$ just
$k$-Artinian, then the linked ideal $\Hom_A (C, A) \subset A$ and the
Macaulay dual $C^*$ of $C$ coincide up to tensor product with a certain
invertible $k$-module; see (<ref>).

In general, when $\bar s < s$, the recursion proceeds by replacing a
given homogeneous $k$-Artinian $C$ by the unique graded quotient $C'$
with socle diameter $s - \bar s -1$, provided $C'$ is $k$-flat,
when $\bar s +1 < s$, similarly replaceable.  Thus we are led to form
the sequence of quotients $\Lambda^mC$ of $C$ where $(\Lambda^mC)^*$ is
the submodule of $C^*$ generated by all the homogeneous elements of
degree at most $-m$.  So $C' = \Lambda^{\bar s+1}C$.  Furthermore, every
$\Lambda^mC$ must be $k$-flat; when so, we call $C$ multilevel,
LamC@$\Lambda^mC$ multilevel
and denote the Hilbert function of $\Lambda^mC$ by $\mb h_m$.  Note that
$\mb h_{\bar s}$ is the Hilbert function of $C$.
hmbzm@$\mb h_m$
Of course, when $k$ is a field, flatness is automatic, and so every $C$ is

Then all the $(\Lambda^mC)^*$ are flat too by (<ref>)(3).  Note
$\Lambda^m\Lambda^nC = \Lambda^nC$ for $m \le n$ and
$\Lambda^m\Lambda^nC = \Lambda^mC$ for $m \ge n$ for any $n$.  Take $n
:= \bar s+1$.  Thus $C'$ is multilevel of socle type $\mb t'$.  Also,
set $\mb h'_m := \mb h_{\bar s+1}$ for $m\le \bar s+1$ and $\mb h'_m :=
\mb h_m$ for $m\ge \bar s+1$.  Then $\mb h'_m$ is the Hilbert function
of $\Lambda^m C'$. We call $\{\mb h'_m\}$ the attendant to
$\{\mb h_m\}$.  As to a filtered module $C$, we call $C$
multilevel if $G_\bu C$ is multilevel.

The algebraic theory of multilevel modules is treated in
(<ref>)–(<ref>), and the geometric theory, in
Secs. 7–10.  Most of the effort concerns graded modules.  For example,
in (<ref>), we prove that, as $\{\mb h_m\}$ varies with $\mb
h_{0}$ fixed, the locus of all graded and all filtered multilevel
quotients of $B$ are parameterized by (possibly empty) subschemes $\bb
H\Lambda_{B}^{\{\mb h_m\}}$ and $\bb F\Lambda_{B}^{\{\mb h_m\}}$ of
$\bb H\Psi_{B}^{\mb h_0}$ and $\bb F\Psi_{B}^{\mb h_0}$, stratifying
HbbLamhm@$\bb H\Lambda_{B}^{\{\mb h_m\}}$
FbbLamhm@$\bb F\Lambda_{B}^{\{\mb h_m\}}$
them.  We use flattening stratification to construct $\bb
H\Lambda_{B}^{\{\mb h_m\}}$, and we take $\bb F\Lambda_{B}^{\{\mb
h_m\}}$ to be the preimage of $\bb H\Lambda_{B}^{\{\mb h_m\}}$ under the
retraction $\bb F\Psi_{\mc B}^{\mb h_0} \onto \bb H\Psi_{B}^{\mb h_0}$.

A recursive construction of the $\bb H\Lambda_{B}^{\{\mb h_m\}}$ is
given in (<ref>)–(<ref>).  It's harder, but leads to
three new notions, treated in (<ref>)–(<ref>).  First, we
call $\{\mb h_m\}$ recursively maximal for $\mb t$ and $T/k$ if
recursively maximal
(1) when $\bar s < s$, the attendant $\{\mb h'_m\}$ is recursively
maximal for the attendant $\mb t'$; and (2) the $\mb h_m$ termwise bound
the Hilbert functions of any multilevel quotient $C^\di$ on any
$T^\di/T$ of $T^\di$-socle type $\mb t$, provided $\Lambda^{\bar s+1}\mc
C^\di$ has the $\mb h'_m$ as its Hilbert functions when $\bar s < s$;
and (3) there's some multilevel quotient $C^\sh$ on some $T^\sh/T$ with
socle type $\mb t$ and Hilbert functions the $\mb h_m$.

Second, we call $\mb t$ quasi-permissible if $\bar s = s$ and
$\mb b(s) \ge \mb t(s)$, or if $\bar s < s$ and (1) the attendant $\mb
t'$ to $\mb t$ is quasi-permissible, and (2) given any $T^\ft/S$ and
$\{\mb h^\ft_m\}$ recursively maximal for $\mb t'$ and $T^\ft$,
necessarily $\mb b(\bar s) - \mb h^\ft_{\bar s+1}(\bar s) \ge \mb t(\bar
s)$.  Third, given a multilevel quotient $C$ on some $T/k$, we call $C$
recursively compressed  for $\mb t$ if $C$ is of $T$-socle type
recursively compressed
$\mb t$; if $\Lambda^{\bar s+1}C$ is recursively compressed for $\mb t'$
when $\bar s < s$; and if, for any multilevel quotient $C^\di$ on any
$T^\di/S$ of $T^\di$-socle type $\mb t$, with $\Lambda^{\bar s+1} C^\di$
recursively compressed for $\mb t'$ if $\bar s < s$, necessarily $\rank
(C^\di) \le \rank (C)$.

Those three notions have these properties.  If a recursively maximal
sequence for $\mb t$ and $T/k$ exists, then it's unique and $\mb t$ is
quasi-permissible.  If $\mb t$ is quasi-permissible and $k$ is a domain,
then a recursively maximal sequence $\{\mb h_m\}$ for $\mb t$ and $T/k$
exists; moreover, $\bb H\Lambda_{B}^{\{\mb h_m\}}$ is nonempty, reduced,
irreducible, and covered by open subschemes, with each one isomorphic to
an open subscheme of the affine space over $k$ of fiber dimension $N$
where $\mb N :=\sum_p\mb t(p)\bigl(\mb b(p) - \mb h_{\bar s}(p)\bigr)$.
Every recursively compressed $C$ for $\mb t$ is of the same rank.  If
$\mb t$ is quasi-permissible, then (1) there's a recursively compressed
$C$ for $\mb t$ on some $T/k$, and (2) given a recursively maximal
sequence $\{\mb h_m\}$ for $\mb t$ and some $T/k$, and given on some
$T^\sh$ a $C^\sh$ of socle type $\mb t$, each $\mb h_m$ is the Hilbert
function of $\Lambda^mC^\sh$ iff $\mc C^\sh$ is recursively compressed
for $\mb t$.

Those three notions are particularly important when $k$ is an infinite
field, $A$ is a polynomial ring in two or more variables, and $B = A$.
This fact was discovered by Iarrobino, and treated in <cit.>.  We
abstract, clarify and extend his work from (<ref>) on.  Our main
results(<ref>), (<ref>), (<ref>),
(<ref>)extend his main results Thms. IIA, IIB,
IIC on p. 351 in <cit.>.  His results describe the geometry of
$\bb H\Lambda_A^{\{\mb h_m\}}$, of $\bb F\Lambda_A^{\{\mb h_m\}}$, and
of the retraction $\bb F\Lambda_A^{\{\mb h_m\}}\onto \bb H\Lambda_A^{\{\mb
h_m\}}$ when $\{\mb h_m\}$ is recursively maximal for $\mb t$ and $\mb
t$ is permissible; these issues are discussed next.  In particular, in
our extensions, $k$ can be any Noetherian base ring, and $A =
\bigoplus_p A_p$, any finitely generated, graded $k$-algebra with $A_p$
a locally free $k$-module of rank, say, $\mb a(p)$.

In general, set $g_m(p) := \ts \sum_{q = m}^s\mb t(q)\,\mb a(q-p)$ and
$\mb h^{\rm I}_m(p) := \min\{\,\mb g_m(p),\, \mb b(p)\,\}$ and $\beta^{\rm I}_m := \ts \sum_p \mb h^{\rm I}_m(p)$ for all $m$, $p$;
hmbzIm@$\mb h^{\rm I}_m(p)$  gmp@$g_m(p)$
babetaIm@$\beta^{\rm I}_m$
see (<ref>); the I's honor Iarrobino's work.  Then given any $m$
and any homogeneous multilevel quotient $C$ of $B$ of socle type $\mb
t$, the Hilbert function $\mb h_m$ of $\Lambda^mC$ satisfies $\mb h_m(p)
\le \mb h^{\rm I}_m(p)$ for all $p$, and also $\rank(\Lambda^mC) \le
\beta^{\rm I}_m$, with equality iff $\mb h_m = \mb h^{\rm I}_m$, as
$C^*$ is of local generator type $\mb t^*$; see (<ref>).

We call the $C$ above I-compressed if $\mb h_m = \mb h^{\rm
I}_m$ for all $m$, that is, if $C$ is represented by a point of $\bb
H\Lambda_B^{\{\mb h^{\rm I}_m\}}$; in addition, $\mb t$ must be
permissible, where permissible means that $\mb t(p) = 0$ if $\mb
b(p) < \mb g_{\bar s}(p)$ and if (<ref>)(d) holds; the latter
holds, for instance, when $\mb a(p)$ and $\mb b(p)$ are nondecreasing in
$p$.  In (<ref>), the meaning of “permissible”
is extended to rather arbitrary $A$ and $B$.

For instance, let $k$ be a field, and $B := A := k[X_1,X_2,X_3]$ the
polynomial ring (with the standard grading).  Let the socle vector be
$(0,0,0,1,2)$; that is, the nonzero values of $\mb t$ are $\mb t(q) = 1$
if $q = 3$ and $\mb t(q) = 2$ if $q = 4$.  Then $s = 4$ and ${\bar s} =
3$; further, for $m \ge 3$ the nonzero $g_m$ are $g_4(p)=2\mb a(4-p)$
and $g_3(p)=g_4(p)+\mb a(3-p)$.  So $\mb h^{\rm I}_4(p) = g_4(p)$ and
$\mb h^{\rm I}_3(p) = g_3(p)$ for $p \ge 3$; further, the coordinates of
the vectors $(1,3,6,6,2)$ and $(1,3,6,7,2)$ given by $\mb h^{\rm I}_4$
and $\mb h^{\rm I}_3$ are at least those of the Hilbert vectors of
$\Delta^4 C$ and $C$.  The latter are of dimensions at most 18 and 19.
Thus any $C$ of socle type $\mb t$ and dimension 19 has Hilbert vector
$(1,3,6,7,2)$, and if $\Delta^4 C$ is 18-dimensional too, then $C$ is
I-compressed and recursively compressed.

Iarrobino proved (and Fröberg and Laksov reproved) in essence that,
remarkably, there exists an I-compressed $C$, or a $k$-point of $\bb
H\Lambda_A^{\{\mb h^{\rm I}_m\}}$, if $\mb t$ is permissible, $k$ is an
infinite field, $A$ is a polynomial ring, and $B = A$ (Iarrobino also
assumed $\mb h^{\rm I}_{\bar s}(1) = \mb a(1)$, but this assumption is
easily removed); see (<ref>).  Consequently, there's such a $C$
if $k$ is replaced by any (Noetherian) local ring with an infinite
residue field; see (<ref>).  Furthermore, see (<ref>),
then $\{\mb h^{\rm I}_m\}$ is recursively maximal for $\mb t$.
Moreover, a permissible $\mb t$ is, by (<ref>)(4),

Preserve Iarrobino's setup.  So his existence theorem holds.  It
follows, see (<ref>), that a homogeneous quotient $C$ of $A$ of
socle type $\mb t$ is I-compressed iff it's recursively compressed;.
Also, $\bb H\Psi_{A}^{\mb h^{\rm I}_{\bar s}, \mb t}$ is
irreducible, and $\bb H\Lambda_A^{\{\mb h^{\rm I}_m\}}$ is open in it;
see (<ref>).

Further, in essence, Iarrobino, just after Prop. 3.4 of
<cit.>*p. 339, constructed an irreducible scheme $P$ whose
$k$-points represent, with repetition, all the homogeneous $C$ of socle
type bounded (termwise) by $\mb t$, and he asserted $P$ has a nonempty
open subset whose $k$-points represent most compressed $C$ of socle type
$\mb t$.  There's a more refined assertion in (<ref>): all
the compressed $C$ of socle type $\mb t$ correspond to an open subset of
$P$, and those recursively compressed, to a nonempty, smaller open
subset.  To put it more informally, a general $C$ of socle type $\mb t$
is recusively compressed.

Conversely, when $k$ is any Noetherian ring, $A$ is a polynomial ring,
and $B = A$, if there exists a $C$ with $\rank(\Lambda^mC) = \beta^{\rm
I}_m$ for all $m$, then $\mb t$ is permissible by (<ref>).
On the other hand, the literature contains many interesting examples
with $k$ a field, $A$ a proper quotient of a polynomial ring, $B = A$,
and $\mb t$ permissible; sometimes there are no $C$ with
$\rank(\Lambda^mC) = \beta^{\rm I}_m$ for all $m$, and sometimes, there
are some.

Published examples of the latter case are reviewed in (<ref>).
There all the $C$ are Gorenstein; so the rank condition reduces to $\dim
C = \beta^{\rm I}_s$.  Moreover, suppose $A$ too is Artinian and
Gorenstein.  Then $\dim C = \beta^{\rm I}_s$ when both $A$ and $C$ are
suitably general for their socle degrees; see (<ref>).  On the
other hand, (<ref>) constructs examples where the Hilbert
function $\mb h$ of $C$ has a nontrivial “plateau"; the construction is
largely distilled and suitably adapted from <cit.>*pp. 9–14.

When, as above, $\{\mb h^{\rm I}_m\}$ is recursively maximal for a
permissible $\mb t$, more can be proved via careful work with the spaces
of local generators of $C^*$.  First, by (<ref>), whether or not
$k$ is a domain, $\bb H\Lambda_{B}^{\{\mb h^{\rm I}_m\}}$ is an open
subscheme of $\bb H\Psi_{B}^{\mb h^{\rm I}_{\bar s}}$, and it's
covered by open subschemes, with each one isomorphic to an open
subscheme of the affine space over $k$ of fiber dimension $\mb H
:=\sum_p\mb t(p)\bigl(\mb b(p) - \mb h^{\rm I}_{\bar s}(p)\bigr)$.
Second, if $A$ and $B$ satisfy certain mild conditions, which hold when
$A$ is a polynomial ring and $B = A$, then by (<ref>) the
retraction map makes $\bb F\Lambda_{\mc B}^{\mb h^{\rm I}_{\bar s}, \mb t}$
an affine-space bundle over $\bb H\Lambda_{\mc B}^{\mb h^{\rm I}_{\bar s},
\mb t}$ of fiber dimension $\mb R := \sum_p \mb t(p)\big(\sum_{q<p}(\mb
b(q) - \mb h^{\rm I}_{\bar s}(q))\bigr)$.  Finally, by (<ref>),
which is a corollary of (<ref>) and (<ref>), then $\bb
F\Lambda_{\mc B}^{\{\mb h^{\rm I}_m\}}$ is covered by open subschemes,
each one isomorphic to an open subscheme of the affine space over $k$ of
fiber dimension $\mb F := \mb H + \mb R = \sum_p \mb t(p)\big(\sum_{q\le
p}(\mb b(q) - \mb h^{\rm I}_{\bar s}(q))\bigr)$; moreover, if $S$ is
irreducible, then so are $\bb H\Lambda_{\mc B}^{\{\mb h^{\rm I}_m\}}$
and $\bb F\Lambda_{\mc B}^{\{\mb h^{\rm I}_m\}}$.

An interesting issue, raised by the referee, is to determine when the
retraction map is an algebraic vector bundle.  The referee noted that
Iarrobino <cit.>*Thm. 1, p. 229; Thm. 2, p. 232 proved in
two variables that it's not for the (nonmaximal) Hilbert vector
$(1,1,1,\dotsc,1)$ of length $n\ge 4$ and that his result was
generalized by Haboush and Hyeon <cit.>*Thm. 9.2, p. 4309 to
$r$ variables.  On the other hand, Jelisiejew proved in
<cit.>*Claim 2, p. 280 that it is a vector bundle in six
variables for the Hilbert vector $(1,6,6,1)$.

The formula for $\mb F$ and the smoothness and irreducibility of
$\smash{\bb F\Lambda_{\mc B}^{\{\mb h^{\rm I}_m\}}}$ are useful in the
study of $d$-dimensional quotients $C$ of the polynomial ring $A$ in $r$
variables over a field $k$, particularly in showing that certain $C$
aren't smoothable within $\Hilb^d_{\bb A^r/k}$ and in finding elementary components of $\Hilb^d_{\bb A_k^r/k}$ (ones parameterizing
elementary component $C$ with absolutely irreducible support).
Examples are discussed in (<ref>).  Notably, there's a new
example of a small elementary component (one of dimension less
than $dr$) with $k =\bb Q$ and with Hilbert vector
small elementary component
$(1,5,6,1)$.  It's produced via a novel method of reduction to the case
$k = \bb Z/p$ for just one $ p > 0$, see (<ref>), accompanied by
computations over $\bb Z/p$ with Macaulay 2.

The study of elementary components is generally based on the method of
small tangent spaces or trivial negative tangents, introduced in 1978 by
Iarrobino and Emsalem <cit.> and put in definitive form in 2019 by
Jelisiejew <cit.>.  Versions of Jelisiejew's results,
adequate for the examples in (<ref>), are, in (<ref>),
given self-contained alternative proofs, largely inspired by

Examples abound of proper quotients $A$ of a polynomial ring
over a field $k$ and of permissible-like $\mb t$ such that every
homogeneous quotient $C$ of $A$ of $k$-socle type $\mb t$ has $\dim_kC
<\beta^{\rm I}_{\bar s}$; in particular, no $C$ has $\rank(\Lambda^mC) =
\beta^{\rm I}_m$ for all $m$, and $\bb H\Psi_A^{\mb h^{\rm I}_{\bar
s},\mb t} = \emptyset$.  Some examples were given by Cho and Iarrobino
in <cit.>*Prp. 2.7, p. 754, and were analyzed by Geramita
et al. in <cit.>*Ex. 7.3, pp. 70–72.  The present second
author made an infinitesimal analysis of many classes of examples of
this sort; it suggests the geometry of $\bb H\Psi_{A}^{\mb h, \mb t}$
and $\bb F\Psi_{A}^{\mb h, \mb t}$ too is essentially as above.

More examples were given by Iarrobino <cit.>*Ex. 2.9,
p. 283, by Zanello <cit.>*Rmk. 6, p. 1090, and by
Migliore, Miró-Roig, and Nagel in <cit.>*Exs. 2.14–2.16,
pp. 343–344.  The last ones are discussed in some detail and refined
in (<ref>), as they nicely illustrate (<ref>), which is
the main result of Section <ref>.

Section 11 develops a theory like that in Sections 9 and 10, but now for
a $k$-Gorenstein, $k$-Artinian $A$ over any Noetherian base $k$.  The
maximal Hilbert function of a quotient $C$ is often not $\mb h^{\rm
I}_{\bar s}$, but is still a predictable function $\mb h$.  The main
results (<ref>)(1a)–(1c) are geometric: if $\mb t$ vanishes
when it should and if $\bb H\Psi_{A}^{\mb h, \mb t} \neq \emptyset$,
then $\bb H\Psi_{A}^{\mb h, \mb t}$ is covered by nonempty open
subschemes, each one isomorphic to an open subscheme of the affine space
over $S$ of fiber dimension $\mb H:= \sum_p \mb t(p)\bigl(\mb a(p) - \mb
h(p)\bigr)$; further, $\bb F\Psi_{A}^{\mb h, \mb t}$ is an
affine-space bundle over $\bb H\Psi_{A}^{\mb h, \mb t}$ of fiber
dimension $\mb R := \sum_p \mb t(p)\bigl(\sum_{p'<p} (\mb a(p') - \mb
h(p')) \bigr)$; finally, $\bb F\Psi_{A}^{\mb h, \mb t}$ is covered by
nonempty open subschemes, each one isomorphic to an open subscheme of
the affine space over $S$ of fiber dimension $\mb F := \mb H + \mb R =
\sum_p \mb t(p)\big(\sum_{p'\le p}(\mb a(p') - \mb h(p')) \bigr)$.

The key to Section 11 is further work with local generators of $C^*$ for
a given $C \in \bb H\Psi_{A}^{\mb h^\di, \mb t}$ for some $\mb
h^\di$.  Here's the idea.  Set $E := \bigoplus_{q\in\Z} A(q)^{\oplus \mb
t(q)}$.  Now, $C^*$ is of local generator type $\mb t^*$.  So given any
maximal ideal $\go m$ of k, we may replace $k$ by the localization $k_h$
for an $h \notin \go m$ so that $C^*$ has a minimal set of homogeneous
generators $f_1,\dotsc,f_m$, which define a surjection $E \onto C^*$ of
degree $0$.  Then $\mb h^\di = \mb h^{\rm I}_{\bar s}$ iff there's $n <
0$ with $E_p \risom (C^*)_p$ for $p < n$ and $(C^*)_p = A^\dg_p$ for
$p\ge n$.  To proceed, for $p < n$, we let the kernel of $E_p \onto
(C^*)_p$ be nonzero in a controlled, but general, fashion.

Say $A^\dg = Af$ with $f\in A^\dg_{-a}$.  So $f_i = g_if$ for some
$g_i\in A_{d_i}$ with $d_i>0$.  Then $\mb h^\di = \mb h$ iff the Koszul
homology group $H_1(g_1,\dotsc,g_m;A^\dg)$ is $0$ in degrees below $n$;
see (<ref>).  If so and if $k_{red}$ is a domain, then
$g_1,\dotsc,g_m$ are general; see (<ref>).  When $k$ is an
infinite field and $A$ is a general complete intersection, Fröberg's
conjecture asserts that, if $g_1,\dotsc,g_m$ are general, then indeed
$\mb h^\di = \mb h$; see (<ref>).
Fröberg's conjecture

In brief, Section 2 treats preliminaries about the category $\bb F_A$ of
FbbA@$\bb F_A$
filtered modules over a filtered ring $A$.  However, (<ref>) makes
a novel observation: the Rees functor embeds $\bb F_A$ into the Abelian
category of graded modules over the Rees algebra of $A$ as an exact
subcategory in Quillen's sense <cit.>*p. 99.  Section 3 develops
Macaulay Duality over any Noetherian base ring $k$, providing the tool
used to transform loci of quotients into loci of submodules.  Section 4
discusses a key invariant, the $k$-socle type $\mb t$ of an $A$-module
$C$.  Its importance stems from (<ref>): if $C$ is $k$-Artinian, that is, locally free of finite rank over $k$, then the
“reverse” $\mb t^*$ of $\mb t$ is the local generator type of the dual
$C^*$.  Section 5 discusses $k$-Gorenstein $k$-Artinian $A$; those
are the $A$ with $A^*$ an invertible $A$-module.  Most of the results in
Section 5 are familiar when $k$ is a field, but these $A$ are used in
some interesting new examples.

From Section 6 on, the focus is geometry. Section 6 constructs various
schemes of quotients of a fixed graded $A$-module $B$ when $A$ is a
graded $k$-algebra with $A_0 = k$. Notably, there are several different
constructions of the schemes $\bb F\Lambda_B^{\mb h}$ and $\bb
H\Lambda_B^{\mb h}$ of filtered and of homogeneous, level quotients with
Hilbert function $\mb h$.  Section 7 constructs the schemes $\bb
F\Lambda_{B}^{\{\mb h_m\}}$ and $\bb H\Lambda_{B}^{\{\mb h_m\}}$ of
multilevel quotients, and treats three new notions: recursively maximal
$\{\mb h_m\}$, quasi-permissible $\mb t$, and recursively compressed
multilevel $C$.  Section 8 treats, in greater generality, Iarrobino's
notions of a permissible $\mb t$ and of the I-set (our term) $\{\mb
h^{\rm I}_m\}$ of any $\mb t$.  Sections 9 and 10 extend Iarrobino's
main results in <cit.>, which describe the geometry of $\bb
H\Lambda_A^{\{\mb h^{\rm I}_m\}}$, of $\bb F\Lambda_A^{\{\mb h^{\rm
I}_m\}}$, and of the retraction $\bb F\Lambda_A^{\{\mb h^{\rm
I}_m\}}\onto \bb H\Lambda_A^{\{\mb h^{\rm I}_m\}}$ when $\mb t$ is
permissible and $A$ is a polynomial ring, thus achieving our main goal.
Finally, Section 11 develops a similar theory for maximal quotients of a
graded Gorenstein Artinian algebra.

§ PRELIMINARIES

Fix a Noetherian base ring $k$ for use throughout this paper.  (In this
paper, all rings are commutative with 1.)

Recall for frequent, but implicit, use that (even when $k$ is not
Noetherian) the following three conditions on a $k$-module $C$ are

* $C$ is locally free of finite (but not necessarily constant) rank
over $k$;

* $C$ is projective and finitely generated over $k$;

* $C$ is flat and finitely presented over $k$.

Given a $k$-algebra $A$, denote the (Abelian) category of $A$-modules
by $\bb M_A$.
MA@$\bb M_A$

Given any $k$-module $C$, define its $k$-dual  $C^*$ by

\begin{equation*}\label{eq1.1}
C^* : = \Hom_k(C,\,k).
\end{equation*}

If $C$ is a module over a given $k$-algebra $A$, then $C^*$ is
canonically an $A$-module too, via the adjoint action: $(af)\psi
:= f(a\psi)$ for all $a\in A$ and $f\in C^*$ and $\psi\in C$.
adjoint action

A map of $A$-modules $\gamma\: C\to D$ induces its dual map of
dual map
$A$-modules $\gamma^*\: D^*\to C^*$ by $\gamma^*g := g\circ\gamma$.  And
$C\mapsto C^*$ defines a left exact $A$-linear functor $*\:\bb M_A\to
\bb M_A$.
dual map

[Filtered modules]
Fix a $k$-algebra $A$ with a (descending) filtration $F^\bu A$;
namely, $F^\bu A$ is a descending chain of $A$-submodules $F^m A$ for
$m\in \Z$ with $F^m A = A$ for $m\le 0$ and with $F^m A\cdot F^nA
\subset F^{m+n}A$ for all $m$ and $n$.

Given an $A$-module $C$, by a filtration $F^\bu C$, let us mean a
descending chain of $A$-submodules $F^n C$ for $n\in \Z$ with $F^m
A\cdot F^nC \subset F^{m+n}C$ for all $m$ and $n$.  Set

\begin{equation*}\label{eqFm1}\ts
G_nC := F^nC/F^{n+1}C\enspace \text{for all }n \and
G_\bu C := \bigoplus_{n\in\Z} G_nC.
\end{equation*}

Then $G_m A\cdot G_nA \subset G_{m+n}A$ and $G_m A\cdot G_nC \subset
G_{m+n}C$ for all $m$ and $n$.  Also, $G_nA = 0$ for $n<0$.  Thus $G_\bu
A$ is a graded $k$-algebra, $G_\bu C$ a graded $G_\bu A$-module.  As
usual, call them the associated graded algebra and module.
Gbu@$G_\bu C$
associated graded

Notice that $A$ always has a trivial filtration, where $F^m A := A$
for $m\le 0$ and $F^m A := 0$ for $m > 0$.  Then $G_0A = A$ and $G_nA =
0$ for $n\neq 0$.  Similarly, $C$ has a trivial filtration.  Thus it's
really no restriction to assume $A$ and $C$ are filtered.

Given $p\in \Z$, let $C[p]$ denote $C$ with its shifted
filtration, $F^n(C[p]) := F^{n+p}C$.
shifted filtration FnCp@$F^n(C[p])$

Call $F^\bu C$ exhaustive if $\,\bigcup_n F^nC =C$, separated if $\,\bigcap_n F^nC =0$, discrete if $F^nC =0$ for
$n\gg0$, and finite if $F^nC =C$ for $n\ll0$ and $F^nC =0$ for

The filtered $A$-modules form a category $\bb F_A$.  Its maps $\beta\:
\indn{FbbA@$\bb F_A$}
(B,\,F^\bu B)\to (C,\,F^\bu C)$ are the $A$-maps $\beta\: B\to C$ with
$\beta(F^nB) \subset F^nC$ for all $n$.

Denote by $\bb{EF}_A$ the full subcategory of $\bb F_A$ of all
$(B,\,F^\bu B)$ with $F^\bu B$ exhaustive, and by $\bb{DEF}_A$ its full
subcategory of all $(B,\,F^\bu B)$ with $F^\bu B$ discrete too.

Given $(B,\,F^\bu B)$ and $(C,\,F^\bu C)$ in $\bb F_A$, filter the
$A$-module $\Hom_A(B,\,C)$ by

\begin{align*}
F^p\Hom_A(B,\,C) &:= \Hom_A\bigl(B,\,C[p]\,\bigr)\\
&:= \{\, \beta\: B\to C \mid\beta(F^nB)
\subset F^{n+p}C \enspace\text{for all }n\,\}.
\end{align*}

Similarly, give the $k$-module $k$ the trivial filtration, and filter
$B^*$ by

\begin{equation*}  \label{eqFm3}
F^p(B^*) := F^p\Hom_k(B,\,k) := \{\, \beta\: B\to k \mid\beta(F^nB)
\subset F^{n+p}k\enspace\text{for all }n\,\}.\\
\end{equation*}

Then it's easy to see that

\begin{equation} \label{eqFm4}
F^p(B^*) = \{\, \beta\: B\to k \mid\beta(F^nB) = 0
\enspace\text{for }n + p =1\,\} = (B/F^{1-p}B)^*.
\end{equation}
Notice, however, that even if $F^\bu B$ is exhaustive,
$F^\bu (B^*)$ need not be exhaustive.

Given a map of filtered modules $\beta\: B\to C$, its dual $\beta^*\:
C^* \to B^*$ is, plainly, a map of filtered modules.  Thus  $C \mapsto
C^*$ is a contravariant, additive functor from $\bb F_A$ to itself.

Filter the tensor product $B\ox_AC$ in $\bb M_A$ by setting

\begin{equation*}\label{eqFm5}\ts
F^p(B\ox_AC) := \Im(\tau) \text{\enspace where }
\tau\: \bigoplus_{m+n=p}F^m(B)\ox_AF^n(C) \to B\ox_AC.
\end{equation*}

Then it's easy to see that this adjoint associativity formula
adjoint associativity
preserves filtrations: $\Hom_k(B\ox_AC,\,k) = \Hom_A\bigl(B,\,
\Hom_k(C,k)\bigr)$.  In other words,
\begin{equation}\label{eqeqFm6}
(B\ox_AC)^* = \Hom_A(B,\,C^*)\text{\quad in } \bb F_A.
\end{equation}

In particular, taking $C:=A$ yields the following (duality) formula:

\begin{equation}\label{eqeqFm7}
B^* = \Hom_A(B,\,A^*)\text{\quad in } \bb F_A.
\end{equation}

Let now $A$ be a graded $k$-algebra, and $C$ a graded $A$-module.  Say
$A = \bigoplus_{p\in\Z} A_p$ with $A_p = 0$ for $p<0$, and $C =
\bigoplus_{p\in\Z} C_p$.  For all $n$, set $F^nA := \bigoplus_{p\ge
n}A_p$ and $F^nC := \bigoplus_{p\ge n}C_p$.  Plainly, the $F^nA$ turn
$A$ into a filtered $k$-algebra, and the $F^nC$ turn $C$ into a filtered
$A$-module; moreover, $G_pA = A_p$ and $G_pC = C_p$ for all $p$.

Also, an $A$-module map $\gamma\: C\to D$ is homogeneous (that
is, $D$ is graded, say $D = \bigoplus D_p$, and $\gamma C_p\subset D_p$
for all $p$) iff $\gamma\in \Hom_{\,\bb F_A}(B,\,C)$.  Thus the category
$\bb G_A$ of graded modules and homogeneous maps is a full subcategory
GbbA@$\bb G_A$
of $\bb F_A$.  And the functor $G_\bu\: \bb F_A\to \bb G_A$, given by
$C\mapsto G_\bu C$, is a retraction; that is, $G_\bu\big| \bb

Let $B$ be the kernel of $\gamma$ in $\bb M_A$.  Then $B$ is naturally
graded: $B = \bigoplus_{p\in\Z} B_p$ where $B_P := B\cap C_p$.  So $B$
is the kernel of $\gamma$ in $\bb G_A$.  Moreover, $F^nB = B\cap F^n C$
for all $n$; so $B$ is the kernel of $\gamma$ in $\bb F_A$ by
(<ref>).  Similarly, the cokernel of $\gamma$ in $\bb M_A$ is
also its cokernel in $\bb G_A$ and in $\bb F_A$.  In particular, $\bb
G_A$ is, therefore, Abelian.

[Strict maps]
Fix a filtered $k$-algebra $(A,\,F^\bu A)$.  Following
<cit.>*p. II-2 (and <cit.>*(1.1.5), p. 7, call a map
$\gamma\: (C,\,F^\bu C)\to (D,\,F^\bu D)$ in $\bb F_A$ strict
strictly compatible with the filtrations) if $\gamma(F^nC) =
\gamma(C)\cap F^nD$ for all $n$.

For example, if $\gamma\: C\to D$ is injective, then $\gamma$ is strict
iff $F^\bu C$ is induced by $F^\bu D$ in the sense that $F^nC =
\gamma^{-1}(F^nD)$.  Instead, if $\gamma\: C\to D$ is surjective, then
$\gamma$ is strict iff $F^\bu D$ is induced by $F^\bu C$ in the
sense that $F^nD = \gamma(F^nC)$.  In general, $\gamma$ is strict iff both
$F^\bu C$ and $F^\bu D$ induce the same filtration on $\gamma(C)$.

Let's see $\gamma\: (C,\,F^\bu C)\to (D,\,F^\bu D)$ has a kernel
$\Ker(\gamma)$ and a cokernel $\Cok(\gamma)$ in $\bb F_A$, whose
underlying $A$-modules are the usual $\Ker(\gamma)$ and $\Cok(\gamma)$
in $\bb M_A$, and whose filtrations are induced by $F^\bu C$ and $F^\bu
Given $\beta\: (B,\, F^\bu B) \to (C,\,F^\bu C)$ with $\gamma\beta =
0$, there's a unique $\alpha\: B\to \Ker(\gamma)$ in $\bb M_A$ with
$\ker(\gamma)\alpha = \beta$ where $\ker(\gamma)\: \Ker(\gamma)\into C$.
But $\ker(\gamma)(\alpha(F^nB)) = \beta(F^nB) \subset F^nC$.  Thus, as
desired, $\alpha(F^nB)\subset \ker(\gamma)^{-1}(F^nC)$.  The analysis of
$\Cok(\gamma)$ is dual.

Consider the following sequence in $\bb F_A$ associated to $\gamma\:
(C,\,F^\bu C)\to (D,\,F^\bu D)$:

\begin{equation*}\label{eqSm1}
\Ker(\gamma)\xto{\ker(\gamma)} C\xto{\coim(\gamma)} \Coim(\gamma)
\xto{\theta(\gamma)} \Im(\gamma) \xto{\im(\gamma)} D
\xto{\cok(\gamma)} \Cok(\gamma)
\end{equation*}

where $\coim(\gamma) := \cok(\ker(\gamma))$ and $\im(\gamma) :=
\ker(\cok(\gamma))$.  Note $\theta(\gamma)$ is bijective in $\bb M_A$,
and it's an isomorphism in $\bb F_A$ iff $F^\bu C$ and $F^\bu D$ induce
the same filtration on $\Im(\gamma)$, so iff $\gamma$ is strict.  Thus,
since not every $\gamma$ is strict, $\bb F_A$ fails to be Abelian.

Given $(E,\, F^nE)$ in $\bb F_A$, let's see $\Hom_A(\bu,\,E)$ is left
exact in the sense that

\begin{equation}\label{eqSm2}
\Hom_A(\,\Cok(\gamma),\,E\,) = \Ker(\,\Hom_A(\gamma,\,E)\,)
\tqn{in} \bb F_A.
\end{equation}

First, (<ref>) holds in $\bb M_A$, as each side can be
canonically identified with the set of $A$-maps $\delta\: D\to E$ with
$\delta\gamma = 0$.  Second, (<ref>) holds in $\bb F_A$, as the
$p$th term of the filtration of each side is identified with the subset
of all $\delta\: D\to E[p]$ in $\bb F_A$.

Finally, it is clear from the construction of $\Cok(\gamma)$ that it's
not only the cokernel in $\bb F_A$, but also in $\bb F_k$.  Hence
$\Cok(\gamma)^* = \Ker(\gamma^*)$ in $\bb F_k$ by (<ref>) with $A
:= k$ and $E := k$.  In particular, given $n$, then $F^n(\Cok(\gamma)^*)
= F^n(\Ker(\gamma^*))$ in $\bb M_k$; so, this equation holds in $\bb
M_A$ too, as it plainly respects multiplication by each $a\in A$.  Thus

\begin{equation}\label{eqSm3}
\Cok(\gamma)^* = \Ker(\gamma^*)  \tqn{in} \bb F_A.
\end{equation}

[Rees modules]
Fix a filtered $k$-algebra $(A,\,F^\bu A)$ and a variable $t$.  Form
the (extended) Rees algebra $\Rees(A) := \bigoplus_{n\in
\indt{Rees algebra}
\Z}F^nA\cdot t^{-n}$ and the Rees functor
Rees functor

\begin{equation*}\label{eqRf}\ts
\Rees\: \bb F_A \to \bb G_{\Rees(A)} \tq{by} (B,\,F^\bu B) \mapsto
\Rees(B) := \bigoplus_{n\in \Z}F^n B\cdot t^{-n}.
\end{equation*}

Note that $\bb F_A$ and $\bb{EF}_A$ have the same image $\bb R_A$,
which (see <cit.>*p. vi) is the full subcategory of graded
$\Rees(A)$-modules on which $t$ is regular (that is,
multiplication by $t$ is injective); an $M := \bigoplus M_n$ in $\bb
R_A$ is equal to the image of $ (B,\,F^\bu B)$ where $B := M/(1-t)M$ and
$F^nB := (M_n+(1-t)M )/ (1-t)M$ is the image of $M_n$ in $B$.  Moreover,
the Rees functor embeds $\bb{EF}$ as $\bb R_A$ in the Abelian category
$\bb G_{\Rees(A)}$.

Notice that $\bb R_A$ is closed under extensions; that is, given a short
exact sequence $0\to M\to M'\to M''\to 0$ in $\bb G_{\Rees(A)}$, if
$M,\ M'' \in \bb R_A$, then $M'\in \bb R_A$; indeed, apply the Snake
Lemma to this diagram, where the $\tau$'s denote multiplication by $t$,

\begin{equation}\label{eqRm1}
\begin{CD}
0@>>> M  @>>>  M'  @>>>   M''  @>>> 0\\
@. @V\tau VV @V\tau' VV @V\tau''VV\\
0@>>> M  @>>>  M'  @>>>   M''  @>>> 0
\end{CD}
\end{equation}

in order to see that $t$ is regular on $M'$ because it is on $M$ and

Thus $\bb R_A$ is the seminal example of an exact category in
exact category
<cit.>*p. 99.  Accordingly, call a sequence $0\to (B,\,F^\bu B)
\xto\beta (C,\,F^\bu C) \xto\gamma (D,\,F^\bu D)\to 0$ in $\bb {EF}_A$
exact if its image in $\bb G_{\Rees(A)}$ is exact.  Plainly, the
latter condition holds iff each sequence $0\to F^nB\to F^nC\to F^nD\to
0$ is exact in $\bb M_A$; so iff $F^\bu B$ and $F^\bu D$ are induced by
$F^\bu C$; so iff $\beta = \ker(\gamma)$ and $\gamma = \cok(\beta)$ in
$\bb {EF}_A$, or equivalently in $\bb F_A$.

Given  $(B,\,F^\bu B)$ in $\bb F_A$, notice that

\begin{equation}\label{eqRm2}
G_\bu B = \Rees(B)\bigm/ t\cdot \Rees(B)
= \Rees(B)\ox_{\Rees(A)}G_\bu A.
\end{equation}

Thus the functor $G_\bu\: \bb F_A \to \bb G_{G_\bu A}$ factors through
$\bu\ox G_\bu A\: \mathbb{G}_{\Rees(A)} \to \bb G_{G_\bu A}$.

The latter functor is, of course, right exact.  Moreover, its
restriction to $\bb R_A$ is exact owing to the 9-Lemma applied to
(<ref>).  Thus $G_\bu$ is exact on $\bb {EF}_A$ in the sense that
it preserves short exact sequences (compare with <cit.>*Prp. 2,
p. 25).

Let's say that the restriction of $G_\bu$ to a subcategory $\bb S$ of
$\bb{EF}_A$ is biexact if a sequence $0\to (B,\,F^\bu B)
\indt{biexact}
\xto\beta (C,\,F^\bu C) \xto\gamma (D,\,F^\bu D)\to 0$ in $\bb S$ with
$\gamma\beta = 0$ is exact in $\bb {EF}_A$ when its image under $G_\bu$
is exact in $\bb G_{G_\bu A}$.

By (<ref>) below, the restriction of $G_\bu$ to $\bb S$ is
biexact if $\bb S$ is fully separated in the sense that $\bb S$
fully separated
is closed under taking kernels and cokernels, and if $(B,\,F^\bu B)\in
\bb S$, then $F^\bu B$ is separated.  For example, $\bb {DEF}_A$ is,
plainly, fully separated.

For another example, take $A$ to be a Zariski ring (see
<cit.>* 4, p. 261 and <cit.>*p. 83); namely,
$\Rees(A)$ is a Noetherian ring, and $F^1A$ lies in the Jacobson radical
$\Jac(A$).  For instance, $A$ could be a weighted power series ring over
$k$.  Let $\bb{ZF}_A$ be the full subcategory of $\bb{EF}_A$ of
$(B,\,F^\bu B)$ such that $\Rees(B)$ is a finitely generated
Plainly, $\bb{ZF}_A$ is closed under taking kernels and cokernels.

Given $(B,\,F^\bu B)$ in $\bb ZF_A$ and $b\in \bigcap_nF^nB$, let's see
$b=0$.  Inspired by Rees (see <cit.>*p. 563), set $M^{(n)} :=
\Rees(A)\cdot bt^{-n} \subset \Rees(B)$.  The $M^{(n)}$ form an
increasing chain.  As $\Rees(B)$ is Noetherian, $M^{(p)}=M^{(p+1)}$ for
some $p$.  So $ubt^{-p} = bt^{-(p+1)}$ for some $u\in \Rees(A)$.  Say $u
= \sum u_it^{-i}$ with $u_i\in F^iA$.  Then $u_1b = b$.  So $(1-u_1)b =
0$.  But $F^1A\subset \Jac(A)$.  So $b=0$.  Thus $\bb{ZF}_A$ is fully

Let $A$ be a filtered $k$-algebra, and $\bb S\subset \bb{EF}_A$ a fully
separated subcategory.  Then $G_\bu\: \bb{EF}_A \to \bb G_{G_\bu A}$ is
exact, and its restriction to $\bb S$ is biexact.

It remains to see that a sequence $0\to M\xto\mu M'\xto{\mu'} M''\to 0$
in the image of $\bb S$ in $\bb R_A$ is exact if $\mu'\mu = 0$ and if
$0\to M/tM \xto{\bar\mu} M'/tM'\xto{\bar\mu'} M''/tM''\to 0$ is exact in
$\bb G_{G_\bu A}$.  Consider the following commutative diagram with
exact rows:

\begin{equation}\label{eqBiex1}
\begin{CD}
0@>>> M  @>\tau>>  M  @>>>  M/tM   @>>> 0\\
@. @V\mu VV     @V\mu VV @V\bar\mu VV\\
0@>>> M'  @>\tau'>> M' @>>> M'/tM' @>>> 0
\end{CD}
\end{equation}

where  the $\tau$'s denote multiplication by $t$.

Set $L := \Ker(\mu)$.  Since $\bar\mu$ is injective, multiplication by
$t$ on $L$ is bijective by the Snake Lemma; so if say $L = \bigoplus
L_n$, then $L_{n+1}\risom L_n$ for all $n$.  But $L$ is in the image of
$\bb S$, and $\bb S$ is fully separated.  Thus $L = 0$.

Set $N := \Cok(\mu)$.  As $\mu'\mu = 0$, there's a commutative diagram
with exact rows

\begin{equation}\label{eqBiex2}
\begin{CD}
0@>>> N @>\theta>> N @>>> N/tN @>>> 0\\
@. @V\nu VV     @V\nu VV @V\bar\nu VV\\
0@>>> M''  @>\tau''>> M'' @>>> M''/tM''' @>>> 0
\end{CD}
\end{equation}

where $\theta$ and $\tau''$ denote multiplication by $t$.  Note that,
applied to (<ref>), the Snake Lemma yields $\Cok(\bar\mu)= N/tN$.

By Hypothesis, $\Cok(\bar\mu)= M''/tM''$.  Hence $\bar\nu$ is bijective.
So applied to (<ref>), the Snake Lemma yields that
multiplication by $t$ is bijective on $\Ker(\nu)$ and on $\Cok(\nu)$.
So arguing as for $L$ yields $\Ker(\nu)=0$ and $\Cok(\nu)=0$.  Thus
$\nu$ is a isomorphism.  Thus $0\to M\to M'\to M''\to 0$ is exact, as

[$k$-Artinian modules] Fix a $k$-algebra $A$.

Call an $A $-module $C$ $k$-Artinian if $C$ is locally free of
constant finite rank over $k$.  Let $\bb {AM}_A\subset \bb M_A$ denote
AMAbb@$\bb {AM}_A$
the full subcategory of such $C$.  Note that $\bb {AM}_A$ is $A$-linear,
but not Abelian.  Given $C\in\bb {AM}_A$, note, plainly, $C^*\in\bb
{AM}_A$ too. Thus $*\:\bb M_A\to \bb M_A$ restricts to an $A$-linear
functor $*\:\bb {AM}_A\to \bb {AM}_A$.

When $k$ is a field, every $C\in \bb M_A$ is free.  So $C$ is
$k$-Artinian iff $\dim_kC <\infty$.

Given a function $\mb h\:\Z\to \Z$, define $\mb h^*\:\Z\to \Z$ by $\mb
h^*(n) := \mb h(-n)$.  Also, set
hmbzst@$\mb h^*(n)$

\begin{equation*}\label{eqinfsup}
i(\mb h) := \inf\{\,n\mid \mb h(n)\neq 0\,\}\and
s(\mb h) := \sup\{\,n\mid \mb h(n)\neq 0\,\}.
\end{equation*}

ihmb@$i(\mb h)$ shmb@$s(\mb h)$
Call $\mb h$ finite if $i(\mb h)$ and $s(\mb h)$ are finite (so
if $h\neq 0$, then $\{\,n\mid \mb h(n)\neq 0\,\}$ is finite).
finite function

Assume $A$ is graded.  Given $C\in \bb G_A$, say $C = \bigoplus C_n$.
Notice that, if $C$ is $k$-Artinian, then $C_n = 0$ for almost all $n$,
and each $C_n$ is locally free of finite, but not necessarily constant,
rank; the converse holds if also the rank of each $C_n$ is constant.
Call $\mb h\:\Z\to \Z$ the Hilbert function of $C$ if each
Hilbert function
$C_n$ is locally free of (constant) rank $\mb h(n)$; let $\bb
{AG}_A^{\mb h}$ denote the collection of all such $C$.
AGbbh@$\bb {AG}_A^{\mb h}$ AGbb@$\bb {AG}_A$

Let $\bb {AG}_A\subset \bb G_A$ denote the full subcategory of all $C
\in \bb {AG}_A^{\mb h}$ for all finite $\mb h$.

Assume $A$ is just filtered.  Given $(C,\,F^\bu C)\in \bb F_A$, call $C$
$k$-Artinian with Hilbert function $\mb h$ if $G_\bu C \in \bb
{AG}_A^{\mb h}$ and if $F^\bu C$ is finite; let $\bb {AF}_A^{\mb h}$
denote the collection of all such $(C,\,F^\bu C)$.  Now, $F^nC =
F^{n+1}C$ for $n < i(\mb h)$ and $n > s(\mb h)$; so $F^\bu C$ is finite
iff $F^nC = C$ for $n\le i(\mb h)$ and $F^nC = 0$ for $n> s(\mb h)$.
AFbbA@$\bb {AF}_A^{\mb h}$

Let $\bb {AF}_A\subset \bb F_A$ denote the full subcategory of all
$(C,\,F^\bu C)$ with $G_\bu C\in \bb {AG}_A$.  Trivially, the functor
$G_\bu \: \bb F_A \to \bb G_{G_\bu A}$ restricts to a functor $G_\bu \:
\bb {AF}_A \to \bb {AG}_{G_\bu A}$.  Trivially, $\bb {AF}_A$ is
exhaustive and discrete.  Thus, (<ref>) implies $G_\bu$ is

As a consequence, a $(C,\,F^\bu C) \in \bb {AF}_A$ is generated over $A$
by a $k$-submodule $N$ with the induced filtration if $G_\bu N$
generates $G_\bu C$ over $G_\bu A$.  Indeed, first note $G_\bu N \subset
G_\bu C$ as $G_\bu$ is exact.  Now, form the multiplication map $\mu\:
A\ox_k N \to C$.  The induced map $G_\bu\mu\: G_\bu A\ox_k G_\bu N \to
G_\bu C$ is surjective by hypothesis.  Thus, as $G_\bu$ is biexact,
$\mu$ is surjective, as desired.

Finally, call $A$ $k$-Gorenstein if $A^*$ is a locally free of
rank 1 (that is, invertible).

Let $A$ be a filtered $k$-algebra, and $(C,\,F^\bu C)\in \bb{AF}_A^{\mb h}$.

\(1) Then $C$ and $C/F^nC$ and $F^mC$ and $F^mC/F^nC$ are, for all $m\le
n$, locally free of ranks $\sum_p\mb h(p)$ and $\sum_{p<n}\mb h(p)$ and
$\sum_{m\le p}\mb h(p)$ and $\sum_{m\le p<n}\mb h(p)$.

\(2) Then $F^n((C^*)^*) = F^nC$ for all $n$.\hfill

\(3) Then $G_n(C^*) = (G_{-n}C)^*$ for all $n$.

\(4) Then $C^*\in \bb{AF}_A^{\mb h^*}$.
\end{proposition}

\begin{proof}
{\bf For (1),} fix $m$.  If $m=n$, then $F^mC/F^nC = 0$.  Suppose $m<n$.
Form the exact sequence $0\to G_nC\to F^mC/F^{n+1}C\to F^mC/F^nC\to 0$.
By hypothesis, $G_nC$ is locally free of rank $\mb h(n)$.  By induction,
assume $F^mC/F^nC$ is locally free of rank $\sum_{m\le p<n}\mb h(p)$.
Thus $F^mC/F^{n+1}C$ is locally free of rank $\sum_{m\le p<n+1}\mb
h(p)$.  But $F^mC = C$ for $m\le i(\mb h)$ and $F^nC = 0$ for $n> s(\mb
h)$.  Thus (1) holds.

{\bf For (2),} note $F^n((C^*)^*) := (C^*/F^{1-n}(C^*))^*$ and
$F^{1-n}(C^*) := (C/F^nC)^*$.  So form the exact sequence $0\to F^nC\to
C\to C/F^nC\to 0$ of $k$-modules.  It splits, as $C/F^nC$ is projective
by (1).  So $0\to (C/F^nC)^* \to C^*\to (F^nC)^*\to 0$ is exact.  Hence
$(F^nC)^* = C^*/F^{1-n}(C^*)$.  Thus $F^n((C^*)^*) = ((F^nC)^*)^*$.  But
$((F^nC)^*)^* = F^nC$ as $F^nC$ is locally free of finite rank by (1).
Thus (2) holds.

{\bf For (3),} form the exact sequence $0\to G_nC\to C/F^{n+1}C\to
C/F^nC\to 0$. Note that $C/F^nC$ is projective by (1).  So replace $n$
by $-n$, and dualize.  Thus $0\to (C/F^{-n}C)^*\to (C/F^{1-n}C)^*\to
(G_{-n}C)^*\to 0$ is exact.  But the first term is equal to
$F^{n+1}(C^*)$, and the second, to $F^n(C^*)$.  Thus (3) holds.

{\bf For (4),} note (3) yields $G_\bu(C^*) \in \bb{AG}_A^{\mb h^*}$.  Also,
$F^n(C^*) := (C/F^{1-n}C)^*$ for all $n$ by \eqref{eqFm4}; so $F^\bu
(C^*)$ is finite as $F^\bu C$ is.
\end{proof}

\begin{proposition}\label{prSes}
Let $A$ be a filtered $k$-algebra, and $0 \to B \xto\beta C\xto\gamma
D \to 0$ an exact sequence in $\bb{EF}_A$.

\(1) Assume $D\in \bb{AF}_A$.  Then $B\in \bb{AF}_A$ iff $C\in
\bb{AF}_A$.

\(2) Assume $B,\,C,\,D\in \bb{AF}_A$.  Then $0 \to D^* \xto{\gamma^*}
C^* \xto{\beta^*} B^* \to 0$ is exact in $\bb{EF}_A$.
\end{proposition}

\begin{proof}
{\bf For (1),} given $n$, note $G_n(D )$ is locally free of constant finite
rank.  Hence, $G_n(B)$ is so iff $G_n(C)$ is so, since $0 \to G_n(B) \to
G_n(C) \to G_n(D )\to 0$ is exact by \eqref{prBiex}.  Also, $F^\bu B$ is
finite iff $F^\bu C$ is, since $F^\bu D$ is.  Thus (1) holds.

{\bf For (2),} given $n$, set $p := 1-n$, and form the following commutative
\begin{equation}\label{eqSes1}
\begin{CD}
0 @>>>   D^*    @>>>   C^*   @>>>   B^*  @>>> 0\\
@.      @VVV          @VVV         @VVV\\
0 @>>> (F^pD)^* @>>> (F^pC)^* @>>> (F^pB)^* @>>> 0
\end{CD}
\end{equation}
The upper sequence is exact, since $0 \to B \to C\to D\to 0$ splits as
$D$ is projective by \eqref{prFilt}(1).  Similarly, the lower sequence
is exact.

The sequence $0\to F^pD\to D\to D/F^pD\to 0$ splits as $D/F^pD$ is
projective by \eqref{prFilt}(1).  So the left map in \eqref{eqSes1} is
surjective, and its kernel is $(D/F^pD)^*$, or $F^n(D^*)$ by
\eqref{eqFm4}.  Similarly, the middle and right maps are surjective, and
their kernels are $F^n(C^*)$ and $F^n(B^*)$.  Hence $0\to F^n(D^*)\to
F^n(C^*)\to F^n(B^*)\to 0$ is exact by the 9-Lemma.  Thus (2) holds.
\end{proof}

\section{Dualities}\label{seDu}

\begin{setup}\label{se3}
Keep the setup of \eqref{se2}.  For a $k$-algebra $K$ and a $k$-module
$C$, set $$C_K := C\ox_k K.$$
\indn{CK@$C_K$}

Let $A$ be a filtered $k$-algebra, and $C \in \bb F_A$.  Given a
$k$-algebra $K$, note that $C_K$ carries an induced filtration; by
definition, $F^n(C_K)$ is the image of $(F^nC)_K$ in $C_K$.

Given a contravariant functor $\mb D$ from an additive category $\bb S$
to itself, call $\mb D$ {\it dualizing\/} if $\mb D^2$ is naturally
\indt{dualizing}
isomorphic to the identity functor.  Trivially, $\mb D$ is an
equivalence of $\bb S$ with its opposite category.

Call an object $D$ of $\bb S$ {\it dualizing\/} if $\Hom_{\bb S}(C,\,
D)$ is in $\bb S$ for any $C\in\bb S$ and if the functor $C\mapsto
\Hom_{\bb S}(C,\, D)$ is dualizing.  More generally, given a forgetful
functor from $\bb S$ to another additive category $\bb T$, call an
object $D$ of $\bb S$ {\it dualizing for\/} $(\bb S,\,\bb T)$ if
$\Hom_{\bb T}(C,\, D)$ is in $\bb S$ for any $C\in\bb S$ and if
$C\mapsto \Hom_{\bb T}(C,\, D)$ is dualizing for $\bb S$.
\end{setup}

\begin{proposition}\label{*duality}
Let $A$ be a $k$-algebra. Then $*\:\bb{AM}_A\to \bb{AM}_A$ is a
dualizing functor.  If $A$ is filtered, then $*$ induces two more
dualizing functors: $*\:\bb{AF}_A\to \bb{AF}_A$ and\/ $* \:
\bb{AG}_{G_\bu A} \to \bb{AG}_{G_\bu A}$;  the latter two $*$'s commute
with $G_\bu \: \bb {AF}_A \to \bb {AG}_{G_\bu A}$.

Moreover, if $A$ is filtered, then for any $B,\,C\in \bb{AF}_A$,
\begin{equation}\label{eqduality1}
\Hom_A(B,\,C) = \Hom_A(C^*,\,B^*) \tqn{in} \bb F_A.
\end{equation}

\end{proposition}

\begin{proof}
Given $C\in\bb{AM}_A$, note that the natural map $C\to (C^*)^*$ is an
isomorphism.  Thus $*\:\bb{AM}_A\to \bb{AM}_A$ is a dualizing functor.

Assume $A$ is filtered.  Given $C\in\bb{AF}_A$, note $C\in\bb{AM}_A$ by
\eqref{prFilt}(1); moreover, $C^*\in \bb{AF}_A$ by \eqref{prFilt}(4).
Thus the functor $*\:\bb{AM}_A\to \bb{AM}_A$ induces a functor
$*\:\bb{AF}_A\to \bb{AF}_A$, which is dualizing by \eqref{prFilt}(2).

Given $B,\,C\in \bb{AF}_A$, note that, as $*\:\bb{AF}_A\to \bb{AF}_A$ is
\begin{equation}\label{eqduality2}
\Hom_{\,\bb F_A}(B,\,C) = \Hom_{\,\bb F_A}(C^*,\,B^*) \tqn{in} \bb F_A.
\end{equation}
Given $p$, note  $C^*[p] = (C[-p])^*$.  Hence, replacing $C$ by $C[p]$
in \eqref{eqduality2} yields
\begin{equation*}\label{eqduality3}
F^p\Hom_A(B,\,C) = F^p\Hom_A(C^*,\,B^*).
\end{equation*}
But $F^\bu B$ and $F^\bu C$ are finite; so $F^\bu\Hom_A(B,\,C)$ and $
F^\bu\Hom_A(C^*,\,B^*)$ are finite too.  Thus \eqref{eqduality1} holds.

Given $C\in \bb {AG}_{G_\bu A}$, say $C = \bigoplus_p C_p$.  Note $C^* =
\bigoplus_q (C_{-q})^*$; so $C^*\in \bb{AG}_{G_\bu A}$.  Thus
$*\:\bb{AM}_A\to \bb{AM}_A$ restricts to a dualizing functor
$*\:\bb{AG}_A\to \bb{AG}_A$.  These two functors commute with $G_\bu$
owing to \eqref{prFilt}(3).
\end{proof}

\begin{proposition}[Relative Matlis Duality]\label{prRMD}
Let $A$ be a $k$-Artinian $k$-algebra.  Then $A^*$ is dualizing and
injective for $\bb{AM}_A$.

Assume $A\in \bb{AF}_A$.  Then $A^*$ is dualizing for
$(\bb{AF}_A,\,\bb{M}_A)$, and $(G_\bu A)^*$ is dualizing for
$(\bb{AG}_{G_\bu A},\,\bb{M}_{G_\bu A})$.  Moreover, $\Hom_{G_\bu
A}\bigl(G_\bu C, \,(G_\bu A)^*\bigr) = G_\bu \Hom_A(C, \,A^*)$ for any
$C$ in $\bb{AF}_A$, and $(G_\bu A)^* = G_\bu(A^*)$.  Lastly, if also
$G_0A=k$, then $k$ embeds in $A^*$ as the smallest nonzero submodule $D$
with $A^*/D\in \bb{AF}_A$.
\end{proposition}

\begin{proof}
Note $A^*\in \bb{AM}_A$ and $\Hom_{\bb{AM}_A}(C, \,A^*) =
\Hom_A(C,\,A^*)$ for $C\in \bb{AM}_A$; see \eqref{sbArt}.  However,
$\Hom_A(C,\,A^*) = C^*$ by \eqref{eqeqFm7} with $A$ and $C$ filtered
trivially, and $*$ is dualizing for $\bb{AM}_A$ by \eqref{*duality}.
Thus $A^*$ is a dualizing module for $\bb{AM}_A$.  In $\bb{AM}_A$,
moreover, $A^*$ is injective, as $C\mapsto C^*$ converts projectives
into injectives.

Assume $A\in \bb{AF}_A$.  Then, similarly, \eqref{*duality} and
\eqref{eqeqFm7} imply that $A^*$ is dualizing for $(\bb{AF}_A,\,\bb
M_A)$, and $(G_\bu A)^*$ is dualizing for $(\bb{AG}_{G_\bu A},\,\bb
M_{G_\bu A})$.

Moreover, $(G_\bu C)^* = G_\bu(C^*)$ by \eqref{prFilt}(3).  Apply
\eqref{eqeqFm7} to $G_\bu A$ and to $A$.  Thus $\Hom_{G_\bu A}(G_\bu C,
\,(G_\bu A)^*) = G_\bu \Hom_A(C, \,A^*)$.  On the other hand, taking $C
:= A$ yields $(G_\bu A)^* = G_\bu(A^*)$.

Lastly, let $\delta\: D\into A^*$ be the inclusion of a nonzero
submodule with $A^*/D$ in $\bb{AF}_A$.  Then $D\in \bb{AF}_A$ by
\eqref{prSes}(1).  Set $C:= D^*$.  Then $C\in \bb{AF}_A$ by
\eqref{prFilt}(4).  So $G_0C$ is locally free.  Moreover, $\delta^*\:
A\to C$ is surjective and $C$ has the induced filtration by
\eqref{prSes}(2) and \eqref{sbSm}.  So $G_0A\to G_0C$ is well defined
and surjective.  But $G_0C\neq0$, as $1\in G_0C$ and as $1\neq 0$ since
$G_\bu C$ had the same rank as $C$.

Assume $G_0A=k$.  Then, by the above, $G_0A\risom G_0C$. Hence the
quotient map $\kappa\: A\onto k$ factors via $\delta^*\: A\onto C$.  So
$\kappa^*: k\into A^*$ factors via $\delta\: D\into A^*$.  Now, $F^1A\in
\bb{AF}_A$; so $A^*/k = (F^1A)^*$ by \eqref{prSes}(2), and $(F^1A)^* \in
\bb{AF}_A$ by \eqref{prFilt}(4).  Thus $\kappa^*$ embeds $k$ in $A^*$ as
the smallest nonzero submodule $D$ with $A^*/D\in \bb{AF}_A$.
\end{proof}

\begin{sbs}[Apolarity]\label{sbApl}
Assume $A$ is a graded $k$-algebra.  Let $\mb b\:\Z\to \Z$ a function,
and $B$ a graded $A$-module whose graded pieces $B_p$ are locally free
of rank $\mb b(p)$ for all $p$.  Set $(B^\dg)_q := (B_{-q})^*$ and
\indn{Badg@$B^\dg$}
$B^\dg := \bigoplus_q (B^\dg)_q$. Note that $B^* = \prod(B^\dg)_{q}$,
that $B^\dg$ is an $A$-submodule of $B^*$, that $B^\dg$ is graded, and
that $(B^\dg)_q$ is locally free of rank $\mb b^*(q)$ for all $q$.
Often, $B^\dg$ is called the {\it graded dual\/} of $B$.
\indt{graded dual}

Given any $A$-submodule $I$ of $B$, its {\it apolar annihilator}
\indt{apolar annihilator}
$(0:_{B^\dg}I)$ is defined by
\begin{equation*}\label{eqApN}
(0:_{B^\dg}I) := \{\, f\in B^\dg \mid \psi\cdot f = 0 \text{ for all }
\psi \in I \,\}.
\end{equation*}
\indn{0colI@$(0:_{B^\dg}I)$}
Note that $(0:_{B^\dg}I)$ is an $A$-submodule of $B^\dg$.

Note $B^{\dg\dg} = B$.  Hence, given any $A$-submodule $D$ of
\begin{equation*}\label{eqApN2}
(0:_BD) =  \{\, \psi\in B \mid \psi\cdot f = 0 \text{ for all } f \in D
\,\}.
\end{equation*}

Equip $B$ and $B^\dg$ with the filtrations arising from their
gradings; equip every subquotient of $B$ and $B^\dg$ with the
induced filtration.  Fix a finite function $\mb h\:\Z\to \Z$.

Denote by $\Psi_B$ the set of $A$-submodules $I \subset B$ with $B/I
\in \bb{AM}_A$.  Denote by $\Delta_{B^\dg}$ the set of $A$-submodules
$D\subset B^\dg$ with $D \in \bb{AM}_A$ and with $B^\dg/D$ flat over
\indn{Psibm@$\Psi_B$} \indn{Delbm@$\Delta_{B^\dg}$}

Denote by $\mb F\Psi_B^{\mb h} \subset \Psi_B$ the subset of $I$
\indn{FmbPsihm@$\mb F\Psi_B^{\mb h}$}
\indn{FmbDelhstB@$\mb F \Delta_{B^\dg}^{\mb h^*}$}
with $B/I \in \bb{AF}_A^{\mb h}$, and by $\mb F \Delta_{B^\dg}^{\mb
h^*} \subset \Delta_{B^\dg}$ the subset of $D$ with $D \in
\bb{AF}_A^{\mb h^*}$ and with $G_\bu(E)$ flat over $k$ where $E :=
B^\dg/D$.  Note that $E$ is flat over $k$ when $G_\bu(E)$ is, as $E =
\bigl(\bigoplus_{p\notin [i,\,s]} B^\dg_{-p}\bigr) \oplus
(F^{-s}E/F^{-i+1}E)$ with $i := i(\mb h)$ and $s := s(\mb h)$ and as
$F^{-s}E/F^{-i+1}E$ is flat by the proof of \eqref{prFilt}(1).

Denote by $\mb H\Psi_B^{\mb h} \subset \mb F\Psi_B^{\mb h}$ and
$\mb H\Delta_{B^\dg}^{\mb h^*} \subset \mb
F\Delta_{B^\dg}^{\mb h^*}$ the subset of homogeneous $I$ and the
subset of homogeneous $D$.
\indn{HmbPsihm@$\mb H\Psi_B^{\mb h}$}
\indn{HmbDelhstB@$\mb H\Delta_{B^\dg}^{\mb h^*}$}
\end{sbs}

\begin{theorem}[Macaulay Duality]\label{thGMD3}
Keep the setup of \eqref{sbApl}.

\(1) Then $I\mapsto (0:_{B^\dg}I)$ gives a bijection $\Psi_B
\risom \Delta_{B^\dg}$, and $D\mapsto (0:_BD)$, its inverse.  Also,
$(0:_{B^\dg}I) = (B/I)^*$ and $B/(0:_BD) = D^*$.

Further, if $B = A$ and if $I$ and $D$ correspond, then $A/I$ is a
$k$-algebra, and $D$ is a dualizing module for $\bb{AM}_{A/I}$; also,
then $A/I$ is $k$-Gorenstein iff, for every maximal ideal $\go m$ of
$k$, there's a field $K$ containing $k/\go m$ such that $D_K$ is

\(2) The bijection in \(1) induces another, $\mb F\Psi_B^{\mb h}
\risom \mb F\Delta_{B^\dg}^{\mb h^*}$.  Further, if $A = B$ and
if $I$ and $D$ correspond, then $D$ is a dualizing module for

The second bijection restricts to a third, $\mb H\Psi_B^{\mb h}
\risom \mb H\Delta_{B^\dg}^{\mb h^*} $.  These two bijections
commute with taking associated graded modules.  Furthermore, if $A = B$
and if $I$ and $D$ correspond, then $D$ is a dualizing module for
\end{theorem}

\begin{proof}
{\bf For (1),} given $I\in \Psi_B$, set $C := B/I$.  Note $0\to I\to B\to
C\to 0$ splits over $k$ as $C\in \bb{AM}_A$.  So $0\to C^* \to B^* \to
I^*\to 0$ splits over $k$.  Correspondingly, there's a retraction
$\rho\: B^*\onto C^*$.

As $C\in \bb{AM}_A$, there are $m$ and $n$ with $G_pC = 0$ for $p< m$
and $p>n$.  Hence $B_p$ maps to $0$ in $C$ for $p< m$ and $p > n$.  So
$\bigoplus_{p=m}^n B_p$ surjects onto $C$.  Therefore,
$$\ts C^*\subset \bigoplus_{p=m}^n (B_p)^* \subset B^\dg
\subset B^*.$$

Note $\rho|C^* = 1$.  So $\rho|B^\dg$ splits $0\to C^* \to B^\dg
\to B^\dg/C^*\to 0$ over $k$.  Thus $B^\dg/C^*$ is flat over
$k$, as $B^\dg$ is.  But $B^\dg/C^* \subset B^*/C^* = I^*$.  So
$f\in B^\dg$ maps to $0$ in $I^*$ iff $f\in C^*$; thus $C^* =

However, $C\in \bb{AM}_A$.  So plainly $C^*\in \bb{AM}_A$ .  Thus
$C^*\in \Delta_{B^\dg}$.

Conversely, given $D\in \Delta_{B^\dg}$, note $D$ is finitely
generated over $k$ as $D \in \bb{AM}_A$.  So there are $m,\,n$ with
$D\subset E:= \bigoplus_{p=m}^n (B^\dg)_p$.  Set $E' :=
\bigoplus_{p\notin[m,\,n]} (B^\dg)_p$, so $B^\dg = E\oplus E'$.
Then $B^\dg/D = (E/D) \oplus E'$.  But $B^\dg/D$ is flat.  So
$E/D$ is flat, so projective.  Thus the inclusion $D\into E$ induces a
surjection $E^*\onto D^*$.

Plainly the surjection $(B^\dg)^*\onto E^*$ restricts to a
surjection $(B^\dg)^\dg \onto E^*$.  But $(B^\dg)^\dg =
B$.  Thus the the inclusion $D\into B^\dg$ induces a surjection
$B\onto D^*$; also, it's not just a $k$-map, but an $A$-map.

Set $I := \Ker(B\onto D^*)$.  Then $I$ is an $A$-submodule of $B$, as
$B\onto D^*$ is an $A$-map.  Further, $B/I = D^*$.  But $D \in
\bb{AM}_A$, so $D^* \in \bb{AM}_A$. Thus $I\in \Psi_B$.  Also, $\psi
\in B$ maps to $0$ in $D^*$ iff $\psi\in I$; so $\psi\cdot f = 0$ for
all $f \in D$.  Thus $I = (0:_BD)$.

Next, let's check that the functions $I\mapsto (B/I)^*$ and $D\mapsto
\Ker(B\onto D^*)$ are inverse.  Given $I$, set $D:= (B/I)^*$.  Then $D^*
= B/I$ as $B/I \in \bb{AM}_A$.  Thus $\Ker(B\onto D^*) = I$.
Conversely, given $D$, set $I := \Ker(B\onto D^*)$.  Then $B/I = D^*$.
So $(B/I)^* = D^{**}$.  But $D^{**} = D$ as $D \in \bb{AM}_A$.  Thus
$(B/I)^* = D$, as desired.

Further, assume $A = B$ and $I$ and $D$ correspond.  Set $C:= A/I$.
Then $C$ is, plainly, a $k$-algebra, and $D$ is a dualizing module for
$\bb{AM}_C$ by \eqref{prRMD}, as $D = C^*$.

Also, by \eqref{prGor}, the $k$-algebra $C$ is $k$-Gorenstein iff, for
every $K$, the $C_K$-module $(C^*)_K$ is cyclic, or equivalently,
$(C^*)_K$ is $A_K$-cyclic.  But $D = C^*$.  Thus (1) holds,

{\bf For (2),} given $I \in \mb F\Psi_B^{\mb h}$, set $C:=B/I \in
\bb{AF}_A^{\mb h}$.  Note $C^*\in \bb{AF}_A^{\mb h^*}$ by
\eqref{prFilt}(4).

Fix $p$.  As $G_pC$ is projective, the quotient map $B_p\onto G_pC$ has
a right inverse.  So $G_pC$ is a direct summand of $B_p$; hence
$(B_p)^*/(G_pC)^*$ is one of $(B_p)^*$, so is projective.  But $(B_p)^*
=: (B^\dg)_{-p}$.  And $(G_pC)^* = G_{-p}(C^*)$ by
\eqref{prFilt}(3).  Also, $(B^\dg)_{-p} / G_{-p}(C^*) =
G_{-p}(B^\dg/C^*)$ by \eqref{prBiex}.  Thus $G_{-p}(B^\dg/C^*)$
is projective, so flat; hence, $G_{\bu}(B^\dg/C^*):= \bigoplus
G_{-p}(B^\dg/C^*)$ is flat too.  Thus $C^* \in \mb
F\Delta_{B^\dg}^{\mb h^*}$.

Conversely, given $D\in \mb F\Delta_{B^\dg}^{\mb h^*}$, note
$D^*\in \bb{AF}_A^{\mb h}$ by \eqref{prFilt}(4).  The first assertion of
(2) now follows from (1), and the second, from \eqref{prRMD}.

Plainly, if $I$ and $D$ are homogeneous, then so are $(0:_{B^\dg}I)$ and
$(0:_BD)$.  Thus the second correspondence restricts as asserted.
Finally, \eqref{prRMD} now yields the last two assertions of (2).  Thus
(2) holds.
\end{proof}

\begin{remark}\label{reMacI}
If $B\in\bb{AM}_A$, but $A$ and $B$ are not graded, then \eqref{thGMD3}
is generalized by the statement below.  Its proof is omitted, as it is
similar and simpler.  If $A$ and $B$ are graded, then the statement
below is a corollary of \eqref{thGMD3}.

First, some notation.  Denote by $\mb Q_B$ the set of quotient
$A$-modules $C$ of $B$ such that $C\in\bb{AM}_A$.  If $A$ is filtered
and $B\in \bb{AF}_A$, denote by $\mb {FQ}_B^{\mb h}$ the subset of
$C\in \mb Q_B$ such that, with the induced filtration,
$C\in\bb{AF}_A^{\mb h}$.

Denote by $\mb S_B$ the set of $A$-submodules $D$ of $B$ with $B/D \in
\bb{AM}_A$.  If $A$ is filtered and $B\in \bb{AF}_A$, denote by $\mb
{FS}_B^{\mb h}$ the subset of $D\in \mb S_B$ such that, with the induced
filtrations, $D \in \bb{AF}_A^{\mb h}$ and $B/D \in \bb{AF}_A$.

If $A$ is graded and $B\in\bb{AG}_A$, denote by $\mb {HQ}_B^{\mb h}$
and $\mb {HS}_B^{\mb h}$ the analogues of $\mb {FQ}_B^{\mb h}$ and $\mb
{FS}_B^{\mb h}$ where $C$ and $D$ are homogeneous.

Here's the statement in the corresponding two parts:{\it

\(1) Then $C\mapsto C^*$ gives a bijection $\mb Q_B\risom \mb S_{B^*}$,
and $D\mapsto D^*$, its inverse.

\(2) Assume $A$ is filtered and $B\in\bb{AF}_A$.  Then there's a
commutative square:
\begin{equation*}\label{eqGMD2cd}
\begin{CD}
\mb {FQ}_B^{\mb h} @>\sim>> \mb {FS}_{B^*}^{\mb h^*}\\
@VVV                       @VVV\\
\mb {HQ}_{G_\bu B}^{\mb h} @>\sim>> \mb {HS}_{G_\bu B^*}^{\mb h^*}
\end{CD}
\end{equation*}
The horizontal maps are, as indicated, bijective; they're given by
$C\mapsto C^*$, and their inverses, by $D\mapsto D^*$.  The vertical
maps are given by $C\mapsto G_\bu C$ and $D\mapsto G_\bu D$.}
\end{remark}

\begin{example}\label{exPolyRg}
Fix variables $X_1,\dotsc,X_r$; take $A := k[X_1,\dotsc,X_r]$.  Weight
$X_i$ by a positive integer $w_i$; grade the polynomial ring $A$ by
weighted degree.  Thus the graded piece $A_n$ is the free $k$-module on
the monomials $M$ of weighted degree $n$.

The graded dual $A^\dg$ can be usefully viewed as the free
$k$-module on all the Laurent monomials $1/M$, namely as
$k[X_1^{-1},\dotsc,X_r^{-1}]$, provided the product of a monomial $L\in
A$ with $1/M$ is taken to be $L/M$ if $L\mid M$ and to be $0$ if not.
Then the graded piece $(A^\dg)_{-n}$ is the free $k$-module on the
$1/M$ of weighted degree $-n$.

As an abstract $A$-module, $A^\dg$ is, plainly, independent of the
choice of the $w_i$.  Moreover (cf.\ \cite{KES}*{Subsec.\,3.2}), $A^\dg$
consists of the maps $A\to k$ that are continuous when $k$ is given the
discrete topology and $A$ is given the topology defined by the
filtration with $F^pA$ generated by the monomials of degree at least
$p$; notice that two choices of the $w_i$ give rise to cofinal
filtrations, and so the same topology on $A$.  Thus, a better name for
$A^\dg$ would be the {\it continuous dual}.
\indt{continuous dual}

If $w_i=1$ for all $i$, then $A^\dg$ is equal to the $k$-algebra of
divided powers; see \cite{Eis}*{Sec.\,A2.4} or \cite{JPAA217}*{Sec.\,3,
pp.\,2258--59}.  If $k=\bb C$ too, then $A^\dg$ is the usual algebra of
polynomial differential operators on $A$; see \cite{Eis}*{Prp.\,A2.8}.
Remarkably, in this case, $A$ can as well be viewed as the algebra of
polynomial differential operators on $A^\dg$, viewed as a polynomial
ring in the $X_i^{-1}$; see \cite{Eis}*{Ex.\,21.7}.  These additional
structures certainly have their charm (and in \eqref{sbApl}, justify the
term ``apolarity''), but in the present work, they'd just add an
unnecessary layer of complication.

In \eqref{thGMD3}(2), take $B := A$.  Note that an $A$-submodule $D$ of
$B^\dg$ is finitely generated over $A$ iff it's finitely generated over
$k$.  Also, note that, if $0\neq C:= A/I$, then $I\subset F^1(A):= \la
X_1,\dotsc,X_r\ra$ (and conversely) because $G_0C = k$ since $G_0A\to
G_0C$ is surjective, $G_0A =k$, and $G_0C$ is locally free and is
nonzero as $1\in G_0C$ and $G_\bu C$ is of the same rank as $C$, so is
nonzero.  Thus \eqref{thGMD3}(2) recovers Macaulay Duality essentially
as is stated in \cite{Eis}*{Thm.\,21.6}, but $k$ is any Noetherian ring
in \eqref{thGMD3}(2), not just a field, and the weights $w_i$ needn't be
1.  In addition, \eqref{thGMD3}(2) asserts the compatibility of duality
with forming associated graded modules.

Finally, let $\vf\: C\to A^*$ be an $A$-map.  Say $A_n\subset I$ for $n>
n_0$.  Then, given any monomial $M\in A_n$ with $n> n_0$, note $M\cdot C
= 0$; hence, $M\cdot \vf(C) = 0$.  Now, given any $c\in C$, say $\vf(c)
= \prod_{n\ge0} \bigl(\bigoplus_i (a_{in}/N_{in})\bigr)$ where
$a_{in}\in k$ and $N_{in}\in A_n$.  If $a_{jp}\neq 0$, then $N_{jp}\cdot
\vf(c) \neq 0$, and so $p\le n_o$.  Thus $\vf(c)\in
\bigoplus_{n=0}^{n_0}(A_n)^*$.  Thus $\vf(C)\subset A^\dg$.  Thus
$\Hom_A(C,\,A^*) = \Hom_A(C,\,A^\dg)$.  But $C^* = \Hom_A(C,\,A^*)$ by
\eqref{eqeqFm7}.  Thus $C^* = \Hom_A(C,\,A^\dg)$, as shown in a
different way assuming $k$ is a field in the course of the proof of
\cite{Eis}*{Thm.\,21.6, p.\,531}.
\end{example}

\section{The  {\it k}-socle}\label{seSoc}

\begin{setup}\label{se4}
Keep the setup of \eqref{se3}

Fix a filtered $k$-algebra $A$ with $A/F^1A =k$; that is, the
composition $k\to A \to k$ is the identity.
Fix a finite function $\mb h \:\Z\to \Z$, and $C\in \bb{AF}_A^{\mb h}$.
\begin{equation*}\label{eqse4}
i := i(\mb h) \and s:= s(\mb h) \and t:= \mb h(s).
\end{equation*}
Note $C\big/((F^1A)\cdot C) = C\ox_Ak$, and  $C\ox_A k$ has the
filtration induced from $C$.
\indn{ihmb@$i(\mb h)$} \indn{shmb@$s(\mb h)$} \indn{tmi@$t$}

Given a $k$-algebra $K$, note that $C_K$ carries an induced filtration;
by definition, $F^n(C_K)$ is the image of $(F^nC)_K$ in $C_K$.
\end{setup}

\begin{definition}\label{deSoc}
\indt{kso@$k$-socle}
The {\it $k$-socle\/} of $C$ and its induced filtration are given by
\indt{kso@$k$-socle}
\begin{gather*}\label{eqDefSoc}
\Soc_k(C) := \Hom_A(k,\,C) = \{\,c\in C\mid (F^1A)\cdot c = 0\,\}
=: (0:_C F^1A) \subset C,\\
F^n(\Soc_k(C)) := F^n\Hom_A(k,\,C)
= \Soc_k(C)\bigcap F^nC \quad\text{for all }n.
\end{gather*}
\end{definition}

\begin{lemma}\label{leSoc}
In $\bb F_A$, there's a canonical isomorphism: $\Soc_k(C) = (C^*\ox_A
\end{lemma}

\begin{proof}
Taking $B:=k$ in \eqref{eqduality1} and taking $B:= C^*$ and $C:=k$
in \eqref{eqeqFm6} yield
\begin{equation*}\label{eqSoc2}
\Hom_A(k,\,C) = \Hom_A(C^*,\,k^*) = (C^*\ox_A k)^* \tqn{in} \bb F_A.
\end{equation*}
Thus \eqref{deSoc} yields the assertion.
\end{proof}

\begin{lemma}\label{leSocLev}
\(1) Then $G_iC \risom G_i(C\ox_A k)$.\hfill
\(2) Then $G_s(\Soc_k(C)) = G_sC$.

\(3) Choose a $k$-submodule $N\subset C$ that maps isomorphically onto
$G_iC$ (such an $N$ exists because $G_iC$ is projective).  Then $A\cdot
N = C$ iff $C\ox_Ak = G_iC$.

\(4) Then \(a) $\Soc_k(C) = G_sC$ and $C^*\ox_A k\in \bb{AM}_A$ iff\/
\(b) $C^*\ox_Ak = G_{-s} (C^*)$.
\end{lemma}

\begin{proof}
{\bf For (1),} recall $C\ox_A k = C/(F^1A)C$.  Let $\kappa\:C\onto
C\ox_A k$ be the quotient map.  Then $F^n(C\ox_A k) := \kappa(F^nC)$ for
all $n$; so $\kappa^{-1}(F^n(C\ox_A k)) = F^nC+\Ker(\kappa)$.  But
$\Ker(\kappa) = (F^1A)C$ and $(F^1A)C \subset F^{1+i}C$.  So
$\kappa^{-1}(F^{1+i}(C\ox_A k)) = F^{1+i}C $.  Thus $C/ (F^{1+i}C)
\risom (C\ox_A k)/ (F^{1+i}(C\ox_A k)) $.  Thus (1) holds.

{\bf For (2),} note $F^{s+1}C=0$.  So $F^{s+1}(\Soc_k(C)) =0$.  But
$(F^1A)(F^sC)\subset F^{1+s}C$.  So $F^sC \subset \Soc_k(C)$.  So $F^sC
= F^s(\Soc_k(C))$.  Thus (2) holds.

{\bf For (3),} set $B := A/F^{s-i+1}A$.  Then $C$ is a $B$-module.  Set $\go N
:= F^1B$.  Then $\go N^{s-i+1} = 0$, and $B/\go N = k$.  So by
Nakayama's lemma, $B\cdot N= C$ iff $(B\cdot N)\ox_Bk$ maps onto
$C\ox_{B}k$, iff $N \to C\ox_{B} k$ is surjective.  However, $B\cdot N=
A\cdot N$ and $C\ox_B k = C\ox_A k$.  Thus $A\cdot N = C$ iff $N\to
C\ox_A k$ is surjective.

By hypothesis, $N\risom G_iC$.  But $G_iC = G_i(C\ox_A k)$ by (1).  So
$N\risom G_i(C\ox_A k)$.  The latter factors through $C\ox_A k$.  So
$N\to C\ox_A k$ is injective; so surjective iff it's an isomorphism; so
iff $C\ox_A k\to G_iC$ is an isomorphism.  Thus (3) holds.

{\bf For (4),} first assume (a).  Then $\Soc_k(C)^* = (G_sC)^*$ and
$(C^*\ox_Ak)^{**} = C^*\ox_Ak$.  Thus \eqref{leSoc} and
\eqref{prFilt}(3) yield (b).  Conversely, assume (b). Then $C^*\ox_A
k\in \bb{AM}_A$ as $G_{-s} (C^*)\in \bb{AM}_A$ by \eqref{prFilt}(4).
Also, $(C^*\ox_Ak)^* = (G_{-s} (C^*))^*$.  So \eqref{leSoc} and
\eqref{prFilt}(3) yield $\Soc_k(C)= ((G_sC)^*)^* = G_sC$.  Thus (a)
holds.  Thus (4) holds.
\end{proof}

\begin{definition}\label{deType}
Fix a finite function $\mb t\:\Z\to \Z$.

Call $\mb t$ the {\it local generator type\/} of $C$ if $C\ox_A k$ is in
\indt{local generator type}
$\bb{AF}_A$ with Hilbert function $\mb t$.
\indt{local generator type}

Call $\mb t$ the {\it $k$-socle type\/} of $C$ if $C^*$ has local
\indt{ksot@$k$-socle type} \indt{type (level of)}
generator type $\mb t^*$.  (Then $\Soc_k(C)$ is in $\bb{AF}_A$ and has
Hilbert function $\mb t$ by \eqref{leSoc} and \eqref{prFilt}(4), but
perhaps not conversely.)

Call $C$ {\it level\/} if $C^*\ox_Ak = G_{-s} (C^*)$, or equivalently by
\indt{level}
\eqref{leSocLev}(4), if $\Soc_k(C) = G_sC$ and $C^*\ox_A k\in
\bb{AM}_A$.  If $C$ is graded, then $C$ is level iff $(C^*)_{-s}$
generates $C^*$ by \eqref{leSocLev}(3).  If $C$ is level, call $s$ its
{\it $k$-socle degree\/} and $t$ its {\it type}.
\indt{ksod@$k$-socle degree}
\end{definition}

\begin{lemma}\label{leDiag}
There's a canonical commutative diagram of surjective maps:
\begin{equation}\label{eqprGAA1}
\begin{CD}
G_\bu(C^*) @>{\kappa_1}>> G_\bu(C^*)\ox_{G_\bu A} k @>{\lambda_1}>> G_{-s}(C^*) \\
@VV=V                            @VV\mu V               @VV=V  \\
G_\bu(C^*) @>{\kappa_2}>> G_\bu(C^*\ox_A k) @>{\lambda_2}>> G_{-s}(C^*)
\end{CD}
\end{equation}
\end{lemma}

\begin{proof}
Plainly, in $G_\bu(C^*)$, there's this nested sequence of ideals:
\begin{equation*}\label{eqleDiag} \ts
\bigl(\bigoplus_{j>0}G_jA\bigr)\cdot G_\bu(C^*) \subset
G_\bu\bigl((F^1A)\cdot C^*\bigr) \subset \bigoplus_{j>-s} G_j(C^*).
\end{equation*}
A few applications of Noether's Third Isomorphism Theorem now yields
\eqref{eqprGAA1}.
\end{proof}

\begin{proposition}\label{prGAA}
Assume $G_\bu C$ is level.  Then so is $C$.
\end{proposition}

\begin{proof}
Consider\eqref{eqprGAA1}.  As $G_\bu C$ is level, $\lambda_1$ is
bijective.  So $\mu$ is injective.  But $\mu$ is surjective.  So $\mu$
is bijective.  So $\lambda_2$ is bijective.  So $G_j(C^*\ox k)=0$ for
$j\neq -s$ by \eqref{leSocLev}(1) with $C^*$ for $C$.  So $G_\bu(C^*\ox k) =
C^*\ox k$.  Thus $C^*\ox k = G_{-s}(C^*)$; that is, $C$ is level.
\end{proof}

\begin{lemma}\label{leBC1}
Let $V$ and $W$ be $k$-modules, with $V$ finitely generated.

\(1) Let $K$ be a $k$-algebra.  Assume $V$ is flat.  Then $\Hom(V,W)_K =
\Hom(V_K,W_K)$ and $(V^*)_K = (V_K)^*$.

\(2) Assume that, for every maximal ideal $\go m$ of $k$, there's a
field $K$ containing $k/\go m$ such that $V_K = 0$.  Then $V = 0$.

\(3) Let $\vf\: V\to W$ be a map.  Assume $W$ is flat and finitely
generated.  Assume, for every maximal ideal $\go m$ of $k$, there's a
field $K$ containing $k/\go m$ such that the induced map $\vf_K\: V_K\to
W_K$ is an isomorphism.  Then $\vf$ is an isomorphism.
\end{lemma}

\begin{proof}
{\bf In (1),} as $V$ is locally free of finite rank, $\Hom(V,W)_K\to
\Hom(V_K,W_K)$ is an isomorphism.  And so, taking $W:=k$ yields $(V^*)_K
= (V_K)^*$.

{\bf For (2),} note $V = 0$ if, for every $\go m$, the localization $V_{\go
m}$ is 0.  By Nakayama's Lemma, $V_{\go m} = 0$ if $V_{\go m}\ox_{k_\go
m}(k_\go m/\go m k_\go m) = 0$; so, if $V\ox_k(k/\go m) = 0$;
so, if $V\ox_kK = 0$.  Thus (2) holds.

{\bf For (3),} set $Q := \Cok \vf$.  Then $V_K\xto{\vf_K} W_K\to Q_K\to 0$ is
exact for every $K$. But $\vf_K$ is an isomorphism.  So $Q_K = 0$.  But
$W$ is finitely generated; so $Q$ is too.  Thus (2) yields $Q = 0$.

Set $P := \Ker \vf$.  As $W$ is flat, $0\to P_K\to V_K\xto{\vf_K} W_K
\to 0$ is exact for every $K$.  But $\vf_K$ is an isomorphism. So $P_K =
0$.  But $V$ is finitely generated; so $P$ is too. So $P = 0$ by (2).
Thus (3) holds.
\end{proof}

\begin{lemma}\label{leBC1a}
Let $B$ and $K$  be $k$-algebras, and $M$ and $N$ two $B$-modules.  Then
\begin{equation}\label{alBaseChg6}
(M\ox_BN)_K = (M_K)\ox_{B_K}(N_K).
\end{equation}
\end{lemma}

\begin{proof}
By Associativity, $(M\ox_BN)_K = M\ox_BN_K$.  But $N_K$ is a
$B_K$-module, and it's an $B$-module via restriction of scalars.  So
$M\ox_BN_K = (M\ox_BB_K)\ox_{B_K}N_K$ by Cancellation.  But $M\ox_BB_K =
M_K$ also by Cancellation.  Thus \eqref{alBaseChg6} holds.
\end{proof}

\begin{lemma}\label{leBaseChg}
Let $K$ be a $k$-algebra, and $n\in\Z$.

\smallskip

\(1) Then the defining surjection is an isomorphism: $(F^nC)_K
\risom F^n(C_K)$.

\(2) Then there's a canonical isomorphism:  $G_n(C_K) = (G_nC)_K$.

\(3) Then the conditions of \eqref{se4} hold for $A_K$ and $C_K$ in
place of $A$ and $C$.

\(4) Then there are canonical isomorphisms of filtered modules:
\begin{align}
(C^*)_K &= (C_K)^*                     \label{alBaseChga}\\
(C^*\ox_Ak)_K &= (C_K)^*\ox_{A_K}K      \label{alBaseChgb}
\end{align}
\end{lemma}

\begin{proof}
{\bf For (1),} tensor the sequence $0\to F^{n+1}C \to F^nC\to G_nC\to
0$ with $K$.  The resulting sequence is exact, because $G_nC$ is flat.
So $(F^{n+1}C)_K \subset (F^nC)_K$ and
\begin{equation}\label{eqBaseChg3}
(F^nC)_K \big/ (F^{n+1}C) _K = (G_nC)_K.
\end{equation}
But $F^nC = C$ for $n\ll0$.  So $(F^nC)_K \subset C_K$ for all
$n$.  Thus (1) holds.

{\bf Note (2)} follows from \eqref{eqBaseChg3} and (1).

{\bf For (3),} tensor $0\to F^1A\to A\to k\to 0$ with $K$, getting an exact
sequence.  So $(F^1A)_K = F^1(A_K)$ and $A_K\big/(F^1A)_K = K$, as

Recall $G_nC$ is locally free of constant rank $\mb h(n)$ over $k$;
hence, $G_n(C_K)$ is so over $K$ by (2).  Hence $C_K$ is in
$\bb{AF}_{A_K}$ and has Hilbert function $\mb h$. Thus (3) holds.

{\bf For (4),} note \eqref{alBaseChga} results from these canonical
isomorphisms, justified after:
\begin{align}
F^n((C^*)_K)   &= (F^n(C^*))_K \label{alBaseChg1}\\
&= ((C/F^{1-n}C)^*)_K\label{alBaseChg2}\\
&= ((C/F^{1-n}C)_K)^*\label{alBaseChg3}\\
&= (C_K\big/(F^{1-n}C)_K)^*\label{alBaseChg4}\\
&= (C_K\big/F^{1-n}(C_K))^*\label{alBaseChg4a}\\
&= F^n((C_K)^*)\label{alBaseChg5}.
\end{align}

First, \eqref{alBaseChg1} holds by (1), as $C^* \in \bb{AF}_A$ by
\eqref{prFilt}(4).  Next, \eqref{alBaseChg2} and \eqref{alBaseChg5} hold
by \eqref{eqFm4}.  Next, $C/F^{1-n}C$ is locally free of finite rank by
\eqref{prFilt}(1); so \eqref{leBC1}(1) yields \eqref{alBaseChg3}. Next,
\eqref{alBaseChg4} holds by the right exactness of $\bu\ox_kK$.
Finally, \eqref{alBaseChg4a} holds by (1).  Thus \eqref{alBaseChga}

Take $B:=A$ and $N := k$ in \eqref{alBaseChg6}.  Note $N_K = k_K$ where
$k$ is viewed as a $k$-module via the composition $k\to A\to k$.  But
that's the identity.  Thus $N_K = K$.

Take $M := C^*$ also.  Then \eqref{alBaseChg6} becomes $(C^*\ox_Ak)_K =
(C^*)_K \ox_{A_K}K$.  This canonical isomorphism preserves the
filtrations, because the $n$th term on each side is, by definition, the
image of $(F^n(C^*))_K$.  Thus \eqref{alBaseChga} now yields
\eqref{alBaseChgb}.
\end{proof}

\begin{proposition}\label{prFib}
Given a function $\mb t\:\Z\to \Z$, these conditions are equivalent:
\begin{enumerate}
\item The $A$-module $C$ has $k$-socle type $\mb t$.

\item For every $k$-algebra $K$, the $A_K$-module $C_K$ has $K$-socle
type $\mb t$.

\item
\begin{enumerate} \def\thenenumii{\alpha{enumii}}
\item The $k$-module $G_n(C^*\ox_Ak)$ is flat for $-s< n \le -i$.

\item For every maximal ideal $\go m$ of $k$, there's a field $K$
containing $k/\go m$ such that $\Soc_K(C_K)$ has Hilbert function
$\mb t$.
\end{enumerate}
\end{enumerate}
\end{proposition}

\begin{proof}
Assume (1).  By \eqref{leBaseChg}(3), the conditions of \eqref{se4}
hold for the pair $A_K,\,C_K$ and also, owing to (1), for the pair
$A_K,\,(C^*\ox_Ak)_K$ with $\mb t^*$ for $\mb h$.  Hence,
\eqref{alBaseChgb} implies that $(C_K)^*\ox_{A_K}K$ is in
$\bb{AF}_{A_K}$ and has Hilbert function $\mb t^*$.  Thus (2) holds.

Assume (2).  Then (3) holds as it's a special case of (2).

Assume (3).  Fix $n$.  Note $C^*\in \bb{AF}_A$ with Hilbert polynomial
$\mb h^*$ by \eqref{prFilt}(4).  So \eqref{prFilt}(1) with $C^*$ for $C$
implies $C^*$ is finitely generated over $k$.  So $C^*\ox_A k$ is too.
Thus $G_n(C^*\ox_A k)$ is.

If $n\le-s$, then $F^n(C^*) = C^*$, so $F^n(C^*\ox_Ak) = C^*\ox_Ak$.  If
$n>-i$. then $F^{n}(C^*) = 0$, so $F^{n}(C^*\ox_Ak) = 0$.  Thus
$G_n(C^*\ox_Ak) = 0$ if $n<-s$ or $n>-i$.

However, $G_{-s}(C^*\ox_Ak) = G_{-s}(C^*)$ by \eqref{leSocLev}(1); so
$G_{-s}(C^*\ox_Ak)$ is flat.  Thus (a) implies that, whatever $n$ is,
$G_n(C^*\ox_Ak)$ is flat, so locally free of finite rank.  It remains to
show that $G_n(C^*\ox_Ak)$ has constant rank $\mb t^*(n)$.

So given any $\go m$ and $K$ as in (b), we have to see
$\dim_K((G_n(C^*\ox_Ak))_K)= \mb t^*(n)$.  But $(G_n(C^*\ox_Ak))_K =
G_n((C^*\ox_Ak)_K)$ by the proof of \eqref{leBaseChg}(2) with
$C^*\ox_Ak$ for $C$, as that proof doesn't require the rank to be

So by \eqref{alBaseChgb}, we have to see $(C_K)^*\ox_{A_K}K$ has Hilbert
function $\mb t^*$, or by \eqref{prFilt}(3), equivalently,
$((C_K)^*\ox_{A_K}K)^*$ has Hilbert function $\mb t$.  Apply
\eqref{leSoc} with $ C_K$ for $C$, noting $C_K\in \bb{AF}_{A_K}$ as $K$
is a field.  Thus (b) now yields (1).
\end{proof}

\begin{corollary}\label{CoLevel}
The following conditions are equivalent:
\begin{enumerate}
\item The $A$-module $C$ is level of type $t$.

\item For every $k$-algebra $K$, the $A_K$-module $C_K$ is level of type

\item For every maximal ideal $\go m$ of $k$, there is a field $K$
containing $k/\go m$ such that $$\dim_K(\Soc_K(C_K)) = t.$$
\end{enumerate}
\end{corollary}

\begin{proof}
Define $\mb t\:\Z\to \Z$ by $\mb t(s) := t$ and $\mb t(j) := 0$ for
$j\neq s$.  Then \eqref{CoLevel} results from \eqref{prFib}, because
\eqref{CoLevel}(3) implies $G_n(C^*\ox_Ak)=0$ for $-s< n$, as shown

We have to show $F^{1-s}(C^*\ox_Ak)=0$, or equivalently, $C^*\ox_Ak
\risom G_{-s}(C^*\ox_Ak)$.  But $G_{-s}(C^*\ox_Ak) = G_{-s}(C^*)$ by
\eqref{leSocLev}(1) with $C^*$ for $C$.  Thus there's a canonical surjection
$\vf\: C^*\ox_Ak \onto G_{-s}(C^*)$, and we have to show it's an

So given any $\go m$ and $K$ as in (3), it suffices by \eqref{leBC1}(3)
to show $\vf\ox_k K$ is an isomorphism, as $G_{-s}(C^*)$ is projective.
Since $\vf\ox_k K$ is surjective and $K$ is a field, it suffices to show
its source and target have the same dimension over $K$.

The source is equal to $(C_K)^*\ox_{A_K}K$ by \eqref{alBaseChgb}.  Its
dimension is equal to that of $((C_K)^*\ox_{A_K}K)^*$, as $K$ is a
field; so to that of $\Soc_K(C_K)$ by \eqref{leSoc}; so to $t$ by (3).
Finally, the target is $(G_{-s}(C^*))_K$, and $G_{-s}(C^*)$ is locally
free of rank $\mb h^*(-s)$ by \eqref{prFilt}(4); so the dimension of the
target is $\mb h(s) =: t$, as desired.
\end{proof}

\begin{sbs}[Multilevel modules]\label{sbMLev}
First, assume $A$ and $C$ are graded.  Set $D :=C^*$,
\begin{equation}\label{eqMlev}\ts
\Delta^mD := A\bigl(\bigoplus_{j= m}^sD_{-j}\bigr) \subset D
\and \Lambda^mC := (\Delta^mD)^* \quad\text{for all }m.
\end{equation}
Fix $\mb h_m\:\Z\to \Z$ for all $m$.  If $\Delta^mD \in
\mb{H\Delta}_D^{\mb h_m^*}$ for all $m$, call $C$ \textit{multilevel}
and the $\mb h_m$ \textit{its Hilbert functions}.
\indt{multilevel} \indn{hmbzmbm@$\mb h_m$} \indn{LamC@$\Lambda^mC$}

Given such a $C$, note each $\Lambda^mC$ is a flat quotient of $C$.
Moreover, $\Delta^mD = D$ for $m\le i$, and $\Delta^mD = 0$ for $m > s$;
so $\mb h_m = \mb h$ for $m\le i$, and $\mb h_m =0$ for $m > s$.  Also,
$C$ is level iff $h_m = h$ for $m\le s$, iff $\Lambda^mC = C$ for $m\le
s$.  Lastly, for all $m$ and $p \ge m$, plainly $(\Lambda^mC)_p =C_p$,
and so $\mb h_m(p) = \mb h(p)$.

Next, assume $C$ is filtered.  If the associated graded module $G_\bu C$
is multilevel (but not necessarily of the same $k$-socle type as $C$), call
$C$ \textit{multilevel}.  And if the $\mb h_m$ are the Hilbert functions
of $G_\bu C$, call them the \textit{Hilbert functions} of $C$ as well.

Notice that, if $C$ is level filtered, $C$ need not be multilevel, as
$G_\bu C$ may be of a different $k$-socle type.  For example, below, the
algebra $C$ in \eqref{exArt} is $k$-Gorenstein, so level of type 1 by
\eqref{prArt}, but $G_\bu C$ isn't $k$-Gorenstein when $C$ is filtered
via the standard weighting, although $G_\bu C$ is $k$-Gorenstein via a
different weighting.

Finally, given a graded $A$-module $B$, denote the sets of all
multilevel homogeneous and of all multilevel filtered quotients of $B$
by $\mb{H\Lambda}_B^{\{\mb h_n\}}$ and by $\mb{F\Lambda}_B^{\{\mb
\indn{HmbLamhBrm@$\mb{H\Lambda}_B^{\{\mb h_m\}}$}
\indn{FmbLamhBrm@$\mb{F\Lambda}_B^{\{\mb h_m\}}$}
\end{sbs}

\begin{proposition}\label{prMLev}
In the setup of \eqref{sbMLev}, fix $C \in \mb{H\Lambda}_B^{\{\mb
h_n\}}$, fix a $k$-algebra $K$, and fix an integer $n$. For all $p$, set
$\mb t(p) := \mb h_p(p) - \mb h_{p+1}(p)$.

\(1) Then $C_K \in \mb{H\Lambda}_{B_K}^{\{\mb h_n\}}$ and
$\Lambda^m(C_K) = (\Lambda^mC)_K$ and $\Delta^m(C^*_K) =

\(2) Then $\mb t$ is the $k$-socle type of $C$.

\(3) Set $\mb h^\di_m := \mb h_n$ for $m\le n$ and $\mb h^\di_m := \mb h_m$
for $m\ge n$.  Then $\Lambda^nC \in \mb{H\Lambda}_{B_K}^{\{\mb h^\di_n\}}$.

\(4) Assume $\mb t(n-1)\neq0$, and set $\?C := \bigl(\Delta^{n-1}D
\big/ \Delta^nD\bigr)^*$ and\/ $\?{\mb h} := \mb h_{n-1} - \mb h_n$.
Then $\?C$ is level, and $\?{\mb h}$ is its Hilbert function.
\end{proposition}

\begin{proof}
{\bf  In (1),} the assertions are easy to check.

{\bf For (2),} by \eqref{deType}, we have to show that $D_{-p}\big/(F^1A\cdot
D)_{-p}$ is locally free of rank $\mb t(p)$.  But $(F^1A\cdot D)_{-p} =
(\Delta^{p+1}D)_{-p}$, and $\Delta^{p+1}D \in \mb {H\Delta}_D^{\mb
h_{p+1}^*}$.  Thus (2) holds.

{\bf For (3),} note $\Delta^m\Delta^nD = \Delta^nD$ for $m\le n$ and
$\Delta^m\Delta^nD= \Delta^mD$ for $m\ge n$.  For $m \ge n$, form the
following exact sequence:
\begin{equation}\label{eqprMLev}
0 \to \Delta^nD \big/\Delta^mD \to D\big/ \Delta^mD \to D\big/ \Delta^nD \to 0.
\end{equation}
For all $p$, recall $\Delta^pD \in \smash{\mb{ H\Delta}}_D^{\mb
h_p^*}$.  Hence $\Delta^m\Delta^nD \in \smash{\mb
{H\Delta}}_{\Delta^nD}^{\mb h^{\di*}_m}$.  Thus (3) holds.

{\bf For (4),} note $\?C \in \bb {AG}_A^{\?{\mb h}}$ by \eqref{eqprMLev} with
$n-1$ for $n$ and with $n$ for $m$.  However, plainly,
$(\Delta^{n-1}D)_m = (F^1A\cdot \Delta^{n-2}D)_m$ for $m>-n+1$, and
$(\?C^*)_m = 0$ for $m<-n+1$; so $\?C^*\ox k = (\?C^*)_{-n+1}$.  Thus
(4) holds.
\end{proof}

\begin{proposition}\label{prMLevCon}
Keep the setup of \eqref{sbMLev}, and fix $n$.  Set $\?C := \bigl(D
\big/ \Delta^nD\bigr)^*$ and\/ $\?{\mb h} := \mb h - \mb h_n$.  Set $\mb
h'_m := \mb h_n$ for $m\le n$ and $\mb h'_m := \mb h_m$ for $m\ge n$.
Assume $\?C$ is level, and $\?{\mb h}$ is its Hilbert function.  Assume
$\Lambda^nC \in \mb{H\Lambda}_B^{\{\mb h'_n\}}$ and  $\mb h_m = \mb
h$ for $m< n$.  Then $C \in \mb{H\Lambda}_B^{\{\mb h_n\}}$.
\end{proposition}

\begin{proof}
We have to show $\Delta^mD \in \mb{H\Delta}_D^{\mb h_m^*}$ for all $m$.

Assume $m < n$.  Set $\?D := D/\Delta^nD = \?C^*$.  Since $\?C$ is
level, $\Delta^m\?D = \?D$.  But $\Delta^m\?D = \Delta^m D/\Delta^nD$.
Hence $\Delta^mD = D$.  By hypothesis, $\mb h_m = \mb h$; so $\mb h_m^*
= \mb h^*$.  Thus $\Delta^mD \in \mb{H\Delta}_D^{\mb h_m^*}$.

Assume $m\ge n$.  Then $\Delta^m\Delta^nD = \Delta^mD$.  But
$\Lambda^nC$ is multilevel, and the $\mb h'_m$, its Hilbert functions.
Thus $\Delta^nD \big/ \Delta^mD$ is flat, and $\Delta^mD$ has $\mb
h_m^*$ as Hilbert function.

As $\?C$ is level, $D\big/ \Delta^nD$ is flat.  But \eqref{eqprMLev} is
exact.  Thus $D/\Delta^mD$ is flat.
\end{proof}

\section{{\it k}-Gorenstein algebras}\label{seGor}

\begin{setup}\label{se5}
Keep the setup of \eqref{se3}.  In addition, recall from \eqref{sbArt},
that we call $A$ {\it k-Gorenstein\/} if $A^*$ is a locally free of
\indt{kGor@$k$-Gorenstein}
rank 1 (that is, invertible).  \end{setup}

\begin{proposition}\label{prGor}
Let $A$ be a $k$-Artinian $k$-algebra. Then these are equivalent:

\begin{enumerate}
\item The $k$-algebra $A$ is $k$-Gorenstein.

\item For every $k$-algebra $K$, the $K$-algebra $A_K$ is

\item For every maximal ideal $\go m$ of $k$, there is a field $K$
containing $k/\go m$ such that the $K$-algebra $A_K$ is $K$-Gorenstein.

\item For every maximal ideal $\go m$ of $k$, there is a field $K$
containing $k/\go m$ such that the $A_K$-module $(A^*)_K$ is cyclic.
\end{enumerate}
\end{proposition}

\begin{proof}
Assume (1).  Then $(A^*)_K$ is an invertible $A_K$-module for every
$k$-algebra $K$.  But $(A^*)_K = (A_K)^*$ by \eqref{leBC1}(1), and
plainly, $A_K$ is $K$-Artinian.  Thus (2) holds.

Trivially, (2) implies (3).

Assume (3).  Then, given $K$ as in (3), the $A_K$-module $(A_K)^*$ is
invertible.  But $A_K$ is semilocal.  Hence, $(A_K)^*$ is free of rank
1; so, cyclic.  But $(A^*)_K = (A_K)^*$ by \ref{leBC1}(1).  Thus (4)

Assume (4).  Given $\go m$, set $L := k/\go m$.  Below we find an $f\in
A^*$ whose image in $(A^*)_L$ generates it over $A_L$.  Given $f$,
define $\vf\: A\to A^*$ by $\alpha\mapsto \alpha f$.  Then the induced
$A_L$-map $\vf_L\: A_L\to (A^*)_L$ is surjective.  But $A$ and $A^*$ are
locally free of the same rank at $\go m$.  So $A_L$ and $(A^*)_L$ have
the same $L$-dimension.  So $\vf_L$ is an isomorphism.  So the
localization $\vf_\go m$ is an isomorphism by \eqref{leBC1}(3).  Thus
(1) holds.

To find $f$, let $\?K$ be the algebraic closure of $K$.  By
Cancellation, $A_{\?K} = A_K \ox_K\?K$ and $(A^*)_{\?K} = (A^*)_K
\ox_K\?K$.  Thus $(A^*)_{\?K}$ is a cyclic $A_{\?K}$-module.

Note $A_L$ is a finite product of Artinian local $L$-algebras $A_i$.  In
turn, each $A_i\ox_L\?K$ is a finite product of Artinian local
$\?K$-algebras $B_{ij}$.  As $\?K$ is algebraically closed, the residue
field of each $B_{ij}$ is $\?K$.

Note $A^*\ox_AB_{ij}$ is a quotient of $(A^*\ox_AA_i)\ox_L\?K$, which is
a quotient of $(A^*)_{\?K}$.  So $A^*\ox_AB_{ij}$ is a cyclic
$B_{ij}$-module.  So $(A^*\ox_AB_{ij})\ox_{B_{ij}}\?K$ is a
1-dimensional $\?K$-vector space.

Fix $i$ and $j$.  Take $f_i\in A^*\ox_AA_i$ whose image in
$(A^*\ox_AB_{ij})\ox_{B_{ij}}\?K$ is nonzero, so generates it.  By
Nakayama's Lemma, the image of $f_i$ in $A^*\ox_AB_{ij}$ generates it.

Note $A_i\ox_L\?K$ is flat over $A_i$. Hence $B_{ij}$ is flat over
$A_i$, so faithfully flat since these rings are local.  Hence $f_i$
generates $A^*\ox_AA_i$ over $A_i$, as its image generates
$A^*\ox_AB_{ij}$ over $B_{ij}$.  So $\prod f_i$ generates $(A^*)_L$ over
$A_L$.  Lift $\prod f_i$ to some $f\in A^*$.  Then $f$ is the desired
\end{proof}

\begin{proposition}\label{prArt}
Assume $A$ is $k$-Artinian with (finite) Hilbert function $\mb h$.

\(A) Then these five conditions are equivalent:
\begin{enumerate}
\item The $k$-algebra $A$ is $k$-Gorenstein (namely, the $A$-module
$A^*$ is invertible).

\item  The  $k$-module  $A^*\ox_Ak$ is invertible.

\item  The  $k$-module  $A^*\ox_Ak$ is flat, and the  $k$-module
$\Soc_k(A)$ is invertible.

\item The $A$-module $A$ is level of type $1$.

\item In $\bb{M}_A$, there's an isomorphism $A\cong L\ox_k A^*$ for
some $k$-module $L$.
\end{enumerate}
Moreover, if \(1)--\(5) hold, then $L$ is invertible, and $L\cong
\Soc_k(A) = G_sA$.

\(B) If $A\in \bb {AG}_A$, say $A = \bigoplus A_n$, then \(1)--\(5) are
equivalent to this condition:
\begin{enumerate}
\item[$(6)$] In $\bb{AG}_A$, there's a canonical (homogeneous)
isomorphism $A = A_s\ox_kA^*$.
\end{enumerate}
(Here $(A_s\ox_kA^*)_n := A_s\ox_k(A^*)_{n-s}$ for all $n$.)  Moreover,
$\mb h(n) = \mb h(s-n)$.
\end{proposition}

\begin{proof}
Plainly, (1) implies (2).

Note (2) and (3) are equivalent owing to \eqref{leSoc}.  Indeed, in
general, an invertible $k$-module is flat, and its dual is invertible.
Conversely, in general, a locally free $k$-module of finite rank has
rank 1 everywhere if its dual does.

Assume (2).  Now, $G_{-s}(A^*) = G_{-s}(A^*\ox_Ak)$ by \eqref{leSocLev}(1)
with $ A^*$ for $C$.  However, $G_{-s}(A^*\ox_Ak)$ is a quotient of
$A^*\ox_A k$.  Let $\kappa\: A^*\ox_A k\onto G_{-s}(A^*)$ be the
quotient map.  Now, $G_{-s}(A^*)$ is locally free of rank $\mb h(s)$,
and $\mb h(s)\neq0$.  So (2) implies $\kappa$ is an isomorphism.  Thus
(4) holds.

Assume (4).  Then $\Soc_k(A) = G_sA$.  Also, $t=1$; that is,
$G_{-s}(A^*)$ is invertible.  But $G_{-s}(A^*) = (G_sA)^*$ by
\eqref{prFilt}(3).  Thus $(G_sA)\ox_k G_{-s}(A^*) \cong k$.

Choose a $k$-submodule $N\subset A^*$ that maps isomorphically onto
$G_{-s}(A^*)$.  As $A$ is level, \eqref{leSocLev}(3) implies $A\cdot N =
A^*$.  And, by the above, $(G_sA)\ox_k N\cong k$.

So the inclusion $N\into A^*$ yields a $k$-map $k\to (G_sA)\ox_k A^*$.
Say 1 maps to $\sum d_i\ox e_i$.  Define an $A$-map $\vf\: A\to
(G_sA)\ox_k A^*$ by $\vf(a) := \sum d_i\ox ae_i$.  Then $\vf$ is
surjective, as $A\cdot N = A^*$.  But $A$ and $A^*$ are locally free of
the same rank, and $G_{s}(A)$ is invertible.  So $\vf$ is an
isomorphism.  Thus (5) holds with $L:= G_sA$.

Assume (5).  Then $k\cong (L\ox_k A^*)\ox_Ak = L\ox_k (A^*\ox_Ak)$.  So
$L$ is invertible, and $L\cong (A^*\ox_Ak)^*$.  But $A\cong L\ox_k A^*$
in $\bb{M}_A$ by (5), and $L\ox_k A^*= (L\ox_kA)\ox_AA^*$.  Thus (1)
holds. Also, $L\cong \Soc_k(A)$ by \eqref{leSoc}.  Thus (A) holds.

In (B), trivially (6)\implies(5).  Conversely, take $N := (A^*)_{-s}$ in
the proof of (4)$\Rightarrow$(5).  Then $\vf\: A\risom A_s\ox_kA^*$ is
canonical, and $\vf(A_n) = A_s\ox_k (A^*)_{n-s}$.  Thus (4)\implies(6),
and as $A_s$ is invertible, $\mb h(n) = \mb h^*(n-s) = \mb h(s-n)$.
Thus (B) holds.
\end{proof}

\begin{example}\label{exSymNonArt}
Let's see how a nonstandard weighting can be used to show that a given
graded algebra is not $k$-Gorenstein, thus suggesting a general

Fix variables $X,\,Y$; take $A := k[X,\,Y]$.  Set $I := \la
X^2,\,XY,\,Y^3\ra$ and $C:= A/I$.  Let $x,\,y \in C$ denote the residues
of $X,\,Y$.  Plainly, $1,\,x,\,y,\,y^2$ form a free $k$-basis of $C$.
So $C$ is $k$-Artinian.  Notice that $I$ is monomial, so homogeneous
under both the standard and the nonstandard weightings.

Let's find $\Soc_k(C)$.  Given $a,\,b,\,c,\,d \in k$, set $\psi :=
a+bx+cy+dy^2$.  Then $x\psi = ax$ and $y\psi = ay+cy^2$.  So $\psi \in
\Soc_k(C)$ iff $a = 0$ and $c = 0$.  Thus $\Soc_k(C) = kx+ky^2$.  Thus
$C$ is, by \eqref{prArt}(A)(1)$\Rightarrow$(3), not $k$-Gorenstein.

First, give $X,\,Y$ their standard weights of 1.  Then the Hilbert
function $\mb h(n)$ has values $1,\,2,\,1$ for $n=0,\,1,\,2$; also,
$s(\mb h)=2$.  Thus $\mb h(n) = \mb h(s-n)$ although $C$ is not

Instead, give $X,\,Y$ weights of 3,\,2.  Then the Hilbert function $\mb
h(n)$ has values $1,\,0,\,1,\,1,\,1$ for $n=0,\,1,\,2,\,3,\,4$; also,
$s(\mb h)=4$.  Thus $\mb h(n) \neq \mb h(s-n)$ for $n=1$, and so $C$ is,
by \eqref{prArt}(B), not $k$-Gorenstein.

This example suggests the following question: given a $k$-Artinian
graded $k$-algebra $C$ with Hilbert function $\mb h(n)$ where $C$ is the
quotient of a polynomial ring by a monomial ideal, but $C$ is not
$k$-Gorenstein, then is there always a way to weight the variables so
that $\mb h(n) \neq \mb h(s-n)$ where $s := s(\mb h)$?

Likely the answer is yes, as Craig Huneke pointed out privately.
Namely, $\Soc_k(C)$ is generated by residues of monomials $M$, and there
must be at least two $M$ by \eqref{prArt}. Likely we can choose weights
so that, with $m := \max\{\deg(M)\}$, at least two $M$ have degree $m$;
then $m = s$ by \eqref{leSocLev}(2), and so $\mb h(s) > 1 = \mb h(0)$.
\end{example}

\begin{example}\label{exArt}
Here's a (fairly standard) example of a $k$-Gorenstein algebra $C$ such
that, under the standard weighting, $G_\bu C$ is not $k$-Gorenstein.
However, under a nonstandard weighting, $C$ is graded, and so $G_\bu C =
C$; thus $G_\bu C$ is $k$-Gorenstein.

Keep the setup of \eqref{exPolyRg}.  Take $r:=2$, so $A:=
k[X_1,\,X_2]$.  Form the ideal $\go M := \la X_1,\,X_2\ra$, so $A/\go M = k$.
Identify $A^\dg$ with $k[X_1^{-1},\,X_2^{-1}]$.

Set $F:= X_1^{-2}+X_2^{-3}\in A^\dg$ and $D := AF\subset A^\dg$ and
$I := (0:_AD)\subset A$.  Note
\begin{equation*}\label{eqArt1}\ts
\bigl(\sum a_{ij}X_1^iX_2^j\bigr)\cdot F = a_{00}F + a_{10}X_1^{-1}
+ a_{01}X_2^{-2} + a_{02}X_2^{-1} + (a_{20} + a_{03}).
\end{equation*}
for any $a_{ij}\in k$.
So $\bigl(\sum a_{ij}X_1^iX_2^j\bigr)\cdot F = 0$ iff
$a_{00},\,a_{10},\,a_{01},\,a_{02},\,(a_{20} + a_{03}) = 0$.
Thus $I$ is the ideal generated by $X_1^{2}-X_2^{3},\ X_1X_2$.

Also $\go MF = kX_1^{-1} + kX_2^{-2} + kX_2^{-1} + k$ and $D = kF \oplus
\go MF$.  Denote the residues in $D/\go MD$ of $X_2^{-1}$ and $F$ by
$x_2^{-1}$ and $f$. Thus $D/\go MD = kf = kx_2^{-3}$.  But $D/\go MD =
D\ox_Ak$.  Thus $D\ox_Ak \cong k$.  Furthermore, $D\in \bb{AM}_A$.
However, $kF+kX_2^{-3} = kX_1^{-2} + kX_2^{-3}$.  Thus $A^\dg/D$ is
free over $k$, so projective.

Set $C:= A/I$.  Then $C = D^*\in \bb{AM}_A$ by \eqref{thGMD3}(1).  So $D
= D^{**} = C^*$.  However, $D := AF$.  So for every maximal ideal $\go
m$ of $k$ and for every field $K$ containing $k/\go m$, the $A_K$-module
$D_K$ is cyclic.  Thus by \eqref{thGMD3}(1), the $k$-algebra $C$ is

Since $D\ox_Ak\cong k$, if $C\in \bb{AF}_A$, then by
\eqref{prArt}(A)(2)$\Rightarrow$(1), again $C$ is $k$-Gorenstein.

To find $\Soc_k(C)$, set $E := k + kX_1 + kX_1^2 + kX_2 + kX_2^2$.  Then
$E\oplus I = A$.  Given $\psi := a+ bX_1+ cX_1^2+ dX_2+ eX_2^2\in E$,
note $X_1\psi\in I$ iff $a,\,b=0$ and $X_2\psi \in I$ iff $a,\,d,\,e =
0$.  For $i = 1,2$, let $x_i\in C$ be the residue of $X_i$.  Thus
whatever the weights $w_i$ of the $X_i$ may be, $\Soc_k(C)= kx_1^2=
kx_2^3$.  Thus, since $D\ox_Ak \cong k$, so is flat, by
\eqref{prArt}(A)(3)$\Rightarrow$(1), if $C\in \bb{AF}_A$, then, once
again, $C$ is $k$-Gorenstein.  Alternatively, $\Soc_k(C)= kx_2^3$ holds
because of \eqref{leSoc}, as $D/\go MD = kx_2^{-3}$.

First, weight the $X_i$ as usual: $w_1,\,w_2 :=1$.  From \eqref{sbArt},
recall $C\in \bb{AF}_A$ iff $G_\bu C\in \bb{AG}_A$.  Note $G_\bu C =
G_\bu A\big/G_\bu I$ by \eqref{prBiex}.  But $A$ is graded; so $G_\bu A
= A$.  Correspondingly, $G_\bu I$ is generated by the initial forms of
all the elements of $I$.  So $X_1X_2,\,X_1^2\in G_\bu I$.  But $X_2^4\in
I$; so $X_2^4\in G_\bu I$.

It's not immediately obvious that $X_1X_2,\,X_1^2,\,X_2^4$ generate
$G_\bu I$.  However, if so, then $G_\bu I$ is the $k$-span of the
monomials divisible by at least one of $X_1X_2,\,X_1^2,\,X_2^4$, and so
$G_\bu C\in \bb{AG}_A$.  Moreover, if so, then $x_1$ and $x_2^3$ lie in
$\Soc_k(G_\bu C)$, and they are independent over $k$; thus, $G_\bu C$
is, by \eqref{prArt}(A)(1)$\Rightarrow$(3), {\it not} $k$-Gorenstein.

Instead, consider $G_\bu D$.  Similarly, it's the $k$-span of the
initial forms of the elements of $D$.  So $G_\bu D = kX_2^{-3} +
kX_2^{-2} + (kX_1^{-1} + kX_2^{-1}) + k$.  Hence $D \in \bb{AF}_A$ and
$G_\bu(A^\dg/D)$ is free over $k$.  Thus \eqref{thGMD3}(2) yields
$G_\bu C\in \bb{AG}_A$.  So $C\in \bb{AF}_A$, and thus \eqref{prArt}(A)
implies, in two different ways, that $C$ is $k$-Gorenstein.

The expression for $G_\bu D$ above also shows that the residues in
$G_\bu D$ of $X_1^{-1}$ and $X_2^{-3}$ form a free basis of it.  Thus
\eqref{leSoc} yields $\Soc_k(G_\bu C) = kx_1\oplus kx_2^3$.

Also, $G_\bu I := (0 :_A G_\bu D)$.  But the monomials in $A$ and the
inverse (reciprocal) monomials in $A^\dg$ form dual bases.  So $G_\bu
I$ is the $k$-span of all the monomials of $A$ except
$1,\,X_1,\,X_2,\,X_2^2,\,X_2^3$.  Thus $X_1X_2,\,X_1^2,\,X_2^4$ do
generate $G_\bu I$, as expected.

In passing, note $\go M\cdot G_\bu D = kX_2^{-2} + kX_2^{-1} + k$. So
$G_\bu D\ox_Ak$ is free of rank 2 over $k$.  Thus, again, $G_\bu C$ is not
$k$-Gorenstein, this time by \eqref{prArt}(A)(1)$\Rightarrow$(2), So, by
\eqref{prGor}(4)$\Rightarrow$(1), although $D$ is, by construction,
generated by $F$, nonetheless $G_\bu D$ is not generated by the initial
form $X_2^{-3}$ of $F$, nor by any other single element.

Moreover, no isomorphism $C\cong D$ in $\bb{M}_A$ can preserve the
filtrations up to shift; else, $G_\bu C$ would be isomorphic in $\bb
G_A$ to a shift of $G_\bu D$.  In particular, the isomorphism $\vf\:
C\risom D$ given by $\vf(\tau):=\tau f$ doesn't do so.  In fact,
$$(F^2C)F = kX_2^{-1} + k \subsetneqq
F^{-1}D = kX_1^{-1} + kX_2^{-1} + k.$$

Lastly, take $w_1 := 3$ and $w_2 := 2$.  Then $F$ is weighted homogeneous
of degree $-6$.  So the analysis simplifies, as there's no separate
associated graded case to consider in order to apply \eqref{prArt}.
Indeed, plainly, $(A^\dg)_{-6} = D_{-6} \oplus X_1^{-2}$ and
$(A^\dg)_{n} = D_{n}$ for $-4\le n \le 0$, with $D_{-5} = 0$ and
$D_{-1} = 0$.  So $D$ and $A^\dg/D$ are free over $k$, and $D\ox_Ak
= k$.  Thus \eqref{prArt}(A)(2)$\Rightarrow$(1) yields $C$ is

Alternatively, note that, for every maximal ideal $\go m$ of $k$ and for
every field $K$ containing $k/\go m$, the generators $X_1^{2}-X_2^{3},\
X_1X_2$ of $I$ map to a regular sequence in $K[X_1,\,X_2]$, because
$C_K$ is Artinian.  So $C_K$ is Gorenstein; see
\cite{Eis}*{Cor.\,21.19}.  Furthermore, $C$ is $k$-flat by
\cite{EGAIII0}*{(10.2.4), p.\,19} applied twice. So $C$ is $k$-Artinian.
Thus \eqref{prGor}(2)$\Rightarrow$(1) yields that $C$ is $k$-Gorenstein.
\end{example}

\begin{proposition}[Linkage]\label{coLink}
Assume $A$ is $k$-Artinian with Hilbert function $\mb h$, and is
$k$-Gorenstein.  Let $I\subset A$ be an ideal with $C := A/I\in
\bb{AM}_A$.  Form the ideal $I'\subset A$ {\bf directly linked} to $I$,
namely $I' := \Hom_A( C,\, A)$, and form the $A$-submodule of $A^*$ {\bf
Macaulay dual} to $C$, namely $C^*$.  Fix an isomorphism $A\cong
\Soc_k(A)\ox_k A^*$ in $\bb{M}_A$, as provided by \eqref{prArt}\(A).
Set $C':=A/I'$.

\(1) Then $I$ is directly linked to $I'$; that is, $I = \Hom_A( C',\,

\(2) Then $I' = \Soc_k(A)\ox_k C^*$ and $C' = \Hom_A(I,\, A)$; moreover,
$C'\in \bb{AM}_A$.

\(3) Give $C$ the induced filtration, and assume $C\in \bb{AF}_A$.  Then
$C$ is $k$-Gorenstein iff there's an invertible $k$-module $L$ such that
$L\ox_k I'$ is a cyclic $A$-module.  If so, then necessarily $L \cong
\Soc_k(C)\ox_k\Soc_k(A)^*$ and $L\ox_k I' \cong C$.
\end{proposition}

\begin{proof}
{\bf For (1),} given any $k$-module $L$ and $A$-modules $M,\,N,\,P$,
form these two canonical maps:
\begin{gather}
\Hom_A( M,\,N) \To
\Hom_A\bigl( L\ox_kM,\ L\ox_kN\bigr),\label{gab2}\\
L\ox_k\Hom_A( P,\,N) \To \Hom_A( P,\ L\ox_kN).\label{gab3}
\end{gather}
Assume $L$ is invertible.  Then, plainly, both maps are isomorphisms.

Take $M := \Hom_A( P,\,N)$ in \eqref{gab2}.  Then \eqref{gab3} yields a
canonical isomorphism:
\begin{equation}\label{eqga4}
\Hom_A\bigl( \Hom_A( P,\,N),\ N\bigr) =
\Hom_A\bigl(\Hom_A( P,\,L\ox_kN),\ L\ox_kN\bigr).
\end{equation}
Plainly, \eqref{eqga4} respects the two canonical maps from $P$, one to
each side.

Given another $A$-module $Q$ and any $A$-isomorphism $\vf\:Q\risom
L\ox_kN$, note $\vf$ and $\vf^{-1}$ yield an isomorphism:
\begin{equation}\label{eqga5}
\psi\: \Hom_A\bigl(\Hom_A( P,\,L\ox_kN),\ L\ox_kN\bigr)
\risom \Hom_A\bigl( \Hom_A( P,\,Q),\ Q)\bigr) .
\end{equation}
Let $\nu_s$ and $\nu_t$ denote the canonical maps from $P$ to the
source and target of $\psi$.  Plainly, $\nu_t= \psi\nu_s$.  Hence, if
$\nu_s$ is an isomorphism, so is $\nu_t$. (Also, $\psi =
\nu_t\nu_s^{-1}$; so $\psi$ is independent of the choice of $\vf$.)

Assume $P\in \bb{AM}_A$; take $N:=A^*$.  Then $P=\Hom_A\bigl( \Hom_A(
P,\,N),\ N)\bigr)
$ by \eqref{prRMD}.  Take $L:=\Soc_k(A)$; it's
invertible by \eqref{prArt}(A)(1)\implies(3).  Then $\nu_s$ is an

Take $Q:=A$ and $\vf\:Q\risom L\ox_kN$ to be the isomorphism fixed in
the hypotheses.  Then $\nu_t$ is an isomorphism; that is,
$P=\Hom_A\bigl( \Hom_A( P,\,A),\ A)\bigr)$.

Form the exact sequence $0\to I\to A\to C\to 0$.  Since $A,\,C\in
\bb{AM}_A$, plainly $I\in \bb{AM}_A$.  So taking $P:=I$ above yields $I
= \Hom_A\bigl( \Hom_A( I,\,A),\ A\bigr)$.

Note $0\to \Hom_A(C,\,A)\to \Hom_A(A,\,A)\xto\alpha \Hom_A(I,\,A)$ is
exact.  But below $\alpha$ is shown to be surjective.  Also
$\Hom_A(C,\,A) =: I'$ and $\Hom_A(A,\,A) =A$.  Thus $C' := A/I'
=\Hom_A(I,\,A)\in \bb{AM}_A$; hence, $\Hom_A(C',\, A) = I$.

The map $\Hom_A(A,\,A^*)\to \Hom_A(I,\,A^*)$ is surjective, as $A^*$ is
dualizing for $\bb{AM}_A$ by \eqref{prRMD}.  And it remains surjective
after applying $L\ox_k\bu$, as $L$ is invertible.  But \eqref{gab3} is
functorial in $P$.  Hence $\Hom_A(A,\,L\ox_kA^*)\to
\Hom_A(I,\,L\ox_k\,A^*)$ is surjective.  Hence the isomorphism
$\vf^{-1}\: L\ox_k\,A^*\risom A$ induces an isomorphism from that
surjection to $\alpha$.  Thus $\alpha$ is surjective, as asserted.  Thus
(1) holds.

{\bf For (2),} note that the quotient map $\kappa\: A\onto C$ induces
these two embeddings:
\begin{gather*}\label{gs2}
L\ox_k\Hom_A( C,\, A^*)\into L\ox_k\Hom_A( A,\, A^*) = L\ox_kA^*\\
\Hom_A( C,\, L\ox_kA^*)\into \Hom_A( A,\, L\ox_kA^*) = L\ox_kA^*.
\end{gather*}
As \eqref{gab3} is an isomorphism that's compatible with $\kappa$, the
two images are the same submodule $G$ of $L\ox_kA^*$.

Set $F := \vf^{-1}G$.  Note $F$ is independent of the choice of $\vf$,
because any two choices differ by an automorphism of $A$, namely, by
multiplication by a unit of $A$ as $\Hom_A(A,A) = A$.  Now, $\Hom_A(
C,\, A^*) = C^*$ by \eqref{eqeqFm6}.  And $I' := \Hom_A( C,\, A)$.  Thus
\eqref{gab3} induces a canonical isomorphism $I' = \Soc_k(A)\ox_k C^*$,
which is independent of the choice of $\vf$,.  Thus (2) holds.

{\bf For (3),} assume $C$ is $k$-Gorenstein.  Then
\eqref{prArt}(A)\(1)\implies\(5) gives an isomorphism $C\cong
\Soc_k(C)\ox_kC^*$, and $\Soc_k(C)$ is invertible by
\eqref{prArt}(A)(1)\implies(3).  So (2) gives $C\cong L\ox_k I'$ with $L
:= \Soc_k(C)\ox_k\Soc_k(A)^*$.  Thus $L\ox_k I'$ is a cyclic $A$-module.

Conversely, assume there's an invertible $k$-module $L$ with $L\ox_k I'$
cyclic over $A$.  Then there's a surjection $A\onto L\ox_k I'$, and it
factors via a surjection $B\onto L\ox_k I'$, as $L\ox_k I'$ is a
$B$-module.  But $L\ox_k I' = M\ox_k B^* $ with $M:= L\ox_k\Soc_k(A)$
invertible by (2).  And $B^*$ is locally free over $k$ of the same rank
as $B$.  Hence $B\risom M\ox_kB^*$.  So $B$ is $k$-Gorenstein, and
$M\cong \Soc_k(B)$ by \eqref{prArt}(A).  Thus (3) holds.
\end{proof}

\section{Basic Geometry}\label{seBasicGeo}

\begin{setup}\label{se6}
Fix a nonempty Noetherian base scheme $S$, and a nonzero, finitely
generated, graded, quasi-coherent $\mc O_S$-algebra $\mc A$ with $\mc
A_p = 0$ for $p < 0$.  Given an $S$-scheme $T$, denote by $\mc A_T$ the
\indn{AcalT@$\mc A_T$}
pullback of $\mc A$, and by $\bb M_{\mc A_T}$ the category of
quasi-coherent $\mc A_T$-modules.  For $\mc V\in \bb M_{\mc A_T}$, set
$\mc V^* := \sHom_T(\mc V, \,\mc O_T)$.

Likewise, generalize all the theory of Sections 2--4 in the standard way
using analogous notation, so that the old theory is recovered over each
open, affine subscheme $U :=\Spec(k)$ of $S$ with $A := H^0(U, \,\mc A)$
and over each open, affine subscheme $V :=\Spec(K)$ of $T$ mapping into

Thus generalizing \eqref{sbFilt} and \eqref{sbArt} yields the categories
$\bb F_{\mc A_T}$ and $\bb G_{\mc A_T}$, their full subcategories
$\bb{AF}_{\mc A_T}$ and $\bb{AG}_{\mc A_T}$, the (associated-graded)
functor $G_\bu\: \bb F_{\mc A_T} \to \bb G_{\mc A_T}$, and its
restriction $G_\bu\: \bb{AF}_{\mc A_T} \to \bb{AG}_{\mc A_T}$.

Fix nonzero functions $\mb b,\,\mb h\:\Z\to \Z$ with $\mb b(p) \ge \mb
h(p)\ge 0$ for all $p$, and with $\mb h$ finite.  Set $s : = s(\mb h) :=
\sup\{\,p\mid \mb h(p)\neq 0\,\}$.

Fix a graded, locally finitely generated $\mc A$-module $\mc B$ with
graded pieces $\mc B_p$ locally free of rank $\mb b(p)$ for all $p$.
\indn{bbmb@$\mb b$}
Then generalizing \eqref{sbApl} yields the graded dual $\mc B^\dg$ of
$\mc B$ and the following sets of subsheaves of $\mc B_T$ and $\mc
\begin{equation}\label{eqse6}
\mb F\Psi_{\mc B_T}^{\mb h}
\index[notation]{FbmPsi@$\mb F\Psi_{\mc B_T}^{\mb h}$}
\and \mb H\Psi_{\mc B_T}^{\mb h}
\index[notation]{HmbPsi@$\mb H\Psi_{\mc B_T}^{\mb h}$}
\and \mb F\Delta_{\mc B^\dg_T}^{\mb h^*}
\index[notation]{FmbDel@$\mb F\Delta_{\mc B^\dg_T}^{\mb h^*}$}
\and \mb H\Delta_{\mc B^\dg_T}^{\mb h^*}.
\index[notation]{HmbDel@$\mb H\Delta_{\mc B^\dg_T}^{\mb h^*}$}
\end{equation}