# Cohomological Blow Ups of Graded Artinian Gorenstein Algebras Along
Surjective Maps

Anthony Iarrobino
Department of Mathematics, Northeastern University, Boston, MA 02115, USA.
<EMAIL_ADDRESS>
Pedro Macias Marques
Departamento de Matemática, Escola de Ciências e Tecnologia, Centro de
Investigação
em Matemática e Aplicações, Instituto de Investigação e Formação Avançada,
Universidade de Évora, Rua Romão Ramalho, 59, P–7000–671 Évora, Portugal
<EMAIL_ADDRESS>
Chris McDaniel
Department of Mathematics, Endicott College, 376 Hale St Beverly, MA 01915,
USA.
<EMAIL_ADDRESS>
Alexandra Seceleanu
Department of Mathematics, University of Nebraska-Lincoln,
203 Avery Hall, Lincoln NE 68588, USA.
<EMAIL_ADDRESS>
Junzo Watanabe

Department of Mathematics, Tokai University,
Hiratsuka, Kanagawa 259-1292 Japan
<EMAIL_ADDRESS>
###### Abstract

We introduce the cohomological blow up of a graded Artinian Gorenstein (AG)
algebra along a surjective map, which we term BUG (Blow Up Gorenstein) for
short. This is intended to translate to an algebraic context the cohomology
ring of a blow up of a projective manifold along a projective submanifold. We
show, among other things, that a BUG is a connected sum, that it is the
general fiber in a flat family of algebras, and that it preserves the strong
Lefschetz property. We also show that standard graded compressed algebras are
rarely BUGs, and we classify those BUGs that are complete intersections. We
have included many examples throughout this manuscript.

††Keywords: Artinian Gorenstein, cohomology ring, blow up, cohomological blow-
up algebra, connected sum, flat family, complete intersection, Hilbert
function, Lefschetz property, Macaulay dual generator.††2020 Mathematics
Subject Classification: Primary: 13H10; Secondary: 13E10, 13M05,14D06, 14F45,
55R10.

###### Contents

  1. 1 Introduction
  2. 2 AG Algebras, Orientations, and Thom Classes
  3. 3 Cohomological Blow Ups
  4. 4 Macaulay Dual Generators
  5. 5 Relation With Connected Sums
    1. 5.1 The Cohomological Blow-up Algebra as a Connected Sum
    2. 5.2 The Blow Down as a Connected Sum
  6. 6 Minimal Generating Sets and Complete Intersections
    1. 6.1 Minimal Generating Sets
    2. 6.2 Complete Intersection Blow-up Algebras and Exact Zero Divisors
    3. 6.3 Application: Watanabe’s Bold Conjecture
  7. 7 Restrictions on Hilbert Functions
  8. 8 The Lefschetz Property
  9. 9 Geometric View and Examples
  10. A A Guide To Our Examples

##  1 Introduction

Given two graded Artinian Gorenstein (AG) algebras $A$ and $T$ over a field
${\mathbb{F}}$, of socle degrees $d>k$, respectively, and a surjective algebra
map between them $\pi\colon A\rightarrow T$, we construct a new graded AG
algebra $\hat{A}$ of socle degree $d$ called the cohomological blow up of of
$A$ along $\pi$; we shall sometimes refer to $\hat{A}$ as BUG, short for Blow
Up Gorenstein. As the name suggests, our construction is based on the blow up
operation in complex geometry, and particularly its effect on the (singular)
cohomology rings of the spaces involved. The purpose of the present paper is
to extend this blow up operation on cohomology rings to the more general class
of graded AG algebras, to study its interactions with other familiar algebraic
constructs, and to draw parallels to corresponding geometric ones. Our
cohomological blow ups are very different from the much-studied blow-up
algebras (Rees algebras and related rings), which correspond to the coordinate
ring (not the cohomology ring) of the blow-up variety [45].

Generally speaking, graded Artinian Gorenstein algebras are algebraic
analogues of cohomology algebras (in even degrees) of smooth, compact, even-
dimensional manifolds, e.g. complex manifolds or symplectic manifolds. For
compact complex (or symplectic) manifolds $\pi\colon Y\hookrightarrow X$ of
(complex) dimensions $k<d$, respectively, the blow-up of $X$ along $\pi$ is
another compact complex $d$-dimensional manifold $\tilde{X}$ obtained from $X$
by removing $Y$ and gluing in its place a codimension one submanifold
$\tilde{Y}$ called the exceptional divisor, which can be realized as the
projectivization of the normal bundle $\mathcal{N}_{Y/X}$. The cohomology
algebras (over ${\mathbb{Q}}$) of these spaces $Y$, $X$, $\tilde{Y}$ and
$\tilde{X}$ satisfy the following three algebraic properties. For notational
convenience, let $A=H^{2\bullet}(X)$, $T=H^{2\bullet}(Y)$,
$\tilde{A}=H^{2\bullet}(\tilde{X})$, and $\tilde{T}=H^{2\bullet}(\tilde{Y})$,
graded so that $A_{i}=H^{2i}(X)$, and so on.

First if $\beta\colon\tilde{X}\rightarrow X$ is the blow down map,
$\beta_{0}\colon\tilde{Y}\rightarrow Y$ its restriction to the exceptional
divisor, and $\hat{\pi}\colon\tilde{Y}\hookrightarrow\tilde{X}$ the natural
inclusion map, then the obvious commutative diagram of spaces induces a
commutative diagram of cohomology algebras:

$\textstyle{X}$$\textstyle{\tilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\tilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\pi}}$$\scriptstyle{\beta_{0}}$
$\displaystyle\hskip 85.35826pt\Rightarrow$ (9)

Second, as a projective space bundle over $Y$, the cohomology algebra of the
exceptional divisor $\tilde{Y}$ is a free extension over the cohomology of $Y$
generated by the Euler class
$\xi=e\left(\mathcal{N}_{\tilde{Y}/\tilde{X}}\right)\in H^{2}(\tilde{Y})$ of
the normal bundle $\mathcal{N}_{\tilde{Y}/\tilde{X}}$, in symbols

$\tilde{T}\cong\frac{T[\xi]}{\left(\xi^{n}+t_{\xi}^{n-1}+\cdots+t_{n-1}\xi+t_{n}\right)}$
(10)

where $t_{i}=(-1)^{i}\cdot c_{i}(Y)\in T_{i}$ are the Chern classes of the
normal bundle $\mathcal{N}_{Y/X}$, the top one of which is the Euler class
$c_{n}(Y)=e(\mathcal{N}_{Y/X})\in T_{n}$, ($n=d-k=$ codimension of $Y$ in
$X$).

Third, the induced map $\beta^{*}$ is injective and fits into a short exact
sequence of $A$-modules:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{*}}$$\textstyle{\tilde{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\pi}^{*}}$$\textstyle{\tilde{T}/\beta_{0}^{*}\left(T\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$
(11)

Equations (10) and (11) imply that the Hilbert function of the cohomology of
the blow up is

$H(\tilde{A})=H(A)+H(T)[1]+\cdots+H(T)[n-1].$ (12)

In fact, one can show that Equations (9), (10), and (11) uniquely determine
the algebra structure of $\tilde{A}$. Moreover, if the restriction map
$\pi^{*}\colon A\rightarrow T$ is surjective, one can show that the cohomology
algebra of the blow up is given by

$\tilde{A}=\frac{A[\xi]}{\left(\xi\cdot K,\
\xi^{n}+a_{1}\xi^{n-1}+\cdots+a_{n-1}+a_{n}\right)}$ (13)

where $K\subset A$ is the kernel of $\pi^{*}$, $\pi^{*}(a_{i})=t_{i}$, and
$a_{n}=(-1)^{n}\cdot\tau(Y)\in A_{n}$ where $\tau(Y)$ is the Thom class of the
normal bundle $\mathcal{N}_{Y/X}$. See [14, 17, 21, 23, 32, 37] for further
details on geometric and topological aspects of blow ups, specifically [17,
Proposition 6.4], [37, Proposition 2.4], and [14, Theorem 3.11]. See also [41]
for further details on characteristic classes.

The novelty of this paper is to show that Equations (9), (10), (11), and (13)
can be extended to arbitrary AG algebras over any field ${\mathbb{F}}$ to
define a new construction on these rings called the cohomological blow up.
Specifically, given any surjective degree preserving map of graded AG algebras
$\pi\colon A\rightarrow T$, we define (Definition 3.10) a _cohomological blow
up of $A$ along $\pi$_ as in Equation (13), leaving $a_{1},\ldots,a_{n-1}$ as
free parameters and setting $a_{n}$ to be the algebraic analogue of the Thom
class.

The algebraic analogue of the Thom class for a map of AG algebras $\pi\colon
A\rightarrow T$ is defined as a certain annihilator of the kernel of $\pi$,
determined by choices of socle generators, called orientations, of $A$ and $T$
(Definition 2.1). Assuming that $\pi$ is surjective, and $A=R/I$ where
$I\subset R$ is a homogeneous ideal in a graded polynomial ring, an
alternative characterization of the Thom class is an element $\tau\in R$ for
which $T=R/(I\colon\tau)$. In terms of Macaulay duality, if $R$ is acting on
its dual divided power algebra $Q$ by contraction and if $F\in Q$ and $G\in Q$
are dual generators of $A=R/\operatorname{Ann}(F)$ and
$T/\operatorname{Ann}(G)$, then $\tau\in R$ is a contraction operator
satisfying $\tau\circ F=G$. Each of these interpretations of the Thom class
leads to distinct, but equivalent descriptions of the cohomological blow up.
We frame these in terms of three general constructions on AG algebras,
starting from the given data consisting of two oriented AG algebras $A$ and
$T$ of socle degrees $d$ and $k$ with $n=d-k\geq 2$ and a surjective degree
preserving algebra map between them $\pi\colon A\rightarrow T$.

Our first construction (Construction 3.1) fixes an indeterminate $\xi$, then
chooses any homogeneous monic polynomial $f_{A}(\xi)\in A[\xi]$ (we use the
subscript to indicate the coefficient algebra) and uses Equation (13) to
construct a new algebra

$f_{A}(\xi)=\xi^{n}+a_{1}\xi^{n-1}+\cdots+a_{n}\ \ \ \ \Rightarrow\ \ \ \
\hat{A}=\frac{A[\xi]}{\left(\xi\cdot K,\hat{f}(\xi)\right)}.$

where $K$ is the kernel of $\pi$, and $a_{i}\in A_{i}$ are any homogenous
elements. We show that $\hat{A}$ is an AG algebra if and only if
$a_{n}=\lambda\cdot\tau$ is a non-zero scalar multiple of the Thom class
$\tau\in A_{n}$ of $\pi$ (Theorem 3.6). In this case we replace the “hat” with
a “tilde” and call the resulting algebra $\tilde{A}$ the _cohomological blow
up_ of $A$ along $\pi$ with parameters
$(\pi(a_{1}),\ldots,\pi(a_{n-1}),\lambda)$, and we call $A$ the _cohomological
blow down_ of $\tilde{A}$ (Definition 3.10). We show that $\tilde{A}$ is
essentially characterized by analogues of Equations (9), (10) and (11) above
(Theorem 3.14). We then use a standard result in commutative algebra to show
that $\tilde{A}$ is isomorphic to the general fiber in a flat family of
algebras (Theorem 8.3). Using this flat family, we further show that if both
$A$ and $T$ have the strong Lefschetz property (SLP) and ${\mathbb{F}}$ is
infinite of characteristic either zero or sufficiently large, then $\tilde{A}$
must also have SLP (Theorem 8.5). This result has parallels in complex
geometry; if $Y\subset X$ are projective manifolds, then the blow up
$\tilde{Y}\subset\tilde{X}$ is also projective, and projective manifolds
always satisfy the hard Lefschetz theorem, see e.g. [17, 21, 23]. We remark
that the essential result of Theorem 8.5, that blow ups preserve SLP, seems to
be well known in various other contexts and guises, e.g. [2, 9, 10, 31, 38,
44]; on the other hand, to our knowledge, the generality of Theorem 8.5 is
new, and we hope it will be a welcome reference for this useful result. We
further show that over an infinite field, if $A$ and $T$ both have the weak
Lefschetz property (WLP) and the difference between their socle degrees is
two, then $\tilde{A}$ also has WLP (Theorem 8.9). We give examples that show
general cohomological blow ups may not preserve WLP (Example 8.8), and blow
downs may not preserve SLP or WLP in general (Example 8.6). In a sequel [34]
we plan to investigate the behavior of the related Hodge-Riemann bilinear
relations (HRR) in relation with cohomological blow ups and blow downs.

Our second construction (Construction 4.2) starts with Macaulay dual
generators $F\in Q_{d}$ and $G\in Q_{k}$ for $A=R/\operatorname{Ann}(F)$ and
$T=R/\operatorname{Ann}(G)$, and a contraction operator $\tau\in R_{n}$ such
that $\tau\circ F=G$. Then one again fixes an indeterminate $\xi$ and chooses
a monic homogeneous polynomial $f_{R}(\xi)\in R[\xi]$, and from this
constructs a new dual generator $\hat{F}\in Q[\Xi]_{d}$ and its associated AG
algebra

$f_{R}(\xi)=\xi^{n}+r_{1}\xi^{n-1}+\cdots+r_{n}\ \ \ \ \Rightarrow\ \ \ \
\hat{A}_{MD}=\frac{R[\xi]}{\operatorname{Ann}(\hat{F})}$

where the $r_{i}\in R_{i}$ are homogeneous of degree $i$, $1\leq i\leq n$.
Applying the projection map $R[\xi]\rightarrow A[\xi]$ to $f_{R}(\xi)\mapsto
f_{A}(\xi)=\overline{f_{R}(\xi)}$, one can also construct $\hat{A}$ from
above, but in general the algebras $\hat{A}$ and $\hat{A}_{MD}$ will be non-
isomorphic. However we show that they are isomorphic exactly when they are
equal to some cohomological blow up of $A$ along $\pi$, and we give necessary
and sufficient conditions for this to occur (Theorem 4.6); in this case we
shall replace the “hat” with “tilde”, drop the subscript, and simply write
$\tilde{A}$. We use this result to show that every cohomological blow up along
a surjective map can be realized as a connected sum in the sense of [30]
(Theorem 5.4). We show that one can also obtain the cohomological blow down as
a connected sum (Theorem 5.6). This latter result implies that every AG
algebra $A$ can be realized as a connected sum (over some
$\tilde{T}\neq{\mathbb{F}}$) by blowing up then blowing down.

Our third construction (Construction 6.2) starts with presentations $A=R/I$
and $T=R/(I:\tau)$. One then fixes an indeterminate $\xi$ and chooses a monic
homogeneous polynomial $f_{R}(\xi)\in R[\xi]$ and constructs a new ideal
$\hat{I}\subset R[\xi]$ by

$f_{R}(\xi)=\xi^{n}+r_{1}\xi^{n-1}+\cdots+r_{n}\ \ \ \ \Rightarrow\ \ \ \
\hat{I}=I+\xi\cdot(I:\tau)+(f_{R}(\xi))$

where the $r_{i}\in R_{i}$ are homogeneous of degree $i$, $1\leq i\leq n$. It
is clear from the definition that $R[\xi]/\hat{I}\cong\hat{A}$ from
Construction 3.1 above, and hence we deduce that $\hat{I}$ is a Gorenstein
ideal (i.e. irreducible and primary to the maximal ideal of $R[\xi]$) if and
only if $a_{n}=\lambda\cdot\tau$, a non-zero scalar multiple of the Thom class
of $\pi$. In this case we replace the “hat” with “tilde” and call $\tilde{I}$
a _cohomological blow up ideal_ because the quotient algebra
$R[\xi]/\tilde{I}$ is isomorphic to the cohomological blow up algebra
$\tilde{A}$ defined above. We compute a minimal generating set of $\tilde{I}$
in terms of minimal generating sets of $I$ and $(I:\tau)$ (Theorem 6.5). Using
this result in conjunction with the notion of exact pairs of zero divisors
borrowed from [22, 33], we classify BUGs which are complete intersections
(Theorem 6.12). We also use cohomological blow ups to prove a special case of
Watanabe’s Bold Conjecture (WBC) [47], namely that any standard graded
Artinian complete intersection cut out by products of linear and/or quadratic
forms can be embedded as a subalgebra of another standard graded Artinian
complete intersection cut out by quadratic forms and of the same socle degree
(Theorem 6.16); see [35] for a different proof of a related result, and see
[36] for a proof of other cases of WBC.

It is interesting to note that our cohomological blow up along a surjective
map preserves the standard grading, i.e. if $A$ and $T$ are standard graded,
then the cohomological blow up $\tilde{A}$ is also standard graded (Remark
5.5). However we show by examples that neither the cohomological blow down,
nor, what one might call, the cohomological blow up along a non-surjective
map, necessarily preserve standard grading at all (Example 5.8, Remark 9.4).
We say that a standard graded AG algebra $\tilde{A}$ has a _standard BUG
structure_ if it is a cohomological blow up of a standard graded AG algebra
$A$ along some surjective map $\pi$. Using Equation (12) above, we deduce that
some Hilbert functions cannot occur with standard BUG structures, and we call
such Hilbert functions _inaccessible_. Specifically we show that the
compressed Hilbert functions of [24] with embedding dimension at least three
and socle degree at least six are always inaccessible (Theorem 7.2). We plan
for a sequel [29] with further results in this direction, including a complete
description of cohomological blowup algebras of codimension two - where all AG
Hilbert functions are accessible - and a study of inaccessible AG Hilbert
functions in codimension at least three.

This paper is organized as follows. In Section 2 we give a description of the
algebraic analogue of the Thom class of a map of graded AG algebras. In
Section 3 we give Construction 3.1 and necessary and sufficient conditions for
it to yield a Gorenstein algebra. Then we define the cohomological blow up
algebra, and give a description of its Hilbert function. In Section 4 we
discuss Macaulay duality, we give Construction 4.2, and we give necessary and
sufficient conditions for it to yield the cohomological blow up algebra. In
Section 5, we show that every cohomological blow-up algebra is a connected
sum. We also describe the blow-down in terms of connected sums, and show that
every AG algebra has a connected sum decomposition, obtained by blowing up
then blowing down. In Section 6, we give Construction 6.2, introduce the
cohomological blow ups ideal, and compute its minimal generating set. In
Section 7 we define inaccessible Hilbert functions, and show that almost all
compressed AG algebras are inaccessible. In Section 8 we show that
cohomological blowing up preserves SLP. In Section 9 we discuss further the
connection of our work with geometry and other areas, give some geometrically
motivated examples, and suggest problems for future work. In this paper, we
have made a special effort to include a wide array of examples, many of which
were found as counter-examples to conjectures or questions that arose in our
discussions and preparations of this manuscript, and we hope the reader will
find them useful. In the appendix we have included a list of all examples and
brief descriptions of them for the readers convenience.

##  2 AG Algebras, Orientations, and Thom Classes

Let ${\mathbb{F}}$ be any field. A _graded AG algebra_ means a commutative
${\mathbb{Z}}_{\geq 0}$-graded connected Artinian Gorenstein
${\mathbb{F}}$-algebra; in particular if $A$ is a graded AG algebra of socle
degree $d$, then $A=\bigoplus_{i=0}^{d}A_{i}$ with $A_{0}={\mathbb{F}}$ and
$\operatorname{socle}(A)=A_{d}\cong{\mathbb{F}}$. Since most algebraic objects
in this paper are graded, we may sometimes drop the adjective and refer simply
to an AG algebra. Unless explicitly stated otherwise (as Remark 3.11) we will
not restrict the ground field ${\mathbb{F}}$ except in our study of Lefschetz
properties (Section 8). We say that $A$ is _standard graded_ if it is
generated as an algebra by its linear forms, i.e. ${\mathbb{F}}[A_{1}]=A$.
Although many of our examples are standard graded, we will not assume this,
except in our discussion of compressed algebras (Section 7). An _orientation_
on the AG algebra $A$ is a choice of linear isomorphism $\int_{A}\colon
A_{d}\overset{\cong}{\rightarrow}{\mathbb{F}}$ or, equivalently, a choice of
socle generator, which we shall denote by $0\neq a_{soc}\in A_{d}$. An
_oriented AG algebra_ is a pair $\left(A,\int_{A}\right)$ consisting of a
graded Artinian Gorenstein algebra $A$ and an orientation $\int_{A}$; when the
orientation is understood we shall drop the $\int_{A}$ and just speak of an
oriented AG algebra $A$, and its distinguished socle generator $a_{soc}$.

Suppose that $A$ and $T$ are two oriented AG algebras of socle degrees $d$ and
$k$, respectively, with $d>k$, and suppose that $\pi\colon A\rightarrow T$ is
a degree-preserving algebra map between them; we may occasionally drop the
adjectives and simply refer to the map $\pi$. Since $A$ is Gorenstein,
multiplication defines a perfect pairing for all $0\leq i\leq d$:

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In particular, for any
$\phi\in\operatorname{Hom}_{\mathbb{F}}(A_{i},{\mathbb{F}})$ there exists a
unique $a\in A_{d-i}$ such that $\phi(b)=\int_{A}a\cdot b$ for every $b\in
A_{i}$. Pulling back the orientation on $T$ by $\pi$ defines a homomorphism
${\int_{T}}\circ{\pi}\colon A_{k}\rightarrow{\mathbb{F}}$ and hence, as shown
in [30, Lemma 2.1], there exists a unique element $\tau\in A_{d-k}$ for which

$\int_{T}\pi(a)=\int_{A}\tau\cdot a,\ \ \ \forall\ a\in A.$ (14)

######  Definition 2.1 (Thom class, Euler class).

The unique element $\tau\in A_{d-k}$ defined by Equation (14) is called the
_Thom class of $\pi$_. The image of the Thom class $\pi(\tau)\in T_{d-k}$ is
called the _Euler class of $\pi$_.111Topologically, if $A=H^{2\bullet}(X)$ and
$T=H^{2\bullet}(Y)$ are the cohomology rings of a manifold $X$ and a
submanifold $\pi\colon Y\hookrightarrow X$ and $\pi^{*}\colon
H^{2\bullet}(X)\rightarrow H^{2\bullet}(Y)$ is the induced restriction map on
cohomology algebras, then the Thom class of $\pi^{*}$ is exactly the Thom
class of the normal bundle of $Y$ in $X$, also known as the _Poincaré dual
class of $Y\subset X$_, and its image under the restriction map is the Euler
class of the normal bundle, e.g. [41].

Note that the Thom class is zero if and only if the socle of $T$ does not
belong to the image of $\pi$. We call $\pi\colon A\rightarrow T$ a
_restriction map_ if its Thom class is non-zero. In particular, if $\pi$ is
surjective, then it is a restriction map, but not every restriction map is
surjective. In complex geometry, if $X$ is a Kähler manifold and $\pi\colon
Y\subset X$ is a codimension $n$ Kähler submanifold, then the induced map on
cohomology $\pi^{*}\colon H^{2\bullet}(X)\rightarrow H^{2\bullet}(Y)$ is a
restriction map [23, Exercise 3.3.9], though it need not be surjective, e.g.
Example 9.3. Here is a useful characterization of the Thom class of a
restriction map.

######  Lemma 2.2.

Let $\pi\colon A\rightarrow T$ be a restriction map between two oriented AG
algebras of socle degrees $d>k$, respectively, and let $K=\ker(\pi)\subset A$
be its kernel. Let $a\in A$ be any homogeneous element of degree $n=d-k$. Then
$a\cdot K=0$ if and only if $a=\lambda\cdot\tau$ is a multiple of the Thom
class of $\pi$.

###### Proof.

Let $\int_{A}\colon A_{d}\rightarrow{\mathbb{F}}$ and $\int_{T}\colon
T_{k}\rightarrow{\mathbb{F}}$ be orientations on $A$ and $T$, respectively.
Assume first that $a\in A_{n}$ and $a\cdot K=0$. Consider the short exact
sequence of vector spaces $0\rightarrow K_{k}\rightarrow A_{k}\rightarrow
T_{k}\rightarrow 0$. Since $T$ is Gorenstein of socle degree $k$, $T_{k}$ is
one-dimensional, and hence $K_{k}\subset A_{k}$ is a codimension-one subspace.
Therefore the set of homomorphisms
$\phi\in\operatorname{Hom}_{{\mathbb{F}}}(A_{k},{\mathbb{F}})$ which vanish on
$K_{k}$ is one-dimensional. Since $\phi_{1}(u)=\int_{A}\tau\cdot
u=\int_{T}\pi(u)=0$ for every $u\in K$, and also $\phi_{2}(u)=\int_{A}a\cdot
u=0$ for every $u\in K$, we must have $\phi_{1}=\lambda\cdot\phi_{2}$, which
implies that $\tau=\lambda\cdot a$. The converse is clear. ∎

If $\pi\colon A\rightarrow T$ is surjective, which will be a standing
assumption throughout this paper, then the Thom class has some nice
alternative descriptions. First, let $R={\mathbb{F}}[x_{1},\ldots,x_{r}]$ be a
graded polynomial ring with homogeneous maximal ideal
$\mathfrak{m}=(x_{1},\ldots,x_{r})\subset R$, and let $I\subset R$ be a
homogeneous $\mathfrak{m}$-primary ideal. Then $R/I$ is graded Artinian and it
is Gorenstein if and only if $I$ is irreducible, meaning that it cannot be
written as an intersection of two strictly larger ideals [39, Lemma I.1.3].
Suppose that $J\subset R$ is another homogeneous $\mathfrak{m}$-primary
irreducible ideal with $I\subset J$, let $T=R/J$ and let $\pi\colon
A\rightarrow T$ be the natural projection map. The following fact is well
known; see [46, Lemma 4] and [39, Theorem I.2.1].

######  Lemma 2.3.

With the above assumptions, let $\bar{\tau}\in A=R/I$ be the Thom class of
$\pi$ and let $\tau\in R$ any homogeneous lift. Then we have

$J=(I\colon\tau)\ \ \text{and}\ \ (I:J)=(\tau)+I.$

In [39], the authors refer to the the homogeneous lift of the Thom class
$\tau\in R$ in Lemma 2.3 as a transition element for $I\subseteq J$.

Next let $Q={\mathbb{F}}[X_{1},\ldots,X_{r}]$ be a divided power algebra on
which $R$ acts by contraction, e.g. [25, Appendix A] or [12, Appendix A.2.4]:

$x_{i}\circ X_{1}^{a_{1}}\cdots X_{i}^{a_{i}}\cdots
X_{r}^{a_{r}}=\begin{cases}X_{1}^{a_{1}}\cdots X_{i}^{a_{i}-1}\cdots
X_{r}^{a_{r}}&\text{if}\ a_{i}>0\\\ 0&\text{if}\ a_{i}=0\\\ \end{cases}.$

Then it is well known that a homogeneous ideal $I\subset R$ is
$\mathfrak{m}$-primary irreducible of socle degree $d$ (meaning the socle
degree of $A=R/I$) if and only if there exists a homogeneous form $F\in Q_{d}$
for which $I=\operatorname{Ann}(F)=\\{r\in R\ |\ r\circ F=0\\}$. In this case,
$F$ is called a Macaulay dual generator of $A$. Suppose that $G\in Q_{k}$
($k<d$) is another homogeneous form for which $I=\operatorname{Ann}(F)\subset
J=\operatorname{Ann}(G)$, let $T=R/J$ and let $\pi\colon
A=R/\operatorname{Ann}(F)\rightarrow R/\operatorname{Ann}(G)$ be the natural
projection map. Note that $A$ and $T$ have natural orientations coming from
$F$ and $G$: $\int_{A}a=(a\circ F)(0)$ and $\int_{T}t=(t\circ G)(0)$.

######  Lemma 2.4.

Under the assumptions of Lemma 2.3, if $\tau\in R$ is any homogeneous lift of
the Thom class of $\pi$, then

$\tau\circ F=G.$

###### Proof.

Indeed the Thom class condition (14) translates to the condition

$\displaystyle(a\circ G)(0)$ $\displaystyle=(\tau\cdot a\circ
F)(0)=(a\circ(\tau\circ F))(0),\ \ \forall\ \ a\in R.$ or, equivalently
$\displaystyle 0$ $\displaystyle=a\circ\left(G-\tau\circ F\right)(0),\ \
\forall\ \ a\in R.$ (15)

but because the pairing $R_{i}\times Q_{i}\rightarrow{\mathbb{F}}$,
$(a,H)=(a\circ H)(0)$ is non-degenerate, we see that Condition (15) is
equivalent to the claimed condition that $G=\tau\circ F$. ∎

######  Remark 2.5.

If $\tau$ is the Thom class of a map of AG algebras $\pi\colon A\rightarrow
T$, and if $\lambda\in{\mathbb{F}}^{\times}$ is a non-zero constant, we can
get a new Thom class $\tau^{\prime}=\lambda\cdot\tau$ by either scaling the
distinguished socle generator of $A$ by $\lambda$, i.e.
$a_{soc}\mapsto\lambda\cdot a_{soc}=a^{\prime}_{soc}$, or by scaling the
distinguished socle generator of $T$ by $\lambda^{-1}$, i.e.
$t_{soc}\mapsto\lambda^{-1}\cdot t_{soc}=t^{\prime}_{soc}$.

In our proofs we make frequent use of the following result: it is valid for an
arbitrary Artinian Gorenstein algebra $A$ (not necessarily graded) and an
arbitrary Artinian algebra $B$.

######  Lemma 2.6.

Let $\varphi:A\to B$ be a homomorphism of Artinian algebras so that $A$ is
Gorenstein and $\varphi$ restricted to the socle of $A$ is injective. Then
$\varphi$ is injective.

In particular, if $\varphi:A\to B$ is a surjective homomorphism of graded AG
rings of the same socle degree, then $\varphi$ is an isomorphism.

###### Proof.

Let $\varphi:A\to B$ be a ring homomorphism with $A$ AG. Let $a$ be an element
of the kernel of $\varphi$. If $a\neq 0$ then there exists $0\neq
a^{\prime}\in A$ such that $0\neq aa^{\prime}\in\operatorname{soc}(A)$.
However $\varphi(aa^{\prime})=\varphi(a)\varphi(a^{\prime})=0$ since
$\varphi(a)=0$ by assumption. This contradicts the restriction of $\varphi$ to
$\operatorname{soc}(A)$ being injective. Thus it must be the case that $a=0$
and consequently $\varphi$ is injective.

Now assume $\varphi:A\to B$ is a surjective homomorphism of graded AG rings of
the same socle degree. Then
$\varphi:\operatorname{soc}(A)\to\operatorname{soc}(B)$ is a vector space
isomorphism. Applying the first assertion, $\varphi$ is injective, hence
bijective. ∎

##  3 Cohomological Blow Ups

Let $A$ and $T$ be oriented AG algebras of socle degrees $d$ and $k$
respectively with $d>k$ and $\pi\colon A\rightarrow T$ a surjective degree
preserving algebra map with Thom class $\tau\in A_{d-k}$.

######  Construction 3.1.

Set $n=d-k$ and let $\xi$ be an indeterminate of degree one. Choose any
homogeneous elements $a_{i}\in A$ for $1\leq i\leq n$, and define the monic
homogeneous polynomial of degree $n$, $f_{A}(\xi)\in A[\xi]$, by

$f_{A}(\xi)=\xi^{n}+a_{1}\xi^{n-1}+\cdots+a_{n-1}\xi+a_{n}.$ (16)

Then let $K\subset A$ be the kernel of $\pi$, and construct the algebra
$\hat{A}$ by

$\hat{A}=\frac{A[\xi]}{(\xi\cdot K,f_{A}(\xi))}.$ (17)

We further set $t_{i}=\pi(a_{i})$ for $1\leq i\leq n$, define the monic
homogeneous polynomial $f_{T}(\xi)\in T[\xi]$

$f_{T}(\xi)=\xi^{n}+t_{1}\xi^{n-1}+\cdots+t_{n},$ (18)

and construct the algebra $\tilde{T}$ by

$\tilde{T}=\frac{T[\xi]}{(f_{T}(\xi))}.$ (19)

Before giving the formal definition of the cohomological blow up (Definition
3.10), we shall discuss some of the fundamental properties of the algebras
$\hat{A}$ and $\tilde{T}$ from Construction 3.1. First, note that since
$\xi\cdot K=0$ in $\hat{A}$, the algebra structure of $\hat{A}$ depends only
on the images $t_{1},\ldots,t_{n-1}$, as well as $a_{n}$. There is a natural
degree preserving algebra map $\beta\colon A\rightarrow\hat{A}$ induced from
the natural inclusions $A\hookrightarrow A[\xi]$ which makes $\hat{A}$ into
and $A$-algebra. Also, the surjective tensor product map $\pi\otimes 1\colon
A[\xi]\rightarrow T[\xi]$ passes to a surjective degree preserving map on
quotients $\hat{\pi}\colon\hat{A}\rightarrow\tilde{T}$. Moreover these maps
$\pi:A\to T$, $\beta\colon A\rightarrow\hat{A}$ and
$\hat{\pi}:\hat{A}\to\tilde{T}$ together with the natural inclusion
$\beta_{0}\colon T\rightarrow\tilde{T}$ fit together in a commutative diagram

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\pi}$$\textstyle{\hat{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\pi}}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{0}}$$\textstyle{\tilde{T}.}$
(20)

The following elementary observation will be useful and we record it as a
lemma. We offer our own proof here for completeness, but it can also be
deduced from [27, Lemma 1.8] and [43, Lemma 2.1]. Recall that for graded
Artinian algebras $A$, $B$, and $C$, we say that $C$ is a _free extension of
$A$ with fiber $B$_ if there are maps $\iota\colon A\rightarrow C$ making $C$
into a free $A$-module and $\pi\colon C\rightarrow B$ with kernel
$\ker(\pi)=\mathfrak{m}_{A}\cdot C$ where $\mathfrak{m}_{A}$ is the maximal
ideal of $A$.

######  Lemma 3.2.

The algebra $\tilde{T}$ from Construction 3.1 in Equation (19) is a free
extension of $T$ with fiber $F\cong{\mathbb{F}}[\xi]/(\xi^{n})$. In
particular, since $T$ and $F$ are Gorenstein, $\tilde{T}$ is also
Gorenstein222A more general result established in a 2008 unpublished note
“Coexact Sequences of Poincaré Duality Algebras”, by L. Smith and R.E. Stong
states that if $\tilde{T}$, $T$ and $F$ are graded Artinian algebras in which
$\tilde{T}$ is a free extension of $T$ with fiber $F$, and any two are
Gorenstein, then so is the third..

###### Proof.

We first claim that the map $\beta_{0}\colon T\rightarrow\tilde{T}$ makes
$\tilde{T}$ into a free $T$-module with basis $\\{1,\xi,\ldots,\xi^{n-1}\\}$.
Clearly this set generates $\tilde{T}$ as a $T$-module, since powers of $\xi$
generate $T[\xi]$ as a $T$-module, and all powers greater than $n-1$ can be
eliminated with the relation $\xi^{n}\equiv-(t_{1}\xi^{n-1}+\cdots+t_{n})$. A
$T$-dependence relation in $\tilde{T}$ lifts to a $T$-relation in $T[\xi]$ of
the form $c_{0}\cdot
1+c_{1}\xi+\cdots+c_{n-1}\xi^{n-1}=t\cdot\left(\xi^{n}+t_{1}\xi^{n-1}+\cdots+t_{n}\right)$
for some $c_{i},r\in T$. Comparing $\xi$-coefficients, we conclude that $t=0$,
and hence $c_{i}=0$ for all $i$ since $\\{1,\xi,\ldots,\xi^{n-1}\\}$ are
$T$-linearly independent in $T[\xi]$. Finally note that the natural projection
map $\phi\colon T\rightarrow{\mathbb{F}}$ extends to a projection map
$\tilde{\phi}\colon\tilde{T}\rightarrow{\mathbb{F}}[\xi]/(\xi^{n})$ with
kernel $\ker(\tilde{\phi})=\mathfrak{m}_{T}\cdot\tilde{T}$. Therefore
$\tilde{T}$ is a free extension of $T$ with fiber
$F={\mathbb{F}}[\xi]/(\xi^{n})$ as claimed.

For the last statement, it suffices to show that the socle of $\tilde{T}$
satisfies $\operatorname{soc}(\tilde{T})=\operatorname{soc}(T)\cdot\xi^{n-1}$.
One containment is obvious, and for the other, assume that
$\hat{t}\in\operatorname{soc}(\tilde{T})$ is a homogeneous socle element. From
our arguments above, we may decompose it as
$\hat{t}=s_{0}+s_{1}\xi+\cdots+s_{n-1}\xi^{n-1}$ for some unique $s_{i}\in T$.
Since $\hat{t}$ is in the socle, we must have $t\cdot\hat{t}=0$ for any
positive degree element $t\in T_{+}$, which implies that
$s_{i}\in\operatorname{soc}(T)$ for all $i$, by linear independence of
$\\{1,\ldots,\xi^{n-1}\\}$. On the other hand, since $\hat{t}$ is homogeneous,
and $\deg(s_{i})=\deg(\hat{t})-i$ it follows that there is only one non-zero
$s_{i}$, and since $\xi\cdot\hat{t}=0$ it follows that $i=n-1$. Therefore we
have shown that $\hat{t}=s_{n-1}\cdot\xi^{n-1}$ where
$s_{n-1}\in\operatorname{soc}(T)$, which implies that
$\operatorname{soc}(\tilde{T})=\operatorname{soc}(T)\cdot\xi^{n-1}$, as
desired. ∎

Lemma 4.3 implies that $\tilde{T}$ is always Gorenstein. On the other hand,
the algebra $\hat{A}$ may not always be Gorenstein, as the following example
shows.

######  Example 3.3.

Let

$A=\frac{{\mathbb{F}}[x,y]}{(x^{3},y^{3})}\ \overset{\pi}{\rightarrow}\
T=\frac{{\mathbb{F}}[x,y]}{(x^{2},y)}$

where $\pi(x)=x$ and $\pi(y)=0$ (here $d=4$, $k=1$, and $n=d-k=3$). Note
$K=\ker(\pi)=(x^{2},y)$. Orient $A$ and $T$ with socle generators
$a_{soc}=x^{2}y^{2}$ and $t_{soc}=x$; then the Thom class of $\pi$ is
$\tau=xy^{2}\in A_{3}$. Set $f_{T}(\xi)=\xi^{3}+x\xi^{2}\in T[\xi]$, so that
$t_{1}=x$ and $t_{2}=t_{3}=0$, and let $\tilde{T}$ be the associated free
extension:

$\tilde{T}=\frac{T[\xi]}{(f_{T}(\xi))}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{2},y,\xi^{3}+x\xi^{2})}.$

Below are different algebras $\hat{A}$ for different choices of $\pi$-lifts
$f_{A}(\xi)=\xi^{3}+a_{1}\xi^{2}+a_{2}\xi+a_{3}$ of $f_{T}(\xi)$, according to
Construction 3.1:

  1. 1.

$a_{1}=x$, $a_{2}=0$, and $a_{3}=(x^{2}y+xy^{2})$. Then $f_{A}(\xi)\in A[\xi]$
is $f_{A}(\xi)=\xi^{3}+x\xi^{2}+(x^{2}y+xy^{2})$, and

$\hat{A}=\frac{A[\xi]}{(\xi\cdot
K,f_{A}(\xi))}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{3},y^{3},x^{2}\xi,y\xi,\xi^{3}+x\xi^{2}+x^{2}y+xy^{2})}$

Then an ${\mathbb{F}}$-basis for $\hat{A}$ is

$\left\\{1,x,y,\xi,x^{2},xy,y^{2},x\xi,\xi^{2},x^{2}y,xy^{2},x\xi^{2}\right\\}$

from which it follows that the Hilbert function for $\hat{A}$ is

$H(\hat{A})=(1,3,5,3)$

and hence $\hat{A}$ is not Gorenstein. Note that in this case, the socle
generator $a_{soc}=x^{2}y^{2}$ is actually in the ideal $(\xi\cdot
K,\hat{f}(\xi))$, hence $\beta(a_{soc})=0$ and thus $\beta$ is not injective.

  2. 2.

$a_{1}=x$, $a_{2}=0$, and $a_{3}=xy^{2}=\tau$. Then
$f_{A}(\xi)=\xi^{3}+x\xi^{2}+xy^{2}\in A[\xi]$ and

$\hat{A}=\frac{A[\xi]}{(\xi\cdot
K,f_{A}(\xi))}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{3},y^{3},x^{2}\xi,y\xi,\xi^{3}+x\xi^{2}+xy^{2})}$

has basis

$\left\\{1,x,y,\xi,x^{2},xy,y^{2},x\xi,\xi^{2},x^{2}y,xy^{2},x\xi^{2},x^{2}y^{2}\right\\}$

and Hilbert function

$H(\hat{A})=(1,3,5,3,1).$

Here the socle of $\hat{A}$ is generated by
$\hat{a}_{soc}=a_{soc}=x^{2}y^{2}$, hence $\hat{A}$ is Gorenstein.

  3. 3.

$a_{1}=x+y$ and $a_{2}=a_{3}=0$. Here we chose a different $\pi$-lift of
$t_{1}=x$, but the reader will see it does not affect $\hat{A}$; the important
choice is $a_{3}=0$. Then $f_{A}(\xi)=\xi^{3}+x\xi^{2}\in A[\xi]$ and

$\hat{A}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{3},y^{3},x^{2}\xi,y\xi,\xi^{3}+(x+y)\xi^{2})}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{3},y^{3},x^{2}\xi,y\xi,\xi^{3}+x\xi^{2})}$

with basis

$\left\\{1,x,y,\xi,x^{2},xy,y^{2},x\xi,\xi^{2},x^{2}y,xy^{2},x\xi^{2},x^{2}y^{2}\right\\}$

and Hilbert function

$H(\hat{A})=(1,3,5,3,1)$

However, note that in this case the socle of $\hat{A}$ is the two-dimensional
${\mathbb{F}}$-vector space generated by $x\xi^{2}$ and $x^{2}y^{2}$,
therefore $\hat{A}$ is not Gorenstein. In contrast to case (1), in this case
the image of the socle of $A$ in $\hat{A}$ via $\beta$ is nonzero and in fact
the map $\beta\colon A\rightarrow\hat{A}$ is injective.

The example suggests a strong dependence of the Gorenstein property on the
choice of $a_{n}$ from Equation (16). Before we give our main result in this
direction, we need some lemmas. The following lemma computes the Hilbert
function of $\hat{A}$ in terms of those of $A$ and $T$.

######  Lemma 3.4.

For each $\hat{a}\in\hat{A}$, there exist unique elements $b_{0}\in A$ and
elements $s_{1},\ldots,s_{n-1}\in T$ for which

$\hat{a}=\beta(b_{0})+\beta(b_{1})\xi^{1}+\cdots+\beta(b_{n-1})\xi^{n-1},\ \
\text{for some}\ \ b_{i}\in A,\ \ \text{where}\ \ \pi(b_{i})=s_{i}.$

In particular the Hilbert function of $\hat{A}$ satisfies

$H(\hat{A})=H(\beta(A))+H(T)[1]+\cdots+H(T)[n-1].$ (21)

###### Proof.

Existence of such a decomposition is easy: every element $\hat{a}\in\hat{A}$
has a representative in the polynomial ring $A[\xi]$, and high powers of $\xi$
can be reduced via the relation

$f_{A}(\xi)=\xi^{n}+a_{1}\xi^{n-1}+\cdots+a_{n}\equiv 0.$

As for uniqueness, suppose that there are some other elements $c_{0}\in A$ and
other $s_{1}^{\prime},\ldots,s_{n-1}^{\prime}\in T$ with $\pi$-lifts
$c_{1},\ldots,c_{n-1}\in A$ such that

$\hat{a}=\beta(c_{0})+\beta(c_{1})\xi+\cdots+\beta(c_{n-1})\xi^{n-1}.$

Comparing their decompositions in $\tilde{T}$ via the projection $\hat{\pi}$
we find that

$0=\beta_{0}(\pi(b_{0}-c_{0}))+\beta_{0}\left(\pi(b_{1}-c_{1})\right)\xi+\cdots+\beta_{0}\left(\pi(b_{n-1}-c_{n-1})\right)\xi^{n-1}.$

It follows from Lemma 4.3 that $\tilde{T}$ is a free $T$-module with basis
$\\{1,\xi,\ldots,\xi^{n-1}\\}$ via the injective map $\beta_{0}\colon
T\rightarrow\tilde{T}$, and hence we conclude that
$s_{i}=\pi(b_{i})=\pi(c_{i})=s_{i}^{\prime}$ for all $i=1,\ldots,n-1$, and
that $\pi(a_{0})=\pi(c_{0})$, hence $b_{0}-c_{0}\in K$. But we also have the
relation in $\hat{A}$

$0=\beta\left(b_{0}-c_{0}\right)+\beta\left(b_{1}-c_{1}\right)\xi+\cdots+\beta\left(b_{n-1}-c_{n-1}\right)\xi^{n-1}.$

Since $\beta\left(b_{i}-c_{i}\right)\xi^{i}\in\left(\xi\cdot K\right)$, the
identity displayed above simplifies to

$0=\beta(b_{0}-c_{0})$

and hence $\beta(b_{0})=\beta(c_{0})$ in $\hat{A}$. Thus the decomposition is
unique in the desired sense. Equation (21) follows immediately from this
decomposition. ∎

######  Lemma 3.5.

With $A$, $T$, $\pi$, $K$, and
$f_{A}(\xi)=\xi^{n}+a_{1}\xi^{n-1}+\cdots+a_{n}\in A[\xi]$ as in Construction
3.1, we have in $A[\xi]$

$\left(\xi\cdot K,f_{A}(\xi)\right)\cap\beta(A)=a_{n}\cdot K.$

###### Proof.

Let $b\in\left(\xi\cdot K,f_{A}(\xi)\right)\cap\beta(A)$. Then there is a
polynomial $g(\xi)\in A[\xi]$ for which $b-g(\xi)f_{A}(\xi)\in\left(\xi\cdot
K\right)$ in $A[\xi]$. Writing $g(\xi)=g_{m}\xi^{m}+\cdots+g_{1}\xi+g_{0}$ for
$g_{i}\in A$, since $f_{A}(\xi)$ is monic we must therefore have in $A[\xi]$

$b-g_{m}\xi^{m+n}+\left(\text{lower order terms}\right)\in\left(\xi\cdot
K\right).$ (22)

Since $m+n\geq n\geq 1$ we can compare coeffients on the left hand side and
right hand side of (22) to deduce that $g_{m}\in K$. If $m\geq 1$, one we can
combine the term $g_{m}\xi^{m}\cdot\hat{f}(\xi)$ with the other
$\left(\xi\cdot K\right)$ terms, and lower the $\xi$-degree of $g(\xi)$.
Repeating this procedure, we may assume that the $\xi$-degree is $m=0$, that
$g_{m}=g_{0}\in K$, and hence that $b-g_{0}\cdot f_{A}(\xi)\in\left(\xi\cdot
K\right)$. Since $g_{0}\in K$, it follows that
$g_{0}\left(f_{A}(\xi)-a_{n}\right)\in\left(\xi\cdot K\right)$ as well, and
therefore that $b-g_{0}\cdot a_{n}\in\left(\xi\cdot K\right)$ which implies
that $b-g_{0}\cdot a_{n}=0$, and hence that $b\in a_{n}\cdot K$, as desired.
The reverse containment is obvious. ∎

######  Theorem 3.6.

Let $A$, $T$, $\pi$, $\tau$, $f_{A}(\xi)$, $\hat{A}$, and $\beta$ be as above.
Then

  1. 1.

the algebra map $\beta\colon A\rightarrow\hat{A}$ is injective if and only if
the constant coefficient of $f_{A}(\xi)$ satisfies $a_{n}=\lambda\cdot\tau$
for some $\lambda\in{\mathbb{F}}$ (possibly $\lambda=0$), and

  2. 2.

the algebra $\hat{A}$ is Gorenstein if and only if $a_{n}=\lambda\cdot\tau$
for some non-zero $\lambda\in{\mathbb{F}}^{\times}$.

###### Proof.

For (1.), observe that $\beta\colon A\rightarrow\hat{A}$ is injective if and
only if $\left(\xi\cdot K,f_{A}(\xi)\right)\cap\beta(A)=0$, which by Lemma 3.5
is equivalent to $a_{n}\cdot K=0$, which is in turn, equivalent to
$a_{n}=\lambda\cdot\tau$ for some $\lambda\in{\mathbb{F}}$ by Lemma 2.2.

For (2.), assume first that $\hat{A}$ is Gorenstein. Then
$\operatorname{soc}(\hat{A})=\hat{A}_{e}$ where $e$ is the largest integer for
which $\hat{A}_{e}\neq 0$. It follows from Lemma 3.4 that $e\leq d$ and that
$e<d$ if and only if $\beta(A_{d})=0$. But if $e<d$, then by the surjectivity
of the algebra map $\hat{\pi}\colon\hat{A}\rightarrow\tilde{T}$ we must have
$\hat{\pi}(\hat{A}_{d-1})=\tilde{T}_{d-1}\neq 0$, and hence
$0\neq\hat{A}_{d-1}=\operatorname{soc}(\hat{A})$ and $e=d-1$. By Lemma 2.6,
$\hat{\pi}$ must be an isomorphism, i.e.

$\hat{\pi}\colon\hat{A}=\frac{A[\xi]}{(\xi\cdot
K,f_{A}(\xi))}\overset{\cong}{\rightarrow}\frac{A[\xi]}{(K,f_{A}(\xi))}=\tilde{T}.$

In particular, we see that $K\subseteq(\xi\cdot K,f_{A}(\xi))$ in $A[\xi]$,
which by Lemma 3.5 implies that $K\subseteq a_{n}.K$, which is impossible for
degree reasons (since we are assuming $d>k$). Therefore we must have
$\operatorname{soc}(\hat{A})=\hat{A}_{d}\neq 0$, which by Lemma 3.4 must be
the image of $A_{d}$, and hence $\beta(a_{soc})\neq 0$. By (1) this implies
that $a_{n}=\lambda\cdot\tau$ for some $\lambda\in{\mathbb{F}}$. Next we claim
that $\lambda\neq 0$. Indeed assume that $\lambda=0$ so that $a_{n}=0$. Let
$b_{0}\in A$ be any $\pi$-lift of $t_{soc}\in T_{k}$. Then note that
$\xi^{n-1}\beta(b_{0})\in\operatorname{soc}(\hat{A})$. Indeed in $A[\xi]$ we
have $\xi\cdot\left(\xi^{n-1}b_{0}\right)-b_{0}f_{A}(\xi)\in\left(\xi\cdot
K\right)$, hence $\xi\cdot\left(\xi^{n-1}b_{0}\right)\in(\xi\cdot
K,f_{A}(\xi))$, but also for any $b\in A$ of positive degree $b_{0}\cdot b\in
K$ and hence $b\cdot\left(\xi^{n-1}\cdot b_{0}\right)\in\left(\xi\cdot
K\right)$ as well. We further claim that
$\left\\{\xi^{n-1}\cdot\beta(b_{0}),\beta(a_{soc})\right\\}$ are linearly
independent in $\hat{A}$. Indeed,
$\hat{\pi}(\xi^{n-1}\cdot\beta(b_{0}))=\xi^{n-1}t_{soc}$ is a socle generator
of $\tilde{T}$ (hence is non-zero), whereas $\hat{\pi}(\beta(a_{soc}))=0$.
Since $\beta(a_{soc})\neq 0$, this shows that the socle of $\hat{A}$ has
dimension at least two, contradicting our assumption that $\hat{A}$ is
Gorenstein.

Conversely assume that $a_{n}=\lambda\cdot\tau$ for some non-zero
$\lambda\in{\mathbb{F}}^{\times}$. Then by (1.) $\beta\colon
A\rightarrow\hat{A}$ is injective, and hence
$\beta(a_{soc})\in\operatorname{soc}(\hat{A})$. We want to show that
$\beta(a_{soc})$ generates the socle. To that end, suppose that
$\hat{a}\in\hat{A}$ is any other socle element, and as in Lemma 3.4 write

$\hat{a}=\beta(b_{0})+\beta(b_{1})\xi+\cdots+\beta(b_{n-1})\xi^{n-1}.$ (23)

Note that since $\hat{\pi}$ is surjective, it must map socle elements of
$\hat{A}$ to socle elements of $\tilde{T}$. Thus applying $\hat{\pi}$ to (23)
we find that there is a constant $c$ (possibly $c=0$) such that

$\hat{\pi}(\hat{a})=\pi(b_{0})+\pi(b_{1})\xi+\cdots+\pi(b_{n-1})\xi^{n-1}=c\cdot
t_{soc}\cdot\xi^{n-1}.$

Since $\tilde{T}$ is a free $T$-module with basis
$\\{1,\xi,\ldots,\xi^{n-1}\\}$, it follows that $\pi(b_{i})=0$ in $T$ for
$0\leq i\leq n-2$ and $\pi(b_{n-1})=c\cdot t_{soc}$. This implies that
$b_{0},\ldots,b_{n-2}\in K$, and therefore that $\beta(b_{i})\xi^{i}=0$ in
$\hat{A}$ for $i=1,\ldots,n-2$. Thus $\hat{a}$ reduces to

$\hat{a}=\beta(b_{0})+\beta(b_{n-1})\xi^{n-1}.$

Since $\hat{a}\in\operatorname{soc}(\hat{A})$ we have $\xi\cdot\hat{a}=0$ in
$\hat{A}$ and therefore

$\displaystyle 0=\xi\cdot\hat{a}=$
$\displaystyle\xi\cdot\beta(b_{0})+\beta(b_{n-1})\xi^{n}$
$\displaystyle(\text{since}\ b_{0}\in K,\ \text{and}\
\xi^{n}\equiv-\left(a_{1}\xi^{n-1}+\cdots+a_{n}\right))$ $\displaystyle\equiv$
$\displaystyle-\beta(b_{n-1})\cdot\left(a_{1}\xi^{n-1}+\cdots+a_{n-1}\xi+a_{n}\right)$
$\displaystyle(\text{since}\ \pi(b_{n-1})=c\cdot t_{soc},\ \text{and}\
a_{i}\in A_{i}\ \text{for}\ 1\leq i\leq n-1)$ $\displaystyle\equiv$
$\displaystyle\beta(b_{n-1})\cdot a_{n}$ $\displaystyle(\text{since}\
b_{n-1}=c\cdot t_{soc}\ \text{and}\ a_{n}=\lambda\cdot\tau)$
$\displaystyle\equiv$ $\displaystyle\lambda\cdot c\cdot\beta(a_{soc}).$

Since $\lambda\neq 0$ and $\beta(a_{soc})\neq 0$ we must conclude that $c=0$
and hence $b_{n-1}\in K$ as well, and therefore that $\hat{a}=\beta(b_{0})$.
But since $\beta(b_{0})=\hat{a}\in\operatorname{soc}(\hat{A})$ and $\beta$ is
injective, it follows that $b_{0}\in\operatorname{soc}(A)$, as desired.
Therefore $\operatorname{soc}(\hat{A})\subseteq\beta(\operatorname{soc}(A))$,
from which it follows that $\hat{A}$ must be Gorenstein. ∎

######  Corollary 3.7.

If $a_{n}=\lambda\cdot\tau$ with $\lambda\in{\mathbb{F}}$ then

$H(\hat{A})=H(A)+H(T)[1]+\cdots+H(T)[n-1]$

and there is a split exact sequence of $A$-modules

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\hat{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\pi}}$$\textstyle{\tilde{T}/\beta_{0}(T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

###### Proof.

Injectivity for $\beta$ follows from Theorem 3.6 part (1) and surjectivity of
$\hat{\pi}$ is by definition. That the displayed sequence is a complex follows
from the commutative diagram (20). Exactness of the sequence above viewed as a
sequence of vector spaces follows from Lemma 3.4 and the identity

$H(\hat{A})=H(\beta(A))+H(T)[1]+\cdots+H(T)[n-1]=H(A)+H(\tilde{T}/\beta_{0}(T)).$

Finally, since $A$ is a Gorenstein ring, $A$ is injective as an $A$-module,
thus the above sequence splits. ∎

######  Remark 3.8.

It follows from Theorem 3.6 and Corollary 3.7, that if $a_{n}=0$ then
$\beta\colon A\rightarrow\hat{A}$ is injective and

$H(\hat{A})=H(A)+H(T)[1]+\cdots+H(T)[n-1],$

but $\hat{A}$ is not Gorenstein. We call such algebras _boundary Gorenstein
algebras_ , indicating that they are in the closure of the Gorenstein locus of
the Hilbert scheme; the algebra $\hat{A}$ in Example 3.3(3.) is a boundary
Gorenstein algebra of this type. More precisely, one can show that if
${\mathbb{F}}$ is algebraically closed then the algebra

$\tilde{A}[\lambda]=\frac{A[\xi,\lambda]}{(\xi\cdot
K,\xi^{n}+a_{1}\xi^{n-1}+\cdots+\lambda\cdot\tau)}$

is flat as a module over ${\mathbb{F}}[\lambda]$, where the fibers
$\tilde{A}[c]/(\lambda-c)\cdot\tilde{A}[\lambda]$ are Gorenstein if $c\neq 0$
and not Gorenstein, but boundary Gorenstein, if $c=0$. We shall give another
flat family in which the Gorenstein algebra $\hat{A}$ is a general fiber in
Section 8.

######  Definition 3.9 (Preferred Orientations).

By Lemma 4.3, $\tilde{T}$ from Construction 3.1 is always Gorenstein, and we
define its preferred orientation as the one corresponding to the socle
generator, $\hat{t}_{soc}=\xi^{n-1}\cdot t_{soc}$. If $\hat{A}$ from
Construction 3.1 is Gorenstein, then we define its preferred orientation as
the one corresponding to the socle generator $\hat{a}_{soc}=\beta(a_{soc})$;
hence the preferred orientation on $\hat{A}$ is the one inherited from $A$ via
$\beta$.

######  Definition 3.10 (Cohomological Blow-Up, Exceptional Divisor,
Cohomological Blow Down).

Given oriented AG algebras $A$ and $T$ of socle degrees $d>k$, respectively,
and surjective degree-preserving algebra map $\pi\colon A\rightarrow T$ with
Thom class $\tau\in A_{n}$ where $n=d-k$, and given a homogeneous monic
polynomial $f_{A}(\xi)=\xi^{n}+a_{1}\xi^{n-1}+\cdots+a_{n}\in A[\xi]$ of
degree $n$ with homogeneous elements $a_{i}\in A_{i}$ for $1\leq i\leq n$ and
with $a_{n}=\lambda\cdot\tau$ for some non-zero constant $\lambda$, and
setting $t_{i}=\pi(a_{i})$ for $1\leq i\leq n-1$, we call the corresponding
oriented AG algebra from Construction 3.1, Equation (17) the _cohomological
blow up of $A$ along $\pi$ with parameters $(t_{1},\ldots,t_{n-1},\lambda)$_,
or BUG for short, and write

$\tilde{A}=(\hat{A}=)\frac{A[\xi]}{(\xi\cdot
K,\underbrace{\xi^{n}+a_{1}\xi^{n-1}+\cdots+\lambda\cdot\tau}_{f_{A}(\xi)})}.$

with its preferred orientation $\tilde{a}_{soc}=a_{soc}$. The oriented AG
algebra from Construction 3.1, Equation (19)

$\tilde{T}=(\tilde{T}=)\frac{T[\xi]}{(\underbrace{\xi^{n}+t_{1}\xi^{n-1}+\cdots+\lambda\cdot\pi(\tau))}_{f_{T}(\xi)}}$

with its preferred orientation $\tilde{t}_{soc}=\xi^{n-1}\cdot t_{soc}$ is
called the _exceptional divisor of $T$ with parameters
$(t_{1},\ldots,t_{n-1},\lambda)$_. In this case we refer to $A$ as the
_cohomological blow down of $\hat{A}$ along $\hat{\pi}$_.

######  Remark 3.11.

One can force $\lambda=1$ in Definition 3.10 by scaling orientations on either
$A$ or $T$. Specifically, given $\pi\colon A\rightarrow T$ with Thom class
$\tau$, take new distinguished socle generator either
$a_{soc}^{\prime}=\lambda\cdot a_{soc}$ or $t_{soc}^{\prime}=\lambda^{-1}\cdot
t_{soc}$ so that the same map with one of these scaled orientations
$\pi^{\prime}\colon A^{\prime}\rightarrow T^{\prime}$ will have Thom class
$\tau^{\prime}=\lambda\cdot\tau$, and hence
$a_{n}=\lambda\cdot\tau=\tau^{\prime}$.

If $\lambda\in{\mathbb{F}}$ has an $n^{th}$-root, say $\mu$ (e.g. if
${\mathbb{F}}$ is algebraically closed), then one can also force $\lambda=1$
by rescaling the parameters in the cohomological blow. Specifically if
$\tilde{A}$ is the cohomological blow up of $A$ along $\pi\colon A\rightarrow
T$ with parameters $(t_{1},\ldots,t_{n-1},\lambda)$, then $\tilde{A}$ is
isomorphic to the the cohomological blow up of $A$ along $\pi$ with parameters
$(t^{\prime}_{1},\ldots,t^{\prime}_{n-1},1)$ where
$t^{\prime}_{i}=\mu^{i}\cdot t_{i}$ via the map $\xi\mapsto\mu\cdot\xi$.

The following example shows that the hypothesis on ${\mathbb{F}}$ in Remark
3.11 is necessary.

######  Example 3.12.

Let $A={\mathbb{Q}}[x]/(x^{3})$ and $T={\mathbb{Q}}={\mathbb{F}}$ with
$\pi\colon A\rightarrow T$ the natural projection having $\ker(\pi)=(x)$ and
Thom class $\tau=x^{2}$. Taking $f_{A}(\xi)=\xi^{2}+\lambda x^{2}$ with
$\lambda\in{\mathbb{Q}}^{\times}$, the cohomological blow-up algebra of $A$
along $\pi$ with parameters $(0,\lambda)$ is

$\tilde{A}(\lambda)=\frac{{\mathbb{Q}}[x,\xi]}{(x\xi,\xi^{2}+\lambda x^{2})}.$

We claim that any pair of integers $p,q$ such that $p$ is a prime that does
not divide $q$ yield non isomorphic algebras $\tilde{A}(p)$ and
$\tilde{A}(q)$. Indeed, assume that there is a ${\mathbb{Q}}$-algebra
isomorphism $\psi:\tilde{A}(q)\to\tilde{A}(p)$. Then for some
$a,b,c,d\in{\mathbb{Q}}$ we must have $\psi(x)=ax+b\xi$ and
$\psi(\xi)=cx+d\xi$. Moreover we may assume that $a,b,c,d$ are integers such
that $\operatorname{gcd}(a,b,c,d)=1$ because $\psi$ is an isomorphism if and
only if $\alpha\cdot\psi$ is an isomorphism for any
$\alpha\in{\mathbb{Q}}^{\times}$. We have

$\psi((x\xi,\xi^{2}+qx^{2}))=(acx^{2}+(ad+bc)x\xi+bd\xi^{2},(c^{2}+qa^{2})x^{2}+2(cd+qab)x\xi+(d^{2}+qb^{2})\xi^{2}).$

In order for $\psi$ to be a ${\mathbb{Q}}$-algebra isomorphism the ideal above
must be equal to $(x\xi,\xi^{2}+px^{2})$ and equating the two ideals in
${\mathbb{Q}}[x,\xi]/(x\xi)$ yields the following equations

$\displaystyle ac$ $\displaystyle=pbd$ (24) $\displaystyle c^{2}+qa^{2}$
$\displaystyle=p(d^{2}+qb^{2}).$ (25)

It follows from (24) that $p$ divides $a$ or $c$, and hence from (25) that $p$
divides both $a$ and $c$. Returning to (24), we now deduce that $p$ divides
$bd$ and hence $p$ divides $b$ or $d$. Similarly we must also have
$p\mid(d^{2}+qb^{2})$ which implies that $p$ must divide $b$ and $d$.
Therefore we must have $p$ divides $a$, $b$, $c$, and $d$, contradicting our
assumption that $\operatorname{gcd}(a,b,c,d)=1$.

In particular, we have shown that every prime $p$ gives cohomological blow-up
of $A$ along $\pi$ with parameters $(0,p)$ which is not isomorphic to the
cohomological blow up algebra $\hat{A}(1)$ with parameters $(0,1)$. In fact
this shows that there are infinitely many distinct isomorphism classes of
cohomological blow-up algebras $\tilde{A}(p)$ of $A$ along $\pi$ with
parameters $(0,p)$, one for each prime $p$.

On the other hand, we shall see in Theorem 8.3 that the algebras
$\tilde{A}(\lambda^{2})$ and $\tilde{A}(1)$ are always isomorphic for any
rational number $\lambda$.

The next lemma gives the Thom class of the restriction map from the
cohomological blow up to its exceptional divisor.

######  Lemma 3.13.

With notations as in Definition 3.10, if $\tilde{A}$ is the cohomological blow
up of $A$ along $\pi\colon A\rightarrow T$ with parameters
$(t_{1},\ldots,t_{n-1},\lambda)$, and $\tilde{T}$ is its exceptional divisor
then the Thom class of the projection map
$\tilde{\pi}\colon\tilde{A}\rightarrow\tilde{T}$ is

$\tilde{\tau}=-\lambda^{-1}\xi.$

###### Proof.

The socle degree of $\tilde{T}$ is $d-1$, and it follows from Lemma 3.4 that
the graded component $\tilde{A}_{d-1}$ has a ${\mathbb{F}}$-vector space
decomposition as

${\mathbb{F}}\cdot a_{T}\xi^{n-1}\oplus A_{d-1},\ \ \text{where}\ \
\pi(a_{T})=t_{soc}\in T_{k}.$

We may assume that $k<d-1$, since if $k=d-1$ we clearly have $\tilde{A}=A$ and
$\tilde{T}=T$. Then for each $a\in A_{d-1}$ we have $a\in K$, hence $\xi\cdot
a\equiv 0$ in $\tilde{A}$. Also we have

$\displaystyle\xi\cdot a_{T}\xi^{n-1}$ $\displaystyle\equiv a_{T}\xi^{n}$
$\displaystyle\equiv
a_{T}\left(-\left(a_{1}\xi^{n-1}+\cdots+a_{n}\right)\right)$
$\displaystyle\quad\text{since}\ a_{i}\in A_{+}\ \Rightarrow\ a_{T}\cdot
a_{i}\in K$ $\displaystyle\equiv a_{T}\cdot(-a_{n})\equiv
a_{T}\cdot(-\lambda\cdot\tau)=-\lambda\cdot
a_{soc}=-\lambda\cdot\tilde{a}_{soc},$

from which it follows that $\xi=-\lambda\cdot\tilde{\tau}$, as desired. ∎

The following theorem gives a useful characterization of the blow-up algebra
in terms of some universal properties, analogous to Equations (9), (10), and
(11) from the Introduction.

######  Theorem 3.14.

Suppose that we are given oriented AG algebras $\left(A,\int_{A}\right)$,
$\left(T,\int_{T}\right)$, $\left(\tilde{A},\int_{\tilde{A}}\right)$, and
$\left(\tilde{T},\int_{\tilde{T}}\right)$ with socle degrees $d$, $k$, $d$,
and $d-1$, respectively with $d>k$, and surjective degree preserving algebra
maps $\pi\colon A\rightarrow T$ and
$\tilde{\pi}\colon\tilde{A}\rightarrow\tilde{T}$. Then $\tilde{A}$ is a
cohomological blow up of $A$ along $\pi$ for some parameters
$(t_{1},\ldots,t_{n-1},\lambda)$, and $\tilde{T}$ is its exceptional divisor
if and only if the following conditions are satisfied:

  1. 1.

There are degree preserving algebra maps $\beta\colon A\rightarrow\hat{A}$ and
$\beta_{0}\colon T\rightarrow\tilde{T}$ making the following diagram commute

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\pi}$$\textstyle{\hat{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\pi}}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{0}}$$\textstyle{\tilde{T}}$

  2. 2.

The Euler class $\epsilon=\tilde{\pi}(\tilde{\tau})\in\tilde{T}_{1}$ generates
$\tilde{T}$ as a $T$-algebra (via $\beta_{0}$) with a single homogeneous
relation in degree $n=d-k$:

$\epsilon^{n}+\beta_{0}(t^{\prime}_{1})\epsilon^{n-1}+\cdots+\beta_{0}(t^{\prime}_{n})\equiv
0$

for some homogeneous elements $t^{\prime}_{i}\in T_{i}$ for $1\leq i\leq n$.

  3. 3.

There is a short exact sequence of $A$-modules

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\tilde{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\pi}}$$\textstyle{\tilde{T}/\beta_{0}(T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

###### Proof.

Assume that $\left(\tilde{A},\int_{\tilde{A}}\right)$ is the cohomological
blow up of $\left(A,\int_{A}\right)$ along $\pi$ and that
$\left(\tilde{T},\int_{\tilde{T}}\right)$ is the exceptional divisor, with
parameters $(t_{1},\ldots,t_{n-1},\lambda)$ from Definition 3.10, and their
preferred orientations from Definition 3.9. Then our discussion following
Construction 3.1 shows condition (1.) is satisfied. Also Lemma 3.13 shows that
the Thom class of $\tilde{\pi}$ is $\tilde{\tau}=-\lambda^{-1}\xi$, and its
Euler class is $\epsilon=\tilde{\pi}(-\lambda^{-1}\xi)$. From the presentation
of $\tilde{T}$ in (19) we have
$f_{T}(\xi)=\xi^{n}+t_{1}\xi^{n-1}+\cdots+\lambda\cdot\pi(\tau)\equiv 0$ in
$\tilde{T}$. Therefore setting $t_{i}^{\prime}=(-1)^{i}\lambda^{-i}t_{i}$ for
$1\leq i\leq n-1$ yields
$(-1)^{n}\lambda^{n}\left(\epsilon^{n}+t_{1}^{\prime}\epsilon^{n-1}+\cdots+t_{n-1}^{\prime}\epsilon+\lambda^{1-n}\cdot\tau\right)\equiv
0$ in $\tilde{T}$, which is (2.). Finally, Corollary 3.7 implies condition
(3.).

Conversely, assume that conditions (1.), (2.), and (3.) hold. Define an
algebra map

$\textstyle{\phi\colon
A[\xi]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{A}}$$\textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\beta(a)}$$\textstyle{\xi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hat{\tau}}$
(26)

where $\beta\colon A\rightarrow\tilde{A}$ is the map given by (1.), and
$\tilde{\tau}$ is the Thom class of
$\tilde{\pi}\colon\tilde{A}\rightarrow\tilde{T}$. Then conditions (2.) and
(3.) guarantee that $\phi$ is surjective. Indeed by (2.), the quotient
$\tilde{T}/\beta_{0}(T)$ is generated as an $A$ module by non-zero powers of
the Euler class $\epsilon=\tilde{\pi}(\tilde{\tau})$, and hence by (3.)
$\tilde{A}$ is generated as an $A$ module by $1$ and non-zero powers of the
Thom class $\tilde{\tau}$. Furthermore, note that the ideal generated by
$\xi\cdot u$ for $u\in K=\ker(\pi)$ is contained in $\ker(\phi)$. Indeed,
$\phi(u\cdot\xi)=\beta(u)\cdot\tilde{\tau}$ and for any
$\tilde{a}\in\tilde{A}$ we have

$\displaystyle\int_{\tilde{A}}\tilde{\tau}\cdot\beta(u)\cdot\tilde{a}a$
$\displaystyle=\int_{\tilde{T}}\tilde{\pi}(\beta(u)\cdot\tilde{a})$
$\displaystyle=\int_{\tilde{T}}\beta_{0}(\pi(u))\cdot\tilde{\pi}(\tilde{a})=0$

which implies that $\beta(u)\tilde{\tau}=\phi(u\cdot\xi)=0$ since $\tilde{A}$
is Gorenstein.

Consider the relation on the Euler class from (2.):

$\epsilon^{n}+\beta_{0}(t_{1}^{\prime})\epsilon^{n-1}+\cdots+\beta_{0}(t_{n-1}^{\prime})\epsilon+\beta_{0}(t_{n}^{\prime})\equiv
0.$

For each $1\leq i\leq n-1$, let $a_{i}$ be any $\pi$-lift of $t_{i}^{\prime}$
(which exists since $\pi$ is surjective), and set
$g_{A}(\xi)=\xi^{n}+a_{1}\xi^{n-1}+\cdots+a_{n-1}\xi\in A[\xi]$. Note that
$\tilde{\pi}\left(\phi\left(g_{A}(\xi)\right)\right)=-\beta_{0}(t_{n}^{\prime})\in\beta_{0}(T)$,
and hence according to condition (3.), there exists $a_{n}\in A$ such that
$\beta(a_{n})=\phi(g_{A}(\xi))$ in $\tilde{A}$. Then setting
$f_{A}(\xi)=g_{A}(\xi)+a_{n}$, we see that $f_{A}(\xi)\in\ker(\phi)$ as well.
Thus we have shown the containment of ideals $(\xi\cdot
K,f_{A}(\xi))\subseteq\ker(\phi)$, and in particular, the map $\phi$ induces a
surjective map on the quotient

$\bar{\phi}\colon\tilde{A}^{\prime}=\frac{A[\xi]}{(\xi\cdot
K,f_{A}(\xi))}\rightarrow\tilde{A}.$

Since $\tilde{A}^{\prime}$ follows Construction 3.1, Lemma 3.4 implies its
Hilbert function is as in (21):

$H(\tilde{A}^{\prime})=H(\beta^{\prime}(A))+H(T)[1]+\cdots+H(T)[n-1].$ (27)

where $\beta^{\prime}\colon A\rightarrow\tilde{A}^{\prime}$ is the natural map
described after Construction 3.1. By condition (3.), we know that the Hilbert
function of $\hat{A}$ is

$H(\tilde{A})=H(A)+H(T)[1]+\cdots+H(T)[n-1].$ (28)

Since $H(\beta(A))_{i}\leq H(A)_{i}$ for all $i$, but also
$H(\tilde{A}^{\prime})_{i}\geq H(A)_{i}$ for all $i$ by surjectivity of
$\bar{\phi}$, we deduce that the Hilbert functions (27) and (28) must be
equal, and therefore that
$\bar{\phi}\colon\tilde{A}^{\prime}\rightarrow\tilde{A}$ must be an
isomorphism. Finally since $\tilde{A}$ is Gorenstein, it follows that
$\tilde{A}^{\prime}$ is Gorenstein, and hence must be a cohomological blow up,
and the result follows. ∎

######  Remark 3.15.

As discussed in the Introduction, the conditions of Theorem 3.14 are satisfied
by cohomology algebras. More precisely if $Y\subset X$ are compact complex
manifolds of dimension $k<d$, with cohomology algebras $A=H^{2\bullet}(X)$ and
$T=H^{2\bullet}(Y)$ and $\pi\colon A\rightarrow T$ the induced restriction
map, surjective or not, then the cohomology algebras of the blown up manifolds
$\tilde{T}=H^{2\bullet}(\tilde{Y})$ and $\hat{A}=H^{2\bullet}(\tilde{X})$ with
(possibly non-surjective) restriction map
$\hat{\pi}\colon\hat{A}\rightarrow\tilde{T}$ satisfies the conditions (1.),
(2.) and (3.) of Theorem 3.14. Therefore Theorem 3.14 seems to offer a way to
define the cohomological blow up of an oriented AG algebra $A$ along any,
possibly non-surjective, map $\pi\colon A\rightarrow T$. On the other hand,
without the surjectivity assumption on $\pi$, one must sacrifice, among other
things, the nice presentation given by Construction 3.1; see Example 9.3 and
Remark 9.4 in Section 9.

##  4 Macaulay Dual Generators

Our reference for this section is [25, Appendix A], but see also [12, Appendix
A.2.4]. Let $R={\mathbb{F}}[x_{1},\ldots,x_{r}]$ be a polynomial ring, and let
$Q={\mathbb{F}}[X_{1},\ldots,X_{r}]$ be the dual divided power algebra. Let
$F\in Q_{d}$ and $G\in Q_{k}$ be homogeneous forms of degrees $d>k$,
respectively, and suppose that $\tau\in R_{d-k}$ is a polynomial for which
$G=\tau\circ F$. Then if $A=R/\operatorname{Ann}(F)$ and
$T=R/\operatorname{Ann}(G)$ are the corresponding oriented AG algebras, the
identity map on $R$ induces a surjective map on the quotients

$\pi\colon
A=\frac{R}{\operatorname{Ann}(F)}\rightarrow\frac{R}{\operatorname{Ann}(G)}=T$

for which the Thom class is $\tau$ as shown in Lemma 2.4.

Next, we give another construction, similar to Construction 3.1, that will
lead to yet another characterization of the cohomological blow up (Theorem
4.6). First, we need some notation. Throughout this section we use the
notation $\overline{r}$ for the coset of $r\in R$ in the quotient algebra $T$.

######  Definition 4.1 ($G$-dual polynomial).

Given $G$ and $T$ as above, let $\xi$ be an indeterminate, let $n$ be any
positive integer and let $f_{R}(\xi)=\xi^{n}+a_{1}\xi^{n-1}+\cdots+a_{n}\in
R[\xi]$ be a homogeneous monic polynomial with $a_{i}\in R_{i}$ for $1\leq
i\leq n$. Evaluation at $\xi=1$ gives a non-homogeneous element in
$f_{R}(1)\in R$ and projection to the local ring $T$ gives a (non-homogeneous)
unit $\mu_{f}=\overline{1}+\overline{a_{1}}+\cdots+\overline{a_{n}}\in T$. Let
$\mu_{h}=\overline{1}+\overline{u_{1}}+\cdots+\overline{u_{k}}\in T$ be its
$T$-inverse, i.e. $\mu_{f}\cdot\mu_{h}=\overline{1}$ in $T$, with homogeneous
components $\overline{u_{i}}\in T_{i}$. A homogeneous monic polynomial
$h_{R}(\xi)=\xi^{k}+u_{1}\xi^{k-1}+\cdots+u_{k}$ with homogeneous coefficients
$u_{i}\in R_{i}$ that project to $\overline{u_{i}}\in T_{i}$ for every $1\leq
i\leq n$ is called a _$G$ -dual polynomial for $f_{R}(\xi)$_.

Since $T$ is graded and $\mu_{f}\cdot\mu_{h}=\overline{1}$ in $T$, it follows
that the polynomials $f_{R}(\xi)$ and its $G$-dual polynomial $h_{R}(\xi)$
satisfy

$f_{R}(\xi)\cdot h_{R}(\xi)\equiv\xi^{n+k}\mod{\operatorname{Ann}_{R}(G)\cdot
R[\xi]}.$

######  Construction 4.2.

Set $n=d-k$, let $\xi$ be an indeterminate, and let $\Xi$ be its dual divided
power. Choose homogeneous elements $a_{i}\in R_{i}$ for $1\leq i\leq n$ and
define a homogeneous monic polynomial $f_{R}(\xi)\in R[\xi]$ by

$f_{R}(\xi)=\xi^{n}+a_{1}\xi^{n-1}+\cdots+a_{n}.$ (29)

Let $h_{R}(\xi)\in R[\xi]$ be any $G$-dual polynomial of $f_{R}(\xi)$:

$h_{R}(\xi)=\xi^{k}+u_{1}\xi^{k-1}+\cdots+u_{k}.$ (30)

Construct the $(d-1)$-form $\tilde{G}\in Q[\Xi]$ by

$\tilde{G}=h_{R}(\xi)\circ\left(\Xi^{d-1}\cdot
G\right)=\Xi^{n-1}G+\Xi^{n}(u_{1}\circ G)+\cdots+\Xi^{d-1}(u_{k}\circ G),$
(31)

and construct the oriented AG algebra

$\tilde{T}_{MD}=\frac{R[\xi]}{\operatorname{Ann}(\tilde{G})}.$ (32)

Next fix a parameter $\lambda\in{\mathbb{F}}^{\times}$ and construct the
$d$-form $\hat{F}\in Q[\Xi]$

$\hat{F}=F-\lambda\cdot\Xi\tilde{G}=F-\lambda\left(\Xi^{n}G+\Xi^{n+1}(u_{1}\circ
G)+\cdots+\Xi^{n+k}(u_{k}\circ G)\right)$ (33)

and construct the oriented AG algebra

$\hat{A}_{MD}=\frac{R[\xi]}{\operatorname{Ann}(\hat{F})}.$ (34)

In Construction 4.2 one can easily check that we have

$\operatorname{Ann}_{R[\xi]}(\hat{F})\cap R=\operatorname{Ann}_{R}(F),\ \
\text{and}\ \ \operatorname{Ann}_{R[\xi]}(\tilde{G})\cap
R=\operatorname{Ann}_{R}(G)$

which implies that the inclusion map $R\hookrightarrow R[\xi]$ induces
_injective_ maps

$\beta\colon
A=\frac{R}{\operatorname{Ann}_{R}(F)}\rightarrow\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}(\hat{F})}=\hat{A}_{MD}\quad\text{
and }\quad\beta_{0}\colon
T=\frac{R}{\operatorname{Ann}_{R}(G)}\rightarrow\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}(\tilde{G})}=\tilde{T}_{MD},$

Next we observe that $\xi\circ\tilde{F}=\lambda\cdot\tilde{G}$, and hence the
identity map $R[\xi]\rightarrow R[\xi]$ passes to a surjective map on the
quotient algebras

$\hat{\pi}\colon\hat{A}_{MD}=\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}(\hat{F})}\rightarrow\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}\left(\lambda^{-1}\xi\circ\hat{F}=-\tilde{G}\right)}=\tilde{T}_{MD}$

with Thom class $\tilde{\tau}=-\lambda^{-1}\xi$ (compare with Lemma 3.13).
Since the identity and inclusion maps on $R$ and $R[\xi]$ form a commutative
square, it follows that the maps on quotients do too, resulting in a
commutative diagram

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\pi}$$\textstyle{\hat{A}_{MD}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\pi}}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{0}}$$\textstyle{\tilde{T}_{MD};}$
(35)

compare with condition (1.) of Theorem 3.14. The following result is related
to condition (2.) of Theorem 3.14, namely that $\hat{T}_{MD}$ is a free
extension of $T$; compare with Lemma 3.2.

######  Lemma 4.3.

Let $G\in Q_{k}$ be any homogeneous form, let
$f_{R}(\xi)=\xi^{n}+a_{1}\xi^{n-1}+\cdots+a_{n}$ any monic homogeneous
polynomial with coefficients $a_{i}\in R_{i}$, and let
$h_{R}(\xi)=\xi^{k}+u_{1}\xi^{k-1}+\cdots+u_{k}\in R[\xi]$ be a $G$-dual of
$f_{R}(\xi)$, as in Construction 4.2. Then we have the following equality of
ideals:

$\operatorname{Ann}\left(h_{R}(\xi)\circ\left(\Xi^{d-1}\cdot
G\right)\right)=\operatorname{Ann}_{R}(G)\cdot R[\xi]+(f_{R}(\xi)).$

In particular, the oriented AG algebra $\tilde{T}_{MD}$ from Construction 4.2
Equation (32) is a free extension over $T$, and in fact is equal to the
algebra $\tilde{T}$ from Construction 3.1 Equation (19), i.e.

$\tilde{T}_{MD}=\frac{R[\xi]}{\operatorname{Ann}(\tilde{G}=h_{R}(\xi)\circ\left(\Xi^{d-1}\cdot
G\right))}=\frac{R[\xi]}{\operatorname{Ann}_{R}(G)\cdot
R[\xi]+(f_{R}(\xi))}=\frac{T[\xi]}{(f_{T}(\xi)=\overline{f_{R}(\xi)})}=\tilde{T}.$

###### Proof.

The containment

$\operatorname{Ann}_{R}(G)\cdot
R[\xi]+(f_{R}(\xi))\subseteq\operatorname{Ann}_{R[\xi]}\left(\tilde{G}=h_{R}(\xi)\circ\left(\Xi^{d-1}\cdot
G\right)\right)$

follows from the relation

$f_{R}(\xi)\cdot h_{R}(\xi)\equiv\xi^{d}\ \text{mod}\
\operatorname{Ann}_{R}(G)\cdot R[\xi].$

Therefore the identity map $R[\xi]\rightarrow R[\xi]$ passes to a surjective
map of algebras

$\phi\colon\tilde{T}=\frac{T[\xi]}{(\overline{f(\xi)})}=\frac{R[\xi]}{\operatorname{Ann}_{R}(G)\cdot
R[\xi]+(f_{R}(\xi))}\rightarrow\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}(\tilde{G})}=\tilde{T}_{MD}.$

Let $t_{soc}\in R_{k}$ be any homogeneous polynomial which projects onto the
socle generator of $T$, so that the distinguished socle generator of
$\tilde{T}$ is $\overline{t_{soc}\cdot\xi^{n-1}}$. Since

$t_{soc}\cdot\xi^{n-1}\cdot h_{R}(\xi)\circ\left(\Xi^{d-1}\cdot G\right)=1$

it follows from Lemma 2.6 that $\phi$ must also be injective and hence an
isomorphism, and the result follows. ∎

######  Remark 4.4.

In [43], the authors refer to the monic polynomial $f_{R}(\xi)$ as the
_homogenizing polynomial_ for $T$ and to the $G$-dual $h_{R}(\xi)$ as the
_dual homogenizing polynomial_. We refer also to [8] for more on de-
homoginization . We also remark monic polynomial $f_{R}(\xi)$ and its $G$-dual
polynomial $h_{R}(\xi)$ also satisfy

$f_{R}(\xi)\cdot h_{R}(\xi)\circ\left(\Xi^{d}\cdot
G\right)=f_{R}(\xi)\circ\left(\Xi\cdot\hat{G}\right)=G;$ (36)

in particular, $f_{R}(\xi)$ is the Thom class of the projection map

$\pi_{B}\colon
B:=\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}\left(\Xi\cdot\tilde{G}\right)}\rightarrow\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}(G)}=T.$

The algebra $B$ defined above will show up again when we discuss connected
sums.

Lest the reader think that the AG algebra

$\tilde{T}=\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}\left(h_{R}(\xi)\circ\left(\Xi^{d-1}\cdot
G\right)\right)}$

is always a free extension over $T=R/\operatorname{Ann}_{R}(G)$, the following
example shows otherwise.

######  Example 4.5.

Let $R={\mathbb{F}}[x,y,z]$, $Q={\mathbb{F}}[X,Y,Z]$, and let $G=XYZ$ with
$k=\deg(G)=3$. Suppose that we choose $n=2$, and
$h_{R}(\xi)=\xi^{3}+(xy+xz+yz)\xi+(xyz)$ and define $\tilde{G}$ as in
Construction 4.2 Equation (31):

$\displaystyle\tilde{G}=$ $\displaystyle\Xi
XYZ+\Xi^{3}(X+Y+Z)+\Xi^{4}=h(\xi)\circ\left(\Xi^{4}G\right).$

Note that there is no monic polynomial
$f_{R}(\xi)\in\operatorname{Ann}_{R[\xi]}(\tilde{G})$ of degree $n=2$, and in
fact we have

$\tilde{T}=\frac{R[\xi]}{\operatorname{Ann}(\tilde{G})}=\frac{{\mathbb{F}}[x,y,z,\xi]}{(x^{2},y^{2},z^{2},\xi
xyz-\xi^{3}(x+y+z),\xi^{3}(x+y+z)-\xi^{4},\xi^{5})}$

which is not a free extension over $T=R/\operatorname{Ann}(G)$.

Note, however, that the corresponding unit $\overline{h(1)}\in T$ is
$\mu_{h}=\overline{1}+\overline{(xy+xz+yz)}+\overline{(xyz)}$ and its
$T$-inverse is
$\left(\mu_{h}\right)^{-1}=\overline{1}-\overline{(xy+xz+yz)}-\overline{(xyz)}$,
indicating that the “correct choice” is $f_{R}(\xi)=\xi^{3}-(xy+xz+yz)\xi-xyz$
of degree $n=3$ (not $n=2$). Indeed we see that if we define the homogeneous
form of degree $5$

$\tilde{G}^{\prime}=\Xi^{2}XYZ+\Xi^{4}(X+Y+Z)+\Xi^{5}=h_{R}(\xi)\circ\left(\Xi^{5}G\right)$

then

$\tilde{T}^{\prime}=\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}(\tilde{G}^{\prime})}=\frac{R[\xi]}{\operatorname{Ann}_{R}(G)\cdot
R[\xi]+(f_{R}(\xi))}=\frac{T[\xi]}{\left(f_{T}(\xi)=\xi^{3}-\overline{(xy+xz+yz)}\xi-\overline{(xyz)}\right)}$

which is a free extension over $T=R/\operatorname{Ann}(G)$ with fiber
$F={\mathbb{F}}[\xi]/(\xi^{3})$. The moral of the story here is that given a
$k$-form $G\in Q_{k}$ and $T=R/\operatorname{Ann}(G)$, then in order to
construct the dual generator of a free extension over $T$ by the formula
$h_{R}(\xi)\circ\left(\Xi^{n+k-1}\cdot G\right)\in Q_{d-1}$, we can either
choose the integer $n$ and the monic polynomial $f_{R}(\xi)$ of degree $n$ and
then take $h_{R}(\xi)$ as its $G$-dual, or we can choose the monic polynomial
$h_{R}(\xi)$ of degree $k$, then take its $G$-dual $f_{R}(\xi)$ and take
$n=\deg(f_{R}(\xi))$, but we cannot necessarily choose the integer $n$ and the
polynomial $h_{R}(\xi)$ simultaneously. Of course the $\tilde{G}$ of
Construction 4.2 Equation (31) follows the former procedure and, by Lemma 4.3,
is always the dual generator of a free extension of $T$.

The following result gives necessary and sufficient conditions for
$\hat{A}_{MD}$ to be a cohomological blow-up algebra of $A$ along $\pi\colon
A\rightarrow T$ in the sense of Definition 3.10.

######  Theorem 4.6.

Let $f_{R}(\xi)$, $h_{R}(\xi)$, $\tilde{G}$, $\tilde{T}_{MD}$, $\lambda$,
$\hat{F}$ and $\hat{A}_{MD}$ be as in Construction 4.2. Then the following
statements are equivalent:

  1. 1.

The algebra $\hat{A}_{MD}$ is isomorphic to a cohomological blow up of $A$
along $\pi$ with some parameters $(t_{1},\ldots,t_{n-1},\lambda)$.

  2. 2.

The Hilbert function of $\hat{A}_{MD}$ satisfies

$H(\hat{A}_{MD})=H(A)+H(T)[1]+\cdots+H(T)[n-1].$

  3. 3.

There exists an element $r\in R$ for which
$f_{R}(\xi)-r\in\operatorname{Ann}_{R[\xi]}(\tilde{F})$.

  4. 4.

The constant coefficient $r_{n}$ of $f_{R}(\xi)$ satisfies

$\left(r_{n}-r\right)\circ F=\lambda\cdot G$

for some $r\in\operatorname{Ann}_{R}(G)$.

###### Proof.

(1.) $\Rightarrow$ (2.). Assume that $\hat{A}_{MD}$ is isomorphic to a
cohomological blow up of $\pi\colon A\rightarrow T$. Then Corollary 3.7 gives
(2.).

(2.) $\Rightarrow$ (3.). Assume that
$H(\hat{A}_{MD})=H(A)+H(T)[1]+\cdots+H(T)[n-1]$. Then the sequence of maps

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\hat{A}_{MD}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\pi}}$$\textstyle{\tilde{T}/\beta_{0}(T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

is exact by the argument in the proof of Corollary 3.7. Let $f_{A}(\xi)$ be
the equivalence class of $f_{R}(\xi)\in R[\xi]$ in $\hat{A}_{MD}$. Note that
by Lemma 4.3, $f_{A}(\xi)$ must be in the kernel of $\hat{\pi}$, and hence by
exactness, there exists $\bar{r}\in A=R/\operatorname{Ann}(F)$ for which
$\beta(\bar{r})=f_{A}(\xi)$. Then if $r\in R$ be any homogeneous lift of
$\bar{r}$, we have $f_{R}(\xi)-r\in\operatorname{Ann}(\hat{F})$ which is (3.).

(3.) $\Rightarrow$ (4.). Assume that there exists $r\in R$ for which
$f_{R}(\xi)-r\in\operatorname{Ann}(\hat{F})$. Then we have

$\displaystyle\left(f_{R}(\xi)-r\right)\circ\hat{F}=$
$\displaystyle\left(f_{R}(\xi)-r\right)\circ\left(F-\lambda\cdot
h_{R}(\xi)\circ\left(\Xi^{n+k}\cdot G\right)\right)$ (37) $\displaystyle=$
$\displaystyle f_{R}(\xi)\circ F-r\circ F-\lambda\cdot\left(f_{R}(\xi)\cdot
h_{R}(\xi)\right)\circ(\Xi^{n+k}\cdot G)+\lambda\cdot\Xi\cdot r\circ\tilde{G}$
$\displaystyle=$ $\displaystyle r_{n}\circ F-r\circ F-\lambda\cdot
G+\lambda\cdot\Xi\cdot r\circ\tilde{G}$ $\displaystyle=$
$\displaystyle(r_{n}-r)\circ F-\lambda\cdot G+\lambda\cdot\Xi\cdot
r\circ\tilde{G}=0,$

which implies, by comparing $\Xi$-coefficients, that

$(r_{n}-r)\circ F-\lambda\cdot G=0.$

Moreover since
$f_{R}(\xi)-r\in\operatorname{Ann}(\hat{F})\subseteq\operatorname{Ann}(\tilde{G}=\xi\circ\hat{F})$,
and $f_{R}(\xi)\in\operatorname{Ann}(\tilde{G})$, we see that
$r\in\operatorname{Ann}(\tilde{G})$ too, which implies that
$r\in\operatorname{Ann}_{R[\xi]}(\tilde{G})\cap R=\operatorname{Ann}_{R}(G)$,
and (4.) follows.

(4.) $\Rightarrow$ (1.). Assume that there exists $r\in\operatorname{Ann}(G)$
such that the constant coefficient of $f_{R}(\xi)-r\in R[\xi]$ satisfies
$(r_{n}-r)\circ F=\lambda\cdot G$. Then we claim that
$f_{R}(\xi)-r\in\operatorname{Ann}(\hat{F})$. Indeed as in (37) we have

$\left(f_{R}(\xi)-r\right)\circ\hat{F}=(r_{n}-r)\circ F-\lambda\cdot
G-\lambda\cdot\Xi\cdot r\circ\tilde{G}=-\Xi\left(\lambda
r\circ\tilde{G}\right)=0$

since $r\circ\tilde{G}=h_{R}(\xi)\circ\left(\Xi^{d}\cdot(r\circ G)\right)=0$.

Next observe that for any
$r^{\prime}\in\operatorname{Ann}_{R}(G)=\operatorname{Ann}_{R[\xi]}(\tilde{G})\cap
R$ we have $\xi\cdot r^{\prime}\in\operatorname{Ann}_{R[\xi]}(\hat{F})$. In
particular then, the kernel of the surjective map

$\Phi\colon
R[\xi]\rightarrow\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}(\hat{F})}=\hat{A}_{MD}$

contains $\operatorname{Ann}_{R}(G)\cdot\xi R[\xi]$, and $f_{R}(\xi)-r$, not
to mention the ideal $\operatorname{Ann}_{R}(F)\cdot R[\xi]$. Set $I\subset
R[\xi]$ to be the sum of these components, i.e.
$I=\operatorname{Ann}_{R}(F)\cdot R[\xi]+\operatorname{Ann}_{R}(G)\cdot\xi
R[\xi]+(f_{R}(\xi)-r)$. Then clearly we have $R[\xi]/I\cong A[\xi]/(\xi\cdot
K,f_{A}(\xi)=\overline{f_{R}(\xi)-r})$ where
$K=\ker(\pi)=\operatorname{Ann}_{R}(G)\cdot
R[\xi]/\operatorname{Ann}_{R}(F)\cdot R[\xi]$. Moreover if $t_{i}\in T_{i}$ is
the equivalence class of $r_{i}\in R_{i}$, then since the equivalence class of
$r_{n}-r\in R_{n}$ in $A$ is $\lambda\cdot\tau\in A_{n}$, it follows that
$R[\xi]/I$ is the cohomological blow up of $A$ along $\pi$ with parameters
$(t_{1},\ldots,t_{n-1},\lambda)$. Therefore we have a surjective map of
Gorenstein algebras

$\bar{\Phi}\colon\hat{A}=R[\xi]/I=\frac{A[\xi]}{(\xi\cdot
K,f_{A}(\xi))}\rightarrow\hat{A}_{MD}.$

Since $\hat{A}$ and $\hat{A}_{MD}$ have the same socle degree, $\bar{\Phi}$
must be an isomorphism by Lemma 2.6, which is (1.). ∎

The preceding Theorem 4.6 implies that the algebras $\hat{A}$ from
Construction 3.1 Equation (17) and $\hat{A}_{MD}$ from Construction 4.2
Equation (34) are equal precisely when they are cohomological blow ups; in
this case we shall replace “hat” with “tilde” on all symbols, and write

$\tilde{A}=\hat{A}_{MD}=\frac{R[\xi]}{\operatorname{Ann}\left(\tilde{F}=F-\Xi\cdot\left(\tilde{G}=h_{R}(\xi)\circ\left(\Xi^{d-1}\cdot
G\right)\right)\right)}=\frac{A[\xi]}{\left(\xi\cdot
K,\overline{f_{R}(\xi)}\right)}=\hat{A}.$

The following gives an example in which $\hat{A}\neq\hat{A}_{MD}$.

######  Example 4.7.

Let $F=X^{2}Y^{2}$ and $G=XY$ so that

$\pi\colon
A=\frac{{\mathbb{F}}[x,y]}{\operatorname{Ann}(X^{2}Y^{2})}\rightarrow\frac{{\mathbb{F}}[x,y]}{\operatorname{Ann}(XY)}=T,\
\ \pi(x)=x,\ \pi(y)=y$

is the natural projection map of corresponding AG algebras with Thom class
$\tau=xy$; here $d=4$, $k=2$, and $n=2$. Taking $f_{R}=\xi^{2}$ we see that
its $G$-dual is itself, i.e. $h_{R}(\xi)=\xi^{2}$, and hence setting
$\tilde{G}=h_{R}(\xi)\circ\left(\Xi^{d-1}G\right)=\xi^{2}\circ\left(\Xi^{3}\cdot
XY\right)=\Xi XY$, we get

$\tilde{T}=\frac{{\mathbb{F}}[x,y,\xi]}{\operatorname{Ann}(\Xi
XY)}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{2},y^{2},\xi^{2})}.$

Then taking $\lambda=1$, Construction 4.2 yields
$\hat{F}=F-\Xi\tilde{G}=X^{2}Y^{2}-\Xi^{2}XY$, and hence

$\hat{A}_{MD}=\frac{{\mathbb{F}}[x,y,\xi]}{\operatorname{Ann}(X^{2}Y^{2}-\Xi^{2}XY)}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{3},y^{3},x^{2}\xi,y^{2}\xi,x(\xi^{2}+xy),y(\xi^{2}+xy),\xi^{3})}.$

Note in this case, there is no $r\in\operatorname{Ann}(G=XY)$ for which
$f_{R}(\xi)-r=\xi^{2}-r\in\operatorname{Ann}(\hat{F})$. We can also compute
the Hilbert function $H(\hat{A}_{MD})=(1,3,6,3,1)$ whereas
$H(A)+H(T)[1]=(1,3,5,3,1)$, and hence $H(\hat{A}_{MD})\neq H(A)+H(T)[1]$, and
therefore $\hat{A}_{MD}$ is not a cohomological blow up of $A$ along $\pi$, by
Theorem 4.6 (or by Theorem 3.14). Moreover taking
$f_{A}(\xi)=\overline{f_{R}(\xi)}=\xi^{2}$, Construction 3.1 yields

$\hat{A}=\frac{A[\xi]}{(\xi\cdot
K,f_{A}(\xi)=\xi^{2})}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{3},y^{3},\xi x^{2},\xi
y^{2},\xi^{2})}$

which is not even Gorenstein; in particular, $\hat{A}\neq\hat{A}_{MD}$.

Alternatively, a “correct choice” is $f_{R}(\xi)=\xi^{2}-xy$ for which a
$G$-dual polynomial is $h_{R}(\xi)=\xi^{2}+xy$ yielding
$\tilde{G}=(\xi^{2}+xy)\circ\left(\Xi^{3}XY\right)=\Xi XY+\Xi^{3}$ and

$\tilde{T}=\frac{{\mathbb{F}}[x,y,\xi]}{\operatorname{Ann}(\Xi
XY+\Xi^{3})}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{2},y^{2},\xi^{2}-xy)}.$

In this case, taking $\lambda=-1$ (Theorem 4.6 requires it!), Construction 4.2
yields $\tilde{F}=X^{2}Y^{2}+\Xi\left(\Xi XY+\Xi^{3}\right)$ with

$\tilde{A}=(\hat{A}_{MD}=)\frac{{\mathbb{F}}[x,y,\xi]}{\operatorname{Ann}(\Xi^{2}XY+\Xi^{4}+X^{2}Y^{2})}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{3},y^{3},\xi
x^{2},\xi y^{2},\xi^{2}-xy)}.$

Here we can verify that the conditions of Theorem 4.6 are satisfied, for
example the Hilbert function is $H(\tilde{A})=(1,3,5,3,1)=H(A)+H(T)[1]$. Hence
in this case, $\tilde{A}(=\hat{A}_{MD})$ is the cohomological blow up of $A$
along $\pi$ with parameters $(t_{1},\lambda)=(0,-1)$. In this case, taking
$f_{A}(\xi)=\overline{f_{R}(\xi)}=\xi^{2}-xy$, then Construction 3.1 yields
the same algebra:

$\tilde{A}=(\hat{A}=)\frac{A[\xi]}{(\xi\cdot
K,f_{A}(\xi))}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{3},y^{3},\xi x^{2},\xi
y^{2},\xi^{2}-xy)}.$

##  5 Relation With Connected Sums

###  5.1 The Cohomological Blow-up Algebra as a Connected Sum

In this section we relate cohomologiocal blow-up algebras to a different
algebraic construction termed connected sum, an algebraic analogue of the
better known topological construction by the same name. In the topogical
construction, a connected sum is obtained by gluing two $2d$-dimensional
manifolds $M_{1}$ and $M_{2}$ along diffeomorphic tubular neighborhoods of a
common $2k$-dimensional submanifold $N$. In complex geometry it is well known
that the connected sum of an $n$-dimensional complex manifold $M$ with a
projective space ${\mathbb{P}}^{n}$ is diffeomorphic to the blow up of $M$ at
a point [23, p.101]. Our contribution here is to recognize that any
cohomological blow-up algebra along a surjective map is a connected sum in the
algebraic sense; see Theorem 5.4. We now recall the algebraic connected sum
construction as defined in [30], based on the original construction defined in
[3].

######  Definition 5.1 (Fibered product, connected sum).

Given oriented AG algebras $\left(A,\int_{A}\right)$,
$\left(B,\int_{B}\right)$, another AG algebra $\left(T,\int_{T}\right)$ and
algebra maps $\pi_{A}\colon A\rightarrow T$ and $\pi_{B}\colon B\rightarrow T$
one forms the _fibered product algebra_ as the sub algebra
$A\times_{T}B\subseteq A\times B$ of the direct product algebra given by

$A\times_{T}B=\left\\{(a,b)\in A\times B\ |\ \pi_{A}(a)=\pi_{B}(b)\right\\}.$

If $A,B$ have the same socle degree $d$, $\pi_{A}$ and $\pi_{B}$ have Thom
classes $\tau_{A}$ and $\tau_{B}$ respectively, and the Euler classes
$\pi_{B}(\tau_{B})=\pi_{A}(\tau_{A})$ are equal, then the _total Thom class_
$(\tau_{A},\tau_{B})$ is in the fibered product algebra $A\times_{T}B$ , and
we define the _connected sum algebra_ as the quotient of the fibered product
by the principal generated by the total Thom class, i.e.

$A\\#_{T}B=\frac{A\times_{T}B}{\left\langle(\tau_{A},\tau_{B})\right\rangle}.$

If the projection maps $\pi_{A}$ and $\pi_{B}$ are both surjective, then the
connected sum algebra defined above is an AG algebra of the same socle degree
$d$ as $A$ and $B$, see e.g. [30, Lemma 3.8].

In terms of Macaulay duality, the following result in [30, Theorem 4.6] gives
a useful criterion to recognize an AG algebra as a connected sum.

######  Proposition 5.2.

Let $R={\mathbb{F}}[x_{1},\ldots,x_{r}]$ and let $F,H\in R_{d}$ be two
linearly independent homogeneous forms of degree $d$. Suppose that there
exists $\sigma\in R_{d-k}$ (for some $k<d$) satisfying

  1. 1.

$\sigma\circ F=\sigma\circ H\neq 0$, and

  2. 2.

$\operatorname{Ann}(\sigma\circ F=\sigma\circ
H)=\operatorname{Ann}(F)+\operatorname{Ann}(H)$.

In this case, set

$A=\frac{Q}{\operatorname{Ann}{F}},\ B=\frac{Q}{\operatorname{Ann}(H)},\
T=\frac{Q}{\operatorname{Ann}(\sigma\circ F=\sigma\circ H)},$

and let $\pi_{A}\colon A\rightarrow T$ and $\pi_{B}\colon B\rightarrow T$ be
the natural projection maps. Then the Thom classes of $\pi_{A}$ and $\pi_{B}$
are given by $\tau_{A}=\sigma+\operatorname{Ann}(F)$ and
$\tau_{B}=\sigma+\operatorname{Ann}(H)$, and we have algebra isomorphisms

$A\times_{T}B\cong\frac{Q}{\operatorname{Ann}(F)\cap\operatorname{Ann}(H)},\ \
A\\#_{T}B\cong\frac{Q}{\operatorname{Ann}(F-H)}.$

And, conversely, every connected sum $A\\#_{T}B$ of graded AG algebras of the
same socle degree over graded AG algebra $T$ arises in this way.

We utilize this result to realize a cohomological blow-up algebra defined as
in Construction 4.2 as a connected sum.

######  Example 5.3.

Let $F=X^{2}Y^{2}$ and $G=Y$ and $\pi$ the projection between their
corresponding AG algebras

$\pi\colon
A=\frac{{\mathbb{F}}[x,y]}{\operatorname{Ann}(X^{2}Y^{2})}\rightarrow\frac{{\mathbb{F}}[x,y]}{\operatorname{Ann}(Y)}=T,$

with Thom class $\tau=x^{2}y$. Here $H(A)=(1,2,3,2,1)$, $H(T)=(1,1)$, $d=4$,
$k=1$, and $n=3$. Take $h_{R}(\xi)=\xi+y$ with corresponding unit
$\mu_{h}=1+y\in T$, and inverse $\left(\mu_{h}\right)^{-1}=1-y$, so that
$f_{R}(\xi)=\xi^{3}-\xi^{2}y$. Then

$\tilde{G}=\left(\xi+y\right)\circ\left(\Xi^{3}Y\right)=\Xi^{2}Y+\Xi^{3}.$

Choosing $\lambda=-2$, we have

$\tilde{F}=X^{2}Y^{2}-2\Xi\left(\Xi^{2}Y+\Xi^{3}\right)=X^{2}Y^{2}-2\Xi^{3}Y-2\Xi^{4}.$

Setting $r=-2x^{2}y\in\operatorname{Ann}_{R}(G)$, we have
$\hat{f}_{R}(\xi)=f_{R}(\xi)-r=\xi^{3}-\xi^{2}y+2x^{2}y\in\operatorname{Ann}_{R[\xi]}(\tilde{F})$,
and hence

$\tilde{A}=\frac{{\mathbb{F}}[x,y,\xi]}{\operatorname{Ann}(\tilde{F})}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{3},y^{3},\xi
x,\xi y^{2},\xi^{3}-\xi^{2}y+2x^{2}y)}$

is according to Theorem 4.6 the cohomological blow up of $A$ along $\pi$ with
parameters $(\overline{-y},0,,0,2)$, with Hilbert function
$H(\tilde{A})=H(A)+H(T)[1]+H(T)[2]=(1,3,5,3,1)$.

Setting $H=2\Xi^{3}Y+2\Xi^{4}$, we have $\tilde{F}=F-H$. Moreover the element
$\tau=\hat{f}_{R}(\xi)$ satisfies $\tau\circ H=\tau\circ F=Y$ and
$\operatorname{Ann}_{R[\xi]}(Y)=\operatorname{Ann}_{R[\xi]}(F)+\operatorname{Ann}_{R[\xi]}(H)$.
Hence it follows from Proposition 5.2 that we can also realize $\tilde{A}$ as
a connected sum of $A=R[\xi]/\operatorname{Ann}(F)$ and
$B=R[\xi]/\operatorname{Ann}(H)$ over $T=R/\operatorname{Ann}(Y)$, i.e.
$\tilde{A}\cong A\\#_{T}B.$

In the following we show that Example 5.3 is an instance of a general
phenomenon: any cohomological blow-up algebra is a connected sum. For the
remainder of this section we work under the following set up. Fix a $d$-form
$F\in Q_{d}$ and a $k$-form $G\in Q_{k}$ with $d>k$, and let
$A=R/\operatorname{Ann}(F)$ and $T=R/\operatorname{Ann}(G)$ be the associated
oriented AG algebras. Assume there exists homogeneous polynomial $\tau\in
R_{d-k}$ for which $\tau\circ F=G$, so that the natural projection map
$\pi_{A}\colon A\rightarrow T$ has Thom class $\tau_{A}=\overline{\tau}$. Let
$n=d-k$, let $\xi$ be an indeterminate, $\Xi$ a divided power variable dual to
$\xi$, and fix a monic homogeneous polynomial
$f_{R}(\xi)=\xi^{n}+r_{1}\xi^{n-1}+\cdots+r_{n}\in R[\xi]$ with coefficients
$r_{i}\in R_{i}$ such that $r_{n}=\lambda\cdot\tau$ for some nonzero constant
$\lambda\in{\mathbb{F}}^{\times}$. Let
$h_{R}(\xi)=\xi^{k}+u_{1}\xi^{k-1}+\cdots+u_{k}\in R[\xi]$ be a $G$-dual
polynomial of $f_{R}(\xi)$. Define the $(d-1)$-form $\tilde{G}$ and its
associated oriented AG algebra $\tilde{T}$ by

$\tilde{G}=h_{R}(\xi)\circ\left(\Xi^{d-1}\cdot
G\right),\qquad\tilde{T}=\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}(\tilde{G})}.$

Define the $d$-form $\tilde{F}$ and its associated oriented AG algebra
$\tilde{A}$

$\tilde{F}=F-\lambda\cdot\Xi\cdot\tilde{G},\qquad\tilde{A}=\hat{A}_{MD}=\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}(\tilde{F})}$

as in construction 4.2; note that Theorem 4.6 guarantees that
$\hat{A}_{MD}=\tilde{A}$ here. Define the new $d$-form $H$ and its associated
AG algebra $B$

$H=\lambda\cdot\Xi\cdot\tilde{G}\in Q[\Xi]_{d},\qquad
B=\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}\left(H\right)}$

and recall that

$H=\lambda\cdot\Xi\cdot\tilde{G}=\lambda\cdot
h_{R}(\xi)\circ\left(\Xi^{d}\cdot G\right).$

It follows that $h_{R}(\xi)$ is also a $G$-dual polynomial for $\xi\cdot
f_{R}(\xi)$ of degree $n+1$, and hence by Lemma 4.3 we have

$B=\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}\left(H\right)}=\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}(\operatorname{Ann}_{R}(G)\cdot
R[\xi]+(\xi\cdot f_{R}(\xi))}=\frac{T[\xi]}{\left(\overline{\xi\cdot
f_{R}(\xi)}\right)}.$

Thus the distinguished socle generator of $B$ as
$b_{soc}=\lambda^{-1}\cdot\xi^{n}\cdot t_{soc}$. The evaluation $\xi=0$ passes
to a map on quotients $\pi_{B}\colon B\rightarrow T$, and from the identity
(36) it follows that its Thom class is

$\tau_{B}=\lambda^{-1}\cdot f_{R}(\xi).$

Note that the Euler class of $\pi_{B}$ is $\pi_{B}(\lambda^{-1}\cdot
f_{R}(\xi))=\tau$ which is equal to the Euler class of $\pi_{A}$, and hence it
makes sense to form the connected sum $A\\#_{T}B$.

######  Theorem 5.4.

The connected sum of $A$ and $B$ over $T$ is equal to the cohomological blow
up of $A$ along $\pi$ with parameters
$(\overline{a_{1}},\ldots,\overline{a_{n-1}},\lambda)$, i.e.

$\tilde{A}=A\\#_{T}B.$

###### Proof.

With our setup above, Theorem 4.6 implies that $\tilde{A}$ is the
cohomological blow up of $A$ along $\pi$ with parameters
$(\overline{a_{1}},\ldots,\overline{a_{n-1}},\lambda)$, i.e.

$\hat{A}_{MD}=\tilde{A}=\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}\left(\tilde{F}=F-H\right)}.$

Setting $\sigma=f_{R}(\xi)$ in Proposition 5.2 it then suffice to check that
conditions (1.) and (2.) hold. Condition (1.) holds since we have

$f_{R}(\xi)\circ F=(\xi^{n}+r_{1}\xi^{n-1}+\cdots+\lambda\cdot\tau)\circ
F=\lambda\cdot G=f_{R}(\xi)\circ\left(H\right).$

It remains to see that

$\operatorname{Ann}_{R[\xi]}(G)=\operatorname{Ann}_{R[\xi]}(H)+\operatorname{Ann}_{R[\xi]}(F).$
(38)

By Lemma 4.3 we have

$\operatorname{Ann}_{R[\xi]}(H)=\operatorname{Ann}_{R}(G)\cdot
R[\xi]+(\xi\cdot f_{R}(\xi)).$

Also, since $F$ is independent of $\xi$ we have

$\operatorname{Ann}_{R[\xi]}(F)=\operatorname{Ann}_{R}(F)\cdot R[\xi]+(\xi).$

Since $\operatorname{Ann}_{R}(F)\subset\operatorname{Ann}_{R}(G)$ and

$\operatorname{Ann}_{R[\xi]}(G)=\operatorname{Ann}_{R}(G)\cdot R[\xi]+(\xi),$

(38) follows, and the desired conclusion follows by Proposition 5.2. ∎

######  Remark 5.5.

In general, fibered products and connected sums of standard graded AG algebras
need not be standard graded, even in the simplest cases. Examples illustrating
this appear for the fibered product in [30, Example 4.5], and for the
connected sum in [30, Proposition 5.22]. Theorem 5.4 distinguishes
cohomological blow-up algebras of surjective maps as a class of connected sums
which preserves the standard grading. However there are examples from geometry
where $A$ and $T$ are standard graded, but the restriction map $\pi\colon
A\rightarrow T$ is not surjective, and the cohomological blow up has a non-
standard grading; see Remark 9.4.

###  5.2 The Blow Down as a Connected Sum

Continuing with our set up above, let
$\tilde{\pi}_{A}\colon\tilde{A}\rightarrow\tilde{T}$ be the projection map
with Thom class $\tilde{\tau}_{A}=-\lambda^{-1}\xi$. Set
$H=\lambda\cdot\Xi\cdot\tilde{G}$ and consider the surjective map of AG
algebras

$\tilde{\pi}_{B}:\tilde{B}:=\frac{R[\xi]}{\operatorname{Ann}(-H=-\lambda\cdot\Xi\cdot\tilde{G})}\to\frac{R[\xi]}{\operatorname{Ann}(\tilde{G})}=\tilde{T}.$

The algebra $\tilde{B}$ is the AG algebra $B$ from the previous subsection,
but with orientation reversed, i.e. $\tilde{b}_{soc}=-b_{soc}$. Thus the Thom
class of the map $\tilde{\pi}_{B}$ is $\tilde{\tau}_{B}=-\lambda^{-1}\xi$
which is equal to the Thom class of $\tilde{\pi}_{A}$. Then it makes sense to
form the connected sum $\tilde{A}\\#_{\tilde{T}}\tilde{B}$.

######  Theorem 5.6.

The connected sum of $\tilde{A}$ and $\tilde{B}$ over $\tilde{T}$ is equal to
$A$, i.e.

$A=\tilde{A}\\#_{\tilde{T}}\tilde{B}.$

###### Proof.

Write

$A=\frac{R[\xi]}{\operatorname{Ann}_{R[\xi]}\left(F=\tilde{F}-(-H)\right)}=\frac{R[\xi]}{\operatorname{Ann}_{R}(F)\cdot
R[\xi]+(\xi)}.$

Setting $\sigma=\xi$ in Proposition 5.2, we will show that conditions (1.) and
(2.) hold. Condition (1.) holds because

$\xi\circ\tilde{F}=-\lambda\cdot\tilde{G}=\xi\circ(-H).$

For Condition (2.) note that it follows from Theorem 4.6 and Construction 3.1
that

$\tilde{A}\cong\frac{A[\xi]}{\left(\xi\cdot
K,\overline{f_{R}(\xi)}\right)}=\frac{R[\xi]}{\operatorname{Ann}_{R}(F)\cdot
R[\xi]+\xi\cdot\operatorname{Ann}_{R}(G)\cdot R[\xi]+(f_{R}(\xi))}$

and hence that

$\operatorname{Ann}_{R[\xi]}(\tilde{F})=\operatorname{Ann}_{R}(F)\cdot
R[\xi]+\xi\cdot\operatorname{Ann}_{R}(G)\cdot R[\xi]+(f_{R}(\xi)).$

Also from Lemma 4.3 we have

$\operatorname{Ann}_{R[\xi]}(H)=\operatorname{Ann}_{R}(G)\cdot
R[\xi]+(\xi\cdot f_{R}(\xi)),$

and since
$\operatorname{Ann}_{R}(F)\subseteq\operatorname{Ann}_{R}(G=\tau\circ F)$,
their sum satisfies

$\operatorname{Ann}_{R[\xi]}(\tilde{F})+\operatorname{Ann}_{R[\xi]}(H)=\operatorname{Ann}_{R}(G)\cdot
R[\xi]+(f_{R}(\xi))=\operatorname{Ann}_{R[\xi]}(\tilde{G})$

which is condition (2). Hence the result follows from Proposition 5.2. ∎

One interesting consequence of Theorem 5.6 is that every AG algebra has a
nontrivial connected sum decomposition over some algebra $\tilde{T}$. This
stands in direct contrast to connected sums over the ground field
$T={\mathbb{F}}$, where $\\#_{\mathbb{F}}$-indecomposable AG algebras exist;
see [3, Theorem 8.3] and also [42, Proposition 3.1]. On the other hand,
Theorem 5.6 shows that over general $T$, there may be no
$\\#_{T}$-indecomposable AG algebras.

######  Example 5.7.

Set $F=Z^{2}XY-X^{2}Y^{2}$ so that the corresponding AG algebra is

$A=\frac{{\mathbb{F}}[x,y,z]}{\operatorname{Ann}(Z^{2}XY-X^{2}Y^{2})}=\frac{{\mathbb{F}}[x,y,z]}{(x^{3},y^{3},x^{2}z,y^{2}z,x(z^{2}+xy),y(z^{2}+xy),z^{3})}$

with Hilbert function $H(A)=(1,3,6,3,1)$. Set $G=1$ so that $T={\mathbb{F}}$
and the Thom class for the projection $\pi\colon A\rightarrow{\mathbb{F}}$ is
just the socle generator $\tau=z^{2}xy$. Then we can set
$f_{R}(\xi)=\xi^{4}-z^{2}xy$ which has $G$-dual $h_{R}(\xi)=1$. Therefore we
have $\tilde{G}=\Xi^{3}$, $\tilde{F}=(Z^{2}XY-X^{2}Y^{2})-\Xi^{4}$, and
$H=\Xi^{4}$. Then

$\tilde{T}=\frac{{\mathbb{F}}[x,y,z,\xi]}{\operatorname{Ann}(\Xi^{3})}=\frac{{\mathbb{F}}[x,y,z]}{(x,y,z,\xi^{4})}$

and

$\tilde{A}=\frac{{\mathbb{F}}[x,y,z,\xi]}{\operatorname{Ann}(Z^{2}XY-X^{2}Y^{2}-\Xi^{4})}=\frac{{\mathbb{F}}[x,y,z,\xi]}{(x^{3},y^{3},x^{2}z,y^{2}z,x(z^{2}+xy),y(z^{2}+xy),z^{3},\xi
x,\xi y,\xi z,\xi^{4}-z^{2}xy)}$

and

$\tilde{B}=\frac{{\mathbb{F}}[x,y,z,\xi]}{\operatorname{Ann}(-\Xi^{4})}=\frac{{\mathbb{F}}[x,y,z,\xi]}{(x,y,z,\xi^{5})}.$

Since
$\xi\circ\left(Z^{2}XY-X^{2}Y^{2}-\Xi^{4}\right)=-\Xi^{3}=\xi\circ\left(-\Xi^{4}\right)$,
and also
$\operatorname{Ann}(Z^{2}XY-X^{2}Y^{2}-\Xi^{4})+\operatorname{Ann}(\Xi^{4})=\operatorname{Ann}(\Xi^{3})$,
it follows that $A$ is a connected sum of $\tilde{A}$ and $\tilde{B}$ along
$\tilde{T}$, as guaranteed by Theorem 5.6.

We conclude this section with an example that shows that unlike the blow up
operation, the blow down operation may not always preserve the standard
grading.

######  Example 5.8.

Define the AG algebra

$A=\frac{{\mathbb{F}}[x,y,u]}{(x^{2},u^{2},xy,xu-
yu,xu-y^{3})}=\frac{{\mathbb{F}}[x,y,u]}{\operatorname{Ann}\left(XU+YU+Y^{3}\right)}$

with the non-standard grading $\deg(x)=\deg(y)=1$ and $\deg(u)=2$. We can blow
up along the projection

$\pi\colon A\rightarrow
T=\frac{{\mathbb{F}}[x]}{\operatorname{Ann}(X)}=\frac{F[x]}{(x^{2})}$

with Thom class is $\tau=u-y^{2}$ and kernel $K=(y,u)$. Then setting
$f_{R}(\xi)=\xi^{2}-(u-y^{2})$ (which has $G=X$-dual polynomial
$h_{R}(\xi)=\xi^{k}$), the cohomological blow up of $A$ along $\pi$ is

$\tilde{A}=\frac{{\mathbb{F}}[x,y,u,\xi]}{(x^{2},u^{2},xy,xu-yu,xu-y^{3},\xi
y,\xi
u,\xi^{2}-(u-y^{2}))}=\frac{{\mathbb{F}}[x,y,u,\xi]}{\operatorname{Ann}(\Xi^{2}X-XU-
YU-Y^{3})}.$

Since $u\equiv\xi^{2}+y^{2}$ in $\tilde{A}$, we can eliminate $u$, and get

$\tilde{A}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{2},(\xi^{2}+y^{2})^{2},xy,(x-y)(\xi^{2}+y^{2}),x(\xi^{2}+y^{2})-y^{3},\xi
y,\xi(\xi^{2}+y^{2}))}\cong\frac{{\mathbb{F}}[x,y,\xi]}{(x^{2},\xi^{3},xy,x\xi^{2}-y^{3},y\xi)},$

which has a standard grading.

Incidentally, $\tilde{A}$ is also the cohomological blow-up algebra of a
_standard graded_ AG algebra $A^{\prime}$ along a different $T^{\prime}$,
namely:

$A^{\prime}=\frac{{\mathbb{F}}[x,\xi]}{(x^{2},\xi^{3})}\rightarrow
T^{\prime}={\mathbb{F}}.$

Here the kernel is $K^{\prime}=(x,\xi)$ and the Thom class is the socle
generator $\tau=x\xi^{2}$, and taking $y$ as the “blow up variable” with
$f_{R}(y)=y^{3}-x\xi^{2}$, we find that the blow up

$\tilde{A^{\prime}}=\frac{A[y]}{(yK^{\prime},\overline{f(y)})}=\frac{{\mathbb{F}}[x,\xi,y]}{(x^{2},\xi^{3},yx,y\xi,y^{3}-x\xi^{2})}\cong\tilde{A}.$

##  6 Minimal Generating Sets and Complete Intersections

In this paper, a complete intersection (CI) is a quotient of a polynomial ring
by an ideal generated by a regular sequence of maximal length. At a cursory
glance, the presentation in Construction 3.1 may lead one to believe that
cohomological blow ups cannot be complete intersections, except in embedding
dimension two where all AG algebras are CI, but the following example shows
otherwise.

######  Example 6.1.

Define the AG algebras and the surjective map between them

$A=\frac{{\mathbb{F}}[x,y]}{(x^{3},y^{3})}\overset{\pi}{\longrightarrow}T=\frac{{\mathbb{F}}[x,y]}{(x^{3},y)},\
\pi(x)=x,\ \pi(y)=0.$

Then the kernel of $\pi$ is $K=(y)$ and the Thom class is $\tau=y^{2}$. Here
$d=4$, $k=2$ and $n=2$. Letting $\xi$ be the blow up variable and
$f_{A}(\xi)=\xi^{2}-y^{2}$, define the associated cohomological blow-up
algebra as in Construction 3.1 as

$\tilde{A}=\frac{A[\xi]}{(\xi\cdot
K,f_{A}(\xi))}=\frac{{\mathbb{F}}[x,y,\xi]}{(x^{3},y^{3},\xi
y,\xi^{2}-y^{2})}$

In this presentation the generator $y^{3}$ is redundant, and we see that
$\tilde{A}$ is indeed a complete intersection of Hilbert function
$H(\tilde{A})=(1,3,4,3,1)=H(A)+H(T)[1]$.

In this section we will show that Example 6.1 is prototypical of the class of
BUG which are CI. First we introduce yet another description of the
cohomological blow-up in terms of its defining ideal.

As usual, let $R={\mathbb{F}}[x_{1},\ldots,x_{r}]$ be a graded polynomial ring
with homogeneous maximal ideal $\mathfrak{m}=(x_{1},\ldots,x_{r})$. Recall
from Section 2 a homogeneous ideal $I\subseteq R$ is $\mathfrak{m}$-primary
and irreducible if and only if the quotient $R/I$ is a graded AG algebra. We
abuse notation slightly and call the socle degree of such an ideal the socle
degree of the corresponding quotient.

######  Construction 6.2.

Fix an $\mathfrak{m}$-primary irreducible homogeneous ideal $I\subset R$ of
socle degree $d$, and fix a homogeneous polynomial $\tau\in R$ of degree $n$
where $2\leq n<d-1$ and such that $I\subsetneq(I\colon\tau)\subsetneq R$. Then
the ideal $(I\colon\tau)$ is also homogeneous, $\mathfrak{m}$-primary, and
irreducible of socle degree $k=d-n$; see Lemma 2.3.

Let $\xi$ be an indeterminate, and fix a homogeneous monic polynomial
$f_{R}(\xi)=\xi^{n}+r_{1}\xi^{n-1}+\cdots+r_{n}$ where $r_{i}\in R_{i}$.

Define the ideal $\hat{I}\subset R[\xi]=\hat{R}$ by

$\hat{I}=I\cdot R[\xi]+\xi\cdot(I\colon\tau)\cdot R[\xi]+f_{R}(\xi)\cdot
R[\xi].$ (39)

We now describe some properties of the ideal (39) of Construction 6.2. Since
$I$ and $(I\colon\tau)$ are both $\mathfrak{m}$-primary and irreducible it
follows that $A=R/I$ and $T=R/(I\colon\tau)$ are AG algebras, and since
$I\subseteq(I\colon\tau)$, the identity map on $R$ passes to a surjective map
of quotient algebras $\pi\colon A\rightarrow T$. Moreover it follows from
Lemma 2.2 that one can choose orientations on $A$ and $T$ such that
$\overline{\tau}\in A_{n}$ is the Thom class of $\pi$. Note that the kernel of
$\pi$ is $K=(I\colon\tau)/I\subset R/I=A$. It follows that if
$f_{A}(\xi)=\overline{f_{R}(\xi)}\in A[\xi]=R[\xi]/I$, then $\hat{A}$ from
Construction 3.1 satisfies

$\hat{A}=\frac{A[\xi]}{(\xi\cdot
K,\overline{f_{R}(\xi)})}=\frac{R[\xi]}{I\cdot
R[\xi]+\xi\cdot(I\colon\tau)\cdot R[\xi]+(f_{R}(\xi))\cdot
R[\xi]}=\frac{\hat{R}}{\hat{I}}.$ (40)

It follows therefore from Lemma 3.6 that the ideal $\hat{I}$ from Construction
6.2 is $\hat{\mathfrak{m}}=(x_{1},\ldots,x_{r},\xi)$-primary irreducible if
and only if the constant coefficient $r_{n}\in R_{n}$ of $f_{R}(\xi)$
satisfies $r_{n}-\lambda\cdot\tau\in I$ for some
$\lambda\in{\mathbb{F}}^{\times}$. In this case we shall replace the “hat”
with “tilde” and call $\tilde{I}=\hat{I}$ the _cohomological blow up ideal of
$I$ and $\tau$_, as it is the defining ideal of the cohomological blow-up
algebra $\tilde{A}$ of $A$ along $\pi$ with parameters
$(\overline{r_{1}},\ldots,\overline{r_{n-1}},\lambda)$.

It is also clear from Construction 6.2, and Lemma 3.13, that the colon ideal
$(\hat{I}\colon\xi)\subset\hat{R}$ satisfies

$(\hat{I}\colon\xi)=(I\colon\tau)+(f_{T}(\xi)=\overline{f_{R}(\xi)})$ (41)

and we have

$\tilde{T}=\frac{T[\xi]}{\left(f_{T}(\xi)\right)}=\frac{R[\xi]}{(I\colon\tau)\cdot
R[\xi]+f_{R}(\xi)\cdot R[\xi]}=\frac{\hat{R}}{\left(\hat{I}\colon\xi\right)},$
(42)

which is the the algebra from Construction 3.1 Equation (19).

###  6.1 Minimal Generating Sets

Next we would like to know how the minimal generators of the cohomological
blow up ideal $\tilde{I}$ compare to those of $I$ and of $(I\colon\tau)$. We
also determine the relations among these minimal generators. We start by
providing a lemma that helps explain some of these relations.

######  Lemma 6.3.

Let $B(\xi)=b_{p}\xi^{p}+\cdots+b_{0}$ and $C(\xi)=c_{q}\xi^{q}++\cdots+c_{0}$
be any homogeneous polynomials in $R[\xi]$ with homogeneous coefficients
$b_{i},c_{i}\in R$. Let $J\subset R$ is any homogeneous ideal in $R$ and
assume that $C(\xi)$ is monic, i.e. $c_{q}=1$. If the product of $B$ and $C$
is in the ideal in $R[\xi]$ generated by $J$, i.e.

$B(\xi)\cdot C(\xi)\in J\cdot R[\xi],$

then every coefficient of $B(\xi)$ must lie in $J$, i.e. $b_{i}\in J\cdot R$
for all $0\leq i\leq p$.

###### Proof.

The key observation here is that if $D(\xi)=d_{r}\xi^{r}+\cdots+d_{0}$ is any
polynomial in the ideal $J\cdot R[\xi]$, then its coefficients $d_{i}\in
J\cdot R$. Indeed, if $D(\xi)\in J\cdot R[\xi]$, plug in $\xi=0$ to see that
$D(0)=d_{0}\in J\cdot R$. Then
$D(\xi)-d_{0}=D_{1}(\xi)=\xi\left(d_{r}\xi^{r-1}+\cdots+d_{1}\right)\in J\cdot
R[\xi]$ and hence $d_{r}\xi^{r-1}+\cdots+d_{1}\in\left(J\cdot
R[\xi]:\xi\right)=J\cdot R[\xi]$, since $\xi$ is a non-zero divisor of $J\cdot
R[\xi]$. Then plug in $\xi=0$ to see that $d_{1}\in J\cdot R$, and so on.

Therefore if the product

$B(\xi)\cdot
C(\xi)=\sum_{i=0}^{p+q}\left(b_{0}c_{i}+b_{1}c_{i-1}+\cdots+b_{i}c_{0}\right)\xi^{i}$

is in the ideal $J\cdot R[\xi]$, then each of its coefficients must be in
$J\cdot R$, i.e.

$b_{0}c_{i}+\cdots+b_{i}c_{0}\in J\cdot R\ \ \text{for each}\ 0\leq i\leq
p+q.$

Taking first $i=p+q$, we find that $b_{p}\cdot c_{q}\in J\cdot R$ and since
$c_{q}=1$ it follows that $b_{p}\in J\cdot R$. Inductively assume that
$b_{p-j+1},\ldots,b_{p}\in J\cdot R$. Then taking $i=p+q-j$ we find that

$b_{p}c_{q-j}+b_{p-1}c_{q-j+1}+\cdots+b_{p-j+1}c_{q-1}+b_{p-j}c_{q}\in J\cdot
R$

from which it follows that $b_{p-j}c_{q}=b_{p-j}\in J\cdot R$ as well.
Therefore by induction, all coefficients $b_{j}$ are in $J\cdot R$. ∎

We can now provide a short exact sequence that determines the relations among
the obvious (not necessarily minimal) set of generators of the cohomological
blow up ideal $\tilde{I}$.

######  Proposition 6.4.

Let $\tilde{I}=I\cdot R[\xi]+\xi\cdot(I:\tau)\cdot R[\xi]+(f_{R}(\xi))$ be the
ideal described in Construction 6.2, let $I^{\prime}=I\cdot
R[\xi],K^{\prime}=(I:\tau)\cdot R[\xi]$ and let
$g=(f_{R}(\xi)-\lambda\tau)/\xi$. There is a short exact sequence of graded
$R[\xi]$-modules

$0\to I^{\prime}(-1)\oplus
K^{\prime}(-n)\xrightarrow{\begin{bmatrix}\xi&\lambda\tau\\\ -1&g\\\
0&-1\end{bmatrix}}I^{\prime}\oplus K^{\prime}(-1)\oplus
R[\xi]\xrightarrow{\begin{bmatrix}1&\xi&f_{R}\end{bmatrix}}\hat{I}\to 0.$ (43)

###### Proof.

The definition of $\tilde{I}$ yields surjectivity of the rightmost nonzero
map, and the injectivity of the leftmost nonzero map is clear from its
description (note there is a unit entry in each column). That (43) is a
complex is seen by matrix multiplication and utilization of
$f=g\xi+\lambda\tau$.

It remains to prove exactness in the middle of (43). For this, consider a
triple $(\alpha(\xi),\beta(\xi),\gamma(\xi))\in I^{\prime}\oplus
K^{\prime}(-1)\oplus R[\xi]$ so that

$\alpha(\xi)+\beta(\xi)\xi+\gamma(\xi)f_{R}(\xi)=0.$ (44)

Since $I^{\prime}\subseteq K^{\prime}$ we have $\alpha(\xi)+\beta(\xi)\xi\in
K^{\prime}$, so that also $\gamma(\xi)f_{R}(\xi)\in K^{\prime}$ and by Lemma
6.3 it follows that $\gamma(\xi)\in K^{\prime}$ since $f_{R}(\xi)$ is monic.
Adding the relation

$\lambda\tau\gamma(\xi)+g(\xi)\gamma(\xi)-f_{R}(\xi)\gamma(\xi)=0,$

to (44) yields
$(\alpha(\xi)-\lambda\tau\gamma(\xi))+(\beta(\xi)-g(\xi)\gamma(\xi))\xi=0$.
Since $\gamma(\xi)\in K^{\prime}=(I:\tau)\cdot R[\xi]$, we have
$\lambda\tau\gamma(\xi)\in I\cdot R[\xi]=I^{\prime}$ and thus
$\beta^{\prime}(\xi):=(\beta(\xi)-g(\xi)\gamma(\xi)\in I^{\prime}$. We have
thus obtained the identity

$\begin{bmatrix}\alpha(\xi)\\\ \beta(\xi)\\\
\gamma(\xi)\end{bmatrix}=-\beta^{\prime}(\xi)\begin{bmatrix}\xi\\\ -1\\\
0\end{bmatrix}-\gamma(\xi)\begin{bmatrix}\lambda\tau\\\ g\\\ -1\end{bmatrix},$

where $\beta^{\prime}(\xi)\in I^{\prime}$ and $\gamma(\xi)\in K^{\prime}$,
establishing the desired exactness. ∎

Based on the presentation in Proposition 6.4, one can infer a minimal
generating set for $\tilde{I}$. Setting $\hat{{\mathfrak{m}}}$ to be the
homogeneous maximal ideal of $R[\xi]$, tensoring the short exact sequence (43)
with $R[\xi]/\hat{{\mathfrak{m}}}$, and observing that there are isomorphisms
$I^{\prime}/\hat{{\mathfrak{m}}}I^{\prime}\cong I/{\mathfrak{m}}I$ and
$K^{\prime}/\hat{{\mathfrak{m}}}K^{\prime}\cong(I:\tau)/{\mathfrak{m}}(I:\tau)$
we obtain a new exact sequence of ${\mathbb{F}}$-vector spaces

$\frac{I}{{\mathfrak{m}}I}(-1)\oplus\frac{(I:\tau)}{{\mathfrak{m}}(I:\tau)}(-n)\xrightarrow[\psi_{1}]{\begin{bmatrix}0&\lambda\phi_{2}\\\
-\phi_{1}&0\\\
0&0\end{bmatrix}}\frac{I}{{\mathfrak{m}}I}\oplus\frac{(I:\tau)}{{\mathfrak{m}}(I:\tau)}(-1)\oplus{\mathbb{F}}\xrightarrow[\psi_{2}]{\begin{bmatrix}1&\xi&f_{R}\end{bmatrix}}\frac{\tilde{I}}{\hat{{\mathfrak{m}}}\tilde{I}}\to
0.$ (45)

The zero entries in the first matrix are due to the containments $\tau
I^{\prime}\subseteq\hat{{\mathfrak{m}}}I^{\prime}$,
$gI^{\prime}\subseteq\hat{{\mathfrak{m}}}K^{\prime}$, and $\xi
I^{\prime}\subseteq\hat{{\mathfrak{m}}}I^{\prime}$. We discuss the remaining
maps $\phi_{1}$ and $\phi_{2}$ – the former is induced by inclusion and the
other by multiplication by $\tau$. These maps fit into the sequence

$\textstyle{(I\colon\tau)/\mathfrak{m}(I\colon\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\times\tau}$$\scriptstyle{\phi_{2}}$$\textstyle{I/\mathfrak{m}I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\textstyle{(I\colon\tau)/\mathfrak{m}(I\colon\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{2}}$$\scriptstyle{\times\tau}$$\textstyle{I/\mathfrak{m}I.}$
(46)

Note this sequence of maps (46) forms a complex, i.e.
$\phi_{1}\circ\phi_{2}=0$ and $\phi_{2}\circ\phi_{1}=0$, hence there are
homology groups

$H=\frac{\ker(\phi_{1})}{\operatorname{im}(\phi_{2})}\ \ \text{and}\ \
H^{\prime}=\frac{\ker(\phi_{2})}{\operatorname{im}(\phi_{1})}.$ (47)

We will see that these homology groups $H$ and $H^{\prime}$ measure the
difference between minimal generating sets of $\tilde{I}$ and those of $I$ or
$(I\colon\tau)$. In fact, we will see that these homology groups are
obstructions to $\tilde{I}$ being generated by a regular sequence.

Using (45), we can express a minimal generating set of $\tilde{I}$ in terms of
the minimal generating sets of $I$ and $(I:\tau)$ and the homology groups $H$
and $H^{\prime}$.

######  Theorem 6.5.

Let $\phi_{1},\phi_{2}$ be defined as in (47) and consider the vectors space
decompositions

$\displaystyle\frac{I}{\mathfrak{m}I}=$ $\displaystyle
U\oplus\underbrace{\operatorname{im}(\phi_{2})\oplus H}_{\ker(\phi_{1})}$
$\displaystyle\frac{(I\colon\tau)}{\mathfrak{m}(I\colon\tau)}=$ $\displaystyle
W\oplus\underbrace{\operatorname{im}(\phi_{1})\oplus
H^{\prime}}_{\ker(\phi_{2})}$ (48)

where $U$ and $W$ are some (non canonical) complements for $\ker(\phi_{1})$
and $\ker(\phi_{2})$ respectively. Then the vector space spanned by the
minimal generators for $\hat{I}$ decomposes as

$\frac{\tilde{I}}{\hat{\mathfrak{m}}\tilde{I}}\cong U\oplus H\oplus\xi
W\oplus\xi H^{\prime}\oplus\langle f_{R}(\xi)\rangle.$ (49)

###### Proof.

This follows almost immediately from exactness of Sequence (45): Since
$\ker(\psi_{2})=\operatorname{im}(\psi_{1})=\operatorname{im}(\phi_{2})\oplus\operatorname{im}(\phi_{1})\subset
I/\mathfrak{m}I\oplus(I:\tau)/\mathfrak{m}(I:\tau)\oplus{\mathbb{F}}$ is a
direct summand, and since $\psi_{2}$ is surjective, it passes to an
isomorphism

$\frac{\hat{I}}{\hat{\mathfrak{m}}\hat{I}}\cong\frac{\frac{I}{{\mathfrak{m}}I}\oplus\frac{(I:\tau)}{{\mathfrak{m}}(I:\tau)}(-1)\oplus{\mathbb{F}}}{\ker(\psi_{2})}\cong
U\oplus\oplus H\oplus W\oplus H^{\prime}\oplus{\mathbb{F}}$

and the result follows. ∎

For a homogeneous ideal $J\subset R$ we denote by
$\mu(J)=\dim_{\mathbb{F}}\left(J/\mathfrak{m}J\right)$ the number of minimal
generators of $J$. The following corollary is an immediate consequence of
Theorem 6.5, and we omit the proof.

######  Corollary 6.6.

With $I$, $\tau$, $(I\colon\tau)$ and $\hat{I}$ as above we have

$\mu(\tilde{I})=\mu(I)+\dim(H^{\prime})+1=\mu(I\colon\tau)+\dim(H)+1.$

Note that in our set up of Construction 6.2, $\mu(\tilde{I})\geq
r+1=\dim(\hat{R})$ with equality if and only if $\tilde{I}$ is generated by a
regular sequence. In particular, Corollary 6.6 shows that the vanishing of
homology groups $H$ and $H^{\prime}$ from (47) is necessary for the
cohomological blow up ideal to be generated by a regular sequence, and hence
for the BUG quotient $\tilde{A}=\hat{R}/\tilde{I}$ to be a CI. However, the
following example shows that the vanishing of $H$ and $H^{\prime}$ is not
quite sufficient for that purpose.

######  Example 6.7.

Let $I=\left(x^{4},y^{4},zx^{2},zy^{2},z^{4}-x^{2}y^{2}\right)$, and
$\tau=z^{2}-xy$ so that
$(I\colon\tau)=\left(x^{4},y^{4},zx^{2},zy^{2},z^{2}+xy\right)$. Then taking
$\hat{R}=R[\xi]={\mathbb{F}}[x,y,z][\xi]$ and $f_{R}(\xi)=\xi^{2}-\tau$ the
cohomological blow up ideal is

$\tilde{I}=\left(x^{4},y^{4},zx^{2},zy^{2},\xi(z^{2}+xy),\xi^{4}-(z^{2}-xy)\right),$

Hence the cohomological blow up $\tilde{A}=\hat{R}/\tilde{I}$ is not a CI.
Note that in this case the homology groups $H$ and $H^{\prime}$ are both zero.

Note in Example 6.7 that the colon ideal
$\left(\tilde{I}\colon\xi\right)=\tilde{I}+(z^{2}+xy)$ is principal over
$\tilde{I}$. This is related to exact pairs of zero divisors, which allow a
complete characterization of BUGs that are CI.

###  6.2 Complete Intersection Blow-up Algebras and Exact Zero Divisors

Below we give a necessary and sufficient condition for a cohomological blow up
algebra to be a complete intersection based on exact zero divisors, a notion
introduced by I. Henriques and L. Şega [22], which is defined as follows:

######  Definition 6.8.

A pair of non-unit elements $a,b$ of a ring $A$ is an exact pair of zero
divisors if $(0:_{A}a)=b\cdot A$ and $(0:_{A}b)=a\cdot A$.

######  Example 6.9.

  1. 1.

In Example 6.1 $a=\overline{y^{2}}$ and $b=\overline{y}$ form an exact pair of
zero divisors on

$A=\frac{{\mathbb{F}}[x,y]}{(x^{3},y^{3})}.$

  2. 2.

In Example 6.7, $a=\overline{z^{2}-xy}$ and $b=\overline{z^{2}+xy}$ form an
exact pair of zero divisors on

$A=\frac{{\mathbb{F}}[x,y,z]}{\left(x^{4},y^{4},zx^{2},zy^{2},z^{4}-x^{2}y^{2}\right)}.$

If $A$ is Artinian then it suffices to check only one of the conditions in
Definition 6.8.

######  Lemma 6.10.

If $A$ is a graded Artinian ${\mathbb{F}}$-algebra then homogeneous elements
$a,b\in A$ of positive degree form an exact pair of zero divisors of $A$
provided that $(0:_{A}a)=b\cdot A$.

###### Proof.

From $(0:_{A}a)=b\cdot A$ we deduce there is an $A$-module isomorphism
$A/(b)=A/(0:_{A}a)\cong a\cdot A$ by means of the diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A/(0:_{A}a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\times
a}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{A/(a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a\cdot
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A/(a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Similarly, there is an isomorphism $A/(0:_{A}b)\cong b\cdot A$. Moreover, the
hypothesis yields $ab=0$ and thus $a\cdot A\subseteq(0:_{A}b)$. To see that
this containment is in fact an equality we compute

$\dim_{\mathbb{F}}(A)=\dim_{\mathbb{F}}(A/(b))+\dim_{\mathbb{F}}(b\cdot
A)=\dim_{\mathbb{F}}(a\cdot
A)+\dim_{\mathbb{F}}(A)-\dim_{\mathbb{F}}(0:_{A}b)$

whence $\dim_{\mathbb{F}}(a)=\dim_{\mathbb{F}}(0:_{A}b)$, and hence $a\cdot
A=(0:_{A}b)$ as desired. ∎

The next lemma relates exact zero divisors to the homology group $H$ in (47).

######  Lemma 6.11.

Let $A=R/I$ and $T=R/(I\colon\tau)$ be AG algebras, and suppose that
$\overline{\tau}\in A_{n}$ is part of a pair of exact zero divisors on $A$.
Then the homology groups $H$ and $H^{\prime}$ from (47) vanish.

###### Proof.

Assume that there exists a homogeneous element (of positive degree) $\sigma\in
R$ for which $(I\colon\tau)=I+(\sigma)$ and $(I\colon\sigma)=I+(\tau)$, so
that $\overline{\tau}$ and $\overline{\sigma}$ are an exact pair of zero
divisors in $A$.

Suppose that $s\in I$ is any minimal generator of $I$ for which
$s+\mathfrak{m}I\in\ker(\phi_{1})$. Then
$s\in\mathfrak{m}(I\colon\tau)=\mathfrak{m}(I+(\sigma))$, and hence there
exists elements $g_{1},\ldots,g_{t}\in I$ and
$a_{1},\ldots,a_{t},b\in\mathfrak{m}$ such that

$s=\sum_{i=1}^{t}a_{i}g_{i}+b\cdot\sigma.$

It follows that $b\in(I\colon\sigma)=I+(\tau)$ and hence $b=q+r\cdot\tau$ for
some $q\in I$ and $r\in R$. Therefore we see that

$s-r\cdot\tau\cdot\sigma\in\mathfrak{m}I.$

Then the equivalence class of $s$ in $H$ is

$\left[s\right]\equiv\left[r\tau\sigma\right]\equiv 0$

which shows that $H=0$, as claimed.

Moreover, by [33, Proposition 1.9] we have $\tau\cdot\sigma\in
I\setminus\mathfrak{m}I$, which yields that
$(I:\tau)/{\mathfrak{m}}(I:\tau)=\langle\sigma\rangle$. Consequently, the
second equation in (48) yields $W=\langle\sigma\rangle$,
$\operatorname{im}(\phi_{1})=0$, and $H^{\prime}=0$, also as claimed. ∎

The following is a characterization of complete intersection cohomological
blow-up algebras in terms of exact pairs of zero divisors.

######  Theorem 6.12.

Fix oriented AG algebras $A=R/I$ and $T=R/(I:\tau)$ of socle degrees $d>k$ and
let $\pi\colon A\rightarrow T$ be the natural surjective algebra map between
them with Thom class $\overline{\tau}\in A_{n}$ and kernel $K\subset A$. Let
$f_{R}(\xi)=\xi^{n}+r_{1}\xi^{n-1}+\cdots+\lambda\cdot\tau\in R[\xi]=\hat{R}$
for some homogeneous elements $r_{i}\in R$, and let
$\tilde{A}=\hat{R}/\tilde{I}$ be the associated cohomological blowup of $A$
along $\pi$ with parameters
$(\overline{r}_{1},\ldots,\overline{r}_{n-1},\lambda)$. Then the following are
equivalent.

  1. 1.

$\tilde{A}$ is a complete intersection.

  2. 2.

$A$ is a complete intersection and $\overline{\tau}\in A_{n}$ is part of an
exact pair of zero divisor on $A$.

  3. 3.

$T$ is a complete intersection and $\overline{\tau}\in A_{n}$ is part of an
exact pair of zero divisor on $A$.

###### Proof.

Assume that (1) holds. Then $\tilde{I}$ must be generated by a
$\hat{R}$-sequence, and hence it follows from Corollary 6.6 that both $I$ and
$(I\colon\tau)$ must also be generated by $R$-sequences and hence that $A$ and
$T$ must be complete intersections, and that the homology groups $H$ and
$H^{\prime}$ must vanish. Thus according to Equations (48) and (49) we have

$\displaystyle\frac{I}{\mathfrak{m}I}=U\oplus\phi_{2}(W),$ $\displaystyle\ \ \
\ \ \ \frac{(I\colon\tau)}{\mathfrak{m}(I\colon\tau)}=\phi_{1}(U)\oplus W,$
$\displaystyle\text{and}\ \ \frac{\tilde{I}}{\hat{\mathfrak{m}}\tilde{I}}=$
$\displaystyle U\oplus\xi\cdot W\oplus\langle f_{R}(\xi)\rangle.$

It follows that $\dim_{\mathbb{F}}(W)=1$, since no two elements of a minimal
generating set of $I$ (or $\tilde{I}$) can have a common divisor. Let
$\sigma\in(I\colon\tau)$ be a minimal generator for which
$\sigma+\mathfrak{m}(I\colon\tau)\in W$. Then we can write

$I=(u_{1},\ldots,u_{r-1},\tau\cdot\sigma)\ \text{and}\
(I\colon\tau)=(u_{1},\ldots,u_{r-1},\sigma)=I+(\sigma)$

which implies by Lemma 6.10 that $a=\overline{\tau}$ and $b=\overline{\sigma}$
form an exact pair of zero divisors for $A$. This shows that (1.) implies (2.)
and (3.).

Next assume that (2.) holds: $A$ is a complete intersection and
$\overline{\tau}\in A_{n}$ is an exact zero divisor for $A$. It follows from
Lemma 6.11 that $H=H^{\prime}=0$, and hence by Corollary 6.6 it follows that
$\tilde{I}$ is generated by a $\hat{R}$-sequence and hence that the
cohomological blow up $\tilde{A}$ is a complete intersection, which is (1.).

Finally, assume that (3.) holds: $T$ is a complete intersection and
$\overline{\tau}\in A_{n}$ is an exact zero divisor for $A$. Again, it follows
from Lemma 6.11 that $H=0$, and hence it follows from Corollary 6.6 that the
cohomological blow up $\tilde{A}$ is a complete intersection, and thus (1.)
holds. ∎

######  Remark 6.13.

It follows from the description of the minimal generators of $\tilde{I}$ in
equation (49) together with Theorem 6.12 and Lemma 6.11 that when $\tilde{A}$
is a complete intersection and thus $\overline{\sigma},\overline{\tau}$ is an
exact pair of zero divisors on $A$, a minimal generating set for $\tilde{I}$
can be described as

$\frac{\tilde{I}}{\hat{\mathfrak{m}}\tilde{I}}\cong
U\oplus\langle\xi\sigma\rangle\oplus\langle f_{R}(\xi)\rangle.$

In particular, since $\tilde{I}$ is generated by a regular sequence and
$\xi\sigma$ is a minimal generator for $\tilde{I}$ it follows from [33,
Proposition 1.9] that $\xi$ is an exact zero divisor of $\tilde{A}$. The
following example shows that this condition is not quite sufficient to
identify a complete intersection as a cohomological blow-up algebra.

######  Example 6.14.

Consider the complete intersection

$\tilde{A}=\frac{{\mathbb{F}}[x,y]}{(x^{4}+y^{4},x^{2}y^{2})}.$

Then $\xi=\overline{x}\in\tilde{A}_{1}$ is an exact zero divisor for
$\tilde{A}$, and its Hilbert function is $H(\tilde{A})=(1,2,3,4,3,2,1)$. This
particular CI algebra (or any isomorphic to it) cannot be a cohomological
blow-up algebra by a result of [29] concerning codimension two.

###  6.3 Application: Watanabe’s Bold Conjecture

The following “rather bold” conjecture was put forth by the fifth author [47]
after noticing that many complete intersections arising as invariant rings
could be realized as subrings of complete intersections cut out by quadrics
and of the same socle degree.

######  Conjecture 6.15 (Watanabe’s Bold Conjecture).

For any standard graded Artinian complete intersection $A$ of socle degree
$d$, there is another standard graded Artinian complete intersection $B$ of
the same socle degree $d$ cut out by quadrics and an injective algebra map
from $A$ into $B$, in symbols:

$\phi\colon
A=\frac{{\mathbb{F}}[x_{1},\ldots,x_{r}]}{(f_{1},\ldots,f_{r})}\hookrightarrow\frac{{\mathbb{F}}[X_{1},\ldots,X_{N}]}{(F_{1},\ldots,F_{N})}=B,\
\ \deg(F_{i})=2,\ \forall\ 1\leq i\leq N.$

In 2016, the third author proved [35] Watanabe’s Bold Conjecture in the
special case where $A$ is cut out by polynomials which factor into a product
of linear forms333See also [36, Theorem 1] for another case in which
Watanabe’s Bold Conjecture holds.. Here we give another, much shorter, proof
of this result (in fact something slightly stronger) using cohomological blow
ups. First some notation. Let $A=R/I$ is a complete intersection where
$R={\mathbb{F}}[x_{1},\ldots,x_{r}]$ has the standard grading and $I$ is
minimally generated by some regular sequence $(f_{1},\ldots,f_{r})$. If the
minimal generators $f_{1},\ldots,f_{n}$ can be chosen such that each $f_{i}$
is a product of linear and/or quadratic forms, i.e. $f_{i}=L_{1}\cdots L_{k}$
where $\deg(L_{j})=1\ \text{or}\ 2$, then we shall say $A$ is of class
$\mathcal{W}$. If $d_{i}=\deg(f_{i})>1$ for all $i$, then the degree sequence
is $(d_{1},\ldots,d_{n})$ and we define its _defect_ to be

$\operatorname{def}(A)=d_{1}+\cdots+d_{n}-2n=\operatorname{soc.deg.}(A)-n.$

Let $\mathcal{W}(m)$ denote the subclass consisting of complete intersections
of defect $m$, so that $\mathcal{W}=\bigsqcup_{m=0}^{\infty}\mathcal{W}(m)$.
Complete intersections of class $\mathcal{W}(0)$ are called _quadratic
complete intersections_.

######  Theorem 6.16.

For every complete intersection $A$ of class $\mathcal{W}$ there is a complete
intersection $B$ of class $\mathcal{W}(0)$ of the same socle degree as $A$ and
an injective map of algebras $\phi\colon A\hookrightarrow B$.

###### Proof.

We use induction on $m$ to show that every complete intersection
$A\in\mathcal{W}(m)$ embeds into a complete intersection $B\in\mathcal{W}(0)$
of the same socle degree. For the base case $m=0$ there is nothing to show.
For the inductive step, assume that $m>0$ and that the statement holds for
every $0\leq m^{\prime}<m$. Fix $A\in\mathcal{W}(m)$ with presentation as in
(28) with minimal generators $f_{1},\ldots,f_{n}$ that factor into products of
linear and/or quadratic forms. Since $\operatorname{def}(A)=m>0$, we may
assume without loss of generality that $\deg(f_{n})\geq 3$. Write
$f_{n}=\tau\cdot g$ where $\deg(\tau)=2$ which is possible since we are
assuming that $f_{n}$ factors as a product of linear and quadratic forms. Then
let $\xi$ be an indeterminate, and define the monic quadratic polynomial
$f_{R}(\xi)=\xi^{2}-\tau\in R[\xi]$. Then the cohomological blow up ideal of
$I$ with respect to $\tau$ and $f_{R}(\xi)$ is by (39)

$\tilde{I}=(f_{1},\ldots,f_{n-1},\xi g,f_{R}(\xi))\subset R[\xi]$

and the quotient $\tilde{A}=R[\xi]/\tilde{I}$ is the cohomological blow-up of
$A$ along the map

$\pi\colon
A=\frac{{\mathbb{F}}[x_{1},\ldots,x_{n}]}{(f_{1},\ldots,f_{n-1},\tau
g)}\rightarrow\frac{{\mathbb{F}}[x_{1},\ldots,x_{n}]}{(f_{1},\ldots,f_{n-1},g)}=T.$

Note that the degree sequence of $\tilde{A}$ is
$(d_{1},\ldots,d_{n-1},d_{n}-1,2)$ and its defect is
$d_{1}+\cdots+d_{n}-1+2-2(n+1)=(d_{1}+\cdots+d_{n})-2n-1=m-1$. Finally it is
clear that the minimal generators of $\tilde{I}$ are either quadratic (like
$f_{R}(\xi)$) or factor into a product of linear and/or quadratic forms (since
the minimal generators of $I$ do). Therefore $\tilde{A}\in\mathcal{W}(m-1)$.
For the injection, note that the blow down map will do the trick:

$\beta\colon
A=\frac{{\mathbb{F}}[x_{1},\ldots,x_{n}]}{(f_{1},\ldots,f_{n-1},\tau
g)}\hookrightarrow\frac{{\mathbb{F}}[x_{1},\ldots,x_{n}][\xi]}{(f_{1},\ldots,f_{n-1},\xi
g,f(\xi))}=\tilde{A}.$

To complete the proof note that by induction, there is an embedding of
$\tilde{A}\in\mathcal{W}(m-1)$ into some complete intersection
$B\in\mathcal{W}(0)$ of the same socle degree as $\tilde{A}$ (and hence also
$A$), say

$\iota\colon\tilde{A}\hookrightarrow B.$

Composing $\beta$ and $\iota$ then gives an embedding $\iota\circ\beta\colon
A\hookrightarrow B$, as desired. ∎

##  7 Restrictions on Hilbert Functions

In this section we show that standard graded cohomological blow-up algebras
cannot have arbitrary Hilbert functions. In fact, in the parameter space of AG
algebras of fixed embedding dimension $\geq 3$ and socle degree $4$ or $\geq
6$, cohomological blow-up algebras are quite rare. To justify this assertion
we recall the notion of compressed AG algebra [24]. We use the notation of
Section 4: $R={\mathbb{F}}[x_{1},\ldots,x_{r}]$ is a standard graded
polynomial ring, $Q={\mathbb{F}}[X_{1},\ldots,X_{r}]$ its dual divided power
algebra. For a $d$-form $H$, its associated AG algebra
$C=R/\operatorname{Ann}(H)$ has embedding dimension $r$ if
$\operatorname{Ann}(H)$ contains no linear forms.

######  Definition 7.1.

A standard graded AG algebra $C$ of embedding dimension $r$ and socle degree
$d$ is compressed provided

$\dim_{k}C_{i}=\begin{cases}\dim_{\mathbb{F}}R_{i}=\binom{r-1+i}{r-1}&\text{if
}i\leq\left\lfloor\frac{d}{2}\right\rfloor\\\
\dim_{\mathbb{F}}C_{d-i}&\text{if
}i>\left\lfloor\frac{d}{2}\right\rfloor\end{cases}.$

We can parametrize graded Artinian Gorenstein algebras of fixed socle degree
$d$ and codimension $r$ by their Macaulay dual generators in $Q_{d}$,
essentially by elements of the projective space ${\mathbb{P}}(Q_{d})$. The
compressed algebras form a dense Zariski open set in this parameter space [25,
Proposition 3.12].444This depends on $R_{j}/(I^{2})_{j}$ being the tangent
space to $\ Gor(T)$ at $A=R/I$ and gives perhaps the shortest proof. Original
references are [13, Theorem 3.31] and [16, Theorem 5.1]. We now show that for
sufficiently large parameters the cohomological blow-up algebras do not belong
to this set.

######  Theorem 7.2.

Compressed algebras of embedding dimension $r\geq 3$ whose socle degree $d$
satisfies $d=4$ or $d\geq 6$ are not cohomological blow up algebras.

###### Proof.

Let $C=R/\operatorname{Ann}(H)$ be a compressed algebra of socle degree $d$
and embedding dimension $r$ and assume that $C$ is the blow up along a
surjective morphism $\pi:A\to T$ of some AG algebras of socle degrees $d$ and
$k$ respectively. Then since $C$ is compressed
$\tilde{I}=\operatorname{Ann}(H)$ is a homogeneous ideal with initial degree

$\min\left\\{i\mid\tilde{I}_{i}\neq
0\right\\}=\left\lfloor\frac{d}{2}\right\rfloor+1.$

By [7, Proposition 3.2] the minimal generators of $\tilde{I}$ have degrees
$\left\lfloor\frac{d}{2}\right\rfloor+1$ and possibly
$\left\lfloor\frac{d}{2}\right\rfloor+2$. Recall that the polynomial
$f_{R}(\xi)$ of degree $n=d-k$ is a minimal generator of $\tilde{I}$ by
Theorem 6.5 and thus