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# Generalized Jouanolou duality, weakly Gorenstein rings, and applications to
blowup algebras

Yairon Cid-Ruiz  Department of Mathematics: Algebra and Geometry, Ghent
University, S25, 9000 Gent, Belgium<EMAIL_ADDRESS>,  Claudia Polini
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA
<EMAIL_ADDRESS>and  Bernd Ulrich  Department of Mathematics, Purdue
University, West Lafayette, IN 47907, USA<EMAIL_ADDRESS>
###### Abstract.

We provide a generalization of Jouanolou duality that is applicable to a
plethora of situations. The environment where this generalized duality takes
place is a new class of rings, that we introduce and call weakly Gorenstein.
As a main consequence, we obtain a new general framework to investigate blowup
algebras. We use our results to study and determine the defining equations of
the Rees algebra of certain families of ideals.

###### Key words and phrases:

blowup algebras, Rees algebra, symmetric algebra, Jouanolou duality, weakly
Gorenstein rings, perfect pairing, Morley forms, canonical module,
determinantal rings, rational maps, approximation complex.

###### 2010 Mathematics Subject Classification:

13A30, 13H10, 13D45, 13D07, 14E05.

Y.C.R. was partially supported by an FWO Postdoctoral Fellowship (1220122N).
C.P. was partially supported by NSF grant DMS-1902033. B.U. was partially
supported by NSF grant DMS-1802383. Y.C.R. is grateful to the mathematics
departments of Purdue University and University of Notre Dame, where the
majority of the work was done, for their hospitality and for excellent working
conditions.

##  1\. Introduction

1.1. Generalized Jouanolou duality and weakly Gorenstein rings.

In a series of seminal papers Jouanolou studied Elimination Theory through the
lens of modern algebraic geometry ([Jo, Jo2, Jo3, Jo4, Jo5, Jo6]). One of his
main tools was a new duality, nowadays dubbed _Jouanolou duality_. Let $S$ be
a positively graded Noetherian $T$-algebra, with graded irrelevant ideal
${\mathfrak{m}}$. When $S$ is a (not necessarily flat) complete intersection
over $T$ and $\dim(S)=\dim(T)$, Jouanolou proved that, up to shift in degree,
there are graded $S$-isomorphisms
$\text{H}_{{\mathfrak{m}}}^{i}(S)\cong{}^{*}\text{Ext}_{T}^{i}(S,T)$ for all
$i\geq 0$, where ${}^{*}\text{Ext}$ denotes the graded Ext functor.

Our goal is to extend this duality to algebras that are not complete
intersections, in fact not even Gorenstein. To this end, we introduce a new
generalization of Gorenstein rings. We call these rings _weakly Gorenstein_.
As in the classical case of Gorenstein rings where local duality takes place,
our new notion of weakly Gorenstein rings is the natural environment where
Jouanolou duality exists and can be generalized. Assume $S$ is Cohen-Macaulay
and $\omega_{S}$ is a graded canonical module of $S$. Let $\mathfrak{a}\subset
S$ be a homogeneous ideal and $i\geq 0$ be an integer. We say that $S$ is _$i$
-weakly Gorenstein with respect to $\mathfrak{a}$_ if there exists a fixed
homogeneous element $y\in\omega_{S}$ that generates $\omega_{S}$ generically
and locally in codimension $i$ at every prime that contains $\mathfrak{a}$.
More precisely, we require that

$\dim\big{(}\text{Supp}\left(\omega_{S}/Sy\right)\big{)}\,<\,\dim(S)\quad\text{
and }\quad\dim\big{(}\text{Supp}\left(\omega_{S}/Sy\right)\cap
V(\mathfrak{a})\big{)}\,<\,\dim(S)-i.$

Our first main result says that a generalization of Jouanolou duality holds
for weakly Gorenstein rings. Indeed, let $B=T[x_{1},\ldots,x_{d}]$ be a
positively graded polynomial ring mapping onto $S$ and assume that $S$ is a
perfect $B$-module of codimension $c$. If $S$ is $(i+1)$-weakly Gorenstein
with respect to ${\mathfrak{m}}$, then there is a graded $S$-isomorphism

$\text{H}_{{\mathfrak{m}}}^{i}(S)\,\cong\,{}^{*}\text{Ext}_{T}^{i+c-d}(S,T)\left(\deg(y)\right)$

(see Theorem 2.2). When $i=0$ and $S$ is standard graded with $c=d$, and
$\delta:=-\deg(y)$, we prove that there is an isomorphism
$\omicron:\,\text{H}_{{\mathfrak{m}}}^{0}(S)_{\delta}\xrightarrow{\;\cong\;}T$
and that the multiplication maps
$\,\text{H}_{{\mathfrak{m}}}^{0}(S)_{j}\otimes_{T}S_{\delta-j}\rightarrow\text{H}_{{\mathfrak{m}}}^{0}(S)_{\delta}\,$
followed by $\omicron$ are perfect pairings inducing isomorphisms
$\,\text{H}_{{\mathfrak{m}}}^{0}(S)_{j}\xrightarrow{\;\cong\;}\;{\text{Hom}}_{T}\left(S_{\delta-j},T\right)\,$
as above (see Theorem 2.3). We provide explicit inverses of these
isomorphisms, which we construct from any suitable element $\Delta$ in the
annihilator of the diagonal ideal, the kernel of the multiplication map
$S\otimes_{T}S\rightarrow S$ (see Theorem 2.4). The significance of such
explicit inverses is that they reduce the computation of
$\text{H}_{{\mathfrak{m}}}^{0}(S)$, as an ideal in $S$, to the computation of
${}^{*}{\text{Hom}}_{T}(S,T)$. Jouanolou achieved this by means of Morley
forms, which our construction generalizes. We will elaborate on this for the
special case of symmetric algebras, in the second part of the Introduction.

Surprisingly, many classes of algebras and ideals of interest satisfy the
weakly Gorenstein condition. We prove that determinantal rings tend to have
this property (see Theorem 3.1), and so do symmetric algebras algebras as long
as they are Cohen-Macaulay (see Theorem 4.1). Our approach to showing that
symmetric algebras have the weakly Gorenstein property is by computing
explicitly the canonical module. The formula for the canonical module is
interesting in its own right, and yields several applications.

We now describe a family of symmetric algebras that satisfy the weakly
Gorenstein property, their canonical modules, and some related results that we
obtain along the way. Let $(R,{\mathfrak{m}})$ be a $d$-dimensional Cohen-
Macaulay local ring, and $I=(f_{1},\ldots,f_{n})\subset R$ be an ideal
minimally generated by $n$ elements and of codimension $g=\text{ht}(I)\geq 2$.
One says that $I$ has the property $F_{0}$ if for any
$\mathfrak{p}\in{\text{Spec}}(R)$ the minimal number of generators satisfies
$\mu(I_{\mathfrak{p}})\leq\dim(R_{\mathfrak{p}})+1$. For any $k\geq 0$, one
says that $I$ has the sliding depth property ${\text{SD}}_{k}$ if the depth of
the $i$-th Koszul homology with respect to the sequence $f_{1},\ldots,f_{n}$
is bigger or equal than $d-n+i+k$ for all $i\geq 0$. The condition $F_{0}$ is
necessary for the Cohen-Macaulayness of $\text{Sym}(I)$, and thus unavoidable
to show the weakly Gorenstein condition. Under the additional assumptions that
$R$ is Gorenstein with infinite residue field and that $I$ satisfies
${\text{SD}}_{1}$, in Theorem 4.1, we prove that:

  1. (i)

the canonical module $\omega_{\text{Sym}(I)}$ is isomorphic to an ideal of the
form $\left(f_{1},y_{1}\right)^{g-2}\text{Sym}(I)$.

  2. (ii)

$\text{Sym}(I)$ is $d$-weakly Gorenstein with respect to the ideal
${\mathfrak{m}}\text{Sym}(I)$.

These results are applicable in a wide range of situations since the condition
${\text{SD}}_{1}$ is satisfied by several classes of ideals (for instance,
strongly Cohen-Macaulay ideals, and so in particular, perfect ideals of
codimension two and Gorenstein ideals codimension three). This formula for the
canonical module of the symmetric algebras coincides with a formula for the
canonical module of certain Rees algebras (see [HSV2]), which leads to
interesting consequences in Section missing 4.

Our main tool to compute $\omega_{\text{Sym}(I)}$ is a newly defined complex
that mends one of the main drawbacks of the approximation complex
$\mathcal{Z}_{\bullet}$. The approximation complex $\mathcal{Z}_{\bullet}$ is
ubiquitous in the study of blowup algebras, and it provides a resolution of
the symmetric algebra in many cases of relevance. However, the fact that is
made up of Koszul syzygies, which are typically non-free modules, can be a non
trivial obstacle. To remedy this problem we introduce a halfway resolution
that refines $\mathcal{Z}_{\bullet}$. In Section missing 4, we introduce the
new complex $\mathcal{L}_{\bullet}$ that consists of free modules in the last
$g-1$ positions and that coincides with $\mathcal{Z}_{\bullet}$ in the
remaining positions. The complex $\mathcal{L}_{\bullet}$ is acyclic when
$\mathcal{Z}_{\bullet}$ is.

Furthermore, these halfway free resolutions lead to actual free resolutions of
the symmetric algebra for special families of ideals such as almost complete
intersections and perfect ideals of deviation two (see Theorem 4.2). The
usefulness of the complex $\mathcal{L}_{\bullet}$ came as surprise, since
computing free resolutions of symmetric algebras is a problem of tall order

1.2. Applications to blowup algebras.

Our generalization of Jouanolou duality and the definition of weakly
Gorenstein rings provide a general framework to study blowup algebras. We are
particularly interested in finding the defining equations of the Rees algebra.
Since Rees algebras appear as the coordinate ring of the blowup of a variety
along a subvariety, the significance of finding their defining equations
becomes apparent. This problem has been extensively studied by algebraic
geometers, commutative algebraists, and researchers in applied areas like
geometric modeling (see, e.g., [KPU3, KPU4, KPU5, Buse, KM, CBD14, V_REES_EQ,
COX_MOV_CURV, COX_HOFFMAN_WANG, HONG_SV, KPU_RAT_SC, CBD13, Morey, MU]).

When the symmetric algebra is a complete intersection, the classical Jouanolou
duality is the standard (and the most forceful) tool to compute the defining
equations of the Rees algebra (see [Buse, KPU3, KM]). We now describe and
justify how our generalized Jouanolou duality can and does play a similar role
when the symmetric algebra is only Cohen-Macaulay.

Let $\mathbb{k}$ be a field, $R=\mathbb{k}[x_{1},\ldots,x_{d}]$ be a standard
graded polynomial ring, ${\mathfrak{m}}=(x_{1},\ldots,x_{d})\subset R$ be the
graded irrelevant ideal, and $I=(f_{1},\ldots,f_{n})\subset R$ be an ideal
minimally generated by $n$ forms of degree $D\geq 1$. Let
$T=\mathbb{k}[y_{1},\ldots,y_{n}]$ be a standard graded polynomial ring and
$\mathfrak{n}=(y_{1},\ldots,y_{n})\subset T$ be the graded irrelevant ideal.
Let $B=R\otimes_{\mathbb{k}}T$ be a standard bigraded polynomial ring, and
consider the epimorphism

$\Phi:B\twoheadrightarrow\mathcal{R}(I)=R[It]=\bigoplus_{j=0}^{\infty}I^{j}t^{j}\subset
R[t],\quad x_{i}\mapsto x_{i}\text{ and }y_{i}\mapsto f_{i}t.$

Then $\mathcal{J}=\text{Ker}(\Phi)\subset B$ is the ideal of defining
equations of $\mathcal{R}(I)$. The graph of the rational map

$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{d-1}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{n-1}$

determined by the forms $f_{1},\ldots,f_{n}$ is naturally given as
$\Gamma={\text{BiProj}}(B/\mathcal{J})\subset{\text{BiProj}}(B)={\mathbb{P}}_{\mathbb{k}}^{d-1}\times_{\mathbb{k}}{\mathbb{P}}_{\mathbb{k}}^{n-1}$.

Traditionally one considers the Rees algebra as a natural epimorphic image of
the symmetric algebra $\text{Sym}(I)$ of $I$ and one studies the kernel of
this map,

$0\rightarrow\mathcal{A}\rightarrow\text{Sym}(I)\rightarrow\mathcal{R}(I)\rightarrow
0.$

The kernel $\mathcal{A}$ is the $R$-torsion submodule of $\text{Sym}(I)$. The
defining equations of $\text{Sym}(I)$ are easily read from a minimal
presentation matrix of $I$. An almost unavoidable constraint in the study of
blowup algebras is the $G_{d}$ condition. The ideal $I$ satisfies $G_{d}$ if
$\mu(I_{\mathfrak{p}})\leq\dim(R_{\mathfrak{p}})$ for all $\mathfrak{p}\in
V(I)$ such that $\text{ht}(\mathfrak{p})<d$. Furthermore, we need the
conditions $F_{0}$ and ${\text{SD}}_{1}$ to show that $\text{Sym}(I)$ has the
weakly Gorenstein property with respect to ${\mathfrak{m}}\text{Sym}(I)$.
Under these conditions, in Theorem 5.1, we provide a general framework of
dualities to study blowup algebras. Let $\delta=(g-1)D-d$ and $\beta=d-g+2$.
Suppose that $n=d+1$, $g=\text{ht}(I)\geq 2$, and $I\subset R$ satisfies the
conditions $G_{d}$ and ${\text{SD}}_{1}$. We then show that the following six
statements hold:

  1. (i)

$\mathcal{A}=\text{H}_{{\mathfrak{m}}}^{0}(\text{Sym}(I))$.

  2. (ii)

For all $0\leq i\leq d-1$, there is an isomorphism of bigraded $B$-modules

$\text{H}_{{\mathfrak{m}}}^{i}(\text{Sym}(I))\,\cong\,{}^{*}\text{Ext}_{T}^{i}\left(\text{Sym}(I),T\right)\left(-\delta,-\beta\right).$

In particular,
$\mathcal{A}=\text{H}_{{\mathfrak{m}}}^{0}(\text{Sym}(I))\cong{}^{*}{\text{Hom}}_{T}\left(\text{Sym}(I),T\right)\left(-\delta,-\beta\right)$.

  3. (iii)

For all $i<0$ and $i>\delta$, we have $\mathcal{A}_{(i,*)}=0$. There is an
isomorphism $\mathcal{A}_{(\delta,*)}\cong T(-\beta)$ of graded $T$-modules.
For all $0\leq i\leq\delta$, we have the equality of $\text{Sym}(I)$-ideals

$\mathcal{A}_{(\geq i,*)}\;=\;0:_{\text{Sym}(I)}{\mathfrak{m}}^{\delta+1-i}.$

  4. (iv)

$\mathcal{A}$ is minimally generated in $\mathbf{x}$-degree at most
$(g-2)D-d+1$.

  5. (v)

Let $0\leq i\leq\delta$. The natural multiplication map
$\mu:\mathcal{A}_{(i,*)}\;\otimes_{T}\;\text{Sym}(I)_{(\delta-i,*)}\;\rightarrow\;\mathcal{A}_{(\delta,*)},\;a\otimes
b\mapsto a\cdot b$ is a perfect pairing that induces the abstract isomorphism

$\nu:\mathcal{A}_{(i,*)}\;\xrightarrow{\;\cong\;}\;{\text{Hom}}_{T}\left(\text{Sym}(I)_{(\delta-i,*)},\mathcal{A}_{(\delta,*)}\right)$

seen in part (ii).

  6. (vi)

For all $2\leq i\leq d+1$, there is an isomorphism of bigraded $B$-modules

$\text{H}_{\mathfrak{n}}^{i}(\text{Sym}(I))\,\cong\,{}^{*}\text{Ext}_{R}^{i-1}(\text{Sym}(I),R)\left(-(g-1)D,g-1\right).$

A couple of words regarding the results of Theorem 5.1 are in place. Part (i)
simply means that $I$ is an ideal of linear type on the punctured spectrum on
$R$. Part (ii) comes by applying our generalization of Jouanolou duality and
from the fact that we prove $\text{Sym}(I)$ to be $d$-weakly Gorenstein with
respect to the ideal ${\mathfrak{m}}\text{Sym}(I)$. Part (iii) shows that the
graded components of $\mathcal{A}$ with respect to the $\mathbf{x}$-grading
can be read from the natural filtration

$\mathcal{A}=0:_{\text{Sym}(I)}{\mathfrak{m}}^{\delta+1}\;\;\supset\;\;0:_{\text{Sym}(I)}{\mathfrak{m}}^{\delta}\;\;\supset\;\;\cdots\;\;\supset\;\;0:_{\text{Sym}(I)}{\mathfrak{m}}^{2}\;\;\supset\;\;0:_{\text{Sym}(I)}{\mathfrak{m}}=\mathcal{A}_{(\delta,*)}.$

Part (iv) goes even further, it gives an upper bound for the
$\mathbf{x}$-degree of the minimal generators of $\mathcal{A}$, and it can
actually be sharp (see §missing 6.2). Part (v) implies that the abstract
isomorphism

$\mathcal{A}\,=\,\text{H}_{{\mathfrak{m}}}^{0}(\text{Sym}(I))\,\cong\,{}^{*}{\text{Hom}}_{T}\left(\text{Sym}(I),T\right)\left(-\delta,-\beta\right)$

naturally comes from a multiplication map, and this becomes a fundamental fact
in our approach to study blowup algebras. Part (vi) gives a generalized
Jouanolou duality statement with $R$ taking the role of $T$. To show part
(vi), since $\text{Sym}(I)$ is only $0$-weakly Gorenstein with respect to the
ideal $\mathfrak{n}\text{Sym}(I)$ (see Section missing 4), we instead rely on
the isomorphism of part (ii) and a duality result of Herzog and Rahimi
([HeRa]).

In a similar vein, we relate the freeness of $\mathcal{A}$ with the depth of
$\mathcal{R}(I)$. Explicitly, we prove in Section missing 5 that $\mathcal{A}$
is a free $T$-module if and only if $0pt(\mathcal{R}(I))\geq d$.

A remarkable feature of the classical Jouanolou duality is that it can be made
completely explicit in terms of _Morley forms_. However, the usual notion of
Morley forms is not enough in our setting. For this reason, we devise a new
reduction procedure that makes explicit the perfect pairing of part (v). We
now briefly describe our generalization of the theory of Morley forms. Let
$\mathbb{D}$ be the kernel of the natural multiplication map
$\text{Sym}(I)\otimes_{T}\text{Sym}(I)\rightarrow\text{Sym}(I)$. We choose a
suitable element $\Delta\in\text{Sym}(I)\otimes_{T}\text{Sym}(I)$ of degree
$\delta$ in the annihilator of the diagonal ideal $\mathbb{D}$. Our definition
of Morley forms ${\text{morl}}_{(i,\delta-i)}$ comes by considering the graded
components of this element $\Delta$. By applying Theorem 2.4, we show that
there is an explicit and computable homogeneous $T$-homomorphism

$\xi\,:\,{\text{Hom}}_{T}\left(\text{Sym}(I)_{(\delta-i,*)},\mathcal{A}_{(\delta,*)}\right)\rightarrow\mathcal{A}_{(i,*)},\qquad
u\mapsto\frac{\Phi_{\Delta}(\omicron\circ u)}{\alpha}$

that gives the inverse map of the isomorphism
$\nu:\mathcal{A}_{(i,*)}\rightarrow{\text{Hom}}_{T}\left(\text{Sym}(I)_{(\delta-i,*)},\mathcal{A}_{(\delta,*)}\right)$.
Here $\Phi_{\Delta}$ is a map with target $\mathcal{A}$ and $\alpha\in T$ is a
non zero element, both determined by $\Delta$. For more details on the
notation, see §missing 2.1, §missing 2.2, and §missing 5.1.

Since $\text{Sym}(I)$ is assumed to be Cohen-Macaulay, it is of dimension
$d+1$ and ${\mathfrak{m}}\text{Sym}(I)$ is a minimal prime of $\text{Sym}(I)$.
When $\text{Sym}(I)$ is a complete intersection at the minimal prime
${\mathfrak{m}}\text{Sym}(I)$, we provide a simple and direct method to find
this required element $\Delta$ (see §missing 5.1). Several classes of ideals
satisfy the condition that $\text{Sym}(I)$ is a complete intersection at
${\mathfrak{m}}\text{Sym}(I)$ (see §missing 6.1 and §missing 6.2). If Sym(I)
is a complete intersection, then $\Delta$ coincides with the element used by
Jouanolou in the construction of the classical Morley forms.

The last part of the paper is dedicated to applications. We show that our
results can deal with cases that were unreachable with previously existing
methods. Our presentation here is divided in terms of two families of ideals
that we treat.

_Zero dimensional ideals._ If $I\subset R$ is additionally an
${\mathfrak{m}}$-primary ideal, then we prove that $\mathcal{R}(I)$ can be
even approximated by two better understood algebras. One is the usual
symmetric algebra $\text{Sym}(I)$. And the other, less standard choice is the
symmetric algebra $\text{Sym}(E)$, where $E$ is the module determined by the
Koszul syzygies. In this case, the ideal $I$ satisfies all the conditions of
Theorem 5.1, and so we can use $\text{Sym}(I)$ and all the above results to
approximate and to study $\mathcal{R}(I)$. On the other hand, in Theorem 6.1,
we also show that there is a short exact sequence
$0\rightarrow\mathcal{B}\rightarrow\text{Sym}(E)\rightarrow\mathcal{R}(I)\rightarrow
0$ with $\mathcal{B}=\text{H}_{{\mathfrak{m}}}^{0}(\text{Sym}(E))$, and that
there is an isomorphism of bigraded $B$-modules

$\text{H}_{{\mathfrak{m}}}^{i}(\text{Sym}(E))\,\cong\,{}^{*}\text{Ext}_{T}^{i}\left(\text{Sym}(E),T\right)\left(-dD-d,-1\right)$

for all $0\leq i\leq d-1$. This opens new possibilities to study
$\mathcal{R}(I)$ when $I$ is assumed be a zero dimensional ideal. We plan to
pursue this approach further in a subsequent paper.

_Gorenstein ideals of codimension three._ In addition, suppose that $I\subset
R$ is a Gorenstein ideal of codimension three. This case is covered by Theorem
5.1 and so one can use the above results to study $\mathcal{R}(I)$ (see
Theorem 6.2). Let $h\geq 1$ be the degree of the homogeneous elements in a
minimal alternating presentation matrix $\varphi\in R^{(d+1)\times(d+1)}$ of
$I$. For ease of exposition, we restrict to the case $h=2$ with the extra
assumption that the monomial support of the entries of $\varphi$ is an almost
complete intersection, and we leave the other cases to be studied in a
subsequent paper. For this case, in Theorem 6.3, we give a complete picture of
the problem. We show that the ideal $\mathcal{J}\subset B$ of defining
equations of $\mathcal{R}(I)$ is minimally generated by the ideal of defining
equations of $\text{Sym}(I)$ and three forms of bidegrees $(0,2d-2)$,
$(1,d-1)$ and $(1,d-1)$. In fact, we explicitly compute these minimal
generators of $\mathcal{J}$ in terms of the Jacobian dual of $\varphi$ and
Morley forms! In Theorem 6.3, we also prove that $\deg(\mathcal{G})=2^{d-2}$,
$\deg(Y)=2d-2$, $0pt(\mathcal{R}(I))\geq d$, and $0pt(\text{gr}_{I}(R))\geq
d-1$, where
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{d-1}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{d}$
is the corresponding rational map and $Y\subset{\mathbb{P}}_{\mathbb{k}}^{d}$
is the closure of the image of $\mathcal{G}$.

Outline. The structure of the paper is as follows. In Section missing 2, we
provide an extension of Jouanolou duality and Morley forms. In Section missing
3, we determine determinantal rings that satisfy the weakly Gorenstein
condition. The weakly Gorenstein condition for symmetric algebras is studied
in Section missing 4. In Section missing 5, we provide a general framework of
duality statements that are relevant in the study of blowup algebras. Lastly,
Section missing 6 is dedicated to the study of specific families of ideals
where we apply the methods developed in this paper.

##  2\. An extension of Jouanolou duality and Morley Forms

In this section, we provide a generalization of Jouanolou duality that can be
applied to a plethora of situations. We assume assume the following setup
throughout.

######  Setup .

Let $S$ be a positively graded Noetherian ring with $T=S_{0}$ a local
Gorenstein ring. Choose a positively graded polynomial ring
$B=T[x_{1},\ldots,x_{d}]$ such that we have a graded surjection
$B\twoheadrightarrow S$. Let ${\mathfrak{m}}=(x_{1},\ldots,x_{d})$ be the
graded irrelevant ideal of $B$. Set $b:=\deg(x_{1})+\cdots+\deg(x_{d})$.

For a graded $B$-module $M$, we denote the graded $T$-dual as

${}^{*}{\text{Hom}}_{T}(M,T)\,:=\,\bigoplus_{j\in{\mathbb{Z}}}{\text{Hom}}_{T}\left([M]_{-j},T\right),$

and the corresponding right derived functor as
${}^{*}\text{Ext}_{T}^{i}(M,T)$. Note that ${}^{*}\text{Ext}_{T}^{i}(M,T)$ is
naturally a graded $B$-module for all $i\geq 0$.

The following result yields a version of Jouanolou duality in terms of the
canonical module of $S$, and it is applicable in great generality.

######  Theorem 2.1.

Assume Section missing 2. Suppose that $S$ is perfect over $B$ and of
codimension $c$. Then we have a graded isomorphism of $\,S$-modules

$\text{H}_{{\mathfrak{m}}}^{i}(\omega_{S})\,\cong\,{}^{*}\text{Ext}_{T}^{i+c-d}(S,T)$

for all $i\geq 0$.

###### First proof.

Let $F_{\bullet}:0\rightarrow F_{c}\rightarrow\cdots\rightarrow
F_{1}\rightarrow F_{0}$ be a minimal graded $B$-resolution of $S$. Since $S$
is perfect of codimension $c$, $F_{\bullet}$ has length equal to $c$. We then
have the following isomorphisms of graded $B$-modules

$\displaystyle{}^{*}\text{Ext}_{T}^{i+c-d}(S,T)$
$\displaystyle\;\cong\;\text{H}^{i+c-d}\left({}^{*}{\text{Hom}}_{T}(F_{\bullet},T)\right)$
$\displaystyle\;\cong\;\text{H}^{i+c-d}\left({}^{*}{\text{Hom}}_{T}(F_{\bullet}\otimes_{B}B,T)\right)$
$\displaystyle\;\cong\;\text{H}^{i+c-d}\left({\text{Hom}}_{B}\left(F_{\bullet},{}^{*}{\text{Hom}}_{T}(B,T)\right)\right)\quad\text{(by
Hom-tensor adjointness)}$
$\displaystyle\;\cong\;\text{H}^{i+c-d}\left({\text{Hom}}_{B}(F_{\bullet},B)\otimes_{B}{}^{*}{\text{Hom}}_{T}(B,T)\right)$
$\displaystyle\;\cong\;\text{H}^{i+c-d}\left({\text{Hom}}_{B}(F_{\bullet},B)\otimes_{B}\text{H}_{{\mathfrak{m}}}^{n}(B)(-b)\right)\quad\text{(by
graded local duality)}$
$\displaystyle\;\cong\;\text{H}_{c-(i+c-d)}\left({\text{Hom}}_{B}(F_{\bullet},B)[-c]\otimes_{B}\text{H}_{{\mathfrak{m}}}^{d}(B)(-b)\right)\quad\text{(by
a homological shift)}$
$\displaystyle\;\cong\;\text{Tor}_{d-i}^{B}\left(\omega_{S}(b),\text{H}_{{\mathfrak{m}}}^{d}(B)(-b)\right)\,=\,\text{Tor}_{d-i}^{B}\left(\omega_{S},\text{H}_{{\mathfrak{m}}}^{d}(B)\right).$

The last step in the above sequence of isomorphisms follows from the fact that
${\text{Hom}}_{B}(F_{\bullet},B(-b))[-c]$ provides a minimal graded
$B$-resolution of $\omega_{S}$, because by assumption $S$ is perfect over $B$.

The Čech complex $C_{\mathfrak{m}}^{\bullet}:0\rightarrow
B\rightarrow\bigoplus_{i=1}^{d}B_{x_{i}}\rightarrow\cdots\rightarrow
B_{x_{1}\cdots x_{d}}\rightarrow 0$ with respect to the sequence
$x_{1},\ldots,x_{d}$ is a complex of flat $B$-modules,
$\text{H}^{i}(C_{\mathfrak{m}}^{\bullet})=0$ for $i<d$, and
$\text{H}^{d}(C_{\mathfrak{m}}^{\bullet})\cong\text{H}_{{\mathfrak{m}}}^{d}(B)$.
By computing
$\text{Tor}_{d-i}^{B}\left(\omega_{S},\text{H}_{{\mathfrak{m}}}^{d}(B)\right)$
via this flat resolution of $\text{H}_{{\mathfrak{m}}}^{d}(B)$, we obtain that

$\text{Tor}_{d-i}^{B}\left(\omega_{S},\text{H}_{{\mathfrak{m}}}^{d}(B)\right)\,\cong\,\text{H}^{d-(d-i)}\left(\omega_{S}\otimes_{B}C_{\mathfrak{m}}^{\bullet}\right)\,\cong\,\text{H}_{{\mathfrak{m}}}^{i}(\omega_{S}).$

Finally, by combining all these isomorphisms, we obtain

$\text{H}_{{\mathfrak{m}}}^{i}(\omega_{S})\cong\text{Tor}_{d-i}^{B}\left(\omega_{S},\text{H}_{{\mathfrak{m}}}^{d}(B)\right)\cong{}^{*}\text{Ext}_{T}^{i+c-d}(S,T).\qed$

###### Second proof.

111This proof is _à la Jouanolou_ in the sense that we utilize a spectral
sequence argument that is present in several parts of Jouanolou’s work.

As before, let $F_{\bullet}:0\rightarrow F_{c}\rightarrow\cdots\rightarrow
F_{1}\rightarrow F_{0}$ be a minimal graded $B$-resolution of $S$. Since $S$
is perfect by assumption, it follows that
$G_{\bullet}:={\text{Hom}}_{B}(F_{\bullet},B(-b))[-c]$ yields a minimal graded
$B$-resolution of $\omega_{S}$. From the functorial isomorphism
$\text{H}_{{\mathfrak{m}}}^{d}(B)\cong{}^{*}{\text{Hom}}_{T}({\text{Hom}}_{B}(B,B(-b)),T)$,
we obtain the following isomorphism of graded complexes

$\displaystyle\text{H}_{{\mathfrak{m}}}^{d}(G_{\bullet})$
$\displaystyle\;\cong\;{}^{*}{\text{Hom}}_{T}\left({\text{Hom}}_{B}\left(G_{\bullet},B(-b)\right),T\right)$
$\displaystyle\;\cong\;{}^{*}{\text{Hom}}_{T}\left({\text{Hom}}_{B}\left({\text{Hom}}_{B}\left(F_{\bullet},B(-b)\right)[-c],B(-b)\right),T\right)$
$\displaystyle\;\cong\;{}^{*}{\text{Hom}}_{T}\left(F_{\bullet}[c],T\right).$

The spectral sequences coming from the second quadrant double complex
$G_{\bullet}\otimes_{B}C_{\mathfrak{m}}^{\bullet}$ converge in the second
pages and yield the following isomorphism

$\text{H}_{{\mathfrak{m}}}^{i}(\omega_{S})\,\cong\,\text{H}_{d-i}\left(\text{H}_{{\mathfrak{m}}}^{d}(G_{\bullet})\right)$

for all $i\geq 0$. After putting the above isomorphisms together, we obtain

$\displaystyle\text{H}_{{\mathfrak{m}}}^{i}(\omega_{S})\,\cong\,\text{H}_{d-i}\left(\text{H}_{{\mathfrak{m}}}^{d}(G_{\bullet})\right)$
$\displaystyle\,\cong\,\text{H}_{d-i}\left({}^{*}{\text{Hom}}_{T}\left(F_{\bullet}[c],T\right)\right)$
$\displaystyle\,\cong\,\text{H}^{i+c-d}\left({}^{*}{\text{Hom}}_{T}\left(F_{\bullet},T\right)\right)$
$\displaystyle\,\cong\,{}^{*}\text{Ext}_{T}^{i+c-d}\left(S,T\right).$

This concludes the second proof of the theorem. ∎

To obtain a “true” generalization of Jouanolou’s duality in terms of $S$, we
need to relate the local cohomology modules of $S$ and $\omega_{S}$. For this
purpose, we introduce the following general definition.

######  Definition .

Suppose that $S$ is a Cohen-Macaulay ring. Let $\mathfrak{a}\subset S$ be a
homogeneous ideal and $i\geq 0$ be an integer. We say that $S$ is _$i$ -weakly
Gorenstein with respect to $\mathfrak{a}$_ if there exists a homogeneous
element $y\in\omega_{S}$ such that

$\dim\big{(}\text{Supp}\left(\omega_{S}/Sy\right)\cap
V(\mathfrak{a})\big{)}\,<\,\dim(S)-i$

and $Sy\otimes_{S}S_{\mathfrak{p}}\cong\omega_{S_{\mathfrak{p}}}$ for all
$\mathfrak{p}\in\text{Ass}(S)$.

Under the above definition one obtains an element $y\in\omega_{S}$ that
generates $\omega_{S}$ at the associated primes of $S$ and at all the primes
containing $\mathfrak{a}$ with codimension $\leq i$; in particular, $S$
becomes a Gorenstein ring after localizing at all these primes. Notice that if
$S$ is $1$-weakly Gorenstein with respect to a nilpotent ideal then $S$ is
already Gorenstein. A simple reinterpretation of the weakly Gorenstein
condition is given in the following remark.

######  Remark .

With the hypothesis that $S$ is Cohen-Macaulay, $S$ is $i$-weakly Gorenstein
with respect to a homogeneous ideal $\mathfrak{a}\subset S$ if and only if
there is a homogeneous element $y\in\omega_{S}$ such that

  1. (i)

$\text{ht}\left(\text{Ann}_{S}\left(\omega_{S}/Sy\right)+\mathfrak{a}\right)\geq
i+1$, and

  2. (ii)

$\text{ht}\left(\text{Ann}_{S}\left(\omega_{S}/Sy\right)\right)\geq 1$.

The usefulness of this definition becomes apparent with the following lemma.

######  Lemma .

Assume that $S$ is a Cohen-Macaulay ring. Suppose $S$ is $(i+1)$-weakly
Gorenstein with respect to a homogeneous ideal $\mathfrak{a}\subset S$. Let
$y\in\omega_{S}$ be a homogeneous element as in Section missing 2. Then we
have a graded isomorphism of $\,S$-modules

$\text{H}_{\mathfrak{a}}^{i}(\omega_{S})\,\cong\,\text{H}_{\mathfrak{a}}^{i}(S)(-\deg(y)).$

###### Proof.

As the canonical module is faithful, for any $\mathfrak{p}\in\text{Ass}(S)$ we
get
$\text{Ann}_{S_{\mathfrak{p}}}(Sy\otimes_{S}S_{\mathfrak{p}})=\text{Ann}_{S_{\mathfrak{p}}}(\omega_{S_{\mathfrak{p}}})=0$,
and so it follows that $\text{Ann}_{S}(y)=0$. As a consequence, we have
$Sy\cong S(-\deg(y))$. From the short exact sequence $0\rightarrow
Sy\rightarrow\omega_{S}\rightarrow\omega_{S}/Sy\rightarrow 0$, we obtain the
exact sequence in cohomology

$\text{H}_{\mathfrak{a}}^{i-1}(\omega_{S}/Sy)\rightarrow\text{H}_{\mathfrak{a}}^{i}(Sy)\rightarrow\text{H}_{\mathfrak{a}}^{i}(\omega_{S})\rightarrow\text{H}_{\mathfrak{a}}^{i}(\omega_{S}/Sy).$

Thus, to conclude the proof it suffices to show that
$\text{grade}(\mathfrak{a},\omega_{S}/Sy)\geq i+1$. Equivalently, we need to
prove that $0pt\left((\omega_{S}/Sy)_{\mathfrak{p}}\right)\geq i+1$ for all
$\mathfrak{p}\in\text{Supp}(\omega_{S}/Sy)\cap V(\mathfrak{a})$. From the
definition of weakly Gorenstein we get that
$0pt(S_{\mathfrak{p}})=\dim(S_{\mathfrak{p}})\geq i+2$ for all
$\mathfrak{p}\in\text{Supp}(\omega_{S}/Sy)\cap V(\mathfrak{a})$. It then
follows from the short exact sequence above that
$0pt\left((\omega_{S}/Sy)_{\mathfrak{p}}\right)\geq i+1$ for all
$\mathfrak{p}\in\text{Supp}(\omega_{S}/Sy)\cap V(\mathfrak{a})$, as required.
∎

Finally, we are ready for our promised generalization of Jouanolou duality.

######  Theorem 2.2 (Generalized Jouanolou duality).

Assume Section missing 2. Suppose that $S$ is perfect over $B$ of codimension
$c$, and that $S$ is $(i+1)$-weakly Gorenstein with respect to
${\mathfrak{m}}S$. Let $y\in\omega_{S}$ be a homogeneous element as in Section
missing 2. Then we have a graded isomorphism of $\,S$-modules

$\text{H}_{{\mathfrak{m}}}^{i}(S)\,\cong\,{}^{*}\text{Ext}_{T}^{i+c-d}(S,T)\left(\deg(y)\right).$

###### Proof.

This follows by combining Theorem 2.1 and Section missing 2. ∎

###  2.1. Perfect pairing and Morley forms

In this subsection, our goal is to show that the isomorphism of Theorem 2.2
for $i=0$ and $c=d$ arises from a perfect pairing given by multiplication. In
addition, we want to make this isomorphism explicit via Morley forms.

######  Observation .

Assume Section missing 2. The following statements hold:

  1. (i)

The module ${}^{*}{\text{Hom}}_{T}(S,T)$ is concentrated in non positive
degrees.

  2. (ii)

$\text{H}_{{\mathfrak{m}}}^{0}({}^{*}{\text{Hom}}_{T}(S,T))={}^{*}{\text{Hom}}_{T}(S,T)$.

  3. (iii)

$\left[{}^{*}{\text{Hom}}_{T}(S,T)\right]_{0}\cong T$.

######  Lemma .

If $S$ is standard graded, then for every $i\geq 0$,

$0:_{{}^{*}{\text{Hom}}_{T}(S,T)}\,{\mathfrak{m}}^{i}\;=\;\big{[}{}^{*}{\text{Hom}}_{T}(S,T)\big{]}_{\geq-i+1}.$

###### Proof.

Since the other inclusion is clear, we only need to show that

$0:_{{}^{*}{\text{Hom}}_{T}(S,T)}{\mathfrak{m}}^{i}\;\subset\;\big{[}{}^{*}{\text{Hom}}_{T}(S,T)\big{]}_{\geq-i+1}.$

Let $f$ be a homogeneous element of
$0:_{{}^{*}{\text{Hom}}_{T}(S,T)}{\mathfrak{m}}^{i}$. Assume that $f$ has
degree $j$ with $j\leq-i$. It follows that the map $f$ restricted to $S_{\leq
i-1}$ is zero. On the other hand, ${\mathfrak{m}}^{i}f=0$ implies that $f$
restricted to $S_{\geq i}$ is also zero. We conclude that $f=0$. ∎

The following setup is used for the reminder of this subsection.

######  Setup .

In addition to Section missing 2, assume that $S$ is standard graded, perfect
over $B$ with $\dim(S)=\dim(T)$, and $S$ is $1$-weakly Gorenstein with respect
to ${\mathfrak{m}}S$. Write $\mathcal{A}$ for
$\text{H}_{{\mathfrak{m}}}^{0}(S)$. Let $y\in\omega_{S}$ be a homogeneous
element as in Section missing 2 of degree $-\delta$.

The next result describes the isomorphism of Theorem 2.2 (with $i=0$ and
$c=d$) in terms of a perfect pairing induced by the natural multiplication
map.

######  Theorem 2.3.

Assume §missing 2.1. The following statements hold:

  1. (i)

The $S$-module $\mathcal{A}=\text{H}_{{\mathfrak{m}}}^{0}(S)$ is concentrated
in degree at most $\delta$.

  2. (ii)

The $T$-module $\mathcal{A}_{\delta}$ is free of rank one.

  3. (iii)

For $0\leq i\leq\delta$, the natural multiplication map

$\mu\;:\;\mathcal{A}_{i}\;\otimes_{T}\;S_{\delta-i}\;\xrightarrow{\;{\rm
mult}\;}\;\mathcal{A}_{\delta}$

is a perfect pairing that induces an isomorphism

$\nu\,:\,\mathcal{A}_{i}\;\xrightarrow{\;\cong\;}\;{\text{Hom}}_{T}\left(S_{\delta-i},\mathcal{A}_{\delta}\right)\,.$

In addition, if we write $\mathcal{A}_{\delta}=T\mathfrak{s}$ and fix an
isomorphism $\omicron:T\mathfrak{s}\xrightarrow{\;\cong\;}T$ with
$\mathfrak{s}\mapsto 1$, then the composition of $\mu$ and $\omicron$,

$\mathcal{A}_{i}\;\otimes_{T}\;S_{\delta-i}\;\xrightarrow{\;{\rm
mult}\;}\;\mathcal{A}_{\delta}=T\mathfrak{s}\xrightarrow{\;\omicron\;}\;T$

is also a perfect pairing that induces an isomorphism

$\nu^{\prime}\,:\,\mathcal{A}_{i}\;\xrightarrow{\;\cong\;}\;{\text{Hom}}_{T}\left(S_{\delta-i},T\right)\,$

as in Theorem 2.2.

###### Proof.

According to Theorem 2.2 we have a graded isomorphism of $\,S$-modules

$\mathcal{A}\,=\,\text{H}_{{\mathfrak{m}}}^{0}(S)\,\cong\,{}^{*}{\text{Hom}}_{T}(S,T)\left(-\delta\right).$

Now part (i) and (ii) follow from §missing 2.1. The same graded isomorphism of
$S$-modules identifies the map $\mu$ with the multiplication map

${\text{Hom}}_{T}(S_{\delta-i},T)\;\otimes_{T}\;S_{\delta-i}\;\xrightarrow{\;{\rm
mult}\;}\;{\text{Hom}}_{T}(S_{0},T)=T\;$

which is a perfect pairing since it induces the identity map

${\text{Hom}}_{T}\left(S_{\delta-i},T\right)\;\xrightarrow{\;{\rm
id}\;}{\text{Hom}}_{T}\left(S_{\delta-i},T\right).\qed$

We will now construct explicit inverses of the maps $\nu$ using a
generalization of Jouanolou’s Morley forms. To any homogeneous element of
degree $\delta$ in the annihilator of the diagonal ideal $\mathbb{D}$ we
associate forms that we call Morley forms in honor of Jouanolou. Recall that
the diagonal ideal $\,\mathbb{D}$ of the enveloping algebra
$S^{e}:=S\otimes_{T}S$ is the kernel of the natural multiplication map
$S^{e}\twoheadrightarrow S$. This ideal is generated by the elements
$\overline{x_{i}}\otimes 1-1\otimes\overline{x_{i}}$, where $\overline{x_{i}}$
denotes the image of $x_{i}$ in $S$.

We think of $S^{e}$ and of
${}^{*}{\text{Hom}}_{T}({}^{*}{\text{Hom}}_{T}(S,T),S)$ as $S-S$-bimodules
with $S$ acting on the left and on the right. The largest submodules on which
the left and right $S$-module structures coincide are $0:_{S^{e}}\mathbb{D}$
and ${}^{*}{\text{Hom}}_{S}({}^{*}{\text{Hom}}_{T}(S,T),S)$, respectively.
Consider the homogeneous morphism of $S-S$-bimodules

$S^{e}\to{}^{*}{\text{Hom}}_{T}\big{(}{}^{*}{\text{Hom}}_{T}(S,T),S\big{)}$

given by $s_{1}\otimes s_{2}\mapsto(f\mapsto s_{1}f(s_{2}))$. Restricting this
map we obtain an $S$-linear map

$0:_{S^{e}}\mathbb{D}\to{}^{*}{\text{Hom}}_{S}\big{(}{}^{*}{\text{Hom}}_{T}(S,T),S\big{)}\,$

(see also [SS, proof of Theorem 3.1], [K, Proposition F.9], [EU, proof of
Theorem A.1]). From §missing 2.1(ii) we see that the target of this map is
${}^{*}{\text{Hom}}_{S}({}^{*}{\text{Hom}}_{T}(S,T),\text{H}_{{\mathfrak{m}}}^{0}(S))\,.$
Hence we obtain an homogeneous $S$-linear map

$0:_{S^{e}}\mathbb{D}\to{}^{*}{\text{Hom}}_{S}\big{(}{}^{*}{\text{Hom}}_{T}(S,T),\mathcal{A}\big{)}\,.$

Theorem 2.2 implies

${}^{*}{\text{Hom}}_{S}({}^{*}{\text{Hom}}_{T}(S,T),\text{H}_{{\mathfrak{m}}}^{0}(S))\,\cong{}^{*}{\text{Hom}}_{S}({}^{*}{\text{Hom}}_{T}(S,T),{}^{*}{\text{Hom}}_{T}(S,T))(-\delta)\,,$

and using Hom-Tensor adjointness we obtain

${}^{*}{\text{Hom}}_{S}({}^{*}{\text{Hom}}_{T}(S,T),{}^{*}{\text{Hom}}_{T}(S,T))\cong{}^{*}{\text{Hom}}_{T}(S\otimes_{S}{}^{*}{\text{Hom}}_{T}(S,T),T)\cong{}^{*}{\text{Hom}}_{T}({}^{*}{\text{Hom}}_{T}(S,T),T)\,.$

The last graded $S$-module is concentrated in degrees $\geq 0$ and its degree
$0$ component is $T$. It follows that any homogeneous $S$-automorphism of
${}^{*}{\text{Hom}}_{T}(S,T)$ is the identity map times a unit in $T$. Hence
all homogeneous $S$-isomorphisms of degree $\delta$ from
${}^{*}{\text{Hom}}_{T}(S,T)$ to $\mathcal{A}$ are equal up to multiplication
by a unit in $T$.

The above discussion yields the following remark.

######  Remark .

Every homogeneous element $\Delta=\sum_{i}s_{i,1}\otimes s_{i,2}$ of degree
$\delta$ in $0:_{S^{e}}\mathbb{D}$ yields a homogeneous $S$-linear map

$\Phi_{\Delta}:{}^{*}{\text{Hom}}_{T}(S,T)\,\to\,\mathcal{A},\qquad
u\mapsto\sum_{i}s_{i,1}u(s_{i,2})$

of degree $\delta$. If $\Phi_{\Delta}$ is a bijection, then $\Phi_{\Delta}$ is
independent of the choice of $\Delta$, up to multiplication by a unit in $T$.

Let $\pi:S\twoheadrightarrow T$ be the $T$-algebra map with
$\pi({\mathfrak{m}})=0$ and let
$\varepsilon=\pi\otimes_{T}S:S^{e}\twoheadrightarrow S$. Write
$\eta:S^{e}\twoheadrightarrow S$ for the natural multiplication map and recall
that $\eta(0:_{S^{e}}\mathbb{D})={\mathfrak{d}}_{N}(S/T)$, the Noether
different of $S$ over $T$, which defines the ramification locus of $S$ over
$T$. To identify elements $\Delta$ as in §missing 2.1 that provide the desire
isomorphisms, we need to understand the relationship between the three ideals
${\mathfrak{d}}_{N}(S/T)$, $\varepsilon(0:_{S^{e}}\mathbb{D})$ and
$0:_{S}{\mathfrak{m}}=\mathcal{A}_{\delta}$ (Theorem 2.2 and §missing 2.1).
Experimental evidence supports the following conjecture:

######  Conjecture .

  1. (i)

$\varepsilon(0:_{S^{e}}\mathbb{D})=0:_{S}{\mathfrak{m}}$.

  2. (ii)

If $\,T$ is a domain, then ${\mathfrak{d}}_{N}(S/T)=({\rm
rank}_{T}S)\cdot\varepsilon(0:_{S^{e}}\mathbb{D})$.

The next two results provide further evidence for this conjecture and show, in
particular, that it holds after tensoring with the total ring of quotients of
$\,T$.

######  Remark .

One has $\varepsilon(0:_{S^{e}}\mathbb{D})\subset 0:_{S}{\mathfrak{m}}$.

###### Proof.

This is clear since $\varepsilon(\mathbb{D})={\mathfrak{m}}$. ∎

######  Proposition .

For $L$ the total ring of quotients of $\,T$, we let $S_{L}$ be the standard
graded $L$-algebra $S\otimes_{T}L$, and $S^{e}_{L}$ and $\mathbb{D}_{L}$ be
the corresponding enveloping algebra and diagonal ideal. The following
statements hold:

  1. (i)

$0:_{S^{e}_{L}}\mathbb{D}_{L}\cong S_{L}(-\delta)$.

  2. (ii)

For every associated prime $\mathfrak{p}$ of $\,{\mathfrak{m}}S$, we have
$[0:_{S^{e}_{L}}\mathbb{D}_{L}]_{\delta}\not\subset(\mathfrak{p}\otimes
1)\,S_{L}^{e}\,.$ In particular,

$[0:_{S^{e}}\mathbb{D}]_{\delta}\not\subset(\mathfrak{p}\otimes 1)\,S^{e}\,.$

  3. (iii)

For every associated prime $\mathfrak{q}$ of $\,T$,
$\varepsilon([0:_{S^{e}}\mathbb{D}]_{\delta})\not\subset{\mathfrak{q}}S\,.$ In
fact, if $\Delta\in[0:_{S^{e}}\mathbb{D}]_{\delta}$ with
$\Delta\not\in(\mathfrak{p}\otimes 1)\,S^{e}$ for every associated prime
$\mathfrak{p}$ of ${\mathfrak{m}}S$, then
$\varepsilon(\Delta)\not\in\mathfrak{q}S\,$ for every associated prime
$\mathfrak{q}$ of $\,T$.

  4. (iv)

$\varepsilon(0:_{S^{e}_{L}}\mathbb{D}_{L})=\varepsilon([0:_{S^{e}_{L}}\mathbb{D}_{L}]_{\delta})=0:_{S_{L}}{\mathfrak{m}}_{L}\,;$
in particular,
$\varepsilon(0:_{S^{e}}\mathbb{D})_{\mathfrak{q}}=\varepsilon([0:_{S^{e}}\mathbb{D}]_{\delta})_{\mathfrak{q}}=(0:_{S}{\mathfrak{m}})_{\mathfrak{q}}\,$
for every associated prime $\mathfrak{q}$ of $\,S$.

  5. (v)

If $\,L$ is a field, then $\,{\mathfrak{d}}_{N}(S_{L}/L)=({\rm
dim}_{L}(S_{L}))\cdot\varepsilon(0:_{S^{e}_{L}}\mathbb{D}_{L})\,.$

###### Proof.

The ring $L$ is a finite product of Artinian local Gorenstein rings. Since
$\dim(S)=\dim(T)$, the $B$-module $S$ is perfect of codimension $d$. Therefore
$S_{L}$ is a perfect module of codimension $d$ over
$B_{L}=B\otimes_{T}L=L[x_{1},\ldots,x_{d}]$. Thus $S_{L}$ is a finite module
over $L$ of finite projective dimension. Now the Auslander-Buchsbaum formula,
applied to the factors of $L$, shows that $S_{L}$ is flat over $L$. Since $S$
is generically Gorenstein, $S_{L}$ is Gorenstein. Thus the enveloping algebra
$S_{L}\otimes_{L}S_{L}$ is Gorenstein, necessarily Artinian since it is a
finite module over $L$.

By §missing 2.1, $\omega_{S_{L}}\cong S_{L}\cdot y\cong S_{L}(\delta),$ which
implies $\omega_{S_{L}^{e}}\cong S_{L}^{e}(2\delta)$ as $S_{L}$ is flat over
$L$. Now part (i) follows because

$0:_{S_{L}^{e}}\mathbb{D}_{L}\cong{\text{Hom}}_{S_{L}^{e}}\big{(}S_{L}^{e}/\mathbb{D}_{L},S_{L}^{e}\big{)}\cong{\text{Hom}}_{S_{L}^{e}}\big{(}S_{L},\omega_{S_{L}^{e}}(-2\delta)\big{)}\cong\omega_{S_{L}}(-2\delta)\cong
S_{L}(-\delta)\,.$

As to part (ii), since $[0:_{S^{e}_{L}}\mathbb{D}_{L}]_{\delta}$ generates the
ideal $0:_{S^{e}_{L}}\mathbb{D}_{L}$ by part (i), it suffice to prove
$0:_{S^{e}_{L}}\mathbb{D}_{L}\not\subset(\mathfrak{p}\otimes 1)\,S_{L}^{e}\,.$
Suppose the contrary. As $S^{e}_{L}$ is an Artinian Gorenstein ring, it
follows that

$\mathbb{D}_{L}=0:_{S^{e}_{L}}(0:_{S^{e}_{L}}\mathbb{D}_{L})\supset
0:_{S^{e}_{L}}\left((\mathfrak{p}\otimes
1)\,S^{e}_{L}\right)\supset\left((0:_{S_{L}}\mathfrak{p})\otimes
1\right)\,S^{e}_{L}\,.$

Applying the multiplication map $S^{e}_{L}\to S_{L}$ that has $\mathbb{D}_{L}$
as its kernel, one sees that $0:_{S_{L}}\mathfrak{p}=0$, which is impossible
since $S_{L}$ is Artinian.

Every associated prime $\mathfrak{q}$ of $T$ is the contraction of an
associated prime $\mathfrak{p}$ of ${\mathfrak{m}}S,$ and
$\varepsilon^{-1}(\mathfrak{q}S)=(\mathfrak{p}\otimes 1)S^{e}$. Now use part
(ii) to establish part (iii).

By §missing 2.1, part (iv) follows once we show that the inclusion
$\varepsilon([0:_{S^{e}_{L}}\mathbb{D}_{L}]_{\delta})\subset
0:_{S_{L}}{\mathfrak{m}}_{L}$ is an equality. §missing 2.1 and Theorem 2.3
imply that $0:_{S_{L}}{\mathfrak{m}}_{L}=L\mathfrak{s}$, hence
$\varepsilon([0:_{S^{e}_{L}}\mathbb{D}_{L}]_{\delta})=K\mathfrak{s}$ for some
ideal $K$ of $L$. If $K\not=L$, then $K$ is contained in a prime ideal of $L$,
contradicting part (iii).

Let $\mathfrak{q}$ be an associated, hence minimal, prime of $\,S$. If
$\mathfrak{q}$ contains ${\mathfrak{m}}S$, then $\mathfrak{q}$ contracts to a
minimal prime of $T$ and the asserted equality locally at $\mathfrak{q}$
follows from the one just proved. If on the other hand $\mathfrak{q}$ does not
contain ${\mathfrak{m}}S$, then $(0:_{S}{\mathfrak{m}})_{\mathfrak{q}}=0$ and
we are done by §missing 2.1.

Part (v) follows from part (iv) because $\,{\mathfrak{d}}_{N}(S_{L}/L)=({\rm
dim}_{L}(S_{L}))\cdot(0:_{S_{L}}{\mathfrak{m}}_{L})\,$ by [EU, Theorems A.1
and A.5]. ∎

We choose an element $\Delta\in[0:_{S^{e}}\mathbb{D}]_{\delta}$ with
$\Delta\not\in(\mathfrak{p}\otimes 1)\,S^{e}$ for every associated prime
$\mathfrak{p}$ of ${\mathfrak{m}}S$. If the residue field of $T$ is infinite,
such an element exists by §missing 2.1 and any general element in
$[0:_{S\otimes_{T}S}\mathbb{D}]_{\delta}$ will do. We consider $S^{e}$ as a
standard bigraded $T$-algebra with

$\text{bideg}(x_{i}\otimes 1)=(1,0)\quad\text{ and }\quad\text{bideg}(1\otimes
x_{i})=(0,1).$

Thus we obtain the decomposition

$\Delta\;=\;\sum_{i=0}^{\delta}{\text{morl}}_{(i,\delta-i)}\quad\text{ where
}\quad{\text{morl}}_{(i,\delta-i)}\in\big{[}S^{e}\big{]}_{(i,\delta-i)}.$

We say that ${\text{morl}}_{(i,\delta-i)}$ is the _$i$ -th Morley form
associated to $\Delta$_. We now list some basic properties of these Morley
forms. Recall that by §missing 2.1 and Theorem 2.3,

$0:_{S}{\mathfrak{m}}\,=\,\mathcal{A}_{\delta}\,=\,T\mathfrak{s}\,\cong\,T\,.$

######  Lemma .

The following statements hold:

  1. (i)

${\text{morl}}_{(0,\delta)}=1\otimes\alpha\cdot\mathfrak{s}\in
S_{0}\otimes_{T}S_{\delta}\,$ for some non zerodivisor $\alpha$ in $T$.

  2. (ii)

For any $b\in S_{l}$, we have the equality

$(b\otimes 1)\cdot{\text{morl}}_{(i,\delta-i)}\;=\;(1\otimes
b)\cdot{\text{morl}}_{(i+l,\delta-i-l)}\;\in S_{i+l}\otimes_{T}S_{\delta-i}.$

###### Proof.

As for part (i), notice that $\alpha\cdot{\mathfrak{s}}=\varepsilon(\Delta)$.
Thus §missing 2.1(iii) shows that $\alpha$ is a non zerodivisor in $T$. Part
(ii) is obvious since $\Delta\in 0:_{S^{e}}\mathbb{D}\,.$ ∎

In the following theorem by utilizing our generalized Morley forms, we obtain
an explicit inverse of the isomorphism
$\nu:\mathcal{A}_{i}\rightarrow{\text{Hom}}_{T}(S_{\delta-i},\mathcal{A}_{\delta})$
from Theorem 2.3(iii). For the statement of Theorem 2.4, we observe that
$\mathcal{A}_{i}$ is $T$-torsionfree owing to the isomorphism
$\mathcal{A}_{i}\cong{\text{Hom}}_{T}(S_{\delta-i},T)$ and that the element
$\alpha$ of §missing 2.1 is a non zerodivisor in $T$.

######  Theorem 2.4.

Assume §missing 2.1, let $\mathfrak{s}$ and $\omicron$ be as in Theorem 2.3,
and let $\alpha$ be as in §missing 2.1. Let $0\leq i\leq\delta$. The following
statements hold:

  1. (i)

For any $u\in{\text{Hom}}_{T}(S_{\delta-i},\mathcal{A}_{\delta})$, we have
that $\Phi_{\Delta}({\omicron}\circ u)\in\alpha\cdot\mathcal{A}_{i}$. We have
a $T$-homomorphism

$\xi\,:\,{\text{Hom}}_{T}(S_{\delta-i},\mathcal{A}_{\delta})\rightarrow\mathcal{A}_{i},\qquad
u\mapsto\frac{\Phi_{\Delta}({\omicron}\circ u)}{\alpha}.$

  2. (ii)

The $T$-homomorphisms

$\nu\,:\,\mathcal{A}_{i}\rightarrow{\text{Hom}}_{T}(S_{\delta-i},\mathcal{A}_{\delta})\quad\text{
and
}\quad\xi\,:\,{\text{Hom}}_{T}(S_{\delta-i},\mathcal{A}_{\delta})\rightarrow\mathcal{A}_{i}$

are inverse to each other.

###### Proof.

For any $a\in\mathcal{A}_{i}$ and the corresponding multiplication map

$\nu(a)\in{\text{Hom}}_{T}(S_{\delta-i},\mathcal{A}_{\delta}),$

we have the equalities

$\displaystyle\Phi_{\Delta}({\omicron}\circ\nu(a))\;$
$\displaystyle=\;\big{(}{\rm id}_{S}\otimes\omicron\big{)}\big{(}(1\otimes
a)\cdot{\text{morl}}_{(i,\delta-i)}\big{)}$ $\displaystyle=\;\big{(}{\rm
id}_{S}\otimes\omicron\big{)}\big{(}(a\otimes
1)\cdot{\text{morl}}_{(0,\delta)}\big{)}$ (by §missing 2.1(ii))
$\displaystyle=\;\big{(}{\rm id}_{S}\otimes\omicron\big{)}\big{(}(a\otimes
1)\cdot(1\otimes\alpha\cdot\mathfrak{s})\big{)}$ (by §missing 2.1(i))
$\displaystyle=\;\alpha\cdot a\;\in\;\alpha\cdot\mathcal{A}_{i}.$

Since the map
$\nu\,:\,\mathcal{A}_{i}\rightarrow{\text{Hom}}_{T}(S_{\delta-i},\mathcal{A}_{\delta})$
was shown to be an isomorphism in Theorem 2.3(iii), we obtain that

$\Phi_{\Delta}({\omicron}\circ u)\in\alpha\cdot\mathcal{A}_{i}\quad\text{ for
all }\quad u\in{\text{Hom}}_{T}(S_{\delta-i},\mathcal{A}_{\delta})\,.$

As $\mathcal{A}_{i}$ is $T$-torsionfree and $\alpha$ is a non zerodivisor on
$T$, division by $\alpha$ in $\alpha\cdot\mathcal{A}_{i}$ is well-defined.
This completes the proof of part (i).

Notice that the above computation already shows that
$\xi\circ\nu=\text{id}_{\mathcal{A}_{i}}$. Since $\nu$ is an isomorphism by
Theorem 2.3(iii), $\xi$ is necessarily the inverse map of $\nu$. ∎

###  2.2. The case where the coefficient ring $T$ is graded

In this short subsection, we deal with the case where $T$ is a graded ring.
This case is of particular importance due to its applicability in the study of
blowup algebras that we pursue later. The proofs are exactly the same and one
only needs to indicate the necessary shifts in bidegree.

######  Setup .

Let $T$ be a Gorenstein positively graded ring with canonical module
$\omega_{T}\cong T(a)$ where $a\in{\mathbb{Z}}$. Let $B=T[x_{1},\ldots,x_{d}]$
be a bigraded polynomial ring such that $\text{bideg}(t)=(0,\deg(t))$ for all
homogeneous $t\in T$ and $\text{bideg}(x_{i})=(b_{i},0)$ with $b_{i}>0$ a
positive integer for all $1\leq i\leq d$. Let $S$ be a bigraded algebra given
as a quotient of $B$. Let ${\mathfrak{m}}\subset B$ be the ideal
${\mathfrak{m}}=(x_{1},\ldots,x_{d})$. Set $b:=b_{1}+\cdots+b_{d}$.

We now restate our generalization of Jouanolou duality in the current bigraded
setting.

######  Theorem 2.5.

Assume §missing 2.2. Suppose that $S$ is perfect over $B$ and of codimension
$c$. Then we have a bigraded isomorphism of $\,S$-modules

$\text{H}_{{\mathfrak{m}}}^{i}(\omega_{S})\,\cong\,{}^{*}\text{Ext}_{T}^{i+c-d}\big{(}S,T\big{)}(0,a).$

###### Proof.

Either of the proofs of Theorem 2.1 adapt to this case directly. One only
needs to notice that the bigraded canonical module of $B$ is
$\omega_{B}=B(-b,a)$, and so $\omega_{S}\cong\text{Ext}_{B}^{c}(S,B(-b,a))$. ∎

######  Theorem 2.6 (Generalized Jouanolou duality).

Assume §missing 2.2. Suppose that $S$ is perfect over $B$ of codimension $c$,
and that $S$ is $(i+1)$-weakly Gorenstein with respect to ${\mathfrak{m}}S$.
Let $y\in\omega_{S}$ be a bihomogeneous element as in Section missing 2. Then
we have a bigraded isomorphism of $\,S$-modules

$\text{H}_{{\mathfrak{m}}}^{i}(S)\,\cong\,{}^{*}\text{Ext}_{T}^{i+c-d}\big{(}S,T\big{)}\big{(}\text{bideg}(y)+(0,a)\big{)}.$

###### Proof.

Since Section missing 2 also holds in a bigraded setting, the result follows
from Theorem 2.5. ∎

For the remainder of this subsection we assume, in addition to §missing 2.2,
that $b_{1}=\cdots=b_{d}=1$, $S$ is perfect over $B$ with ${\rm dim}(S)={\rm
dim}(T)$, and $S$ is $1$-weakly Gorenstein with respect to ${\mathfrak{m}}S$.
Let $y\in\omega_{S}$ be a bihomogeneous element as in Section missing 2 of
bidegree $(-\delta,-\gamma)$.

In this case the element $\mathfrak{s}$ of Theorem 2.3 can be chosen to be
bihomogeneous of bidegree $(\delta,\gamma-a)$, the map $\nu$ is homogeneous,
and the map $\nu^{\prime}$ is homogenous of degree $a-\gamma$. If the element
$\Delta$ of §missing 2.1 is bihomogeneous of bidegree $(\delta,\rho)$, then so
is the induced map $\Phi_{\Delta}$.

We consider the enveloping algebra $B\otimes_{T}B$ as a trigraded ring with

$\text{trideg}(x_{i}\otimes 1)=(1,0,0),\;\text{trideg}(1\otimes
x_{i})=(0,1,0),\;\text{trideg}(t)=(0,0,{\rm deg}(t))\text{ for homogeneous
}t\in T.$

This induces a triple grading on $S^{e}$. If the element
$\Delta\in[0:_{S^{e}}\mathbb{D}]_{\delta}$ is bihomogeneous of bidegree
$(\delta,\rho)$, we obtain the decomposition

$\Delta\;=\;\sum_{i=0}^{\delta}{\text{morl}}_{(i,\delta-i)}\quad\text{ where
}\quad{\text{morl}}_{(i,\delta-i)}\in\big{[}S^{e}\big{]}_{(i,\delta-i,\rho)}.$

With this grading the isomorphism $\xi$ and the element $\alpha$ of Theorem
2.4 are homogeneous.

##  3\. Weakly Gorenstein determinantal rings

In this section, we study the weakly Gorenstein condition for determinantal
rings. Our findings show that under reasonable assumptions the weakly
Gorenstein property holds. These results combined with our generalization of
Jouanolou duality (Theorem 2.6) yield new tools to study local cohomology
modules of determinantal rings.

######  Setup .

Let $B$ be a Noetherian ring and $\varphi\in B^{m\times n}$ be an $m\times n$
matrix with $m\leq n$. Let $1\leq t\leq m$ and $S$ be the determinantal ring
$S:=B/I_{t}(\varphi)$. We consider the following submatrices of $\varphi$

$\left[\begin{array}[]{c|ccc}\delta&&*&\\\ \hline\cr&&&\\\ &&*&\\\ &&&\\\
\end{array}\right],\qquad\left[\begin{array}[]{c|ccc}&&&\\\ &&&\\\
\varphi^{\prime}&&*&\\\ &&&\\\
\end{array}\right],\qquad\left[\begin{array}[]{cccc}&&\varphi^{\prime\prime}&\\\
\hline\cr&&&\\\ &&*&\\\ &&&\\\ \end{array}\right]$

where $\delta\in B^{(t-1)\times(t-1)}$, $\varphi^{\prime}\in
B^{m\times(t-1)}$, and $\varphi^{\prime\prime}\in B^{(t-1)\times n}$. Let
$\Delta:=\det(\delta)\in B$.

We first prove an elementary lemma.

######  Lemma .

We have the inclusion $I_{t-1}(\varphi^{\prime})\cdot
I_{t-1}(\varphi^{\prime\prime})S\subset\Delta S$ in $S$.

###### Proof.

We proceed by induction on $t$. The case $t=1$ is vacuous. As we have
$\varphi_{i,1}\varphi_{1,j}-\varphi_{1,1}\varphi_{i,j}\in I_{2}(\varphi)$ for
all $i,j$, the claim follows for the case $t=2$.

Suppose that $t\geq 3$. Since containments are preserved under extensions, we
may assume that $B={\mathbb{Z}}[x_{i,j}]$ where $x_{i,j}$ are independent
variables and $\varphi=(x_{i,j})$ is the generic $m\times n$ matrix. Since $S$
is a Cohen-Macaulay domain, all the associated primes of $\Delta S$ have
height one.

Let $P\in{\text{Spec}}(S)$ be a minimal prime of $(x_{1,1},x_{1,2})S$. Set
$p=P\cap{\mathbb{Z}}$,
$\overline{B}=B\otimes_{\mathbb{Z}}{\text{Quot}}({\mathbb{Z}}/p)$, and
$\overline{S}=S\otimes_{\mathbb{Z}}{\text{Quot}}({\mathbb{Z}}/p)$. Notice that
$\text{ht}(P)\geq\text{ht}(P\overline{S})$. Since $t\geq 3$, the minimal
monomial generators of the initial ideal of $I_{t}(\varphi)\overline{B}$ with
any antidiagonal term order do not involve the variables $x_{1,1}$ and
$x_{1,2}$ (see, e.g., [MS, Theorem 16.28]), and this implies that
$x_{1,1},x_{1,2}$ form a regular sequence on $\overline{S}$ (see, e.g., [E,
Proposition 15.15]). Thus $\text{ht}(P)\geq 2$, and so
$\text{ht}((x_{1,1},x_{1,2})S)=2$. Hence, there is an element $b\in B$ such
that the image of $x_{1,1}+bx_{1,2}$ in $S$ is in no associated prime of
$\Delta S$. After elementary column operations, we obtain that $\varphi_{1,1}$
is not contained in any of the associated primes of $\Delta S$, and so it
suffices to show the containment after localizing at the element
$\varphi_{1,1}$. Once $\varphi_{1,1}$ is invertible, we can apply standard
arguments to reduce to the case $t-1$. ∎

The following result provides families of determinantal rings that have the
weakly Gorenstein property.

######  Theorem 3.1.

In addition to Section missing 3, assume that $B$ is normal, Gorenstein, and
local with infinite residue field. Suppose that

$\text{ht}\left(I_{t}(\varphi)\right)=(m-t+1)(n-t+1)\quad\text{ and
}\quad\text{ht}\left(I_{t-1}(\varphi)\right)>(m-t+1)(n-t+1).$

After elementary row operations, we may assume that
$\text{ht}\left(I_{t-1}(\varphi^{\prime\prime})S\right)>0$. Let
$\mathfrak{a}\subset B$ be an ideal and set
$i=\text{ht}\left(I_{t-1}(\varphi^{\prime\prime})S+\mathfrak{a}S\right)-1$.
Then $S$ is $i$-weakly Gorenstein with respect to $\mathfrak{a}S$.

###### Proof.

Since the row and column spaces of $\varphi\otimes_{B}S$ have rank $t-1$, we
may assume after elementary row and column operations that
$\text{ht}\left(I_{t-1}(\varphi^{\prime\prime})S\right)\geq 1$ and
$\text{ht}\left(I_{t-1}(\varphi^{\prime})S\right)\geq 1$. Let
$B^{\prime}=B[X]$ where $X=(x_{i,j})$ is a generic $m\times n$ matrix with
$x_{i,j}$ independent variables, and set $S^{\prime}=B^{\prime}/I_{t}(X)$.
From [Br] we know the canonical of $S^{\prime}$ satisfies the isomorphism
$\omega_{S^{\prime}}\cong I_{t-1}(X^{\prime})^{n-m}S^{\prime}$, where
$X^{\prime}$ is the submatrix given by the first $t-1$ columns of $X$ (also,
see [BV, Chapter 8]). Since the entries of the matrix $X-\varphi$ form a
regular sequence on $S^{\prime}$ and $S\cong S^{\prime}/I_{1}(X-\varphi)$, it
follows that

$\omega_{S}\,\cong\,\omega_{S^{\prime}}\otimes_{S^{\prime}}S\,\cong\,I_{t-1}(X^{\prime})^{n-m}S^{\prime}\otimes_{S^{\prime}}S\,\twoheadrightarrow\,I_{t-1}(\varphi^{\prime})^{n-m}S.$

Thus we obtain a surjection

$\omega_{S}\twoheadrightarrow I_{t-1}(\varphi^{\prime})^{n-m}S.$

As $\text{ht}\left(I_{t-1}(\varphi)\right)>(m-t+1)(n-t+1)$, $S$ is generically
a complete intersection on the Gorenstein ring $B$, hence generically
Gorenstein. It follows that $\omega_{S}$ has rank $1$. On the other hand,
$I_{t-1}(\varphi^{\prime})^{n-m}S$ is an ideal of positive grade and hence
again a module of rank $1$. It follows that the kernel of the above surjection
is a torsion submodule of $\omega_{S}$, and hence zero because $\omega_{S}$ is
a maximal Cohen-Macaulay module.

Let $y$ be the image of $\Delta^{n-m}$ in $S$. Notice that $y\in\omega_{S}$
under the isomorphism $\omega_{S}\cong I_{t-1}(\varphi^{\prime})^{n-m}S$. From
Section missing 3 it follows that $\text{Ann}_{S}(\omega_{S}/Sy)\supseteq
I_{t-1}(\varphi^{\prime\prime})S$, and the latter ideal has positive height.
Thus, by definition, $S$ is $i$-weakly Gorenstein with respect to
$\mathfrak{a}S$ when
$i=\text{ht}\left(I_{t-1}(\varphi^{\prime\prime})S+\mathfrak{a}S\right)-1$. ∎

We believe that simple combinations of Theorem 2.6 and Theorem 3.1 can lead to
a better understanding of local cohomology modules of determinantal rings in
several cases of interest. To be more precise, we provide a sample that will
be useful later (see Theorem 6.1).

######  Proposition .

Let $R=\mathbb{k}[x_{1},\ldots,x_{d}]$ and $T=\mathbb{k}[y_{1},\ldots,y_{n}]$
be standard graded polynomial rings over a field $\mathbb{k}$. Let
${\mathfrak{m}}=(x_{1},\ldots,x_{d})\subset R$ be the irrelevant ideal. Let
$B$ be the standard bigraded polynomial ring $R\otimes_{\mathbb{k}}T$. Let
$f_{1},\ldots,f_{n}$ be a sequence of forms in $R$ of the same degree $D\geq
1$. Let $\varphi$ be the $2\times n$ matrix given by

$\varphi\;=\;\left(\begin{array}[]{ccc}y_{1}&\cdots&y_{n}\\\
f_{1}&\cdots&f_{n}\end{array}\right),$

and set $S=B/I_{2}(\varphi)$. If $\,\text{ht}(I_{2}(\varphi))=n-1$, then there
is a bigraded isomorphism of $\,S$-modules

$\text{H}_{{\mathfrak{m}}}^{i}(S)\,\cong\,{}^{*}\text{Ext}_{T}^{i+n-1-d}\left(S,T\right)\left(d-(n-1)D,-1\right)$

for all $0\leq i\leq d-1$.

###### Proof.

We first show the result when $\mathbb{k}$ is infinite.

######  Claim 3.1.1.

The claimed isomorphisms exist when $\mathbb{k}$ is infinite.

###### Proof.

Assume $\mathbb{k}$ is infinite. Since
$\text{ht}\left(I_{1}(\varphi)\right)\geq n+1$ and
$\text{ht}\left(I_{1}(\varphi^{\prime\prime})S+{\mathfrak{m}}S\right)=\text{ht}\left((y_{1},\ldots,y_{d})S+{\mathfrak{m}}S\right)=\dim(S)=d+1$,
Theorem 3.1 implies that $S$ is $d$-weakly Gorenstein with respect to
${\mathfrak{m}}S$.

After elementary column operations over the infinite field $\mathbb{k}$, we
may assume that $y_{1}$ is a non zerodivisor on $S$. Notice that
$I_{2}(\varphi)$ is a geometric residual intersection of the ideal
$(y_{1},f_{1})$ because

$I_{2}(\varphi)\,=\,\left(\Delta_{1,2},\ldots,\Delta_{1,n}\right):(f_{1},y_{1}),$

$\text{ht}\left(I_{2}(\varphi)\right)=n-1$ and
$\text{ht}\left(I_{2}(\varphi)+(f_{1},y_{1})\right)\geq n$, where
$\Delta_{1,i}$ denotes the $2\times 2$ minor corresponding to columns $1$ and
$i$ (see [HSCM]). From the formulas for the canonical module of residual
intersections ([HU2, KU]), we obtain that
$\omega_{S}\cong(f_{1},y_{1})^{n-2}S\left(-d+(n-1)D,-1\right)$. As in the
proof of Theorem 3.1, let $y\in\omega_{S}$ be the class of $y_{1}^{n-2}$,
which is the chosen element to make $S$ $d$-weakly Gorenstein with respect to
${\mathfrak{m}}S$. Since
$\text{bideg}(y)=(0,n-2)-(-d+(n-1)D,-1)=(d-(n-1)D,n-1)$, Theorem 2.6 yields

$\text{H}_{{\mathfrak{m}}}^{i}(S)\,\cong\,{}^{*}\text{Ext}_{T}^{i+n-1-d}\left(S,T\right)\left(d-(n-1)D,-1\right)$

for all $0\leq i\leq d-1$, as required. ∎

We now settle the proposition for an arbitrary field $\mathbb{k}$. Let
$\mathbb{k}^{\prime}$ be an infinite field containing $\mathbb{k}$, and set
$S^{\prime}:=S\otimes_{\mathbb{k}}\mathbb{k}^{\prime}$. The computations above
show that
$\dim_{\mathbb{k}^{\prime}}\big{(}\left[\omega_{S^{\prime}}\right]_{(d-(n-1)D,n-1)}\big{)}=1$.
This implies that
$\dim_{\mathbb{k}}\big{(}\left[\omega_{S}\right]_{(d-(n-1)D,n-1)}\big{)}=1$.
Thus, take an non-zero element $0\neq
y\in\left[\omega_{S}\right]_{(d-(n-1)D,n-1)}$. It necessarily follows that, up
to multiplication by a unit in $\mathbb{k}^{\prime}$, the extension of $y$
into $\omega_{S^{\prime}}\cong\omega_{S}
\otimes_{\mathbb{k}}\mathbb{k}^{\prime}$ equals the element chosen in Claim
3.1.1 to make $S^{\prime}$ weakly Gorenstein over ${\mathfrak{m}}S^{\prime}$.
This completes the proof for the general case. ∎

##  4\. Weakly Gorenstein symmetric algebras

The principal motivation of this section is to find a comprehensive family of
ideals whose symmetric algebra has the weakly Gorenstein property. This first
goal will allow us to apply our generalized Jouanolou duality (Theorem 2.6) in
several new cases. However, along the way, we achieve more:

  1. (i)

we compute explicitly the canonical module of a large family of symmetric
algebras,

  2. (ii)

when the approximation complex $\mathcal{Z}_{\bullet}$ is acyclic, we define a
new complex that also resolves the symmetric algebra but has the advantage of
having free modules in the last $g-1$ positions, where $g$ is the grade of the
ideal,

  3. (iii)

when the ideal is an almost complete intersection or perfect of deviation two
we give an explicit free resolution for the symmetric algebra, provided
$\mathcal{Z}_{\bullet}$ is acyclic.

Modules of Koszul cycles were used before in [HSV3] to obtain information
about the free resolution of the symmetric algebra of an ideal $I\subset R$
and about $\,\omega_{\text{Sym}(I)}\otimes_{B}R$.

Throughout this section we assume the following setup.

######  Setup .

Let $(R,{\mathfrak{m}})$ be a $d$-dimensional Cohen-Macaulay local ring. Let
$I=(f_{1},\ldots,f_{n})\subset R$ be an ideal minimally generated by $n$
elements, and set $g=\text{ht}(I)\geq 1$ to be the height of $I$. Let
$B=R[y_{1},\ldots,y_{n}]$ be a standard graded polynomial ring. Let
$K_{\bullet}$ be the Koszul complex associated with to sequence
$f_{1},\ldots,f_{n}$. Let $Z_{\bullet}$ and $H_{\bullet}$ be the cycles and
homologies of $K_{\bullet}$, respectively.

######  Definition .

Let $k\geq 0$ be an integer.

  1. (G)

$I$ satisfies the _condition $G_{k}$_ if
$\mu(I_{\mathfrak{p}})\leq\text{ht}(\mathfrak{p})$ for all $\mathfrak{p}\in
V(I)\subset{\text{Spec}}(R)$ with $\text{ht}(\mathfrak{p})\leq k-1$.

  2. (F)

$I$ satisfies the _condition $F_{0}$_ if
$\mu(I_{\mathfrak{p}})\leq\text{ht}(\mathfrak{p})+1$ for all
$\mathfrak{p}\in{\text{Spec}}(R)$.

  3. (SD)

$I$ has the _sliding depth condition ${\text{SD}}_{k}$_ if $0pt(H_{i})\geq
d-n+i+k$ for all $i$.

  4. (SCM)

$I$ is _strongly Cohen-Macaulay_ if $H_{i}$ is a Cohen-Macaulay module for all
$i$.

One can check that the conditions strongly Cohen-Macaulay and
${\text{SD}}_{k}$ yield the following inequalities

$0pt(Z_{i})\,\geq\,\begin{cases}\min\\{d,d-g+2\\}&\text{ if $I$ is strongly
Cohen-Macaulay}\\\ \min\\{d,d-n+i+k+1\\}&\text{ if $I$ satisfies
${\text{SD}}_{k}$}\end{cases}$

for all $i\geq 0$ (see [HVV]). Recall that perfect ideals of height $2$ and
Gorenstein ideals of height $3$ are strongly Cohen-Macaulay (see [HSCM,
Proposition 0.3]).

Let $\mathcal{Z}_{\bullet}$ be the approximation complex

$\mathcal{Z}_{\bullet}:\quad 0\rightarrow
Z_{n-1}\otimes_{R}B(-n+1)\rightarrow\cdots\rightarrow
Z_{i}\otimes_{R}B(-i)\rightarrow\cdots\rightarrow
Z_{1}\otimes_{R}B(-1)\rightarrow Z_{0}\otimes_{R}B$

corresponding to the sequence $f_{1},\ldots,f_{n}$. For more details regarding
approximation complexes the reader is referred to [HSV]. For the sake of
completeness, we include a couple of well-known results regarding symmetric
algebras.

######  Lemma .

If $\,I\subset R$ has $F_{0}$ on the punctured spectrum of $R$, then
$\dim(\text{Sym}(I))=\max\left\\{n,d+1\right\\}$.

###### Proof.

By the Huneke-Rossi formula (see [HR], [V, Theorem 1.2.1]), we have

$\dim(\text{Sym}(I))=\sup\left\\{\mu(I_{\mathfrak{p}})+\dim(R/\mathfrak{p})\mid\mathfrak{p}\in{\text{Spec}}(R)\right\\}.$

Since $I$ has $F_{0}$ on the punctured spectrum of $R$, we deduce
$\dim(\text{Sym}(I))=\max\left\\{\mu(I),\dim(R)+1\right\\}=\max\left\\{n,d+1\right\\}$.
∎

######  Lemma .

If $\,I\subset R$ satisfies the conditions ${\text{SD}}_{0}$ and $F_{0}$, then
the following statements hold:

  1. (i)

$\text{Sym}(I)$ is Cohen-Macaulay.

  2. (ii)

The approximation complex $\mathcal{Z}_{\bullet}$ is acyclic.

###### Proof.

See for instance [HSV]. ∎

We now construct a new complex that will allow us to compute the canonical
module of $\text{Sym}(I)$ in many cases. Apparently, the existence of this
quite useful complex has been unnoticed. Its construction was inspired to us
by [HSV3, Theorem 5.8].

######  Proposition .

Let $\mathfrak{D}_{p,q}$ be the double complex

${0}$${0}$${0}$${\wedge^{n}R^{n}\otimes_{R}B(-n+g-1)}$${\vdots}$${\vdots}$${\vdots}$${\vdots}$${0}$${\wedge^{n}R^{n}\otimes_{R}B(-n+2)}$${\cdots}$${\wedge^{n-g+3}R^{n}\otimes_{R}B(-n+g-1)}$${\wedge^{n}R^{n}\otimes_{R}B(-n+1)}$${\wedge^{n-1}R^{n}\otimes_{R}B(-n+2)}$${\cdots}$${\wedge^{n-g+2}R^{n}\otimes_{R}B(-n+g-1)}$$\partial_{\mathbf{y}}$$\partial_{\mathbf{y}}$$\partial_{\mathbf{y}}$$\partial_{\mathbf{y}}$$\partial_{\mathbf{y}}$$\partial_{\mathbf{f}}$$\partial_{\mathbf{f}}$$\partial_{\mathbf{f}}$$\partial_{\mathbf{f}}$

where $\mathfrak{D}_{p,q}=\wedge^{n-g+2+p+q}R^{n}\otimes_{R}B(-n+g-1+p)$, and
$\partial_{\mathbf{f}}$ and $\partial_{\mathbf{y}}$ are the differentials of
the Koszul complexes $K(f_{1},\ldots,f_{n};B)$ and $K(y_{1},\ldots,y_{n};B)$,
respectively. Let $\mathfrak{T}_{\bullet}:={\rm
Tot}(\mathfrak{D}_{\bullet,\bullet})$ be the total complex

$\mathfrak{T}_{\bullet}:\quad
0\;\rightarrow\;\bigoplus_{j=1}^{g-1}\left(\wedge^{n}R^{n}\otimes_{R}B(-n+j)\right)\;\rightarrow\;\cdots\;\rightarrow\;\wedge^{n-g+2}R^{n}\otimes_{R}B(-n+g-1)\;\rightarrow\;0$

of $\mathfrak{D}_{\bullet,\bullet}$. We can splice together
$\mathfrak{T}_{\bullet}$ and the truncated complex
$0\rightarrow\mathcal{Z}_{n-g}\rightarrow\cdots\rightarrow\mathcal{Z}_{0}\rightarrow
0$ to obtain a complex

$\mathfrak{L}_{\bullet}:\quad
0\rightarrow\mathfrak{T}_{g-2}\rightarrow\cdots\rightarrow\mathfrak{T}_{1}\rightarrow\mathfrak{T}_{0}\rightarrow\mathcal{Z}_{n-g}\rightarrow\cdots\rightarrow\mathcal{Z}_{0}\rightarrow
0.$

If $\mathcal{Z}_{\bullet}$ is acyclic, then $\mathfrak{L}_{\bullet}$ is.

###### Proof.

Since the Koszul complex is depth sensitive and $g=\text{grade}(I)$, it
follows that $H_{i}=0$ for all $i\geq n-g+1$. Hence, after computing homology
by columns in the double complex $\mathfrak{D}_{\bullet,\bullet}$ we get that
the only non-zero row is the bottom one, which is then given by

$0\rightarrow\mathcal{Z}_{n-1}\rightarrow\cdots\rightarrow\mathcal{Z}_{n-g+2}\xrightarrow{\Psi}\mathcal{Z}_{n-g+1}\rightarrow
0.$

This implies that
$\text{H}_{i}(\mathfrak{T}_{\bullet})=\text{H}_{n-g+1+i}(\mathcal{Z}_{\bullet})$
for all $i\geq 1$ and that
$\text{H}_{0}(\mathfrak{T}_{\bullet})={\text{Coker}}(\Psi)$. By construction
of the $\mathcal{Z}_{\bullet}$ complex, there is a natural map from
${\text{Coker}}(\Psi)$ to $\text{Ker}(\Phi)$, where
$\mathcal{Z}_{n-g}\xrightarrow{\Phi}\mathcal{Z}_{n-g-1}\rightarrow\cdots\rightarrow\mathcal{Z}_{0}$.
Therefore, we can always make the claimed splicing, and
$\mathfrak{L}_{\bullet}$ is acyclic when $\mathcal{Z}_{\bullet}$ is. ∎

The following theorem contains the main results of this section.

######  Theorem 4.1.

Assume Section missing 4 where $R$ is Gorenstein with infinite residue field.
Suppose that $I\subset R$ has height $g=\text{ht}(I)\geq 2$ and that it
satisfies the conditions $F_{0}$ and ${\text{SD}}_{1}$. Then the following
statements hold:

  1. (i)

$\omega_{\text{Sym}(I)}\cong\left(f_{1},y_{1}\right)^{g-2}\text{Sym}(I)(-1)$,
where $f_{1}$ is a minimal generator of $I$ (possibly after changing the
minimal set of generators of $\,I\,$).

  2. (ii)

$\text{Sym}(I)$ is $d$-weakly Gorenstein with respect to the ideal
${\mathfrak{m}}\text{Sym}(I)$.

###### Proof.

The condition $F_{0}$ implies, in particular, that $n\leq d+1$. From Section
missing 4 and Section missing 4 we have that $\text{Sym}(I)$ is a Cohen-
Macaulay ring of dimension $d+1$ and that the approximation complex
$\mathcal{Z}_{\bullet}$ is acyclic. The condition ${\text{SD}}_{1}$ yields the
following lower bounds for the depth of the Koszul cycles

$0pt(Z_{i})\,\geq\,\min\left\\{d,d-n+i+2\right\\}\quad\text{for all $i$}.$

This implies that $0pt(\mathcal{Z}_{i})\geq\min\left\\{d+n,d+i+2\right\\}$. We
consider the complex

$\mathfrak{L}_{\bullet}:\quad
0\rightarrow\mathfrak{T}_{g-2}\rightarrow\cdots\rightarrow\mathfrak{T}_{1}\rightarrow\mathfrak{T}_{0}\rightarrow\mathcal{Z}_{n-g}\rightarrow\cdots\rightarrow\mathcal{Z}_{0}\rightarrow
0$

that was constructed in Section missing 4, which is acyclic under the current
assumptions. To simplify notation, set $S:=\text{Sym}(I)$.

Denote by $(\mathfrak{L}_{\bullet},\partial_{\bullet})$ the complex
$\mathfrak{L}_{\bullet}$ with its differential. Let
$U_{i}={\text{Coker}}(\partial_{i+1})$. For all $0\leq i\leq n-g$, we have the
short exact sequence

$0\rightarrow U_{i+1}\rightarrow\mathcal{Z}_{i}\rightarrow U_{i}\rightarrow 0$

that gives the induced exact sequence

$\text{Ext}_{B}^{n-1-i-1}(U_{i+1},\omega_{B})\rightarrow\text{Ext}_{B}^{n-1-i}(U_{i},\omega_{B})\rightarrow\text{Ext}_{B}^{n-1-i}(\mathcal{Z}_{i},\omega_{B}).$

For $0\leq i\leq n-g$, we have
$\text{Ext}_{B}^{n-1-i}(\mathcal{Z}_{i},\omega_{B})=0$ since
$n-i-1>\max\\{0,n-i-2\\}\geq d+n-0pt(\mathcal{Z}_{i})$. Thus we obtain the
surjection

$\text{Ext}_{B}^{n-1-i-1}(U_{i+1},\omega_{B})\twoheadrightarrow\text{Ext}_{B}^{n-1-i}(U_{i},\omega_{B})$

for all $0\leq i\leq n-g$. By composing these surjections, we obtain

$\text{Ext}_{B}^{g-2}(U_{n-g+1},\omega_{B})\twoheadrightarrow\text{Ext}_{B}^{n-1}(U_{0},\omega_{B})=\text{Ext}_{B}^{n-1}(S,\omega_{B})\cong\omega_{S}.$

The complex
$0\rightarrow\mathfrak{T}_{g-2}\rightarrow\cdots\rightarrow\mathfrak{T}_{1}\rightarrow\mathfrak{T}_{0}\rightarrow
0$ gives a free $B$-resolution of $U_{n-g+1}$.

If $g=2$, then $U_{n-g+1}$ is equal to the free $B$-module $B(-n+1)$. In this
case we have a surjection
${\text{Hom}}_{B}(U_{n-g+1},\omega_{B})\twoheadrightarrow\omega_{S}$, hence
$\omega_{S}$ is generated by one element; this means that $S$ is Gorenstein.
Hence the proof is complete for the case $g=2$.

Assume that $g\geq 3$. The module $\text{Ext}_{B}^{g-2}(U_{n-g+1},\omega_{B})$
is the cokernel of the map

${\text{Hom}}_{B}(\mathfrak{T}_{g-3},\omega_{B})\rightarrow{\text{Hom}}_{B}(\mathfrak{T}_{g-2},\omega_{B}).$

The map $\mathfrak{T}_{g-2}\rightarrow\mathfrak{T}_{g-3}$ is given explicitly
as

$\bigoplus_{j=1}^{g-1}B(-n+g-j)\xrightarrow{\;\;\mathfrak{A}\;\;}\bigoplus_{j=1}^{g-2}B^{n}(-n+g-j).$

For $1\leq j\leq g-1$, let $e_{j}$ be the basis element of
$\mathfrak{T}_{g-2}$ corresponding to the summand $B(-n+g-j)$. With respect to
this basis, the matrix $\mathfrak{A}$ can be written explicitly as

$\mathfrak{A}\;=\;\left[\begin{array}[]{cccccc}f_{1}&y_{1}&0&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
(-1)^{n}f_{n}&(-1)^{n}y_{n}&0&\cdots&0&0\\\ \hline\cr
0&f_{1}&y_{1}&\cdots&0&0\\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&(-1)^{n}f_{n}&(-1)^{n}y_{n}&\cdots&0&0\\\
\hline\cr\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\ \hline\cr
0&0&0&\cdots&f_{1}&y_{1}\\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&\cdots&(-1)^{n}f_{n}&(-1)^{n}y_{n}\\\ \end{array}\right]$

where the $j$-th column corresponds to the basis element $e_{j}$. Let
$\mathfrak{C}=\mathfrak{A}^{T}$ and

$\Omega\;:=\;{\text{Coker}}\left(\bigoplus_{j=1}^{g-2}\omega_{B}^{n}(n-g+j)\xrightarrow{\;\;\mathfrak{C}\;\;}\bigoplus_{j=1}^{g-1}\omega_{B}(n-g+j)\right).$

Denote by $w_{j}$ the generator of $\Omega$ corresponding with the summand
$\omega_{B}(n-g+j)$. Since we have the relations
$\overline{f_{i}y_{j}}=\overline{f_{j}y_{i}}$ in $S$, we obtain the natural
surjective $B$-homomorphism

$\Omega\twoheadrightarrow(f_{1},y_{1})^{g-2}S(-1),\;\;\text{ where
}\;\;w_{1}\mapsto\overline{y_{1}^{g-2}},\;w_{2}\mapsto-\overline{f_{1}y_{1}^{g-3}},\;\ldots,\;w_{g-1}\mapsto(-1)^{g-2}\overline{f_{1}^{g-2}}.$

We reduce module the equations of $S$ and consider the induced surjective
$S$-homomorphisms

$\Omega\otimes_{B}S\,\twoheadrightarrow\,\omega_{S}\qquad\text{ and
}\qquad\Omega\otimes_{B}S\twoheadrightarrow(f_{1},y_{1})^{g-2}S(-1).$

Since $\dim(S/(y_{1},\ldots,y_{n})S)=\dim(R)=d$ and $S$ is a Cohen-Macaulay
ring of dimension $d+1$, it follows that
$\text{grade}\left((y_{1},\ldots,y_{n})S\right)\geq 1$. So, after possibly
changing the generators of $I$, we can assume that $y_{1}$ is a non
zerodivisor on $S$. By the pigeonhole principle, every $(g-1)\times(g-1)$
submatrix of $\mathfrak{A}$ should contain a $2\times(g-1)$ submatrix of the
form

$\left(\begin{array}[]{cccccccc}0&\cdots&0&(-1)^{i_{1}}f_{i_{1}}&(-1)^{i_{1}}y_{i_{1}}&0&\cdots&0\\\
0&\cdots&0&(-1)^{i_{2}}f_{i_{2}}&(-1)^{i_{2}}y_{i_{2}}&0&\cdots&0\end{array}\right).$

Accordingly,
$I_{g-1}(\mathfrak{C})\subset\left(\left\\{f_{i}y_{j}-f_{j}y_{i}\right\\}_{i,j}\right)\subset
B$, hence $I_{g-1}(\mathfrak{C}\otimes_{B}S)=0$. Notice that

$y_{1}^{g-2}S\in I_{g-2}(\mathfrak{C}\otimes_{B}S)$

and $y_{1}$ is a non zerodivisor on $\omega_{B}\otimes_{B}S$, therefore
$\text{rank}_{S}(\mathfrak{C}\otimes_{B}S)=g-2$ and
$\text{rank}_{S}(\Omega\otimes_{B}S)=1$. The canonical module $\omega_{S}$ is
faithful and the epimorphic image of a module of rank $1$, hence $\omega_{S}$
is an $S$-module of rank $1$. The ideal $(f_{1},y_{1})^{g-2}S$ contains the
non zerodivisor $y_{1}^{g-2}S$, and so it is a torsion-free $S$-module of rank
$1$. Finally, since $\omega_{S}$ is always torsion-free, we obtain the
isomorphisms

$\omega_{S}\;\cong\;\frac{\Omega\otimes_{B}S}{\text{tor}\left(\Omega\otimes_{B}S\right)}\;\cong\;(f_{1},y_{1})^{g-2}S(-1).$

This concludes the proof of part (i) of the theorem.

We now concentrate on part (ii) of the theorem, which become straightforward
after computing the canonical module $\omega_{S}$.

Since
$(\overline{y_{1}},\ldots,\overline{y_{n}})\cdot\overline{f_{1}}\subset(\overline{y_{1}})$
in $S$, it follows that

$(\overline{y_{1}},\ldots,\overline{y_{n}})^{g-2}\,\subseteq\,\text{Ann}_{S}\left((\overline{f_{1}},\overline{y_{1}})^{g-2}/(\overline{y_{1}})^{g-2}\right)\,=\,\text{Ann}_{S}\left(\omega_{S}/S\overline{w_{1}}\right).$

As a consequence, we get

$\text{ht}\left(\text{Ann}_{S}\left(\omega_{S}/S\overline{w_{1}}\right)\right)\geq
1\qquad\text{ and
}\qquad\text{ht}\left(\text{Ann}_{S}\left(\omega_{S}/S\overline{w_{1}}\right)+{\mathfrak{m}}S\right)\geq
d+1.$

Therefore, $S$ is $d$-weakly Gorenstein with respect to ${\mathfrak{m}}S$, and
this establishes the remaining part (ii). ∎

######  Remark .

It should be mentioned that the proof of Theorem 4.1 works under the weaker
assumption that $R$ is just a Cohen-Macaulay ring with canonical module
$\omega_{R}$. In that case, we have the formula

$\omega_{\text{Sym}(I)}\,\cong\,\omega_{R}\left(f_{1},y_{1}\right)^{g-2}\text{Sym}(I)(n-1)$

when $g=\text{ht}(I)\geq 2$ and $I\subset R$ satisfies $F_{0}$ and
${\text{SD}}_{1}$.

######  Remark .

Assume the assumptions of Theorem 4.1. A natural question is whether
$\text{Sym}(I)$ is weakly Gorenstein over $\mathfrak{n}S$, where
$\mathfrak{n}=(y_{1},\ldots,y_{n})\subset B$. The natural choice for making
$\text{Sym}(I)$ weakly Gorenstein over $\mathfrak{n}S$ is to choose the
element $f_{1}^{g-2}$ instead of $y_{1}^{g-2}$ in the proof of Theorem 4.1.
However, this choice is not valid. Set $S=\text{Sym}(I)$. Indeed, since
$\text{ht}({\mathfrak{m}}S)=0$ and $y_{1}^{g-2}$ is a non zerodivisor on $S$,
we cannot achieve the condition that $Sf_{1}^{g-2}\otimes_{S}S_{\mathfrak{p}}$
generates
$(f_{1},y_{1})^{g-2}\text{Sym}(I)\otimes_{S}S_{\mathfrak{p}}=S_{\mathfrak{p}}$
for all $\mathfrak{p}\in\text{Ass}(S)$. The same argument shows that no
element in ${\mathfrak{m}}S$ would work. On the other hand, in Theorem 5.1 we
do consider the local cohomology modules $\text{H}_{\mathfrak{n}}^{i}(S)$ for
$i\geq 2$ by using further duality results.

Our computation of the canonical module of $\text{Sym}(I)$ has the following
interesting consequence.

######  Corollary .

Assume Section missing 4 with $R$ Gorenstein. Suppose that:

  1. (a)

$I\subset R$ satisfies the conditions $F_{0}$ and ${\text{SD}}_{1}$, and $I$
is generically a complete intersection with $g=\text{ht}(I)\geq 2$.

  2. (b)

$\text{gr}_{I}(R)$ is a Gorenstein ring.

Then the following statements hold:

  1. (i)

$I$ is an ideal of linear type.

  2. (ii)

If $I$ is Cohen-Macaulay, then it is actually strongly Cohen-Macaulay.

###### Proof.

We may assume that the residue field is infinite. By Section missing 4 we have
$\dim\left(\text{Sym}(I)\right)=d+1=\dim\left(\mathcal{R}(I)\right)$. After
dualizing, the surjection $\text{Sym}(I)\twoheadrightarrow\mathcal{R}(I)$
yields the natural inclusion

$\omega_{\mathcal{R}(I)}\,\hookrightarrow\,\omega_{\text{Sym}(I)}.$

On the other hand, [HSV2, Corollaries 2.5, 2.9] imply that
$\omega_{\mathcal{R}(I)}\cong(f_{1},y_{1})^{g-2}\mathcal{R}(I)(-1)$, and so we
have a natural surjection
$\omega_{\text{Sym}(I)}\cong(f_{1},y_{1})^{g-2}\text{Sym}(I)(-1)\twoheadrightarrow(f_{1},y_{1})^{g-2}\mathcal{R}(I)(-1)\cong\omega_{\mathcal{R}(I)}$.
This shows that $\omega_{\mathcal{R}(I)}$ is a direct summand
$\omega_{\text{Sym}(I)}$. If
$\omega_{\mathcal{R}(I)}\not\cong\omega_{\text{Sym}(I)}$, then it would follow
that $\text{Sym}(I)$ is not connected in codimension one (see, e.g., [HH,
Theorem 3.6]), but this would contradict Hartshorne’s connectedness theorem as
$\text{Sym}(I)$ is Cohen-Macaulay (see Section missing 4). Thus
$\omega_{\mathcal{R}(I)}\cong\omega_{\text{Sym}(I)}$.

Next we prove that $\mathcal{R}(I)$ is Cohen-Macaulay. By localizing at an
associated prime $\mathfrak{p}\in\text{Ass}(R/I)$, $I_{\mathfrak{p}}$ becomes
a complete intersection, and so
$\text{gr}_{I_{\mathfrak{p}}}(R_{\mathfrak{p}}/I_{\mathfrak{p}})\cong(R_{\mathfrak{p}}/I_{\mathfrak{p}})[z_{1},\ldots,z_{\text{grade}(I_{\mathfrak{p}})}]$
where the $z_{i}^{\prime}s$ are new variables. Since $\text{gr}_{I}(R)$ is
Gorenstein,
$a(\text{gr}_{I}(R))=a(\text{gr}_{I_{\mathfrak{p}}}(R_{\mathfrak{p}}/I_{\mathfrak{p}}))<0$,
and this implies that $\mathcal{R}(I)$ is Cohen-Macaulay (see [TrIk], [V,
Theorem 5.1.23]). Since $\text{Sym}(I)$ and $\mathcal{R}(I)$ are Cohen-
Macaulay of the same dimension, after dualizing the isomorphism
$\omega_{\mathcal{R}(I)}\cong\omega_{\text{Sym}(I)}$, we obtain
$\text{Sym}(I)=\mathcal{R}(I)$. This settles part (i).

Since $I$ is of linear type, the $\mathcal{M}_{\bullet}$ approximation complex
is exact and so $I$ is generated by a $d$-sequence (see [HSV, Theorem 12.9]).
Finally, the statement of part (ii) follows from [HSV2, Theorem 1.3]. ∎

Finally, we give an explicit free resolution for the symmetric algebra when
the ideal is an almost complete intersection or a perfect ideal of deviation
two, both under the assumption that $\mathcal{Z}_{\bullet}$ is acyclic. The
_deviation_ of $I\subset R$ is defined as $d(I):=\mu(I)-\text{ht}(I)=n-g$; one
says that $I$ is an almost complete intersection when $d(I)=1$.

######  Theorem 4.2.

Assume Section missing 4. Suppose that the $\mathcal{Z}_{\bullet}$ complex is
acyclic and let

$F_{\bullet}:\;\;\cdots\rightarrow F_{2}\rightarrow
R^{n}\xrightarrow{(f_{1},\ldots,f_{n})}R\rightarrow 0$

be a free $R$-resolution of $\,R/I$. Let $\mathfrak{T}_{\bullet}$ be the
acyclic complex of free $B$-module defined in Section missing 4. The following
statements hold:

  1. (i)

If $I\subset R$ is an almost complete intersection, then a free $B$-resolution
of $\text{Sym}(I)$ is given by

$\mathfrak{F}_{\bullet}:\;\;\cdots\rightarrow\mathfrak{F}_{i}\rightarrow\cdots\rightarrow\mathfrak{F}_{1}\rightarrow\mathfrak{F}_{0}\rightarrow
0$

where $\mathfrak{F}_{0}=B$, $\mathfrak{F}_{1}=F_{2}\otimes_{R}B(-1)$, and
$\mathfrak{F}_{i}=\mathfrak{T}_{i-2}\,\oplus\,\left(F_{i+1}\otimes_{R}B(-1)\right)$
for $i\geq 2$.

  2. (ii)

If $I\subset R$ is a perfect ideal of deviation two and
$F_{\bullet}:0\rightarrow F_{g}\rightarrow\cdots\rightarrow F_{2}\rightarrow
R^{n}\xrightarrow{(f_{1},\ldots,f_{n})}R\rightarrow 0$ is a minimal free
$R$-resolution of $R/I$, then a free $B$-resolution of $\text{Sym}(I)$ is
given by

$\mathfrak{F}_{\bullet}:\;\;\cdots\rightarrow\mathfrak{F}_{i}\rightarrow\cdots\rightarrow\mathfrak{F}_{1}\rightarrow\mathfrak{F}_{0}\rightarrow
0$

where $\mathfrak{F}_{0}=B$, $\mathfrak{F}_{1}=F_{2}\otimes_{R}B(-1)$, and
$\mathfrak{F}_{i}=\mathfrak{T}_{i-3}\oplus\big{(}\big{(}F_{i+1}\oplus
K_{i+1}\oplus F_{g-i+2}^{*}\big{)}\otimes_{R}B(-1)\big{)}$ for $i\geq 2$.

###### Proof.

Let $\mathcal{L}$ be the kernel of the natural surjection
$B\twoheadrightarrow\text{Sym}(I)$.

(i) From Section missing 4 we have the acyclic complex

$\mathfrak{L}_{\bullet}:\quad
0\rightarrow\mathfrak{T}_{g-2}\rightarrow\cdots\rightarrow\mathfrak{T}_{1}\xrightarrow{\Psi}\mathfrak{T}_{0}\rightarrow\mathcal{Z}_{1}\rightarrow
B\rightarrow 0$

that resolves $\text{Sym}(I)$. The map
${\text{Coker}}(\Psi)\rightarrow\mathcal{Z}_{1}$ induces a map of complexes
$f:\mathfrak{T}_{\bullet}\rightarrow G_{\bullet}$, where

$G_{\bullet}:\quad\cdots\rightarrow F_{3}\otimes_{R}B(-1)\rightarrow
F_{2}\otimes_{R}B(-1)\rightarrow 0$

is the resolution of $\mathcal{Z}_{1}=Z_{1}\otimes_{R}B(-1)$ obtained by
truncating $F_{\bullet}\otimes_{R}B(-1)$. Since we have a short exact sequence
$0\rightarrow{\text{Coker}}(\Psi)\rightarrow\mathcal{Z}_{1}\rightarrow\mathcal{L}\rightarrow
0$, the mapping cone $\text{cone}(f)$ of $f$ yields a free $B$-resolution of
$\mathcal{L}$. So, the proof of this part is complete.

(ii) The complex of Section missing 4 is now given by

$\mathfrak{L}_{\bullet}:\quad
0\rightarrow\mathfrak{T}_{g-2}\rightarrow\cdots\rightarrow\mathfrak{T}_{1}\xrightarrow{\Psi}\mathfrak{T}_{0}\rightarrow\mathcal{Z}_{2}\xrightarrow{\Phi}\mathcal{Z}_{1}\rightarrow
B\rightarrow 0.$

Notice that there is a short exact sequence
$0\rightarrow\text{B}_{2}(K_{\bullet})\rightarrow\text{Z}_{2}(K_{\bullet})\rightarrow\text{H}_{2}(K_{\bullet})\rightarrow
0$, the truncated Koszul complex $0\rightarrow
K_{n}\rightarrow\cdots\rightarrow K_{3}\rightarrow 0$ is a free $R$-resolution
of $\text{B}_{2}(K_{\bullet})$, and ${\text{Hom}}_{R}(F_{\bullet},R)[-g]$
gives a free $R$-resolution of $\text{H}_{2}(K_{\bullet})\cong\omega_{R/I}$ as
$R/I$ is perfect $R$-module. By the Horseshoe lemma, a free $B$-resolution of
$\mathcal{Z}_{2}=Z_{2}\otimes_{R}B(-1)$ is given by a complex $P_{\bullet}$
with $P_{i}=\big{(}K_{i+3}\oplus F_{g-i}^{*}\big{)}\otimes_{R}B(-1)$. As in
part (i), a mapping cone construction along the short exact sequence
$0\rightarrow{\text{Coker}}(\Psi)\rightarrow\mathcal{Z}_{2}\rightarrow\text{Im}(\Phi)\rightarrow
0$ yields a free $B$-resolution $Q_{\bullet}$ of $\text{Im}(\Phi)$ with
$Q_{i}=P_{i}\oplus\mathfrak{T}_{i-1}$. Another mapping cone construction along
the short exact sequence
$0\rightarrow\text{Im}(\Phi)\rightarrow\mathcal{Z}_{1}\rightarrow\mathcal{L}\rightarrow
0$ gives a free $B$-resolution $L_{\bullet}$ of $\mathcal{L}$ with
$L_{0}=F_{2}\otimes_{R}B(-1)$ and

$L_{i}\,=\,Q_{i-1}\oplus\big{(}F_{i+2}\otimes_{R}B(-1)\big{)}\,=\,\mathfrak{T}_{i-2}\,\oplus\,\big{(}\big{(}F_{i+2}\oplus
K_{i+2}\oplus F_{g-i+1}^{*}\big{)}\otimes_{R}B(-1)\big{)}$

for all $i\geq 1$. The resolution $\mathfrak{F}_{\bullet}$ is now given by
setting $\mathfrak{F}_{0}=B$ and $\mathfrak{F}_{i}=L_{i-1}$ for all $i\geq 1$.
∎

##  5\. A general framework of dualities to study blowup algebras

In this section, we apply our generalization of Jouanolou’s duality to study
the defining equations of several interesting classes of Rees algebras.
Determining the defining equations of Rees is a problem of utmost importance
with applications in Algebraic Geometry, Commutative Algebra and applied areas
like Geometric Modeling (see [KPU3, KPU4, KPU5, Buse, KM, CBD14, V_REES_EQ,
COX_MOV_CURV, COX_HOFFMAN_WANG, HONG_SV, KPU_RAT_SC, CBD13, Morey, MU]).

######  Setup .

Let $\mathbb{k}$ be a field, $R=\mathbb{k}[x_{1},\ldots,x_{d}]$ be a standard
graded polynomial ring and ${\mathfrak{m}}=(x_{1},\ldots,x_{d})\subset R$ be
the graded irrelevant ideal. Let $\\{f_{1},\ldots,f_{n}\\}\subset R$ be a set
of forms of the same degree $D\geq 1$, and set $I=(f_{1},\ldots,f_{n})\subset
R$ to be the ideal generated by these forms. Let
$T=\mathbb{k}[y_{1},\ldots,y_{n}]$ be a standard graded polynomial ring and
$\mathfrak{n}=(y_{1},\ldots,y_{n})\subset T$ be the graded irrelevant ideal.
Let $B$ be the standard bigraded polynomial ring $B=R\otimes_{\mathbb{k}}T$
(i.e., $\text{bideg}(x_{i})=(1,0)$ and $\text{bideg}(y_{i})=(0,1)$). Since we
are primarily interested in the $\mathbf{x}$-grading, for any bigraded
$B$-module $M$, we denote by $M_{i}$ the graded $T$-module
$M_{i}:=\bigoplus_{j\in{\mathbb{Z}}}[M]_{(i,j)}$.

As customary, we consider the $\mathbb{k}$-algebra homomorphism

$\Phi:B\twoheadrightarrow\mathcal{R}(I)=R[It]=\bigoplus_{j=0}^{\infty}I^{j}t^{j}\subset
R[t],\quad x_{i}\mapsto x_{i}\text{ and }y_{i}\mapsto f_{i}t.$

Our goal is to determine the ideal of defining equations
$\mathcal{J}:=\text{Ker}(\Phi)\subset B$ of the Rees algebra $\mathcal{R}(I)$.
Let
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{d-1}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{n-1}$
be the rational map

$(x_{1}:\cdots:x_{d})\;\mapsto\;(f_{1}(x_{1},\ldots,x_{d}):\cdots:f_{n}(x_{1},\ldots,x_{d}))$

determined by the forms $f_{1},\ldots,f_{n}$ generating $I$. The Rees algebra
$\mathcal{R}(I)$ provides the bihomogeneous coordinate ring of the closure of
the graph of $\mathcal{G}$, and this reinforces the interest on finding the
defining equations of $\mathcal{R}(I)$. Typically, a good way to study the
Rees algebra is by approximating it by the symmetric algebra (which is a much
better understood object). Given a minimal graded presentation
$F_{1}\xrightarrow{\varphi}F_{0}\rightarrow I\rightarrow 0$ of the ideal $I$,
the ideal of defining equations of $\text{Sym}(I)$ is given by

$\mathcal{L}\,:=\,(g_{1},\ldots,g_{u})=I_{1}\left(\left[y_{1},\ldots,y_{n}\right]\cdot\varphi\right);$

that is, $\text{Sym}(I)\cong B/\mathcal{L}$ and $u=\text{rank}(F_{1})$ is the
number of syzygies of $I$. There is a short exact sequence of bigraded
$B$-modules

$0\rightarrow\mathcal{A}\rightarrow\text{Sym}(I)\rightarrow\mathcal{R}(I)\rightarrow
0,$

where $\mathcal{A}=\mathcal{J}/\mathcal{L}$ coincides with the $R$-torsion of
the symmetric algebra. Therefore, one can study $\mathcal{A}$ to determine (or
to obtain information about) the defining equations $\mathcal{J}$ of the Rees
algebra $\mathcal{R}(I)$. Our main contribution in this direction is the
following theorem.

######  Theorem 5.1.

Assume Section missing 5. Suppose that $I=(f_{1},\ldots,f_{n})\subset R$ has
height $g=\text{ht}(I)\geq 2$, that $I$ satisfies the conditions $G_{d}$ and
${\text{SD}}_{1}$, and that $n=d+1$. Let $\delta:=(g-1)D-d$ and
$\beta:=d-g+2$. Then the following statements hold:

  1. (i)

$\mathcal{A}=\text{H}_{{\mathfrak{m}}}^{0}(\text{Sym}(I))$.

  2. (ii)

For all $0\leq i\leq d-1$, there is an isomorphism of bigraded $B$-modules

$\text{H}_{{\mathfrak{m}}}^{i}(\text{Sym}(I))\,\cong\,{}^{*}\text{Ext}_{T}^{i}\left(\text{Sym}(I),T\right)\left(-\delta,-\beta\right).$

In particular,
$\mathcal{A}=\text{H}_{{\mathfrak{m}}}^{0}(\text{Sym}(I))\cong{}^{*}{\text{Hom}}_{T}\left(\text{Sym}(I),T\right)\left(-\delta,-\beta\right)$.

  3. (iii)

For all $i<0$ and $i>\delta$, we have $\mathcal{A}_{i}=0$. There is an
isomorphism $\mathcal{A}_{\delta}\cong T(-\beta)$ of graded $T$-modules. For
all $0\leq i\leq\delta$, we have the equality of $\text{Sym}(I)$-ideals

$\mathcal{A}_{\geq i}\;=\;0:_{\text{Sym}(I)}{\mathfrak{m}}^{\delta+1-i}.$

  4. (iv)

$\mathcal{A}$ is minimally generated in $\mathbf{x}$-degree at most
$(g-2)D-d+1$.

  5. (v)

Let $0\leq i\leq\delta$. The natural multiplication map
$\mu:\mathcal{A}_{i}\;\otimes_{T}\;\text{Sym}(I)_{\delta-i}\;\rightarrow\;\mathcal{A}_{\delta},\;a\otimes
b\mapsto a\cdot b$ is a perfect pairing that induces the abstract isomorphism

$\nu:\mathcal{A}_{i}\;\xrightarrow{\;\cong\;}\;{\text{Hom}}_{T}\left(\text{Sym}(I)_{\delta-i},\mathcal{A}_{\delta}\right)$

seen in part (ii).

  6. (vi)

For all $2\leq i\leq d+1$, there is an isomorphism of bigraded $B$-modules

$\text{H}_{\mathfrak{n}}^{i}(\text{Sym}(I))\,\cong\,{}^{*}\text{Ext}_{R}^{i-1}(\text{Sym}(I),R)\left(-(g-1)D,g-1\right).$

###### Proof.

To simplify notation, set $S:=\text{Sym}(I)$. Recall that $S$ is a Cohen-
Macaulay ring of dimension $d+1$ under the present conditions (see Section
missing 4 and Section missing 4). We may assume that $\mathbb{k}$ is an
infinite field (see Section missing 5).

(i) From the assumed conditions it follows that $I$ is of linear type on the
punctured spectrum of $R$ (see [HSV]). This implies that
$\mathcal{A}=\text{H}_{{\mathfrak{m}}}^{0}(S)$. Alternatively, see [KPU4,
§3.7].

(ii) The existence of the claimed isomorphism follows from Theorem 4.1 and
Theorem 2.6. We only need to compute the precise shifts in bidegree, and for
that it is enough to compute $\text{bideg}(\overline{w_{1}})$, where
$\overline{w_{1}}\in\omega_{S}$ is the element chosen in the proof of Theorem
4.1 to make $S$ weakly Gorenstein with respect ${\mathfrak{m}}S$.

In the current bigraded setting, we have that

$\mathfrak{T}_{g-2}=\bigoplus_{j=1}^{g-1}B\left(-(g-j)D,-n+g-j\right)$

where
$\mathfrak{L}_{\bullet}:\,\mathfrak{T}_{g-2}\rightarrow\cdots\rightarrow\mathfrak{T}_{1}\rightarrow\mathfrak{T}_{0}\rightarrow\mathcal{Z}_{n-g}\rightarrow\cdots\rightarrow\mathcal{Z}_{0}\rightarrow
0$ is the complex introduced in Section missing 4. As in the proof of Theorem
4.1, we have the surjection

${\text{Hom}}_{B}\left(\mathfrak{T}_{g-2},\omega_{B}\right)\otimes_{B}S\;=\;\bigoplus_{j=1}^{g-1}S\left((g-j)D-d,-g+j\right)\;\twoheadrightarrow\;\omega_{S}$

and that $\overline{w}_{1}\in\omega_{S}$ is image of the generator
corresponding with the summand $S\left((g-1)D-d,-g+1\right)$. Thus
$\text{bideg}(\overline{w_{1}})=\left(d-(g-1)D,g-1\right)$, and so Theorem 2.6
gives

$\text{H}_{{\mathfrak{m}}}^{i}(S)\,\cong\,{}^{*}\text{Ext}_{T}^{i}\left(S,T\right)\left(d-(g-1)D,-n+g-1\right)$

for all $0\leq i\leq d-1$.

(iii) The isomorphism
$\mathcal{A}\cong{}^{*}{\text{Hom}}_{T}\left(S,T\right)(-\delta,-\beta)$
implies that $\mathcal{A}_{i}=0$ when $i<0$ and $i>\delta$, and it gives the
isomorphism $\mathcal{A}_{\delta}\cong T(-\beta)$ of graded $T$-modules. The
same isomorphism for $\mathcal{A}$ together with §missing 2.1 shows the
equality $\mathcal{A}_{\geq i}\,=\,0:_{S}{\mathfrak{m}}^{\delta+1-i}.$

(iv) Since the ideal $I\subset R$ satisfies ${\text{SD}}_{1}$, the graded
strands of bidegree $(*,k)$ of the complex of Section missing 4 are acyclic
complexes of $R$-modules that satisfy the assumptions of [KPU5, Theorem 4.3].
The $(d-1)$-th module in each of these complexes is

$\left[\mathfrak{T}_{g-3}\right]_{(*,k)}=\left[\bigoplus_{j=1}^{g-2}B^{n}\left(-(g-1-j)D,-n+g-j\right)\right]_{(*,k)},$

it follows that these $R$-modules are generated in degree at most $(g-2)D$.
According to [KPU5, Theorem 4.3] the $\mathbf{x}$-degree of the minimal
generators of $\mathcal{A}$ is at most $(g-2)D-d+1$.

(v) This follows from the analogue of Theorem 2.3(iii) in the bigraded
setting.

(vi) To conclude the proof of part (v) of the theorem we shall need the
following duality result of Herzog and Rahimi [HeRa]; for the reader’s
convenience, we provide a short direct proof in Theorem 5.2 below. We have the
following isomorphisms

$\displaystyle\text{H}_{\mathfrak{n}}^{d+1-i}(S)$
$\displaystyle\;\cong\;\big{(}\text{H}_{\mathfrak{m}}^{i}(\omega_{S})\big{)}^{\vee_{B}}$
(by Theorem 5.2)
$\displaystyle\;\cong\;\big{(}\text{H}_{\mathfrak{m}}^{i}(S)\big{)}^{\vee_{B}}(\text{bideg}(\overline{w_{1}}))$
(by Section missing 2 and Theorem 4.1)
$\displaystyle\;\cong\;\text{H}_{\mathfrak{n}}^{d+1-i}(\omega_{S})(\text{bideg}(\overline{w_{1}}))$
(by Theorem 5.2)

for all $0\leq i\leq d-1$. By changing the role of $R$ and $T$ in Theorem 2.5
we obtain the isomorphism

$\text{H}_{\mathfrak{n}}^{i}(\omega_{S})\;\cong\;{}^{*}\text{Ext}_{R}^{i-1}(S,R)(-d,0)$

for all $i\geq 0$. Therefore, by combining everything we get

$\text{H}_{\mathfrak{n}}^{i}(S)\,\cong\,{}^{*}\text{Ext}_{R}^{i-1}(S,R)\left(-(g-1)D,g-1\right)$

for all $2\leq i\leq d+1$. ∎

For any bigraded $B$-module, we consider the bigraded Matlis dual functor

$(M)^{\vee_{B}}:=\bigoplus_{i,j\in{\mathbb{Z}}}{\text{Hom}}_{\mathbb{k}}\big{(}\left[M\right]_{(-i,-j)},\mathbb{k}\big{)}.$

The following theorem contains a short proof of Herzog-Rahimi duality. This
duality was needed in the proof of part (vi) of the above theorem.

######  Theorem 5.2 (Herzog-Rahimi duality [HeRa]).

Let $M$ be a finitely generated bigraded $B$-module. Then there are two
converging spectral sequences of bigraded $B$-modules

$\text{H}_{\mathfrak{n}}^{p}\left(\text{Ext}_{B}^{q}(M,\omega_{B})\right)\;\,\Longrightarrow\;\,\left(\text{H}_{{\mathfrak{m}}}^{d+n-p-q}(M)\right)^{\vee_{B}}$

and

$\text{H}_{\mathfrak{m}}^{p}\left(\text{Ext}_{B}^{q}(M,\omega_{B})\right)\;\,\Longrightarrow\;\,\left(\text{H}_{\mathfrak{n}}^{d+n-p-q}(M)\right)^{\vee_{B}}.$

In particular, if $M$ is Cohen-Macaulay, then there are two isomorphisms of
bigraded $B$-modules

$\text{H}_{\mathfrak{n}}^{i}(\omega_{M})\;\cong\;\big{(}\text{H}_{\mathfrak{m}}^{\dim(M)-i}(M)\big{)}^{\vee_{B}}\quad\text{
and
}\quad\text{H}_{\mathfrak{m}}^{i}(\omega_{M})\;\cong\;\big{(}\text{H}_{\mathfrak{n}}^{\dim(M)-i}(M)\big{)}^{\vee_{B}}$

for all $i\in{\mathbb{Z}}$.

###### Proof.

Let $F_{\bullet}:\cdots\rightarrow F_{i}\rightarrow\cdots\rightarrow
F_{1}\rightarrow F_{0}\rightarrow 0$ be a minimal bigraded free $B$-resolution
of $M$. As in the second proof of Theorem 2.1, via the spectral sequences
coming from the second quadrant double complex
$F_{\bullet}\otimes_{R}C_{\mathfrak{m}}^{\bullet}$ we obtain the isomorphisms
$\text{H}_{{\mathfrak{m}}}^{i}(M)\,\cong\,\text{H}_{d-i}\left(\text{H}_{{\mathfrak{m}}}^{d}(F_{\bullet})\right).$
Dualizing with the exact functor $(\bullet)^{\vee_{B}}$, we now get

$\left(\text{H}_{{\mathfrak{m}}}^{i}(M)\right)^{\vee_{B}}\;\cong\;\text{H}^{d-i}\left(\left(\text{H}_{{\mathfrak{m}}}^{d}(F_{\bullet})\right)^{\vee_{B}}\right)\;\cong\;\text{H}^{d-i}\Big{(}\text{H}_{\mathfrak{n}}^{n}\big{(}{\text{Hom}}_{B}(F_{\bullet},\omega_{B})\big{)}\Big{)}$

where the last isomorphism follows from the functorial isomorphism
$\left(\text{H}_{{\mathfrak{m}}}^{d}(B)\right)^{\vee_{B}}\cong\text{H}_{\mathfrak{n}}^{n}\left({\text{Hom}}_{B}(B,\omega_{B})\right)$.
Let $G^{\bullet}:={\text{Hom}}_{B}\left(F_{\bullet},\omega_{B}\right)$ and
consider the first quadrant double complex
$G^{\bullet}\otimes_{T}C_{\mathfrak{n}}^{\bullet}$; the corresponding spectral
sequences are given by

${}^{\text{I}}E_{2}^{p,q}\;=\;\begin{cases}\text{H}^{p}\left(\text{H}_{\mathfrak{n}}^{n}(G^{\bullet})\right)\,\cong\,\left(\text{H}_{{\mathfrak{m}}}^{d-p}(M)\right)^{\vee_{B}}&\text{
if }q=n\\\ 0&\text{ otherwise}\end{cases}$

and

${}^{\text{II}}E_{2}^{p,q}\;=\;\text{H}_{\mathfrak{n}}^{p}\left(\text{H}^{q}(G^{\bullet})\right)\;\cong\;\text{H}_{\mathfrak{n}}^{p}\left(\text{Ext}_{B}^{q}(M,\omega_{B})\right).$

Therefore, we obtain the convergent spectral sequence

$\text{H}_{\mathfrak{n}}^{p}\left(\text{Ext}_{B}^{q}(M,\omega_{B})\right)\;\,\Longrightarrow\;\,\left(\text{H}_{{\mathfrak{m}}}^{d+n-p-q}(M)\right)^{\vee_{B}}.$

The other spectral sequence is obtained by a completely symmetrical argument.
The two claimed isomorphisms follow immediately from the spectral sequences. ∎

######  Remark .

The isomorphisms from Theorem 5.1 are probably best seen in the form of a
diagram. Assume Section missing 5 with all the conditions and notations of
Theorem 5.1. We have the following diagram of bigraded $B$-isomorphisms, where
the label on each arrow specifies the range on $i$ where the isomorphism is
valid:

${\left({}^{*}\text{Ext}_{R}^{d-i}\left(\text{Sym}(I),R\right)\right)^{\vee_{B}}(d,0)}$${\left(\text{H}_{\mathfrak{n}}^{d+1-i}\left(\omega_{\text{Sym}(I)}\right)\right)^{\vee_{B}}}$${\text{H}_{{\mathfrak{m}}}^{i}\left(\text{Sym}(I)\right)}$${\left(\text{H}_{\mathfrak{n}}^{d+1-i}\left({\text{Sym}(I)}\right)\right)^{\vee_{B}}(-\delta,-\beta+n)}$${\text{H}_{{\mathfrak{m}}}^{i}\left(\omega_{\text{Sym}(I)}\right)(-\delta,-\beta+n)}$${{}^{*}\text{Ext}_{T}^{i}\left(\text{Sym}(I),T\right)(-\delta,-\beta)}$$\cong$$i\geq
0$$\cong$$i\geq 0$$\cong$$i\geq 0$$\cong$$0\leq i\leq d-1$$\cong$$i\geq 0$

In particular, when $0\leq i\leq d-1$, we obtain the surprising fact that the
above six $B$-modules are related by bigraded $B$-isomorphisms.

######  Remark .

The results of Theorem 5.1 hold independently of whether the field
$\mathbb{k}$ is infinite or not. Let $\mathbb{k}^{\prime}$ be an infinite
field containing $\mathbb{k}$. Set
$R^{\prime}:=R\otimes_{\mathbb{k}}\mathbb{k}^{\prime}$, $S:=\text{Sym}(I)$ and
$S^{\prime}:=S\otimes_{\mathbb{k}}\mathbb{k}^{\prime}$. Assume the assumptions
of Theorem 5.1. From Theorem 4.1, we obtain that $S^{\prime}$ is $d$-weakly
Gorenstein over ${\mathfrak{m}}S^{\prime}$. As in the proof of Theorem
5.1(ii), we have that
$\dim_{\mathbb{k}^{\prime}}\big{(}\left[\omega_{S^{\prime}}\right]_{\left(d-(g-1)D,g-1\right)}\big{)}=1.$
This implies that
$\dim_{\mathbb{k}}\big{(}\left[\omega_{S}\right]_{\left(d-(g-1)D,g-1\right)}\big{)}=1$.
Let $y\in\left[\omega_{S}\right]_{\left(d-(g-1)D,g-1\right)}$ be a non-zero
element. The element $\overline{w_{1}^{\prime}}\in\omega_{S^{\prime}}$ chosen
to make $S^{\prime}$ $d$-weakly Gorenstein over ${\mathfrak{m}}S^{\prime}$ has
bidegree $(d-(g-1)D,g-1)$, and so it follows that
$\overline{w_{1}}=\alpha\cdot(y\otimes_{R}1_{R^{\prime}})\in\omega_{S^{\prime}}\cong\omega_{S}
\otimes_{R}R^{\prime}$ where $\alpha\in\mathbb{k}^{\prime}$ is a unit.

Below we find a simple condition for the $T$-freeness of $\mathcal{A}$.

######  Proposition .

Assume Section missing 5. Suppose that $\text{Sym}(I)$ is Cohen-Macaulay and
$I$ is of linear type on the punctured spectrum of $R$. Then,
$0pt(\mathcal{R}(I))\geq d$ if and only if $\mathcal{A}$ is a free $T$-module.

###### Proof.

Since $\text{Sym}(I)$ is Cohen-Macaulay, it follows that it is of dimension
$d+1$. We have the equality
$\mathcal{A}=\text{H}_{{\mathfrak{m}}}^{0}(\text{Sym}(I))$ because $I$ is of
linear type on ${\text{Spec}}(R)\setminus\\{{\mathfrak{m}}\\}$. Then, there is
a positive integer $k>0$ such that $\mathcal{A}$ is a module over
$B/{\mathfrak{m}}^{k}B$. By computing depths via the finite homomorphism
$T\rightarrow B/{\mathfrak{m}}^{k}B$, we get

$0pt_{T}(\mathcal{A})\,=\,0pt_{B/{\mathfrak{m}}^{k}B}(\mathcal{A})\,=\,0pt_{B}(\mathcal{A}).$

The short exact sequence
$0\rightarrow\mathcal{A}\rightarrow\text{Sym}(I)\rightarrow\mathcal{R}(I)\rightarrow
0$ yields $\text{depth}_{B}(\mathcal{A})\geq
1+\min\\{d,\text{depth}_{B}(\mathcal{R}(I))\\}$. Thus
$0pt_{T}(\mathcal{A})=d+1$ if and only if $0pt_{B}(\mathcal{R}(I))\geq d$. In
other words, $\mathcal{A}$ is a free $T$-module if and only if
$0pt_{B}(\mathcal{R}(I))\geq d$. ∎

###  5.1. Explicit equations via Morley forms

We are now going to apply the theory of Morley forms developed in §missing 2.1
to symmetric algebras of ideals. Unlike in the classical case (where
$\text{Sym}(I)$ is a complete intersection), here the Morley forms _do not_
give a perfect pairing. We instead need to introduce a division/reduction to
make explicit the perfect pairing seen in Theorem 5.1(v). Throughout this
subsection, we assume the hypotheses of Theorem 5.1.

######  Setup .

Assume Section missing 5. Suppose that $I=(f_{1},\ldots,f_{n})\subset R$ has
height $g=\text{ht}(I)\geq 2$, that $I$ satisfies the conditions $G_{d}$ and
${\text{SD}}_{1}$, and that $n=d+1$. In particular, Theorem 5.1 yields the
isomorphism

$\mathcal{A}=\text{H}_{{\mathfrak{m}}}^{0}(\text{Sym}(I))\cong{}^{*}{\text{Hom}}_{T}\left(\text{Sym}(I),T\right)\left(-\delta,-\beta\right)$

where $\delta:=(g-1)D-d$ and $\beta:=d-g+2$. Assume there is a bihomogeneous
regular sequence $\ell_{1},\ldots,\ell_{d}$ belonging to $\mathcal{L}$ that
generates $\mathcal{L}$ at ${\mathfrak{m}}B\in\text{Min}(\mathcal{L})$, and
satisfies $\sum_{i=1}^{d}\deg_{\mathbf{x}}(\ell_{i})=\delta+d$ and
$\deg_{\mathbf{y}}(\ell_{i})=1$.

To construct an explicit element
$\Delta\in\left[0:_{\text{Sym}(I)^{e}}\mathbb{D}\right]_{\delta}$ that can be
used in the definition of Morley forms, we write

$\left[\ell_{1},\ldots,\ell_{d}\right]\;=\;\left[x_{1},\ldots,x_{d}\right]\cdot
G\,$

where $G\in B^{d\times d}$ is a $d\times d$ matrix whose entries are
bihomogenous with constant bidegrees along the columns. Multiplying with the
adjoint of $G$ yields that

$\left(x_{1},\ldots,x_{d}\right)\cdot\det(G)\,\subset\,\left(\ell_{1},\ldots,\ell_{d}\right)\,\subset\,\mathcal{L}.$

Let ${\text{syl}}\in\text{Sym}(I)$ be the image of $\det(G)$ in
$\text{Sym}(I)$. Notice that ${\text{syl}}\in\mathcal{A}$. Moreover, syl is
non-zero. Indeed, since $\ell_{1},\ldots,\ell_{d}$ generates $\mathcal{L}$
locally at the minimal prime ${\mathfrak{m}}B\in\text{Min}(\mathcal{L})$, it
follows that

$(\ell_{1},\ldots,\ell_{d}):_{B}\mathcal{L}\,\not\subset\,{\mathfrak{m}}B.$

However, by linkage, this is equivalent to
$\mathcal{L}\not\supset(\ell_{1},\ldots,\ell_{d}):_{B}{\mathfrak{m}}B=(\ell_{1},\ldots,\ell_{d},\det(G))$.

Since $\mathcal{A}_{\delta}\cong T(-\beta)$, we choose a generator
${\text{rsyl}}\in\left[\mathcal{A}_{\delta}\right]_{\beta}$ of
$\mathcal{A}_{\delta}$. Consider the graded $T$-isomorphism
$\mu_{\text{rsyl}}:T(-\beta)\xrightarrow{\;\cdot{\text{rsyl}}\;}\mathcal{A}_{\delta}.$
As
$\text{bideg}({\text{syl}})=(\sum_{i=1}^{d}\deg_{\mathbf{x}}(\ell_{i})-d,d)=(\delta,d)$,
we have that ${\text{syl}}\in\mathcal{A}_{\delta}$, and so there is a non-zero
element $\alpha\in T_{d-\beta}=T_{g-2}$ such that

${\text{syl}}=\alpha\cdot{\text{rsyl}}.$

Following standard notation in the literature, we say that
${\text{syl}}\in\mathcal{A}_{\delta}$ is the _Sylvester form of
$\ell_{1},\ldots,\ell_{d}$ with respect to $x_{1},\ldots,x_{d}$ _ (or
_determinant of the Jacobian dual_). As we are interested in the generator of
$\mathcal{A}_{\delta}$, we consider the element
${\text{rsyl}}=\frac{{\text{syl}}}{\alpha}$, which can be dubbed _reduced
Sylvester form_. This element corresponds to the element $\mathfrak{s}$ of
Theorem 2.3.

Consider the enveloping algebra $B^{e}=B\otimes_{T}B$. Because each
$\ell_{i}\otimes 1-1\otimes\ell_{i}$ is in the kernel of the multiplication
map $B^{e}\rightarrow B$, we can write

$\left[\ell_{1}\otimes 1-1\otimes\ell_{1},\;\ldots,\;\ell_{d}\otimes
1-1\otimes\ell_{d}\right]\;=\;\left[x_{1}\otimes 1-1\otimes
x_{1},\;\ldots,\;x_{d}\otimes 1-1\otimes x_{d}\right]\cdot H\,,$

where $H$ is a $d\times d$ matrix with entries in $B^{e}$. Let $\Delta$ be the
image of $\det(H)$ in $\text{Sym}(I)^{e}$. Notice that $\Delta\in
0:_{\text{Sym}(I)^{e}}\mathbb{D}$ and $\Delta$ is bihomogeneous of bidegree
$(\delta,d)$. Moreover, $\varepsilon(H)={\text{syl}}$, which is not zero, and
so $\Delta\not\in({\mathfrak{m}}\otimes 1)\text{Sym}(I)^{e}$. Therefore, after
the previous discussion, we have the following remark.

######  Remark .

Assume §missing 5.1. The element
$\Delta=\overline{\det(H)}\in\text{Sym}(I)^{e}$ can be chosen to define Morley
forms as in §missing 2.1. Thus we obtain the decomposition

$\Delta\;=\;\overline{\det(H)}\;=\;\sum_{i=0}^{\delta}{\text{morl}}_{(i,\delta-i)}\quad\text{
where
}\quad{\text{morl}}_{(i,\delta-i)}\in\big{[}\text{Sym}(I)\otimes_{T}\text{Sym}(I)\big{]}_{(i,\delta-i,d)}.$

Now, applying Theorem 2.4 with $\mathfrak{s}={\text{rsyl}}$, we obtain
explicit isomorphisms between the $T$-modules
${\text{Hom}}_{T}(\text{Sym}(I)_{\delta-i},\mathcal{A}_{\delta})$ and
$\mathcal{A}_{i}$.

The above requirement that $\text{Sym}(I)$ is a complete intersection at the
minimal prime ${\mathfrak{m}}\text{Sym}(I)$ is satisfied in several situations
of interest (see §missing 6.1 and §missing 6.2).

##  6\. Applications to certain families of ideals

In this section, we apply the results and techniques developed in previous
sections. For organizational purposes, we divide the section into two
subsections. Each subsection covers a family of ideals where our results are
applied.

###  6.1. Zero dimensional ideals

In this subsection, we study the Rees algebra of zero dimensional ideals. In
this case, we are able of approximating the Rees algebras with two algebras:
one is the symmetric algebra of the ideal, and the other is the symmetric
algebra of the module determined by the Koszul syzygies. This provides two
methods to study the Rees algebra of these ideals. We now use the following
setup.

######  Setup .

Assume Section missing 5 with $I\subset R$ an ${\mathfrak{m}}$-primary ideal,
and suppose that $n=d+1$. Let $(K_{\bullet},\partial_{\bullet})$ be the Koszul
complex associated to the sequence $f_{1},\ldots,f_{n}$, and $H_{\bullet}$
denote its homologies. Let $E:={\text{Coker}}(\partial_{2})$ be the module
determined by the Koszul syzygies.

The next lemma collects some basic relations between the module $E$ and the
ideal $I$.

######  Lemma .

The following statements hold:

  1. (i)

$\text{Sym}(E)\cong B/\mathcal{K}$ where is $\mathcal{K}$ the determinantal
ideal

$\mathcal{K}\,=\,I_{2}\left(\begin{array}[]{cccc}y_{1}&y_{2}&\cdots&y_{d+1}\\\
f_{1}&f_{2}&\cdots&f_{d+1}\end{array}\right).$

  2. (ii)

$\mathcal{R}(E)\cong\mathcal{R}(I)$, and there is a short exact sequence
$0\rightarrow\text{H}_{{\mathfrak{m}}}^{0}(\text{Sym}(E))\rightarrow\text{Sym}(E)\rightarrow\mathcal{R}(I)\rightarrow
0.$

  3. (iii)

$\dim\left(\text{Sym}(E)\right)=\dim\left(\text{Sym}(I)\right)=d+1$.

###### Proof.

(i) By construction, the presentation of $E$ is given by
$K_{2}\xrightarrow{\partial_{2}}K_{1}\rightarrow E\rightarrow 0.$ If we set
$e_{1},\ldots,e_{d+1}$ to be a basis of the free module $K_{1}$, then
$e_{i}\wedge e_{j}$ with $1\leq i<j\leq d+1$ give a basis for the free module
$K_{2}$. The map $\partial_{2}$ is defined by $\partial_{2}(e_{i}\wedge
e_{j})=f_{i}e_{j}-f_{j}e_{i}.$ Let $\left[\partial_{2}\right]$ be the matrix
corresponding to $\partial_{2}$ with respect to the chosen bases of $K_{1}$
and $K_{2}$. Then the defining equations of $\text{Sym}(E)$ are given by

$\mathcal{K}=\left(y_{1},y_{2},\ldots,y_{d+1}\right)\cdot\left[\partial_{2}\right]=\left({\\{f_{i}y_{j}-f_{j}y_{i}\\}}_{1\leq
i<j\leq
d+1}\right)=I_{2}\left(\begin{array}[]{cccc}y_{1}&y_{2}&\cdots&y_{d+1}\\\
f_{1}&f_{2}&\cdots&f_{d+1}\end{array}\right).$

(ii) This follows from the fact that, for any
$\mathfrak{p}\in{\text{Spec}}(R)\setminus\\{{\mathfrak{m}}\\}$, we have
$E_{\mathfrak{p}}\cong I_{\mathfrak{p}}=R_{\mathfrak{p}}$.

(iii) This part follows from the Huneke-Rossi formula (see [HR], [V, Theorem
1.2.1]). ∎

As a consequence of the above lemma, we now have the following two short exact
sequences

$0\rightarrow\mathcal{A}\rightarrow\text{Sym}(I)\rightarrow\mathcal{R}(I)\rightarrow
0 \quad\text{ and }\quad
0\rightarrow\mathcal{B}\rightarrow\text{Sym}(E)\rightarrow\mathcal{R}(I)\rightarrow
0$

where $\mathcal{A}=\text{H}_{{\mathfrak{m}}}^{0}(\text{Sym}(I))$ and
$\mathcal{B}=\text{H}_{{\mathfrak{m}}}^{0}(\text{Sym}(E))$. The next theorem
makes $\text{Sym}(I)$ and $\text{Sym}(E)$ good candidates to approximate
$\mathcal{R}(I)$ when $I\subset R$ is ${\mathfrak{m}}$-primary.

######  Theorem 6.1.

Assume §missing 6.1. Then the following statements hold:

  1. (i)

For all $0\leq i\leq d-1$, there is an isomorphism of bigraded $B$-modules

$\text{H}_{{\mathfrak{m}}}^{i}(\text{Sym}(I))\,\cong\,{}^{*}\text{Ext}_{T}^{i}\left(\text{Sym}(I),T\right)\left(-dD-
d-D,-2\right).$

In particular,
$\mathcal{A}=\text{H}_{{\mathfrak{m}}}^{0}(\text{Sym}(I))\cong{}^{*}{\text{Hom}}_{T}\left(\text{Sym}(I),T\right)\left(-dD-
d-D,-2\right)$.

  2. (ii)

For all $0\leq i\leq d-1$, there is an isomorphism of bigraded $B$-modules

$\text{H}_{{\mathfrak{m}}}^{i}(\text{Sym}(E))\,\cong\,{}^{*}\text{Ext}_{T}^{i}\left(\text{Sym}(E),T\right)\left(-dD-d,-1\right).$

In particular,
$\mathcal{B}=\text{H}_{{\mathfrak{m}}}^{0}(\text{Sym}(E))\cong{}^{*}{\text{Hom}}_{T}\left(\text{Sym}(E),T\right)\left(-dD-d,-1\right)$.

###### Proof.

Part (i) follows from Theorem 5.1 and part (ii) from Section missing 3. ∎

######  Remark .

After having the abstract duality statements of part (ii) in the above
theorem, all the statements of Theorem 5.1 hold for the $\text{Sym}(E)$ with
shifts in bidegree given by $\delta=dD-d$ and $\beta=1$. Also, since
$\text{Sym}(E)$ is a complete intersection at the minimal prime
${\mathfrak{m}}\text{Sym}(E)$, it follows that the new Morley form techniques
developed in §missing 5.1 can be applied to $\text{Sym}(E)$. We are not
pursuing this approach in the present paper due to length constraints. In a
subsequent paper, we plan to study in great depth the Rees algebra
$\mathcal{R}(I)$ when $I$ is an ${\mathfrak{m}}$-primary ideal; from a
geometrical standpoint it is quite relevant, as this entails studying the
graph of a morphism
${\mathbb{P}}_{\mathbb{k}}^{d-1}\rightarrow{\mathbb{P}}_{\mathbb{k}}^{d}$
parametrizing a hypersurface.

###  6.2. Gorenstein ideals of codimension three

In this subsection, we concentrate on Gorenstein ideals of height $3$ and we
determine the defining equations of the Rees algebra for a particular family.

######  Theorem 6.2.

Assume Section missing 5. Suppose that $I=(f_{1},\ldots,f_{n})\subset R$ is a
Gorenstein ideal of height $3$ with $\mu(I)=n=d+1$ that satisfies $G_{d}$. Let
$h\geq 1$ be the degree of every non-zero entry of an alternating minimal
presentation matrix of $I$ (consequently, $\deg(f_{i})=D=\frac{n-1}{2}h$).
Then we have a bigraded isomorphism of $B$-modules

$\text{H}_{{\mathfrak{m}}}^{i}(\text{Sym}(I))\,\cong\,{}^{*}\text{Ext}_{T}^{i}\left(\text{Sym}(I),T\right)\left(-d(h-1),-d+1\right).$

for all $0\leq i\leq d-1$.

###### First proof.

It is a particular case of Theorem 5.1. ∎

###### Second proof.

To simplify notation, set $S:=\text{Sym}(I)$. From Section missing 4 and
Section missing 4 we have that $S$ is a Cohen-Macaulay ring of codimension $d$
in $B$. Then Theorem 2.5 yields the bigraded isomorphism
$\text{H}_{{\mathfrak{m}}}^{i}\left(\omega_{S}\right)\,\cong\,{}^{*}\text{Ext}_{T}^{i}\left(S,T\right)(0,-d-1)$
for all $i\geq 0$. It remains to determine which weakly Gorenstein condition
with to ${\mathfrak{m}}S$ is satisfied and to compute the shifts in bidegree.

We have that
$\mathcal{L}=(g_{1},g_{2},\ldots,g_{d+1})=I_{1}\left(\left[y_{1},\ldots,y_{d+1}\right]\cdot\varphi\right)\subset
B$ where $\varphi\in R^{(d+1)\times(d+1)}$ is an alternating matrix with
homogeneous entries of degree $h$. We may assume that $\mathbb{k}$ is an
infinite field, and after elementary column and row operations we transform
$\varphi$ into a new alternating matrix such that $(g_{2},\ldots,g_{d+1})$ is
a regular sequence in $B$ (see, e.g., [BH, Proposition 1.5.12]). Set
$\overline{B}:=B/(g_{2},\ldots,g_{d+1})$. The canonical module $\omega_{S}$ is
given by

$\displaystyle\omega_{S}\;\cong\;{\text{Hom}}_{\overline{B}}\left(S,\omega_{\overline{B}}\right)$
$\displaystyle\;\cong\;{\text{Hom}}_{\overline{B}}\left(S,\overline{B}\right)\left(d(h-1),-1\right)$
$\displaystyle\;\cong\;\frac{\left(g_{2},\ldots,g_{d+1}\right):\mathcal{L}}{\left(g_{2},\ldots,g_{d+1}\right)}\left(d(h-1),-1\right)$

(see, e.g., [BH, Corollary 3.6.14]). From [HU85, Proof of Proposition 5.9] we
can compute the link $\left(g_{2},\ldots,g_{d+1}\right):\mathcal{L}$ as

$\left(g_{2},\ldots,g_{d+1}\right):\mathcal{L}\,=\,(y_{1},\text{Pf}_{1},g_{2},\ldots,g_{d+1}),$

where $\text{Pf}_{1}$ is the Pfaffian of $\varphi$ after deleting the first
row and the first column. We then set $y\in\omega_{S}$ to be the class of
$y_{1}$ in $\omega_{S}$. The annihilator $\text{Ann}_{B}(\omega_{S}/Sy)$ is
given by

$\displaystyle\text{Ann}_{B}(\omega_{S}/Sy)$
$\displaystyle\,=\,\left(y_{1},g_{2},\ldots,g_{d+1}\right):_{B}\left(y_{1},\text{Pf}_{1},g_{2},\ldots,g_{d+1}\right)$
$\displaystyle\,=\,\left(y_{1},I_{1}\left(\left[y_{2},\ldots,y_{d+1}\right]\cdot\varphi^{\prime}\right)\right):_{B}\left(\text{Pf}_{1}\right)$

where $\varphi^{\prime}$ is the matrix obtained after deleting the first row
and the first column of $\varphi$. By using the determinantal trick for
Pfaffians, it follows that
$\text{Ann}_{B}(\omega_{S}/Sy)\,\supseteq\,\left(y_{1},\ldots,y_{d+1}\right).$
It is now easy to check the following two conditions

  1. (i)

$\text{ht}\left(\text{Ann}_{S}\left(\omega_{S}/Sy\right)+{\mathfrak{m}}S\right)\geq
d+1$ and

  2. (ii)

$\text{ht}\left(\text{Ann}_{S}\left(\omega_{S}/Sy\right)\right)\geq 1$.

This means that $S$ is $d$-weakly Gorenstein with respect to the ideal
${\mathfrak{m}}S$.

The bidegree of the homogeneous element $y\in\omega_{S}$ is equal to
$\deg(y)=(0,1)-(d(h-1),-1)=(-d(h-1),2)$. Finally, Theorem 2.6 yields the
bigraded isomorphism

$\text{H}_{{\mathfrak{m}}}^{i}(S)\,\cong\,{}^{*}\text{Ext}_{T}^{i}\left(S,T\right)\left(-d(h-1),-d+1\right)$

for all $0\leq i\leq d-1$. ∎

For the rest of this subsection, we consider a particular case that was not
covered before in the literature. This shows the applicability of the methods
developed in this paper.

######  Setup .

Assume Section missing 5. Let $n=d+1$ (i.e.,
$R=\mathbb{k}[x_{1},\ldots,x_{d}]$ and $T=\mathbb{k}[y_{1},\ldots,y_{d+1}]$).
Let $I=(f_{1},\ldots,f_{d+1})$ be a Gorenstein ideal of height $3$ that
satisfies $G_{d}$. Let $\varphi\in R^{(d+1)\times(d+1)}$ be an alternating
minimal presentation matrix of $I$ with homogeneous entries of degree $h\geq
1$. Suppose that the monomial support of the entries of $\varphi$ is given by
at most $d+1$ monomials. Consequently, we may assume that
$I_{1}(\varphi)\subseteq\left(m,x_{1}^{h},\ldots,x_{d}^{h}\right)$ where $m$
is a monomial of degree $h$. Then, the defining equations of $\text{Sym}(I)$
have the following two expressions

$\left[y_{1},\ldots,y_{d+1}\right]\cdot\varphi\;=\;\left[m,x_{1}^{h},\ldots,x_{d}^{h}\right]\cdot
A,$

where $A\in T^{(d+1)\times(d+1)}$ is a $(d+1)\times(d+1)$ matrix

$A\;=\;\left[\begin{array}[]{ccccc}a_{i,j}\end{array}\right]$

with linear entries in $T$. The matrix $A$ is referred to as the _Jacobian
dual_ in the literature. Let $\Delta_{i,j}$ be the minor of $A$ obtained by
deleting the $i$-th row and the $j$-th column multiplied by $(-1)^{i+j}$.

######  Lemma .

The matrix $A$ has all the following properties:

  1. (i)

$\text{rank}(A)=d$.

  2. (ii)

$\text{Ker}(A)$ is a free $T$-module generated by the vector
$[y_{1},\ldots,y_{d+1}]^{t}$.

  3. (iii)

The adjoint of $A^{t}$ is of the form

${\rm
adj}(A^{t})\;=\;\left[\begin{array}[]{cccc}y_{1}\delta_{1}&y_{2}\delta_{1}&\cdots&y_{d+1}\delta_{1}\\\
y_{1}\delta_{2}&y_{2}\delta_{2}&\cdots&y_{d+1}\delta_{2}\\\
\vdots&\vdots&\cdots&\vdots\\\
y_{1}\delta_{d+1}&y_{2}\delta_{d+1}&\cdots&y_{d+1}\delta_{d+1}\end{array}\right]$

where $\delta_{i}=\Delta_{i,j}/y_{j}\in T$.

  4. (iv)

$\text{Ker}(A^{t})$ is a free $T$-module generated by the vector
$\left[\delta_{1}\ldots,\delta_{d+1}\right]^{t}$.

###### Proof.

We first show that $\text{ht}(I_{d}(A))\geq 2$. By contradiction, suppose not.
Then there exists $\mathfrak{p}\in{\text{Spec}}(T)$ of height one containing
$I_{d}(A)$. Since the only minimal primes of $\text{Sym}(I)$ are
${\mathfrak{m}}\text{Sym}(I)$ and $\mathcal{A}$, and $\mathfrak{p}$ is in none
of them, it follows that $\text{ht}(\mathcal{L},\mathfrak{p})\geq d+1.$ Hence
$\text{ht}\left((\mathcal{L},\mathfrak{p})/\mathfrak{p}\right)\geq d$, and so
we have that $\text{ht}\left(\mathcal{L}(R\otimes_{\mathbb{k}}Q)\right)\geq d$
where $Q={\text{Quot}}(T/\mathfrak{p})$. But the image of $A$ in $Q$ has rank
$\leq d-1$. Hence the proper ideal $\mathcal{L}(R\otimes_{\mathbb{k}}Q)$ can
be generated by $d-1$ elements, a contradiction.

Since $\varphi$ is alternating, it follows that

$[m,x_{1}^{h},\ldots,x_{d}^{h}]\cdot
A\cdot[y_{1},\ldots,y_{d+1}]^{t}=[y_{1},\ldots,y_{d+1}]\cdot\varphi\cdot[y_{1},\ldots,y_{d+1}]^{t}=0,$

and so $A\cdot[y_{1},\ldots,y_{d}]^{t}=0$. This shows that $\text{rank}(A)\leq
d$. Since $I_{d}(A)\neq 0$, we the equality. This settles part (i).

By the above paragraph, we have a complex $0\rightarrow
T\xrightarrow{\left[y_{1},\ldots,y_{d+1}\right]^{t}}T^{d+1}\xrightarrow{\;A\;}T^{d+1}$.
This complex is exact by the Buchsbaum-Eisenbud acyclicity criterion since
$\text{ht}(y_{1},\ldots,y_{d+1})\geq 2$ and $\text{rank}(A)=d$. This shows
part (ii).

As $\left[y_{1},\ldots,y_{d+1}\right]\cdot A^{t}=0$ and
$\text{adj}(A^{t})\cdot A^{t}=0$, part (ii) implies that each row of
$\text{adj}(A^{t})$ is a multiple of $[y_{1},\ldots,y_{d+1}]$. This
establishes part (iii).

By part (iii), we have the following complex $0\rightarrow
T\xrightarrow{\left[\delta_{1},\ldots,\delta_{d+1}\right]^{t}}T^{d+1}\xrightarrow{\;A^{t}\;}T^{d+1}$
and the inclusion $(\delta_{1},\ldots,\delta_{d+1})\supseteq I_{d}(A)$. Since
$\text{ht}(I_{d}(A))\geq 2$, again, the Buchsbaum-Eisenbud acyclicity
criterion shows that the latter complex is exact, completing the proof of part
(iv). ∎

######  Remark .

Assume that $\mathbb{k}$ is an infinite field. Then there exists a
bihomogeneous regular sequence $\ell_{1},\ldots,\ell_{d}$ belonging to
$\mathcal{L}$ that generates $\mathcal{L}$ at
${\mathfrak{m}}B\in\text{Min}(\mathcal{L})$, and satisfies
$\text{bideg}(\ell_{i})=(h,1)$ (as required in §missing 5.1).

###### Proof.

Since $\text{rank}(A)=d$ (see §missing 6.2(i)),
$\mu(\mathcal{L}\otimes_{B}B_{{\mathfrak{m}}B})\leq d$. Now the claim follows
because $\mathcal{L}$ is generated in bidegree $(h,1)$. ∎

In the next theorem, we compute explicitly the defining equations of the Rees
algebra of a Gorenstein ideal as in §missing 6.2 with $h=2$.

######  Theorem 6.3.

Assume §missing 6.2 with $h=2$. Without loss of generality we may assume that
$m=x_{1}x_{2}\in R$. Then the following statements hold:

  1. (i)

Under the natural bigraded isomorphism ${}^{*}{\text{Hom}}_{T}(B,T)\cong
T[x_{1}^{-1},\ldots,x_{d}^{-1}]$ of $\,B$-modules, we obtain the isomorphism

$\displaystyle{}^{*}{\text{Hom}}_{T}(\text{Sym}(I),T)$
$\displaystyle\;\cong\;B\cdot\left(\delta_{1}x_{1}^{-1}x_{2}^{-1}+\delta_{2}x_{1}^{-2}+\cdots+\delta_{d+1}x_{d}^{-2}\right)x_{3}^{-1}\cdots
x_{d}^{-1}$ $\displaystyle\quad+B\cdot x_{1}^{-1}x_{3}^{-1}\cdots x_{d}^{-1}$
$\displaystyle\quad+B\cdot x_{2}^{-1}x_{3}^{-1}\cdots
x_{d}^{-1}\;\subset\;T[x_{1}^{-1},\ldots,x_{d}^{-1}].$

In particular,
$\mathcal{A}=\text{H}_{{\mathfrak{m}}}^{0}(\text{Sym}(I))\cong{}^{*}{\text{Hom}}_{T}\left(\text{Sym}(I),T\right)(-d,-d+1)$
is a free $T$-module that is minimally generated as a $B$-module by three
elements of bidegrees $(0,2d-2)$, $(1,d-1)$ and $(1,d-1)$.

  2. (ii)

As in §missing 2.2, consider the Morley forms ${\text{morl}}_{(i,d-i)}$ and
the homogeneous element $0\neq\alpha\in T$. The expressions

$\displaystyle
U_{0}\;=\;\frac{1}{\alpha}\cdot{\text{morl}}_{(0,d)}\cdot\left(1\otimes_{T}\left(\delta_{1}x_{1}^{-1}x_{2}^{-1}+\delta_{2}x_{1}^{-2}+\cdots+\delta_{d+1}x_{d}^{-2}\right)x_{3}^{-1}\cdots
x_{d}^{-1}\right),$ $\displaystyle
U_{1}\;=\;\frac{1}{\alpha}\cdot{\text{morl}}_{(1,d-1)}\cdot\left(1\otimes_{T}x_{1}^{-1}x_{3}^{-1}\cdots
x_{d}^{-1}\right)\text{ and }$ $\displaystyle
U_{2}\;=\;\frac{1}{\alpha}\cdot{\text{morl}}_{(1,d-1)}\cdot\left(1\otimes_{T}x_{2}^{-1}x_{3}^{-1}\cdots
x_{d}^{-1}\right)$

naturally give elements in $\mathcal{A}_{0}$, $\mathcal{A}_{1}$ and
$\mathcal{A}_{1}$, respectively. The $\text{Sym}(I)$-ideal $\mathcal{A}$ is
minimally generated as

$\mathcal{A}\;=\;\left(U_{0},U_{1},U_{2}\right).$

In particular, the ideal $\mathcal{J}\subset B$ of defining equations of
$\mathcal{R}(I)$ is minimally generated by $\mathcal{L}$ and lifts to $B$ of
the elements $U_{0},U_{1},U_{2}$.
Moreover, by §missing 6.2, if $\mathbb{k}$ is an infinite field, then we can
find a regular sequence $\ell_{1},\ldots,\ell_{d}\in\mathcal{L}$ that
satisfies the conditions of §missing 5.1.

  3. (iii)

Let
$\mathcal{G}:{\mathbb{P}}_{\mathbb{k}}^{d-1}\dashrightarrow{\mathbb{P}}_{\mathbb{k}}^{d}$
be the rational map determined by the forms $f_{1},\ldots,f_{n}$ generating
$I$. Let $Y\subset{\mathbb{P}}_{\mathbb{k}}^{d}$ be the closure of the image
of $\mathcal{G}$. The degree of $\mathcal{G}$ is equal to
$\deg(\mathcal{G})=2^{d-2}$ and $\deg(Y)=2d-2$.

  4. (iv)

$0pt\left(\mathcal{R}(I)\right)\geq d$ and
$0pt\left(\text{gr}_{I}(R)\right)\geq d-1$.

###### Proof.

Set $S:=\text{Sym}(I)$. From Theorem 6.2, we have the isomorphism
$\mathcal{A}\cong{}^{*}{\text{Hom}}_{T}(S,T)(-d,-d+1)$. Thus to study
$\mathcal{A}_{i}$ we need to compute ${\text{Hom}}_{T}(S_{d-i},T)$, and this
will be done by considering a carefully constructed (non minimal) presentation
$P_{i}\xrightarrow{\;\psi_{i}\;}Q_{i}\rightarrow S_{i}\rightarrow 0$ of
$S_{i}$. We now proceed to build this presentation.

Let $\mathfrak{a}\subset R$ be the ideal generated by the monomial support of
$I_{1}(\varphi)$. Let $C=\\{x_{1}x_{2},x_{1}^{2},\ldots,x_{d}^{2}\\}$ be the
minimal monomial generating set of $\mathfrak{a}$. Let $C^{j}$ be the minimal
monomial generating set of $\mathfrak{a}^{j}$. For $i\geq 2$, $0\leq
j\leq\frac{i-2}{2}$ and
$\alpha=(\alpha_{1},\ldots,\alpha_{d-2})\in{\mathbb{Z}}_{\geq 0}^{d-2}$ such
that $\alpha_{i}\leq 1$ and $i-2j-3\leq|\alpha|\leq i-2j-2$, we define the set

$D_{i,j,\alpha}\,:=\,\big{\\{}x_{1}^{b_{1}}x_{2}^{b_{2}}x_{3}^{\alpha_{1}}\cdots
x_{d}^{\alpha_{d-2}}c\mid c\in C^{j}\text{ and
}b_{1}+b_{2}=i-2j-2-|\alpha|\big{\\}}.$

Denote by $M_{i}$ the set of monomials of degree $i$ that are not in
$\mathfrak{a}$.

######  Claim 6.3.1.

Write $D_{i,j,\alpha}\cdot C$ for the sets of products of elements in
$D_{i,j,\alpha}$ and $C$. For all $2\leq i\leq d$, the sets
$D_{i,j,\alpha}\cdot C$ give a partition of a monomial generating set of
${\mathfrak{m}}^{i-2}\mathfrak{a}\subset R$.

###### Proof.

For $\gamma\in{\mathbb{Z}}_{\geq 0}^{d}$, let $\mathbf{x}^{\gamma}$ be a
monomial generators of ${\mathfrak{m}}^{i-2}\mathfrak{a}$. By repeatedly
dividing $\mathbf{x}^{\gamma}$ by monomials in $C$, we obtain a decomposition
$\mathbf{x}^{\gamma}=\mathbf{x}^{\gamma^{\prime}}c$ where $c\in C^{j+1}$ for
some $0\leq j\leq\frac{i-2}{2}$ and
$\gamma^{\prime}=(\gamma_{1}^{\prime},\ldots,\gamma_{d}^{\prime})\in\mathbb{N}^{d}$
is a vector such that

$\gamma_{1}^{\prime}+\gamma_{2}^{\prime}=(\gamma_{1}+\gamma_{2})\text{ mod
}2,\quad\gamma_{3}^{\prime}=\gamma_{3}\text{ mod
}2,\quad\ldots,\quad\gamma_{d}^{\prime}=\gamma_{d}\text{ mod }2\quad\text{ and
}\quad|\gamma^{\prime}|=i-2j-2.$

Thus $\mathbf{x}^{\gamma}\in D_{i,j,\alpha}\cdot C$ with
$\alpha=(\gamma_{3}^{\prime},\ldots,\gamma_{d}^{\prime})\in\mathbb{N}^{d-2}$.
This shows that the sets $D_{i,j,\alpha}\cdot C$ give a partition of a
monomial generating set of ${\mathfrak{m}}^{i-2}\mathfrak{a}$. ∎

We consider the (non minimal) presentation of the graded $T$-module $S_{i}$
given by

$P_{i}\xrightarrow{\;\psi_{i}\;}Q_{i}\rightarrow S_{i}\rightarrow 0$

where

$Q_{i}\;:=\;\bigoplus_{j,\alpha}\bigoplus_{u\in D_{i,j,\alpha}}\bigoplus_{c\in
C}T\cdot q_{u,c}\;\;\;\oplus\;\;\bigoplus_{v\in M_{i}}T\cdot q_{v}$

and

$P_{i}\;:=\;\bigoplus_{j,\alpha}\left(\bigoplus_{u\in
D_{i,j,\alpha}}\bigoplus_{l=1}^{d+1}T\cdot p_{u,l}\;\;\;\oplus\bigoplus_{r\in
F_{i,j,\alpha}}T\cdot p_{r}\right)$

where $q_{u,c}$, $q_{v}$, $p_{u,l}$ and $p_{r}$ are the chosen basis elements
for the free $T$-modules $Q_{i}$ and $P_{i}$. The free $T$-module $B_{i}$ has
the natural description
$B_{i}=\bigoplus_{|\gamma|=i}T\cdot\mathbf{x}^{\gamma}\cong
T^{\binom{i+d-1}{d-1}}$. In the free $T$-module $Q_{i}$, $q_{u,c}$ corresponds
to the pair $(u\in D_{i,j,\alpha},c\in C)$ and $q_{v}$ corresponds to the
monomial $v\in M_{i}$. The map $Q_{i}\rightarrow S_{i}$ is defined by sending
$q_{u,c}\mapsto\overline{uc}\in S_{i}$ and $q_{v}\mapsto\overline{v}\in
S_{i}$. By Claim 6.3.1, all the sets $D_{i,j,\alpha}\cdot C$ together with
$M_{i}$ give a partition of the monomials of degree $i$ in $R$. Hence we have
a natural surjection $Q_{i}\twoheadrightarrow S_{i}$. Notice that there can
exist $(u\in D_{i,j,\alpha},c\in C)$ and $(u^{\prime}\in
D_{i,j,\alpha},c^{\prime}\in C)$ such that $uc=u^{\prime}c^{\prime}$, and as a
consequence we need a relation to identify $q_{u,c}$ and
$q_{u^{\prime},c^{\prime}}$. The fact that the sets $D_{i,j,\alpha}\cdot C$
are disjoint (see Claim 6.3.1) implies that there are no further relations
among the chosen basis elements of $Q_{i}$. Denote by $F_{i,j,\alpha}$ a set
indexing all the needed relations (columns in $\psi_{i}$) among the
$q_{u,c}$’s with $u\in D_{i,j,\alpha}$ and $c\in C$. Let $p_{r}\in P_{i}$ be
the basis element that gives the relation $r\in F_{i,j,\alpha}$. For any $r\in
F_{i,j,\alpha}$, we have that

$\psi_{i}(p_{r})\,\in\,\bigoplus_{u\in D_{i,j,\alpha}}\bigoplus_{c\in C}T\cdot
q_{u,c}\;\subset\,Q_{i}.$

It now follows that $B_{i}\cong Q_{i}\big{/}\big{(}\sum_{r\in
F_{i,j,\alpha}}T\cdot\psi_{i}(p_{r})\big{)}$.

The relations of $S_{i}$ come from the generators of the ideal
${\mathfrak{m}}^{i-2}\cdot\mathcal{L}$, and they can be recovered from the
products $D_{i,j,\alpha}\cdot\mathcal{L}$. In the free $T$-module $P_{i}$,
$p_{u,l}$ corresponds to the pair $(u\in D_{i,j,\alpha},g_{l})$. Since
$g_{l}=[x_{1}x_{2},x_{1}^{2},\ldots,x_{d}^{2}]\cdot[a_{1,l},a_{2,l},\ldots,a_{d,l}]^{t}$,
for any $u\in D_{i,j,\alpha}$, we have that

$\big{[}\psi_{i}(p_{u,1}),\psi_{i}(p_{u,2}),\ldots,\psi_{i}(p_{u,d+1})\big{]}\,=\,\big{[}q_{u,x_{1}x_{2}},q_{u,x_{1}^{2}},\ldots,q_{u,x_{d}^{2}}\big{]}\cdot
A.$

By combining the above basic facts, it follows that $\psi_{i}$ is a
presentation matrix of $S_{i}$. See §missing 6.2 for an illustration of this
presentation. We then have the isomorphism
${\text{Hom}}_{T}(S_{i},T)\cong\text{Ker}(\psi_{i}^{t})\subset{\text{Hom}}_{T}(Q_{i},T)$,
and by an abuse of notation, we also denote by $q_{u,c}$ and $q_{v}$ the dual
basis element $(q_{u,c})^{*}\in{\text{Hom}}_{T}(Q_{i},T)$ and
$(q_{v})^{*}\in{\text{Hom}}_{T}(Q_{i},T)$, respectively.

We divide the matrix $\psi_{i}$ into blocks of rows: let $\psi_{i,j,\alpha}$
be the submatrix given by the rows corresponding to the basis vectors
$\\{q_{u,c}\\}_{u\in D_{i,j,\alpha},c\in C}$ and $\eta_{v}$ be the submatrix
given by the row corresponding with the basis vector $q_{v}$. We have
$\text{Ker}(\eta_{v}^{t})=T\cdot q_{v}$ because $\eta_{v}$ is the zero matrix.
If $j=0$ and $|\alpha|=i-2$, then $D_{i,0,\alpha}=\\{z_{\alpha}\\}$ where
$z_{\alpha}:=x_{3}^{\alpha_{1}}\cdots x_{d}^{\alpha_{d}}$, and so there are no
relations among the $q_{u,c}$’s indexed by the pairs $(u\in
D_{i,0,\alpha},c\in C)$. Hence, when $j=0$ and $|\alpha|=i-2$, §missing
6.2(iv) yields that $\text{Ker}(\psi_{i,0,\alpha}^{t})$ is the free $T$-module
given by

$\text{Ker}(\psi_{i,0,\alpha}^{t})\;=\;T\cdot\left(\delta_{1}q_{z_{\alpha},x_{1}x_{2}}+\delta_{2}q_{z_{\alpha},x_{1}^{2}}+\cdots+\delta_{d+1}q_{z_{\alpha},x_{d}^{2}}\right).$

Denote by
$Z_{i,0,\alpha}:=\delta_{1}q_{z_{\alpha},x_{1}x_{2}}+\delta_{2}q_{z_{\alpha},x_{1}^{2}}+\cdots+\delta_{d+1}q_{z_{\alpha},x_{d}^{2}}$
the generator of $\text{Ker}(\psi_{i,0,\alpha}^{t})$. Since there is a direct
sum decomposition of maps

$\psi_{i}\;=\;\bigoplus_{j,\alpha}\psi_{i,j,\alpha}\;\oplus\;\bigoplus_{v\in
M_{i}}\eta_{v},$

we obtain that

$\displaystyle{\text{Hom}}_{T}(S_{i},T)\,\cong\,\text{Ker}(\psi_{i}^{t})$
$\displaystyle\,=\,\bigoplus_{j,\alpha}\text{Ker}(\psi_{i,j,\alpha}^{t})\;\;\;\oplus\;\bigoplus_{v\in
M_{i}}\text{Ker}(\eta_{\nu}^{t})$