# Schur powers of the cokernel of a graded morphism

Jan O. kleppe, Rosa M. Miró-Roig∗ OsloMet-Oslo Metropolitan University,
Faculty of Technology, Art and Design, PB. 4, St. Olavs Plass, N-0130 Oslo,
Norway<EMAIL_ADDRESS>Universitat de Barcelona, Departament de
Matemàtiques i Informàtica, Gran Via de les Corts Catalanes 585, 08007
Barcelona, SPAIN<EMAIL_ADDRESS>
###### Abstract.

Let $\varphi:F\longrightarrow G$ be a graded morphism between free $R$-modules
of rank $t$ and $t+c-1$, respectively, and let $I_{j}(\varphi)$ be the ideal
generated by the $j\times j$ minors of a matrix representing $\varphi$. In
this short note: (1) We show that the canonical module of $R/I_{j}(\varphi)$
is up to twist equal to a suitable Schur power $\Sigma^{I}M$ of
$M=\operatorname{coker}(\varphi^{*})$; thus equal to $\wedge^{t+1-j}M$ if
$c=2$ in which case we find a minimal free $R$-resolution of $\wedge^{t+1-j}M$
for any $j$, (2) For $c=3$, we construct a free $R$-resolution of
$\wedge^{2}M$ which starts almost minimally (i.e. the first three terms are
minimal up to a precise summand), and (3) For $c\geq 4$, we construct under a
certain depth condition the first three terms of a free $R$-resolution of
$\wedge^{2}M$ which are minimal up to a precise summand. As a byproduct we
answer the first open case of a question posed by Buchsbaum and Eisenbud in
[2]; pg. 299.

∗ Partially supported by MTM2016-81735-P and PID2019-104844GB-I00

##  1\. Introduction

The study of Schur functors and determinantal ideals has a long history that
goes back to the seminal work of Lascoux [13] who used Schur functors to
construct (in characteristic 0) a free graded resolution of determinantal
ideals. More precisely, let $\varphi:F\longrightarrow G$ be a graded
$R$-morphism between free $R$-modules of rank $t$ and $t+c-1$, respectively.
Set $M=\operatorname{coker}(\varphi^{*})$. It is an interesting open problem
in commutative algebra to determine a graded free minimal $R$-resolution of
the cokernel of the natural map,

$\varphi_{r,s}^{*}:\bigwedge^{r}G^{*}\otimes\bigwedge^{s}F^{*}\longrightarrow\bigwedge^{r+s}F^{*},\quad
f\otimes g\mapsto\bigwedge^{r}\varphi^{*}(f)\wedge g,$

and in particular to find a graded free minimal $R$-resolution of the
$p$-exterior power,
$\bigwedge^{p}M:=\operatorname{coker}(\varphi^{*}_{1,p-1})$, of $M$, as well
as to describe a graded free minimal $R$-resolution of any Schur power
$\Sigma^{I}M$. So far only few particular cases have been described, and the
most relevant contribution to this problem is due to Lascoux who using Schur
complexes constructed a minimal free resolution of
$\operatorname{coker}(\varphi^{*}_{r,t-r})=R/I_{r}(\varphi)$. In [2],
Buchsbaum and Eisenbud solves the case
$\operatorname{coker}(\varphi^{*}_{p,0})$ and in pg. 299 they predict the
first terms of a graded free minimal $R$-resolution of
$\operatorname{coker}(\varphi^{*}_{p-1,1})$. No contributions have been
achieved until now and we devote this paper to study the first open cases and
solve the case $p=c=2$ as well as the case $p=2$ and $c=3$ up to an explicitly
described summand which seems to split off. Also the case $p=2$, $c\geq 4$ is
solved under a certain depth condition, up to a concrete summand. More
precisely, this note has two goals: (1) To show that the canonical module of
any determinantal ring $R/I_{i}(\varphi)$ is up to twist equal to a suitable
Schur power of the cokernel $M$ of $\varphi^{*}$, thus equal to
$\wedge^{t+1-i}M$ if $c=2$; and (2) To construct a free resolution of
$\wedge^{2}M$ for the cases $c=2,3$ and, hence, solve Buchsbaum-Eisenbud
question for $p=c=2$, and for $p=2$ and $c=3$ up to a concrete summand.
Moreover, for $c=2$, we find a minimal $R$-free resolution of $\wedge^{p}M$
for any $p$.

As a main tool we use the resolution of determinantal ideals $I_{i}(\varphi)$
constructed by Lascoux [13] using Schur functors. For sake of completeness we
quickly review the construction. As a byproduct we obtain the minimal
resolution of the normal module of a standard determinantal ideal of
codimension 3. We end the paper with a couple of conjectures based on our
results and some explicit computations using Macaulay2 (Conjectures 5.1 and
5.3).

Acknowledgment Part of this work was done while the second author was a guest
of the Oslo and Akershus University College of Applied Sciences (which now is
the Oslo Metropolitan University, Faculty of Technology, Art and Design) and
she thanks the Oslo Metropolitan University for its hospitality.

Notation. Throughout this paper $k$ will be an algebraically closed field of
characteristic zero, $R=k[x_{0},x_{1},\cdots,x_{n}]$, and
$\mathfrak{m}=(x_{0},\ldots,x_{n})$. For any graded Cohen-Macaulay quotient
$A$ of $R$ of codimension $c$, we let $I_{A}=\ker(R\twoheadrightarrow A)$ and
$K_{A}=\operatorname{Ext}^{c}_{R}(A,R)(-n-1)$ be its canonical module. If $M$
is a finitely generated graded $A$-module, let $\operatorname{depth}_{J}{M}$
denote the length of a maximal $M$-sequence in a homogeneous ideal $J$ and let
$\operatorname{depth}{M}=\operatorname{depth}_{\mathfrak{m}}{M}$.

##  2\. Background and preparatory results

For convenience of the reader we summarize the basic results on determinantal
ideals and the associated complexes needed in the sequel and we refer to [4],
[5], [13] and [14] for more details.

######  Definition 2.1.

If $\mathcal{A}$ is a homogeneous $q\times r$ matrix, we denote by
$I(\mathcal{A})$ the ideal of $R$ generated by the maximal minors of
$\mathcal{A}$ and by $I_{j}(\mathcal{A})$ the ideal generated by the $j\times
j$ minors of $\mathcal{A}$. Assume $q\geq r$. $I(\mathcal{A})$ (resp.
$R/I(\mathcal{A})$) is called a _standard determinantal_ ideal (resp. ring) if
$\operatorname{depth}_{I(\mathcal{A})}R=q-r+1$, and $I_{j}(\mathcal{A})$ and
$R/I_{j}(\mathcal{A})$ is said to be determinantal if
$\operatorname{depth}_{I_{j}(\mathcal{A})}R=(q-j+1)(r-j+1)$.

Denote by $\varphi:F\longrightarrow G$ the morphism of free graded $R$-modules
of rank $t$ and $t+c-1$, defined by the homogeneous matrix $\mathcal{A}$, and
by ${\mathcal{C}}_{i}(\varphi^{*})$ the (generalized) Koszul complex:

${\mathcal{C}}_{i}(\varphi^{*}):\ 0\rightarrow\bigwedge^{i}G^{*}\otimes
S_{0}(F^{*})\rightarrow\bigwedge^{i-1}G^{*}\otimes
S_{1}(F^{*})\rightarrow\ldots\rightarrow\bigwedge^{0}G^{*}\otimes
S_{i}(F^{*})\rightarrow 0$

where as usual $\bigwedge^{i}F^{*}$ (resp. $S_{i}(F^{*})$) denotes the $i$-th
exterior (symmetric) power of $F^{*}$.

Let ${\mathcal{C}}_{i}(\varphi^{*})^{*}$ be the $R$-dual of
${\mathcal{C}}_{i}(\varphi^{*})$. The map $\varphi$ induces graded morphisms

$\mu_{i}:\bigwedge^{t+i}G^{*}\otimes\bigwedge^{t}F\rightarrow\bigwedge^{i}G^{*},$

which are used to splice the complexes
${\mathcal{C}}_{c-i-1}(\varphi^{*})^{*}\otimes\bigwedge^{t+c-1}G^{*}\otimes\bigwedge^{t}F$
and ${\mathcal{C}}_{i}(\varphi^{*})$ to a complex
${\mathcal{D}}_{i}(\varphi^{*}):$

(2.1) $0\rightarrow\bigwedge^{t+c-1}G^{*}\otimes
S_{c-i-1}(F)\otimes\bigwedge^{t}F\rightarrow\bigwedge^{t+c-2}G^{*}\otimes
S_{c-i-2}(F)\otimes\bigwedge^{t}F\rightarrow\ldots\rightarrow$
$\bigwedge^{t+i}G^{*}\otimes
S_{0}(F)\otimes\bigwedge^{t}F\stackrel{{\scriptstyle\epsilon_{i}}}{{\longrightarrow}}\bigwedge^{i}G^{*}\otimes
S_{0}(F^{*})\rightarrow\bigwedge^{i-1}G^{*}\otimes
S_{1}(F^{*})\rightarrow\ldots\rightarrow\bigwedge^{0}G^{*}\otimes
S_{i}(F^{*})\rightarrow 0.$

The complex ${\mathcal{D}}_{0}(\varphi^{*})$ is called the Eagon-Northcott
complex and the complex ${\mathcal{D}}_{1}(\varphi^{*})$ is called the
Buchsbaum-Rim complex. Let us rename the complex
${\mathcal{C}}_{c}(\varphi^{*})$ as ${\mathcal{D}}_{c}(\varphi^{*})$. Denote
by $I_{j}(\varphi)$ the ideal generated by the $j\times j$ minors of a matrix
$\mathcal{A}$ representing $\varphi$. Call $\varphi$ minimal if
$\varphi(F)\subset\mathfrak{m}G$. Letting
$S_{-1}M:=\operatorname{Hom}(M,R/I(\mathcal{A}))$ we have the following well
known result:

######  Proposition 2.2.

Let $\varphi:F\rightarrow G$ be a graded minimal morphism of free $R$-modules
of rank $t$ and $t+c-1$, respectively. Set
$M=\operatorname{coker}(\varphi^{*})$. Assume
$\operatorname{depth}_{I_{t}(\varphi)}R=c$. Then it holds:

(i) ${\mathcal{D}}_{i}(\varphi^{*})$ is acyclic for $-1\leq i\leq c$.

(ii) ${\mathcal{D}}_{0}(\varphi^{*})$ is a minimal free graded $R$-resolution
of $R/I_{t}(\varphi)$ and ${\mathcal{D}}_{i}(\varphi^{*})$ is a minimal free
graded $R$-resolution of length $c$ of $S_{i}M$, $-1\leq i\leq c$ .

(iii) $K_{R/I_{t}(\varphi)}\cong S_{c-1}M(\mu)$ for some $\mu\in\mathbb{Z}$.
So, up to twist, $\mathcal{D}_{c-1}(\varphi^{*})$ is a minimal free graded
$R$-module resolution of $K_{R/I_{t}(\varphi)}$.

###### Proof.

See, for instance [4]; Theorem 2.20 and [5]; Theorem A2.10 and Corollaries
A2.12-13. ∎

Let us also recall the following useful comparison of cohomology groups. If
$Z\subset X:=\operatorname{Proj}(B)$, $B=R/I_{j}(\varphi)$ is a closed subset
such that $U=X\setminus Z$ is a local complete intersection, $L$ and $N$ are
finitely generated $B$-modules, $\widetilde{N}$ is locally free on $U$ and
$\operatorname{depth}_{I(Z)}L\geq r+1$, then the natural map

(2.2)
$\operatorname{Ext}^{i}_{B}(N,L)\longrightarrow\operatorname{H}_{*}^{i}(U,{\mathcal{H}}om_{\mathcal{O}_{X}}(\widetilde{N},\widetilde{L}))$

is an isomorphism, (resp. an injection) for $i<r$ (resp. $i=r$) cf. [8];
exposé VI. Note that we interpret $I(Z)$ as $\mathfrak{m}$ if $Z=\emptyset$.

##  3\. Schur powers of the cokernel of a graded morphism

Let $\varphi:F:=\oplus_{i=1}^{t}R(b_{i})\longrightarrow
G:=\oplus_{j=1}^{t+c-1}R(a_{j})$ be a graded $R$-morphism between free
$R$-modules of rank $t$ and $t+c-1$, respectively. Set
$M=\operatorname{coker}(\varphi^{*})$. It is a longstanding problem in
commutative algebra to determine a graded free minimal $R$-resolution of the
$p$-exterior power, $\bigwedge^{p}M$, of $M$ or, even more generally, a graded
free minimal $R$-resolution of the cokernel of the natural map

$\varphi_{r,s}^{*}:\bigwedge^{r}G^{*}\otimes\bigwedge^{s}F^{*}\longrightarrow\bigwedge^{r+s}F^{*},\quad
f\otimes g\mapsto\bigwedge^{r}\varphi^{*}(f)\wedge g.$

Note that $\bigwedge^{p}M=\operatorname{coker}(\varphi^{*}_{1,p-1})$ and, in
particular, $M=\operatorname{coker}(\varphi^{*}_{1,0})$. So far, only few
particular cases have been described. To our knowledge, an explicit finite
free $R$-resolution has been computed for
$\operatorname{coker}(\varphi^{*}_{1,0})$ under the assumption that
$\operatorname{depth}_{I_{t}(\varphi)}R$ has the largest possible value,
namely $\operatorname{depth}_{I_{t}(\varphi)}R=c$. It is known that this value
is achieved if the entries of $\varphi$ are general enough. A minimal graded
free $R$-resolution of
$\operatorname{coker}(\varphi^{*}_{r,t-r})=R/I_{r}(\varphi)$ is known under
the assumption that $\operatorname{depth}_{I_{r}(\varphi)}R=(t-r+1)(t-r+c)$ by
[13]. Again we know that this value is achieved if the entries of $\varphi$
are general enough. Finally, in [2]; Theorem 3.1 and Corollary 3.2, an
explicit finite free $R$-resolution of
$\operatorname{coker}(\varphi^{*}_{p,0})$ is given under the hypothesis that
$\operatorname{depth}_{I_{t}(\varphi)}R=c$, and in Proposition 6.1 of the same
paper a minimal free $R$-resolution of $\bigwedge^{2}M$ is given in the case
$c=1$ under the assumption $\operatorname{depth}_{I_{t-1}(\varphi)}R=4$.

In [2], pg. 299, Buchsbaum and Eisenbud predict the first terms of a graded
free minimal $R$-resolution of $\operatorname{coker}(\varphi^{*}_{p-1,1})$
and, in particular, of $\wedge^{2}M$. To state their guess we need to fix some
notation. Given a graded free $R$-module $F$, we define

(3.1) $L_{p}^{q}F=\ker(\delta_{p+1}^{q-1})\ \ \text{ where }\\\
\delta_{k}^{\ell}:S_{k-1}F\otimes\bigwedge^{\ell}F\longrightarrow
S_{k}F\otimes\bigwedge^{\ell-1}F.$

It holds (see, for instance, [2]; Corollary 2.3 and Proposition 2.5):

  * (1)

If $p+q\neq 1$, then
$L_{p}^{q}=\ker(\delta_{p+1}^{q-1})=\operatorname{im}(\delta_{p}^{q})=\operatorname{coker}(\delta_{p-1}^{q+1})$.

  * (2)

$L^{1}_{p}F=S_{p}F$ for all $p$.

  * (3)

$L_{1}^{q}F=\bigwedge^{q}F$ for all $q\neq 0$.

  * (4)

$L_{p}^{0}F=L_{0}^{q}F=0$ for all $q\neq 1$ and all $p$.

  * (5)

$L_{p}^{q}F=0$ for all $q>\operatorname{rk}(F)$.

  * (6)

If $\operatorname{rk}(F)=n$, then $L_{p}^{n}F=S_{p-1}F\otimes\bigwedge^{n}F$.

  * (7)

If $\operatorname{rk}(F)=n$, then
$\operatorname{rk}L_{p}^{q}F=\binom{n+p-1}{q+p-1}\binom{q+p-2}{p-1}$.

Given the graded morphism $\varphi^{*}:G^{*}\longrightarrow F^{*}$ between
free modules of rank $t+c-1$ and $t$, respectively, we set

$A^{p}(F):=\operatorname{coker}(\bigwedge^{t-p}F\stackrel{{\scriptstyle\theta}}{{\longrightarrow}}\bigwedge^{t-p+1}F\otimes
F^{*})$

where $\theta(\beta):=\sum g_{i}\wedge\beta\otimes\gamma_{i}$ being
$\\{g_{1},\cdots,g_{t}\\}$ is a basis of $F$ and
$\\{\gamma_{1},\cdots,\gamma_{t}\\}$ its dual.

######  Conjecture 3.1.

(See [2], pg. 299) Let $\varphi^{*}:G^{*}\longrightarrow F^{*}$ be a graded
morphism between free $R$-modules of rank $t+c-1$ and $t$, respectively, and
suppose $\operatorname{depth}_{I_{t-1}(\varphi)}R=2(c+1)$. A resolution of
$\operatorname{coker}(\varphi^{*}_{p-1,1})$ should always start out as:

$\cdots\longrightarrow L_{2}^{p-1}(G)^{*}\oplus
A^{p}(F)\otimes\bigwedge^{t}G^{*}\longrightarrow\bigwedge^{p-1}G^{*}\otimes
F^{*}\longrightarrow\bigwedge^{p}F^{*}\longrightarrow\operatorname{coker}(\varphi^{*}_{p-1,1})\longrightarrow
0.$

In particular, a resolution of $\bigwedge^{2}M$ should always start out as:

$\cdots\longrightarrow L_{2}^{1}(G)^{*}\oplus
A^{2}(F)\otimes\bigwedge^{t}G^{*}\longrightarrow G^{*}\otimes
F^{*}\longrightarrow\bigwedge^{2}F^{*}\longrightarrow\bigwedge^{2}M\longrightarrow
0.$

(We will see later that $L_{2}^{1}(G)^{*}\oplus
A^{2}(F)\otimes\bigwedge^{t}G^{*}\cong S_{2}(G)^{*}\oplus
S_{2}(F^{*})\otimes\bigwedge^{t}F\otimes\bigwedge^{t}G^{*}$.)

######  Remark 3.2.

The assumption $\operatorname{depth}_{I_{t-1}(\varphi)}R=2(c+1)$ cannot be
skipped. Indeed we have used Macaulay2 to compute the minimal resolution of
$\wedge^{2}M$ for general linear $3\times(c+2)$ matrices with coefficients in
$R={\mathbb{Q}}[x_{0},x_{1},\cdots,x_{n}]$ for different values of $n$ and any
$c\in\\{2,3\\}$ with $\mathbb{Q}$ the field of rational numbers. The graded
Betti numbers for fixed $c$ and various $n\geq 2(c+1)-1$ are the same and
satisfies Conjecture 3.1 while for the choice $n+1=2(c+1)-1$ the minimal
resolution of $\wedge^{2}M$ radically change and the prediction in Conjecture
3.1 does not hold. Note that in the latter case,
$\operatorname{depth}_{I_{t-1}(\varphi)}R=2(c+1)-1$ and $R/I_{t-1}(\varphi)$
is not determinantal.

The main goal of this section is to relate the canonical module of
$I_{i}(\varphi)$ with a suitable Schur power of
$M=\operatorname{coker}(\varphi^{*})$ (Proposition 3.5 and Theorem 3.10) and
prove Conjecture 3.1 for $p=2$ and $c=2$ (Theorems 3.7). Indeed, we will
determine a minimal graded free resolution of
$\bigwedge^{p}M=\operatorname{coker}(\varphi^{*}_{1,p-1})$ for $p\geq 1$ and
$c=2$ (Theorem 3.7). Our proof is based on the fact that the minimal graded
free $R$-resolution of the determinantal ideal $I_{t+1-p}(\varphi)$ is known,
for any $c\geq 2$, under the assumption that
$\operatorname{depth}_{I_{t-p+1}(\varphi)}R=p(p+c-1)$, and Proposition 3.5
which identifies, up to twist, $\bigwedge^{p}M$ with the canonical module of
$R/I_{t+1-p}(\varphi)$ provided $c=2$. In section 4, we will generalize
Theorem 3.7 to the case $c=3$ and we will prove Conjecture 3.1 for $p=2$ and
$c=3$ up to an explicitly described summand which we guess splits off (Theorem
4.6).

For sake of completeness, let us start explaining how we build a minimal
graded free $R$-resolution of the determinantal ideal $I_{t+1-p}(\varphi)$.
This construction rests heavily on the theory of Schur functors and we refer
to [13] and [3] for more details. We know that $R/I_{i}(\varphi)$ is a Cohen-
Macaulay ring of projective dimension $(t-i+1)(t+c-i)$ and let us call
$\mathbb{L}^{\bullet}(i,\varphi)$ a graded minimal free $R$-resolution of
$R/I_{i}(\varphi)$. To define the terms

$\mathbb{L}^{-s}(i,\varphi)\ \text{ for }\ 1\leq s\leq(t-i+1)(t+c-i)$

of the complex $\mathbb{L}^{\bullet}(i,\varphi)$, we need to recall some basic
facts on partitions and Schur functors.

Partitions. Let $r$ be an integer. A partition is an $r$-tuple
$I=(i_{1},\cdots,i_{r})\in\mathbb{Z}^{r}_{\geq 0}$ such that $0\leq i_{1}\leq
i_{2}\leq\cdots\leq i_{r}$. The length of the partition $I$ is the number of
its non-zero elements, and its weight, denoted by $|I|$, is
$i_{1}+\cdots+i_{r}$. Given two partitions
$I=(i_{1},\cdots,i_{r})\in\mathbb{Z}_{\geq 0}^{r}$ and
$I=(j_{1},\cdots,j_{s})\in\mathbb{Z}_{\geq 0}^{s}$ we say that $J\subset I$ if
$j_{v}\leq i_{v}$ for all $v\geq 1$.

We will draw the Young diagram (also called Ferrer diagram) corresponding to a
partition $I$ by putting $i_{j}$ boxes in the $j$-th row and to the left as in
the pictures below. We will also denote by $I$ the corresponding Young
diagram. If we flip this diagram over its diagonal, we get the Young diagram
of the conjugate (or dual) partition $I^{*}$. For example the partition
$I=(2,2,3,5,8)$ corresponds to the Young diagram

and $I^{*}=(1,1,1,2,2,3,5,5)$ corresponds to the Young diagram:

Given a partition $I=(i_{1},\cdots,i_{r})$, we will say that $I$ has Durfee
square $p_{I}$ if $p_{I}$ is the dimension of the greatest square contained in
the Young diagram of $I$. So, if $I=(2,2,3,5,8)$, then $p_{I}=3$,

Schur functors. Given a rank $r$ vector bundle $\mathcal{E}$ on
$\mathbb{P}^{n}$, we will denote by $\Sigma^{I}\mathcal{E}$ the result of
applying the Schur functor $\Sigma^{I}$ with
$I=(i_{1},\cdots,i_{s})\in\mathbb{Z}^{s}_{\geq 0}$ a partition with $s\leq r$
parts, i.e., $0\leq i_{1}\leq i_{2}\leq\cdots\leq i_{s}$. As special cases one
recover symmetric and exterior powers:

$\Sigma^{(0,\cdots,0,m)}\mathcal{E}=S_{m}\mathcal{E}$

is the $m$-symmetric power of $\mathcal{E}$ and

$\Sigma^{(0,\cdots,0,\tiny{\overbrace{1,\cdots,1}^{m}})}\mathcal{E}=\bigwedge^{m}\mathcal{E}$

is the $m$-exterior power of $\mathcal{E}$. We also use the convention
$\Sigma^{(i_{1},\cdots,i_{s})}\mathcal{E}=\Sigma^{(\tiny{\overbrace{0,\cdots,0,i_{1},\cdots,i_{s}}^{r}})}\mathcal{E}$
whenever $s<r$. In particular, for any rank $r$ vector bundle $\mathcal{E}$ on
$\mathbb{P}^{n}$,

$\det(\mathcal{E})=\bigwedge^{r}\mathcal{E}=\Sigma^{(\tiny{\overbrace{1,\cdots,1}^{r}})}\mathcal{E},\text{
and }$
$\Sigma^{(i_{1}+m,\cdots,i_{r}+m)}\mathcal{E}=\Sigma^{(i_{1},\cdots,i_{r})}\mathcal{E}\otimes\det(\mathcal{E})^{\otimes
m}.$

In general, given $I=(i_{1},\cdots,i_{q})$ a partition and
$I^{*}=(j_{1},\cdots,j_{s})$ its dual and writing
$(\otimes^{j_{1}}\mathcal{E})\otimes\cdots\otimes(\otimes^{j_{s}}\mathcal{E})$
as $\bigotimes^{|I|}(\mathcal{E})$, we define the Schur power
$\Sigma^{I}\mathcal{E}$ of $\mathcal{E}$ with exponent $I$ as the image of the
composite map

$d_{I}(\mathcal{E}):\bigwedge^{j_{1}}\mathcal{E}\otimes\cdots\otimes\bigwedge^{j_{s}}\mathcal{E}\stackrel{{\scriptstyle\alpha_{I}}}{{\longrightarrow}}\bigotimes^{|I|}(\mathcal{E})\stackrel{{\scriptstyle\beta_{I}}}{{\longrightarrow}}S_{i_{1}}\mathcal{E}\otimes\cdots\otimes
S_{i_{q}}\mathcal{E}$

where $\alpha_{I}$ is the tensor product of the injections
$\bigwedge^{j_{v}}\mathcal{E}\longrightarrow\otimes^{j_{v}}\mathcal{E}$ which
sends $f_{v}^{1}\wedge\cdots\wedge
f_{v}^{j_{v}}\mapsto\frac{1}{j_{v}!}\sum_{\sigma\in\mathfrak{S}_{j_{v}}}(-1)^{\varepsilon(\sigma)}f_{v}^{\sigma(1)}\otimes\cdots\otimes
f_{v}^{\sigma(j_{v})}$ being $\mathfrak{S}_{j_{v}}$ the permutation group.
Moreover, $\beta_{I}:\bigotimes^{|I|}(\mathcal{E})\longrightarrow
S_{i_{1}}\mathcal{E}\otimes\cdots\otimes S_{i_{q}}\mathcal{E}$ sends
$(f_{1}^{1}\otimes\cdots\otimes
f_{1}^{j_{1}})\otimes(f_{2}^{1}\otimes\cdots\otimes
f_{2}^{j_{2}})\otimes\cdots\otimes(f_{s}^{1}\otimes\cdots\otimes
f_{s}^{j_{s}})\mapsto(f_{1}^{q}f_{2}^{q}\cdots
f_{s}^{q})\otimes(f_{1}^{q-1}f_{2}^{q-1}\cdots
f_{s}^{q-1})\otimes\cdots\otimes(f_{1}^{1}f_{2}^{1}\cdots f_{s}^{1})$ where
$q=j_{s}$ and where we let $f_{i}^{j}=1$ for $j_{i}<j\leq j_{s}$, i.e.
$f_{j}^{i}=1$ if $(i,j)$ is not a position (or box) in the Young diagram of
$I^{*}$. Recall that if $\mathcal{E}$ is a rank $r$ vector bundle on
$\mathbb{P}^{n}$ and $I=(i_{1},\cdots,i_{r})$ a partition ($0\leq i_{1}\leq
i_{2}\leq\cdots\leq i_{r}$) then (see, for example, [6]; Theorem 6.3):

(3.2) $\operatorname{rk}(\Sigma^{I}\mathcal{E})=\prod_{1\leq u<v\leq
r}\frac{i_{v}-i_{u}+v-u}{v-u}.$

For instance, we consider $I=(1,2,4)$ and its dual $I^{*}=(1,1,2,3)$ with
Young diagrams

Then, $\Sigma^{I}\mathcal{E}$ is the image of the map

$\hskip-156.49014ptd_{I}(\mathcal{E}):\mathcal{E}\otimes\mathcal{E}\otimes\bigwedge^{2}\mathcal{E}\otimes\bigwedge^{3}\mathcal{E}\longrightarrow\mathcal{E}\otimes
S_{2}\mathcal{E}\otimes S_{4}\mathcal{E}$ $f_{1}^{1}\otimes
f_{2}^{1}\otimes(f_{3}^{1}\wedge f_{3}^{2})\otimes(f_{4}^{1}\wedge
f_{4}^{2}\wedge
f_{4}^{3})\mapsto\frac{1}{12}\sum_{\overset{\tau\in\mathfrak{S}_{2}}{\sigma\in\mathfrak{S}_{3}}}(-1)^{\varepsilon(\tau)+\varepsilon(\sigma)}f_{4}^{\sigma(3)}\otimes(f_{3}^{\tau(2)}f_{4}^{\sigma(2)})\otimes(f_{1}^{1}f_{2}^{1}f_{3}^{\tau(1)}f_{4}^{\sigma(1)})$

obtained composing the maps

$\hskip-221.93158pt\alpha_{I}:\mathcal{E}\otimes\mathcal{E}\otimes\bigwedge^{2}\mathcal{E}\otimes\bigwedge^{3}\mathcal{E}\longrightarrow\bigotimes^{7}\mathcal{E}$
$f_{1}^{1}\otimes f_{2}^{1}\otimes(f_{3}^{1}\wedge
f_{3}^{2})\otimes(f_{4}^{1}\wedge f_{4}^{2}\wedge
f_{4}^{3})\mapsto\frac{1}{12}\sum_{\overset{\tau\in\mathfrak{S}_{2}}{\sigma\in\mathfrak{S}_{3}}}(-1)^{\varepsilon(\tau)+\varepsilon(\sigma)}f_{1}^{1}\otimes
f_{2}^{1}\otimes(f_{3}^{\tau(1)}\otimes
f_{3}^{\tau(2)})\otimes(f_{4}^{\sigma(1)}\otimes f_{4}^{\sigma(2)}\otimes
f_{4}^{\sigma(3)})$

and

$\hskip
56.9055pt\beta_{I}:\bigotimes^{7}(\mathcal{E})\longrightarrow\mathcal{E}\otimes
S_{2}\mathcal{E}\otimes S_{4}\mathcal{E}$ $f_{1}^{1}\otimes
f_{2}^{1}\otimes(f_{3}^{1}\otimes f_{3}^{2})\otimes(f_{4}^{1}\otimes
f_{4}^{2}\otimes f_{4}^{3})\mapsto
f_{4}^{3}\otimes(f_{3}^{2}f_{4}^{2})\otimes(f_{1}^{1}f_{2}^{1}f_{3}^{1}f_{4}^{1}).$

In addition, if $\mathcal{E}$ has rank 5, then $\Sigma^{I}\mathcal{E}$ has
rank 700.

We will often use Pieri’s formula (see [13]; pg. 209): Let $E$ be a free
$R$-module of rank $r$ and let $I$ be a partition. It holds:

(3.3) $\wedge^{n}E\otimes\Sigma^{I}E\cong\bigoplus_{J}\Sigma^{J}E$

where the index set consists of all partitions $J$ whose Young diagram is
obtained by adding n boxes to the Young diagram of $I$ with no two boxes in
the same row, see [1], Corollary IV.2.6. So, if $I=(i_{1},\cdots,i_{s})$ then
the index set consists of all $J=(j_{0},j_{1},\cdots,j_{s})$ satisfying
$|J|=|I|+n$, $i_{t}\leq j_{t}\leq i_{t}+1$ and $j_{0}=0$ (i.e.
$J=(j_{1},\cdots,j_{s})$) if $s=r$ and $0\leq j_{0}\leq 1$ if $s<r$.

Dually, we have

(3.4) $S_{n}E\otimes\Sigma^{I}E\cong\bigoplus_{J}\Sigma^{J}E$

where the index set consists of all partitions $J$ whose Young diagram is
obtained by adding n boxes to the Young diagram of $I$ with no two boxes in
the same column.

######  Example 3.3.

Let $E$ be a free $R$-module of rank 6. By Pieri’s formula we have

$S_{4}E\otimes\bigwedge^{2}E\cong\Sigma^{(0,0,0,0,1,5)}E\oplus\Sigma^{(0,0,0,1,1,4)}E,$

i.e. $S_{4}E\otimes\bigwedge^{2}E$ is a rank 1890 free $R$-module which
decompose into a direct sum of two free $R$-modules of rank 1050 and 840,
respectively.

Let $\varphi:F\longrightarrow G$ be a graded $R$-morphism of free $R$-modules
of rank $t$ and $t+c-1$, respectively. We define the complex
$\mathbb{L}^{\bullet}(i,\varphi)$ of $R$-modules as follows:

(3.5)
$L^{j}(i,\varphi)=\bigoplus_{\overset{-|I|+n(I)=j}{I,I^{\prime}\neq\emptyset}}\Sigma^{(I)^{*}}(G^{*})\otimes\Sigma^{I^{\prime}}(F),\quad\text{
for }\quad-(t-i+1)(t+c-i)\leq j\leq-1,$

where $I=(i_{1},\cdots,i_{q})$ is a partition, $q=t-i+1$, $p:=p_{I}$ is the
Durfee square of $I$, $n(I)=(i-1)p_{I}$,
$I^{\prime}=(i_{1},\cdots,i_{q-p_{I}},\overbrace{p_{I},\cdots,p_{I}}^{i-1},i_{q-p_{I}+1}-i+1,\cdots,i_{q}-i+1)$
if $i_{q-p_{I}+1}\geq p_{I}+i-1$, otherwise $I^{\prime}=\emptyset$ and
$(I)^{*}$ the dual of the partition $I$.

Now, we have to define a morphism between $L^{j}(i,\varphi)$ and
$L^{j+1}(i,\varphi)$, i.e. for any pair of partitions $(I,H)$ with
$-|I|+n(I)=j=-|H|+n(H)-1$ we must define

$\psi_{I,H}:\Sigma^{(I)^{*}}(G^{*})\otimes\Sigma^{I^{\prime}}(F)\longrightarrow\Sigma^{(H)^{*}}(G^{*})\otimes\Sigma^{H^{\prime}}(F).$

We distinguish two cases:

Case 1: If $H\nsubseteq I$, we set $\psi_{(I,H)}=0$.

Case 2: $H\subset I$. By [13]; Lemma 5.13, if $(I,H)$ is a pair of partitions
such that $H\subset I$, $I^{\prime}\neq\emptyset$, $H^{\prime}\neq\emptyset$
and $-|I|+n(I)=j=-|H|+n(H)-1$, then $I$ and $H$ (resp. $I^{\prime}$ and
$H^{\prime}$) coincide except in one spot, i.e.
$I=(i_{1},\cdots,i_{x-1},a,i_{x+1},\cdots,i_{q})$,
$H=(i_{1},\cdots,i_{x-1},h,i_{x+1},\cdots,i_{q})$,
$(I^{\prime})^{*}=(j_{1},\cdots,j_{y-1},a^{\prime},j_{y+1},\cdots,j_{s})$,
$(H^{\prime})^{*}=(j_{1},\cdots,j_{y-1},h^{\prime},j_{y+1},\cdots,j_{s})$. By
[13]; Lemma 5.13 (see also [3]; Lemma 3.1), we have
$a-h=a^{\prime}-h^{\prime}$ and we set $\rho:=a-h=a^{\prime}-h^{\prime}$.
Since the Young diagram of $I^{*}$ is obtained by adding $\rho$ boxes to the
Young diagram of $H^{*}$ with no two boxes in the same row, we have by Pieri’s
formula (3.3) that $\Sigma^{(I)^{*}}G^{*}$ is a direct summand of
$\bigwedge^{\rho}G^{*}\otimes\Sigma^{(H)^{*}}G^{*}$. Therefore, we have a
canonical injective map

$\delta_{I,\rho}:\Sigma^{(I)^{*}}G^{*}\longrightarrow\bigwedge^{\rho}G^{*}\otimes\Sigma^{(H)^{*}}G^{*}$

and, dually, we have a canonical injective map

$\delta_{I^{\prime},\rho}:\Sigma^{I^{\prime}}F\longrightarrow\bigwedge^{\rho}F\otimes\Sigma^{H^{\prime}}F$

which give us

$\delta_{I,\rho}\otimes\delta_{I^{\prime},\rho}:\Sigma^{(I)^{*}}G^{*}\otimes\Sigma^{I^{\prime}}F\longrightarrow\bigwedge^{\rho}G^{*}\otimes\bigwedge^{\rho}F\otimes\Sigma^{(H)^{*}}G^{*}\otimes\Sigma^{H^{\prime}}F.$

Composing the morphism $\delta_{I,\rho}\otimes\delta_{I^{\prime},\rho}$ with

$\mu\otimes
1:\bigwedge^{\rho}G^{*}\otimes\bigwedge^{\rho}F\otimes\Sigma^{(H)^{*}}G^{*}\otimes\Sigma^{H^{\prime}}F\longrightarrow\Sigma^{(H)^{*}}G^{*}\otimes\Sigma^{H^{\prime}}F$

where $\mu:\bigwedge^{\rho}G^{*}\otimes\bigwedge^{\rho}F\longrightarrow R$ is
the natural map induced by $\varphi^{*}:G^{*}\longrightarrow F^{*}$, we get a
morphism

$\sigma_{(I,H)}:\Sigma^{(I)^{*}}G^{*}\otimes\Sigma^{I^{\prime}}F\longrightarrow\Sigma^{(H)^{*}}G^{*}\otimes\Sigma^{H^{\prime}}F.$

Finally, we define the differential map:

$d_{j}=\oplus\psi_{(i,H)}:L^{j}(i,\varphi)=\bigoplus_{\overset{-|I|+n(I)=j}{I,I^{\prime}\neq\emptyset}}\Sigma^{(I)^{*}}(G^{*})\otimes\Sigma^{I^{\prime}}(F)\longrightarrow
L^{j+1}(i,\varphi)=\bigoplus_{\overset{-|H|+n(H)=j+1}{H,H^{\prime}\neq\emptyset}}\Sigma^{(H)^{*}}(G^{*})\otimes\Sigma^{H^{\prime}}(F).$

We have:

######  Theorem 3.4.

Let $\varphi:F\longrightarrow G$ be a graded $R$-morphism of free $R$-modules
of rank $t$ and $t+c-1$, respectively. Assume that $\varphi$ is minimal and
$\operatorname{depth}_{I_{i}(\varphi)}R=(t-i+1)(t+c-i)$. Then, the complex
$\mathbb{L}^{\bullet}(i,\varphi)$ is exact and
$\operatorname{H}^{0}(\mathbb{L}^{\bullet}(i,\varphi))=R/I_{i}(\varphi)$, i.e.
$\mathbb{L}^{\bullet}(i,\varphi)$ is a graded minimal free $R$-resolution of
$R/I_{i}(\varphi).$

###### Proof.

See [13]; Théorème 3.3 or [3]; Theorem 3.2. ∎

Our next goal will be to apply the above Theorem to solve Conjecture 3.1 for
$p=c=2$ (see Theorem 3.7).

######  Proposition 3.5.

Set $R=k[x_{0},\cdots,x_{n}]$. Let $\varphi^{*}:G^{*}\longrightarrow F^{*}$ be
a graded minimal morphism between free $R$-modules of rank $t+1$ and $t$,
respectively. Set $B_{t-p+1}:=R/I_{t+1-p}(\varphi)$,
$M=\operatorname{coker}(\varphi^{*})$ and
$\ell:=\sum_{j=1}^{t+1}a_{j}-\sum_{i=1}^{t}b_{i}$ and assume that $B_{t+1-p}$
is determinantal. Then, we have:

$K_{B_{t+1-p}}(n+1-p\ell)\cong\bigwedge^{p}M.$

###### Proof.

By hypothesis $M=\operatorname{coker}(\varphi^{*})$. Therefore, a minimal
graded $R$-free resolution of $\bigwedge^{p}M$ starts out as:

$\cdots\longrightarrow
G^{*}\otimes\bigwedge^{p-1}F^{*}\stackrel{{\scriptstyle\overline{\varphi^{*}}}}{{\longrightarrow}}\bigwedge^{p}F^{*}\longrightarrow\bigwedge^{p}M\longrightarrow
0$

with $\overline{\varphi^{*}}(\eta\otimes\rho)=\varphi^{*}(\eta)\wedge\rho$.
So, $\bigwedge^{p}M=\operatorname{coker}(\overline{\varphi^{*}})$.

On the other hand, we know that $B_{t-p+1}$ is a Cohen-Macaulay ring and
$\mathbb{L}^{\bullet}(t-p+1,\varphi)$ is a graded minimal free $R$-resolution
of $B_{t-p+1}$ of length $p(p+1)$ (see Theorem 3.4). Applying
$\operatorname{Hom}_{R}(-,R)$ to $\mathbb{L}^{\bullet}(t+1-p,\varphi)$ we get
a minimal graded free $R$-resolution of
$\operatorname{Ext}_{R}^{p(p+1)}(B_{t-p+1},R)\cong K_{B_{t+1-p}}(n+1)$. Let us
explicitly compute $\mathbb{L}^{-p(p+1)}(t+1-p,\varphi)$,
$\mathbb{L}^{-p(p+1)+1}(t+1-p,\varphi)$ and the morphism linking them.
$\mathbb{L}^{-p(p+1)}(t+1-p,\varphi)$ corresponds to the unique partition
$I=(\overbrace{t+1,\cdots,t+1}^{p})$ for which the Durfee square $p_{I}=p$,
$n(I)=p(t-p)$ and
$I^{\prime}=(\overbrace{p,\cdots,p}^{t-p},\overbrace{p+1,\cdots,p+1}^{p})$.
Thus, we have

$\begin{array}[]{rcl}\mathbb{L}^{-p(p+1)}(t+1-p,\varphi)&:=&\Sigma^{(I)^{*}}G^{*}\otimes\Sigma^{I^{\prime}}F\\\
&\cong&(\bigwedge^{t+1}G^{*})^{\otimes p}\otimes(\bigwedge^{t}F)^{\otimes
p}\otimes\bigwedge^{p}F\\\ &\cong&\bigwedge^{p}F\otimes R(-p\ell).\end{array}$

Analogously, we apply [13]; Corollaire 5.10 and we get that the $R$-free
module $\mathbb{L}^{-p(p+1)+1}(t+1-p,\varphi)$ corresponds to a unique
partition, namely, $I=(t,\overbrace{t+1,\cdots,t+1}^{p-1})$ for which
$p_{I}=p$, $n(I)=p(t-p)$,
$I^{\prime}=(\overbrace{p,\cdots,p}^{t-p+1},\overbrace{p+1,\cdots,p+1}^{p-1})$
and $I^{*}=(p-1,\overbrace{p,\cdots,p}^{t})$. Thus, we have

$\begin{array}[]{rcl}\mathbb{L}^{-p(p+1)+1}(t+1-p,\varphi)&:=&\Sigma^{(I)^{*}}G^{*}\otimes\Sigma^{I^{\prime}}F\\\
&\cong&(\bigwedge^{t+1}G^{*})^{\otimes(p-1)}\otimes\bigwedge^{t}(G^{*})\otimes(\bigwedge^{t}F)^{\otimes
p}\otimes\bigwedge^{p-1}F\\\ &\cong&G\otimes\bigwedge^{p-1}F\otimes
R(-p\ell)\end{array}$

where the last isomorphism follows from the well known fact that
$\bigwedge^{t}(G^{*})\cong G\otimes\bigwedge^{t+1}(G^{*})$. To finish the
proof it is enough to observe that for the unique partitions
$I=(\overbrace{t+1,\cdots,t+1}^{p})$ and
$H=(t,\overbrace{t+1,\cdots,t}^{p-1})$ associated to
$\mathbb{L}^{-p(p+1)}(t+1-p,\varphi)$ and
$\mathbb{L}^{-p(p+1)+1}(t+1-p,\varphi)$, respectively, we have
$\psi_{(I,H)}^{*}\cong\overline{\varphi^{*}}\otimes 1_{R(p\ell)}$, whence
$K_{B_{t+1-p}}(n+1-p\ell)\cong\bigwedge^{p}(M).$ ∎

By [15], Proposition 4, we know that $\bigwedge^{p}M$ is a maximal Cohen-
Macaulay $R/I_{t+1-p}(\varphi)$-module. As a consequence of the above
Proposition we recover Vetter’s result and even more. We have:

######  Corollary 3.6.

Set $R=k[x_{0},\cdots,x_{n}]$. Let $\varphi^{*}:G^{*}\longrightarrow F^{*}$ be
a graded minimal morphism between free $R$-modules of rank $t+1$ and $t$,
respectively, and let $M$ be the cokernel of $\varphi^{*}$. Assume that
$\operatorname{depth}_{I_{t+1-p}(\varphi)}R=(p+1)p$. Then, $\bigwedge^{p}M$ is
a Gorenstein $R/I_{t+1-p}(\varphi)$-module.

###### Proof.

Since $R/I_{t+1-p}(\varphi)$ is a Cohen-Macaulay ring, the result immediately
follows from Proposition 3.5 and [12]; Lemma 5.4. ∎

######  Theorem 3.7.

Set $R=k[x_{0},\cdots,x_{n}]$. Let $\varphi^{*}:G^{*}\longrightarrow F^{*}$ be
a graded minimal morphism between free $R$-modules of rank $t+1$ and $t$,
respectively. Set $\ell:=\sum_{j=1}^{t+1}a_{j}-\sum_{i=1}^{t}b_{i}$ and assume
$\operatorname{depth}_{I_{t+1-p}(\varphi)}R=p(p+1)$. Then, the $R$-dual
$\mathbb{L}^{\bullet}(t+1-p,\varphi)^{*}$ of
$\mathbb{L}^{\bullet}(t+1-p,\varphi)$ is a minimal free $R$-resolution of
$\bigwedge^{p}M(p\ell)$.

In particular, Conjecture 3.1 (with $\varphi$ minimal) is true for $p=c=2$.

###### Proof.

We know that $\mathbb{L}^{\bullet}(t+1-p,\varphi)$ is a minimal graded free
$R$-resolution of $B_{t+1-p}:=R/I_{t+1-p}(\varphi)$. We also know that
$R/I_{t+1-p}(\varphi)$ is a Cohen-Macaulay ring. Therefore, the $R$-dual
$\mathbb{L}^{\bullet}(t+1-p,\varphi)^{*}$ of
$\mathbb{L}^{\bullet}(t+1-p,\varphi)$ gives us a minimal graded free
$R$-resolution of $K_{B_{t+1-p}(\varphi)}(n+1)$ and we now conclude using
Proposition 3.5.

To prove Conjecture 3.1 for $p=c=2$, we only have to identify the last terms
of $\mathbb{L}^{\bullet}(t+1-p,\varphi)$. By the proof of Proposition 3.5 we
know that

$\mathbb{L}^{-p(p+1)}(t+1-p,\varphi)=(\bigwedge^{t+1}G^{*})^{p}\otimes(\bigwedge^{t}F)^{p}\otimes\bigwedge^{p}F\cong\bigwedge^{p}F(-p\ell)$

and

$\mathbb{L}^{-p(p+1)+1}(t+1-p,\varphi)=(\bigwedge^{t+1}G^{*})^{p}\otimes
G\otimes(\bigwedge^{t}F)^{p}\otimes\bigwedge^{p-1}F\cong
G\otimes\bigwedge^{p-1}F(-p\ell).$

Let us compute $\mathbb{L}^{-p(p+1)+2}(t+1-p,\varphi)$.
$\mathbb{L}^{-p(p+1)+2}(t+1-p,\varphi)$ corresponds to two different
partitions:

  * •

$I_{1}:=(p-1,\overbrace{t+1,\cdots,t+1}^{p-1})$ with Durfee square
$p_{I_{1}}=p-1$, $n(I_{1})=(p-1)(t-p)$ and
$I_{1}^{\prime}=(\overbrace{p-1,\cdots,p-1}^{t-p+1},\overbrace{p+1,\cdots,p+1}^{p-1})$.

  * •

$I_{2}:=(t,t,\overbrace{t+1,\cdots,t+1}^{p-2})$ for which $p_{I_{2}}=p$,
$n(I_{2})=p(t-p)$ and
$I_{2}^{\prime}=(\overbrace{p,\cdots,p}^{t-p+2},\overbrace{p+1,\cdots,p+1}^{p-2})$

Therefore, we have

$\begin{array}[]{rcl}\Sigma^{I_{1}^{\prime}}F&=&(\bigwedge^{t}F)^{\otimes
p-1}\otimes\Sigma^{(0,\cdots,0,\overbrace{2,\cdots,2}^{p-1})}F,\\\
\Sigma^{I_{2}^{\prime}}F&=&(\bigwedge^{t}F)^{\otimes
p}\otimes\Sigma^{(0,\cdots,0,\overbrace{1,\cdots,1}^{p-2})}F\\\
&\cong&(\bigwedge^{t}F)^{\otimes p}\otimes\bigwedge^{p-2}F,\end{array}$
$\begin{array}[]{rcl}\Sigma^{(I_{1})^{*}}G^{*}&=&(\bigwedge^{t+1}G^{*})^{\otimes
p-1}\otimes\Sigma^{(0,\cdots,0,\overbrace{1,\cdots,1}^{p-1})}G^{*}\\\
&\cong&(\bigwedge^{t+1}G^{*})^{\otimes p-1}\otimes\bigwedge^{p-1}G^{*},\text{
and }\\\ \Sigma^{(I_{2})^{*}}G^{*}&=&(\bigwedge^{t+1}G^{*})^{\otimes
p-2}\otimes\Sigma^{(0,2,\cdots,2)}G^{*}\\\
&\cong&(\bigwedge^{t+1}G^{*})^{\otimes
p}\otimes\Sigma^{(-2,0,\cdots,0)}G^{*}\\\
&\cong&(\bigwedge^{t+1}G^{*})^{\otimes p}\otimes S_{2}G.\end{array}$

In particular, for $p=c=2$, we obtain:

$\begin{array}[]{rcl}\Sigma^{I_{1}^{\prime}}F&=&\bigwedge^{t}F\otimes
S_{2}F\cong(\bigwedge^{t}F)^{\otimes 2}\otimes\bigwedge^{t}F^{*}\otimes
S_{2}F,\\\ \Sigma^{I_{2}^{\prime}}F&=&(\bigwedge^{t}F)^{\otimes 2},\\\
\Sigma^{(I_{1})^{*}}G^{*}&=&\bigwedge^{t+1}G^{*}\otimes
G^{*}\cong(\bigwedge^{t+1}G^{*})^{\otimes 2}\otimes\bigwedge^{t}G,\text{ and
}\\\ \Sigma^{(I_{2})^{*}}G^{*}&=&(\bigwedge^{t+1}G^{*})^{\otimes 2}\otimes
S_{2}G.\end{array}$

Thus, we have

$\begin{array}[]{rcl}\mathbb{L}^{-p(p+1)+2}(t+1-p,\varphi)&:=&(\Sigma^{(I_{1})^{*}}G^{*}\otimes\Sigma^{I_{1}^{\prime}}F)\oplus(\Sigma^{(I_{2})}G^{*}\otimes\Sigma^{I_{2}^{\prime}}F)\\\
&\cong&\big{(}S_{2}G\oplus\bigwedge^{t}G\otimes
S_{2}F\otimes\bigwedge^{t}F^{*}\big{)}\otimes R(-2\ell).\end{array}$

Since, for $c=p=2$, we have
$A^{2}(F)=\operatorname{coker}(\bigwedge^{t-2}F\longrightarrow\bigwedge^{t-1}F\otimes
F^{*})\cong S_{2}F^{*}\otimes\bigwedge^{t}F$, we have got what we wanted to
prove. ∎

######  Remark 3.8.

(1) The above Theorem solves Conjecture 3.1 for $p=c=2$ and it gives a minimal
resolution of $\operatorname{coker}(\varphi^{*}_{1,p-1})=\bigwedge^{p}M$ for
$c=2$ and any $p$.

(2) For $c\geq 3$ the proof of Theorem 3.7 does not generalize since it is no
longer true that $K_{{R/I_{t+1-p}(\varphi)}}\cong\bigwedge^{p}M(\delta)$ for a
suitable $\delta$ as the following example computed using Macaulay2 ([7])
shows.

######  Example 3.9.

Set $R={\mathbb{Q}}[x_{0},x_{1},\cdots,x_{n}]$ with $n$ either $13$ or $14$.
Let $\mathcal{A}$ be a $3\times 5$ matrix with general enough entries of
linear forms. Take $M=\operatorname{coker}(\varphi^{*})$ where
$\varphi^{*}:R(-1)^{5}\longrightarrow R^{3}$ is the graded morphism associated
to $\mathcal{A}$ and consider $I:=I_{2}(\mathcal{A})$ the ideal defined by the
$2\times 2$ minors of $\mathcal{A}$. Using Macaulay2 we get that the minimal
graded free $R$-resolutions of $R/I$, $K_{R/I}$ and $\bigwedge^{2}M$ have the
following shape:

$0\rightarrow R(-10)^{6}\rightarrow R(-9)^{40}\rightarrow
R(-8)^{105}\rightarrow\begin{array}[]{ccccccc}R(-6)^{50}\\\ \oplus\\\
R(-7)^{120}\end{array}\rightarrow\begin{array}[]{ccccccc}R(-5)^{168}\\\
\oplus\\\ R(-6)^{50}\end{array}\rightarrow R(-4)^{210}$ $\rightarrow
R(-3)^{120}\rightarrow R(-2)^{30}\rightarrow R\rightarrow R/I\rightarrow 0,$

$0\rightarrow R(-11)\rightarrow R(-9)^{30}\rightarrow R(-8)^{120}\rightarrow
R(-7)^{210}\rightarrow\begin{array}[]{ccccccc}R(-5)^{50}\\\ \oplus\\\
R(-6)^{168}\end{array}\rightarrow\begin{array}[]{ccccccc}R(-4)^{120}\\\
\oplus\\\ R(-5)^{50}\end{array}\rightarrow R(-3)^{105}$ $\rightarrow
R(-2)^{40}\rightarrow R(-1)^{6}\rightarrow K_{R/I}\rightarrow 0,$

and

$0\rightarrow R(-10)^{3}\rightarrow
R(-9)^{15}\rightarrow\begin{array}[]{ccccccc}R(-7)^{60}\\\ \oplus\\\
R(-8)^{15}\end{array}\rightarrow R(-6)^{210}\rightarrow R(-5)^{294}\rightarrow
R(-4)^{210}\rightarrow\begin{array}[]{ccccccc}R(-2)^{15}\\\ \oplus\\\
R(-3)^{60}\end{array}$ $\rightarrow R(-1)^{15}\rightarrow
R^{3}\rightarrow\bigwedge^{2}M\rightarrow 0.$

Therefore, $K_{R/I}$ is not isomorphic to a twist of $\bigwedge^{2}M$.
Nevertheless, using again Macaulay2 we compute a graded minimal $R$-resolution
of $S_{2}(\bigwedge^{2}M)$ and we obtain:

$0\rightarrow R(-10)\rightarrow R(-8)^{30}\rightarrow R(-7)^{120}\rightarrow
R(-6)^{210}\rightarrow\begin{array}[]{ccccccc}R(-4)^{50}\\\ \oplus\\\
R(-5)^{168}\end{array}\rightarrow\begin{array}[]{ccccccc}R(-3)^{120}\\\
\oplus\\\ R(-4)^{50}\end{array}\rightarrow R(-2)^{105}$ $\rightarrow
R(-1)^{40}\rightarrow R^{6}\rightarrow S_{2}(\bigwedge^{2}M)\rightarrow 0.$

We also have $K_{R/I}\cong S_{2}(\bigwedge^{2}M)(-1)$ (see Remark 3.11(3)).

The above example shows that, for $c\geq 3$, it is no longer true that
$K_{{R/I_{t+1-p}(\varphi)}}\cong\bigwedge^{p}M(\delta)$ for a suitable
$\delta$. Nevertheless, given $\varphi^{*}:G^{*}\longrightarrow F^{*}$ a
graded morphism between free $R$-modules of rank $t+c-1$ and $t$,
respectively, there is a close relationship between the canonical module of
$I_{t+1-p}(\varphi)$ and the Schur power $\Sigma^{I}M$ of $M$ for a suitable
partition $I$. Indeed, we have

######  Theorem 3.10.

Set $R=k[x_{0},\cdots,x_{n}]$. Let $\varphi^{*}:G^{*}\longrightarrow F^{*}$ be
a graded minimal morphism between free $R$-modules of rank $t+c-1$ and $t$,
respectively. Assume that
$\operatorname{depth}_{I_{t+1-p}(\varphi)}R=p(p+c-1)$. Set
$\ell:=\sum_{j=1}^{t+c-1}a_{j}-\sum_{i=1}^{t}b_{i}$,
$M=\operatorname{coker}(\varphi^{*})$ and
$J=(\overbrace{c-1,\cdots,c-1}^{p})$. Then, we have:

$K_{B_{t+1-p}}(n+1-p\ell)\cong\Sigma^{J}M$

###### Proof.

By hypothesis $M=\operatorname{coker}(\varphi^{*})$ and it has a presentation

$G^{*}\stackrel{{\scriptstyle\varphi^{*}}}{{\longrightarrow}}F^{*}\longrightarrow
M\longrightarrow 0.$

Therefore, by [1]; Proposition V.2.2, $\Sigma^{J}M$ has a presentation of the
following type:

$\Sigma^{J_{1}}F^{*}\otimes
G^{*}\stackrel{{\scriptstyle\Sigma^{J}\varphi^{*}}}{{\longrightarrow}}\Sigma^{J}F^{*}\longrightarrow\Sigma^{J}M\longrightarrow
0$

where $J_{1}=(c-2,\overbrace{c-1,\cdots,c-1}^{p-1})$.

On the other hand, we know that the $\mathbb{L}^{\bullet}(t+1-p,\varphi)$
gives us a minimal free $R$-resolution of $B_{t+1-p}$ of length $p(c+p-1)$ and
its $R$-dual $\mathbb{L}^{\bullet}(t+1-p,\varphi)^{*}$ provides a minimal free
$R$-resolution of $K_{B_{t+1-p}}(n+1)$. Therefore, it is enough to describe
the end of $\mathbb{L}^{\bullet}(t+1-p,\varphi)$.
$\mathbb{L}^{-p(c+p-1)}(t+1-p,\varphi)$ corresponds to the unique partition
$I=(\overbrace{t+c-1,\cdots,t+c-1}^{p})$ for which the Durfee square
$p_{I}=p$, $n(I)=p(t-p)$ and
$I^{\prime}=(\overbrace{p,\cdots,p}^{t-p},\overbrace{p+c-1,\cdots,p+c-1}^{p})$.
Hence, we have

$\begin{array}[]{rcl}\mathbb{L}^{-p(p+c-1)}(t+1-p,\varphi)&:=&\Sigma^{(I)^{*}}G^{*}\otimes\Sigma^{I^{\prime}}F\\\
&\cong&(\bigwedge^{t+c-1}G^{*})^{\otimes p}\otimes(\bigwedge^{t}F)^{\otimes
p}\otimes\Sigma^{J}F\\\ &\cong&\Sigma^{J}F\otimes R(-p\ell).\end{array}$

Analogously, associated to $\mathbb{L}^{-p(p+c-1)+1}(t+1-p,\varphi)$ there is
a unique partition, namely, $I=(t+c-2,\overbrace{t+c-1,\cdots,t+c-1}^{p-1})$
with $p_{I}=p$, $n(I)=p(t-p)$ and
$I^{\prime}=(\overbrace{p,\cdots,p}^{t-p},p+c-2,\overbrace{p+c-1,\cdots,p+c-1}^{p-1})$.
So,, we have

$\begin{array}[]{rcl}\mathbb{L}^{-p(p+c-1)+1}(t+1-p,\varphi)&:=&\Sigma^{(I)^{*}}G^{*}\otimes\Sigma^{I^{\prime}}F\\\
&\cong&(\bigwedge^{t+1}G^{*})^{\otimes(p-1)}\otimes\bigwedge^{t}(G^{*})\otimes(\bigwedge^{t}F)^{\otimes
p}\otimes\Sigma^{J_{1}}F\\\ &\cong&G\otimes\Sigma^{J_{1}}F\otimes
R(-p\ell)\end{array}$

and we easily conclude arguing as in the last part of the proof of Proposition
3.5. ∎

######  Remark 3.11.

(1) For $p=1$, we recover Proposition 2.2 (iii), i.e.
$K_{B_{t}}(n+1-\ell)\cong S_{c-1}M$.

(2) For $c=2$, we recover Proposition 3.5, i.e.
$K_{B_{t+1-p}}(n+1-p\ell)\cong\bigwedge^{p}M$.

(3) For $t=3$ and $c=3$ we have $S_{c-1}(\bigwedge^{2}M)\cong
K_{B_{2}}(n+1-2\ell)$ where the last isomorphism follows from Theorem 3.10 and
the Plethysm formula

$S_{m}(\bigwedge^{2}F)=\bigoplus_{|I|=2m,i_{j}\text{ even for all
j}}\Sigma^{I}F.$

##  4\. The resolution of the exterior powers

In Theorem 3.7, we have found a minimal graded $R$-resolution of
$\bigwedge^{p}M$ where $M=\operatorname{coker}\varphi^{*}$ and
$\varphi^{*}:G^{*}\longrightarrow F^{*}$ a graded minimal morphism between
free $R$-modules of rank $t-c+1$ and $t$, respectively, provided $c=2$. We
already pointed out that our approach does not generalize to $c\geq 3$. In
this section, supposing $\operatorname{depth}_{J}(B_{t})\geq 2$,
$J:=I_{t-1}(\varphi)$ (considered as an ideal of $B_{t}$, i.e. we allow
writing $J$ in place of $JB_{t}$), we will outline an alternative proof that
will provide a (possibly non-minimal) free $R$-resolution of $\wedge^{2}M$ for
$c=2,3$ and an affirmative answer for $c=p=2$, and for $p=2$ and $c=3$ up to a
concrete summand which we guess splits off (In fact, many cases have been
checked with Macaulay2) of Conjecture 3.1. This approach applies several times
the mapping cone construction to the exact sequence

$0\longrightarrow S_{2}M\otimes I_{t}\longrightarrow G^{*}\otimes
M\stackrel{{\scriptstyle\varphi^{*}\otimes
id_{M}}}{{\longrightarrow}}F^{*}\otimes M\longrightarrow M\otimes
M\longrightarrow 0,$

where $I_{t}:=I_{t}(\varphi)$ and then deduces a free resolution of
$\wedge^{2}M$ using the split-exact sequence

$0\longrightarrow S_{2}M\longrightarrow M\otimes
M\longrightarrow\bigwedge^{2}M\longrightarrow 0$

and the free resolution of $S_{2}M$ given in Proposition 2.2(ii). The problem
we encounter is to prove that $\ker(\varphi^{*}\otimes id_{M})\cong
S_{2}M\otimes I_{t}$ where we succeed for $c=2,3$ assuming
$\operatorname{depth}_{J}(B_{t})\geq c+2$, but the method has the potential to
prove Conjecture 3.1 for $p=2$ and every $c\geq 2$ if we can show
$\operatorname{depth}_{J}(S_{2}M\otimes I_{t})\geq 2$. Indeed if $c\geq 2$ and
$\operatorname{depth}_{J}(B_{t})\geq 2$ it is not so difficult is to see that

(4.1) $\ker(\varphi^{*}\otimes
id_{M})\cong\operatorname{H}_{*}^{0}(U,\widetilde{S_{2}M\otimes
I_{t}})\cong\operatorname{Tor}_{1}^{R}(M,M),\ \ \ \ {\rm where}\ \
U=\operatorname{Proj}(B_{t})\setminus V(J).$

Since Macaulay2 computations indicate that
$\operatorname{depth}_{J}(S_{2}M\otimes I_{t})\geq 2$ if $c\geq 2$ and
$\operatorname{depth}_{J}B_{t}\geq c+2$, whence $S_{2}M\otimes I_{t}\
\cong\operatorname{H}_{*}^{0}(U,\widetilde{S_{2}M\otimes I_{t}})$, the method
may apply to any $c\geq 2$ with $B_{t}/J$ determinantal. To have sufficient
depth of $S_{2}M\otimes I_{t}$ is related to problems investigated in [11],
and we finish the paper by setting forth another Conjecture 5.3, seemingly
unrelated to Conjecture 3.1, that will imply the above isomorphism.

Let us prove (4.1). Recall
$\ell_{c}:=\sum_{j=1}^{t+c-1}a_{j}-\sum_{i=1}^{t}b_{i}$ and that we have the
exact sequence

(4.2)
$G^{*}\stackrel{{\scriptstyle\varphi^{*}}}{{\longrightarrow}}F^{*}\longrightarrow
M\longrightarrow 0.$

######  Remark 4.1.

We can continue (4.2) by
$P\stackrel{{\scriptstyle\epsilon}}{{\longrightarrow}}G^{*}\longrightarrow\cdots$
to the left where $P:=\bigwedge^{t+1}G^{*}\otimes\bigwedge^{t}F$ and
$\epsilon$ is the ”splice map” (see Proposition 2.2(ii)). Let $-1\leq i\leq
c$. Since $\epsilon\otimes_{R}id_{S_{i}M}=0$ and
$\operatorname{Tor}_{1}^{R}(M,S_{i}M):=\ker(\varphi^{*}\otimes
id_{S_{i}M})/\operatorname{im}(\epsilon\otimes_{R}id_{S_{i}M})$, we get that
the sequence

$0\longrightarrow\operatorname{Tor}_{1}^{R}(M,S_{i}M)\longrightarrow
G^{*}\otimes S_{i}M\stackrel{{\scriptstyle\varphi^{*}\otimes
id_{S_{i}M}}}{{\longrightarrow}}F^{*}\otimes S_{i}M\longrightarrow M\otimes
S_{i}M\longrightarrow 0$

is exact. Thus $i=1$ takes care of one isomorphism in (4.1). Moreover note
that if $\operatorname{depth}_{J}(B_{t})\geq 2$, the above exact sequence
shows $\operatorname{depth}_{J}\operatorname{Tor}^{R}_{1}(M,S_{i}M)\geq 2$ as
$S_{i}M$ is maximally Cohen-Macaulay.

######  Lemma 4.2.

Let $U=X\setminus V(J)$ where $J=I_{t-1}(\varphi)$,
$X=\operatorname{Proj}(B_{t})$ and suppose $B_{t}$ is standard determinantal
and that $\operatorname{depth}_{J}(B_{t})\geq 2$. It holds

$\widetilde{\operatorname{Tor}_{1}^{R}(M,M)}\cong\widetilde{S_{2}M\otimes_{R}I_{t}}\text{
as sheaves over }U.$

In particular,
$\operatorname{Tor}_{1}^{R}(M,M)\cong\operatorname{H}_{*}^{0}(U,\widetilde{S_{2}M\otimes
I_{t}})$.

###### Proof.

First note that tensoring $0\to I_{t}\to R\to B_{t}\to 0$ with the
$B_{t}$-module $M$ and using the induced long exact sequence of
$\operatorname{Tor}_{j}$ for $j\leq 1$ we easily get
$\operatorname{Tor}^{R}_{1}(M,B_{t})\cong M\otimes_{R}I_{t}$. Then consider
the sheafification over $U$ of the exact sequence in Remark 4.1 for $i=0$.
Since $\widetilde{M_{|U}}$ is invertible, hence flat over $U$, we can tensor
this sequence with $\widetilde{M_{|U}}$, and we get the sheafification of the
exact sequence of Remark 4.1 for $i=1$. This shows the first isomorphism in

$\widetilde{\operatorname{Tor}_{1}^{R}(M,M)}\cong\widetilde{M}\otimes_{\tilde{B}_{t}}\widetilde{\operatorname{Tor}_{1}^{R}(M,B_{t})}\cong\widetilde{M}\otimes_{\tilde{B}_{t}}(\widetilde{M\otimes_{R}I_{t})}\cong\widetilde{S_{2}M\otimes_{R}I_{t}}.$

Since $\operatorname{depth}_{J}\operatorname{Tor}_{1}^{R}(M,M)\geq 2$ by
Remark 4.1, we also get the final conclusion. ∎

Combining Remark 4.1 and Lemma 4.2, we get that (4.1) holds.

######  Lemma 4.3.

Suppose $B_{t}$ is standard determinantal.

(i) If $\operatorname{depth}_{J}B_{t}\geq 4$ and
$c=\operatorname{codim}_{R}B_{t}=2$, then

$\operatorname{Tor}_{1}^{R}(M,M)\cong S_{2}M\otimes_{R}I_{t}.$

(ii) If $\operatorname{depth}_{J}B_{t}\geq 5$ and
$c=\operatorname{codim}_{R}B_{t}=3$, then

$\operatorname{Tor}_{1}^{R}(M,M)\cong
S_{2}M\otimes_{R}I_{t}\cong\operatorname{Ext}^{2}_{R}(I_{t},I_{t})(-\ell_{3})\quad\text{
where }\quad\ell_{3}=\sum_{j=1}^{t+2}a_{j}-\sum_{i=1}^{t}b_{i}.$

###### Proof.

(i) As $c=2$ and $\operatorname{depth}_{J}B_{t}\geq 4$, [11, Corollary 3.7]
applies and we get

$\operatorname{Ext}_{R}^{1}(B_{t},M)\cong\operatorname{Ext}_{R}^{1}(M,S_{2}M)$

by choosing $r=1$, $s=2$ and $a=1$ in Corollary 3.7. Using (5.2) below with
$j=1$ and $(r,s)=(0,2)$, and also with $(r,s)=(1,1)$, we get
$\operatorname{Tor}_{1}^{R}(M,M)\cong\operatorname{Tor}_{1}^{R}(B_{t},S_{2}M)$.
But if we tensor $0\to I_{t}\to R\to B_{t}\to 0$ with the $B_{t}$-module
$S_{2}M$ we deduce $\operatorname{Tor}^{R}_{1}(B_{t},S_{2}M)\cong
S_{2}M\otimes_{R}I_{t}$. And combining the two last isomorphisms we conclude
as claimed.

(ii) In this case the first isomorphism in (ii) follows from [11, Corollary
3.7]. Indeed choosing $r=s=1$ and $a=2$ in the “resp.-assumptions” of
Corollary 3.7, we get

$\operatorname{Ext}_{R}^{2}(M,M)\cong\operatorname{Ext}_{R}^{2}(B_{t},B_{t}).$

But (5.2) below with $j=1$ and $r=s=1$ yields
$\operatorname{Tor}_{1}(M,M)(l)\cong\operatorname{Ext}^{2}(M,M)$ where
$\ell=\ell_{3}$, and (5.2) with $j=1$, $r=2$ and $s=0$ yields
$\operatorname{Ext}^{2}(B_{t},B_{t})\cong\operatorname{Tor}_{1}(S_{2}M,B_{t})(l)$.
Thus we get

$\operatorname{Tor}_{1}^{R}(M,M)\cong\operatorname{Tor}_{1}^{R}(S_{2}M,B_{t})$

which proves what we want since $\operatorname{Tor}_{1}(S_{2}M,B_{t})\cong
S_{2}M\otimes I_{t}$ can be shown as in (i).

As $c=3$, we have $\operatorname{pd}I_{t}=2$, whence
$\operatorname{Ext}^{2}_{R}(I_{t},-)$ right-exact and since $S_{2}M$ is by
Proposition 2.2(iii) the canonical module
$K_{B_{t}}=\operatorname{Ext}^{2}_{R}(I_{t},R)(-n-1)$ of $B_{t}$ (up to
twist), we get

$\begin{array}[]{rcc}0=\operatorname{Ext}^{1}_{R}(I_{t},R)\longrightarrow\operatorname{Ext}^{1}_{R}(I_{t},B_{t})\longrightarrow\operatorname{Ext}_{R}^{2}(I_{t},I_{t})\longrightarrow\operatorname{Ext}^{2}_{R}(I_{t},R)&\longrightarrow&\operatorname{Ext}^{2}_{R}(I_{t},B_{t})\\\
\downarrow\cong&&\downarrow\cong\\\ S_{2}M(\ell)&\cong&S_{2}M(\ell)\otimes
B_{t}.\end{array}$

Indeed to see the twist, if we apply $\operatorname{Hom}_{R}(-,R)$ to the
$R$-free resolution $(P_{\bullet})$ of $I_{t}$ in (4.9) below, we find exactly
the $R$-free resolution of $S_{2}M(\ell)$ given by (2.1) and Proposition
2.2(ii). Moreover as the right-exactness of
$\operatorname{Ext}^{2}_{R}(I_{t},-)$ also implies
$\operatorname{Ext}^{2}_{R}(I_{t},I_{t})\cong\operatorname{Ext}^{2}_{R}(I_{t},R)\otimes
I_{t}$, we get

(4.3)
$\operatorname{Ext}^{1}_{R}(I_{t},B_{t})\cong\operatorname{Ext}^{2}_{R}(I_{t},I_{t})\cong
S_{2}M(\ell_{3})\otimes I_{t},$

as desired. ∎

######  Remark 4.4.

For generic determinantal rings (i.e. where the matrix $\mathcal{A}$ consists
of the indeterminites of the polynomial ring R) we have used Macaulay2 over
the field of rational numbers to show
$\operatorname{pd}(S_{2}M\otimes_{R}I_{t})=2c$ for $2\leq c\leq 5$ and $t=3$
as well as for $2\leq c\leq 9$ and $t=2$. Using notably Auslander-Buchsbaum
formula connecting depth and projective dimension, we get
$\operatorname{depth}_{J}(S_{2}M\otimes_{R}I_{t})\geq 2$ as $\dim B_{t}=c+2$,
hence $\dim R=2c+2$. This indicates that Lemma 4.3 likely extends to any
$c\geq 2$ as Conjecture 5.3 asserts. So these Macaulay2 computations give some
evidence to an important case of that conjecture, namely that
$\operatorname{Tor}^{R}_{1}(S_{2}M,B_{t})\cong\operatorname{Tor}^{R}_{1}(M,M)$
holds. As $\operatorname{Tor}^{R}_{1}(S_{2}M,B_{t})\cong
S_{2}M\otimes_{R}I_{t}$, Conjecture 5.3 asserts

(4.4) $\operatorname{Tor}^{R}_{1}(M,M)\cong S_{2}M\otimes_{R}I_{t}$

provided $\operatorname{depth}_{J}B_{t}\geq c+2$ which by Remark 4.1 implies
the exactness of the desired (4.1). Note that only assuming
$\operatorname{depth}_{J}B_{t}\geq 2$ and $B_{t}$ determinantal, (4.4) is
equivalent to $\operatorname{depth}_{J}(S_{2}M\otimes_{R}I_{t})\geq 2$ by
Lemma 4.2 as $\operatorname{depth}_{J}\operatorname{Tor}^{R}_{1}(M,M)\geq 2$
by Remark 4.1. Thus $\operatorname{pd}(S_{2}M\otimes_{R}I_{t})=2c$ implies
(4.4).

As a consequence of the above results we will be able to prove Conjecture 3.1
for $c=p=2$, and for $p=2$ and $c=3$ up to a concrete direct summand. As a
main tool we will use the following generalization of the mapping cone
process:

######  Lemma 4.5.

Let
$Q_{\bullet}\stackrel{{\scriptstyle\sigma_{\bullet}}}{{\longrightarrow}}P_{\bullet}\stackrel{{\scriptstyle\tau_{\bullet}}}{{\longrightarrow}}F_{\bullet}$
be morphisms of complexes satisfying $Q_{j}=P_{j}=F_{j}=0$ for $j<0$. Denote
by $d_{Q}^{i}:Q_{i}\to Q_{i-1}$ (respectively $d_{P}^{i}$, $d_{F}^{i}$) the
differentials of $Q_{\bullet}$ (respectively $P_{\bullet}$, $F_{\bullet}$).
Assume that all three complexes are acyclic (exact for $j\neq 0$) and that the
sequence

$0\longrightarrow\operatorname{coker}d_{Q}^{1}\longrightarrow\operatorname{coker}d_{P}^{1}\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}\operatorname{coker}d_{F}^{1}$

is exact. Moreover assume that there exists a morphism
$\ell_{\bullet}:Q_{\bullet}\longrightarrow F_{\bullet}[1]$ such that for any
integer $i$:

(4.5) $d_{F}^{i+1}\ell_{i}+\ell_{i-1}d_{Q}^{i}=\tau_{i}\sigma_{i}.$

Then, the complex $Q_{\bullet}\oplus P_{\bullet}[1]\oplus F_{\bullet}[2]$
given by

$Q_{i}\oplus P_{i+1}\oplus F_{i+2}\stackrel{{\scriptstyle
d^{i}_{Q,P,F}}}{{\longrightarrow}}Q_{i-1}\oplus P_{i}\oplus F_{i+1}$

where

$d^{i}_{Q,P,F}:=\begin{pmatrix}d^{i}_{Q}&0&0\\\ \sigma_{i}&-d_{P}^{i+1}&0\\\
\ell_{i}&-\tau_{i+1}&d^{i+2}_{F}\end{pmatrix}$

is acyclic (exact for $i\neq-2$) and
$\operatorname{coker}{d^{-1}_{Q,P,F}}=\operatorname{coker}\alpha$.

###### Proof.

See [11]; Lemma 3.1. ∎

We are now ready to prove the main result of this section.

######  Theorem 4.6.

Let $J=I_{t-1}(\varphi)$ and $B_{t}$ be standard determinantal, and assume
either $\operatorname{depth}_{J}B_{t}\geq 2$ and
$\operatorname{depth}_{J}(S_{2}M\otimes_{R}I_{t})\geq 2$, or
$\operatorname{depth}_{J}B_{t}\geq c+2$ and $2\leq c\leq 3$. Then Conjecture
3.1 (with $\varphi$ minimal) holds for $c=p=2$, and for $p=2$ and $c\geq 3$ up
to a direct summand.

###### Proof.

As we have $\operatorname{Tor}_{1}^{R}(M,M)\cong S_{2}M\otimes_{R}I_{t}$ by
Lemma 4.2, Lemma 4.3 and assumptions, the sequence

(4.6) $0\longrightarrow S_{2}M\otimes I_{t}\longrightarrow G^{*}\otimes
M\longrightarrow F^{*}\otimes M\longrightarrow M\otimes M\longrightarrow 0$

is exact. Using the free resolution of $S_{2}M$ given in Proposition 2.2(ii)

(4.7)
$\cdots\longrightarrow\bigwedge^{t+2}G^{*}\otimes\bigwedge^{t}F\longrightarrow\bigwedge^{2}G^{*}\longrightarrow
G^{*}\otimes F^{*}\longrightarrow S_{2}F^{*}\longrightarrow
S_{2}M\longrightarrow 0$

and the Eagon-Northcott resolution of $I_{t}$ appearing in the horizontal row
below

$\begin{array}[]{cccccccccc}&&\bigwedge^{t}G^{*}\otimes\bigwedge^{t}F\otimes
F^{*}\otimes G^{*}&&&\\\ \downarrow&&\downarrow&&&\\\
\bigwedge^{t+1}G^{*}\otimes F\otimes\bigwedge^{t}F\otimes
S_{2}F^{*}&\longrightarrow&\bigwedge^{t}G^{*}\otimes\bigwedge^{t}F\otimes
S_{2}F^{*}&&&\\\ \downarrow&&\downarrow&&&\\\ (\bigwedge^{t+1}G^{*}\otimes
F\otimes\bigwedge^{t}F)\otimes
S_{2}M&\longrightarrow&(\bigwedge^{t}G^{*}\otimes\bigwedge^{t}F)\otimes
S_{2}M&\longrightarrow&S_{2}M\otimes I_{t}&\longrightarrow 0\\\ \end{array}$

it is easy to see that a (possibly non-minimal) presentation of
$S_{2}M\otimes_{R}I_{t}$ starts out as:

$(G^{*}\otimes
F^{*}\otimes\bigwedge^{t}G^{*}\otimes\bigwedge^{t}F)\oplus(S_{2}F^{*}\otimes\bigwedge^{t+1}G^{*}\otimes
F\otimes\bigwedge^{t}F)\longrightarrow
S_{2}F^{*}\otimes\bigwedge^{t}G^{*}\otimes\bigwedge^{t}F\longrightarrow
S_{2}M\otimes I_{t}\longrightarrow 0.$

Free $R$-resolutions of $G^{*}\otimes M$ and of $F^{*}\otimes M$ are given by
Proposition 2.2(ii). Therefore, applying the generalization of the mapping
cone process (Lemma 4.5) we obtain that a free $R$-resolution of $M\otimes M$
begins as follows:

$\begin{array}[]{ccccccc}F^{*}\otimes\bigwedge^{t}F\otimes
F\otimes\bigwedge^{t+2}G^{*}\\\ \oplus\\\
G^{*}\otimes\bigwedge^{t}F\otimes\bigwedge^{t+1}G^{*}\\\ \oplus\\\
G^{*}\otimes F^{*}\otimes\bigwedge^{t}G^{*}\otimes\bigwedge^{t}F\\\ \oplus\\\
S_{2}F^{*}\otimes\bigwedge^{t+1}G^{*}\otimes
F\otimes\bigwedge^{t}F\end{array}\longrightarrow\begin{array}[]{ccccccc}G^{*}\otimes
G^{*}\\\ \oplus\\\ F^{*}\otimes\bigwedge^{t}F\otimes\bigwedge^{t+1}G^{*}\\\
\oplus\\\
S_{2}F^{*}\otimes\bigwedge^{t}G^{*}\otimes\bigwedge^{t}F\end{array}\longrightarrow(F^{*}\otimes
G^{*})^{\oplus 2}\longrightarrow F^{*}\otimes F^{*}\longrightarrow M\otimes
M\longrightarrow 0.$

Using Pieri’s formula 3.4 and letting $I=(0,\overbrace{1,\cdots,1}^{t-2},3)$
we have

$\begin{array}[]{rcl}S_{2}F^{*}\otimes\bigwedge^{t+1}G^{*}\otimes
F\otimes\bigwedge^{t}F&\cong&(S_{2}F^{*}\otimes\bigwedge^{t-1}F^{*})\otimes\wedge^{t}F\otimes\bigwedge^{t+1}G^{*}\otimes\bigwedge^{t}F\\\
&\cong&(F^{*}\otimes\bigwedge^{t}F^{*}\oplus\Sigma^{I}F^{*})\otimes\bigwedge^{t}F\otimes\bigwedge^{t+1}G^{*}\otimes\bigwedge^{t}F\\\
&\cong&F^{*}\otimes\bigwedge^{t}F\otimes\bigwedge^{t+1}G^{*}\oplus\Sigma^{I}F^{*}\otimes\bigwedge^{t}F\otimes\bigwedge^{t+1}G^{*}\otimes\bigwedge^{t}F.\end{array}$

It is important to point out that the summand
$H:=F^{*}\otimes\bigwedge^{t}F\otimes\bigwedge^{t+1}G^{*}$ will split off if
the corresponding matrix representing the last morphism of the above free
$R$-resolution of $M\otimes M$ is invertible. Macaulay2 computations and
explicit description of the involved morphism for small $t$ give evidences
that it should be true.

Using (4.7), the short exact sequence

$0\longrightarrow S_{2}M\longrightarrow M\otimes
M\longrightarrow\bigwedge^{2}M\longrightarrow 0,$

and applying once more the mapping cone process we get that a free
$R$-resolution of $\bigwedge^{2}M$ looks like

$\begin{array}[]{ccccccc}F^{*}\otimes\bigwedge^{t}F\otimes
F\otimes\bigwedge^{t+2}G^{*}\\\ \oplus\\\
G^{*}\otimes\bigwedge^{t}F\otimes\bigwedge^{t+1}G^{*}\\\ \oplus\\\
G^{*}\otimes F^{*}\otimes\bigwedge^{t}G^{*}\otimes\bigwedge^{t}F\\\ \oplus\\\
\Sigma^{I}F^{*}\otimes\bigwedge^{t}F\otimes\bigwedge^{t+1}G^{*}\otimes\bigwedge^{t}F\\\
\oplus\\\ H\end{array}\longrightarrow\begin{array}[]{ccccccc}S_{2}G^{*}\\\
\oplus\\\ S_{2}F^{*}\otimes\bigwedge^{t}G^{*}\otimes\bigwedge^{t}F\\\
\oplus\\\ H\end{array}\longrightarrow F^{*}\otimes
G^{*}\longrightarrow\bigwedge^{2}F^{*}\longrightarrow\bigwedge^{2}M\longrightarrow
0$

and hence, except for $H$, it starts as predicted in Conjecture 3.1; cf.
Theorem 3.7 for the case $c=p=2$. ∎

The last part of this section has as a main goal to explicitly construct a
full free $R$-resolution (possibly non-minimal) of $\bigwedge^{2}M$ for $2\leq
c\leq 3$ using mapping cone procedures. As an intermediate result we obtain a
free graded resolution of the normal module $N_{B_{t}}$ for $c=3$ (the case
$c=2$ is well known). A problem with the above procedure is that the free
resolution often contains redundant summands. In some cases one may more
easily get rid of all redundant summands using a generalization of the mapping
cone process. We illustrate this in Theorem 4.9 by finding a minimal free
graded resolution of the normal module $N_{B_{t}}$ for $c=3$. We analyze
separately 2 cases:

Case 1: Suppose $c=2$ and $\operatorname{depth}_{J}B_{t}\geq 4$. Applying
$\operatorname{Hom}_{R}(-,S_{2}M)$ to the exact sequence (4.2) continued by
$P\stackrel{{\scriptstyle\epsilon}}{{\longrightarrow}}G^{*}\longrightarrow\cdots$
to the left as in Remark 4.1, $\epsilon$ the ”splice map”, and using
$\operatorname{Ext}^{1}_{R}(M,S_{2}M)\cong S_{2}M\otimes I_{t}(\ell_{2})$
(which follows from (5.2) below) we get the following exact sequence

(4.8) $0\longrightarrow M\longrightarrow F\otimes S_{2}M\longrightarrow
G\otimes S_{2}M\longrightarrow S_{2}M\otimes I_{t}(\ell_{2})\longrightarrow 0$

as $\operatorname{Hom}_{R}(\epsilon,S_{2}M)=0$, (e.g. see [11], Equation (3.6)
for details). Since $M$ and $S_{2}M$ are maximally Cohen-Macaulay, the mapping
cone construction applied to (4.8) yields a free $R$-resolution of
$S_{2}M\otimes I_{t}(\ell_{2})$. Using the exact sequence (4.6), this leads
via mapping cone to a free $R$-resolution of $M\otimes M$ and using the short
exact sequence $0\longrightarrow S_{2}M\longrightarrow M\otimes
M\longrightarrow\bigwedge^{2}M\longrightarrow 0$ we finally get a free
$R$-resolution of $\bigwedge^{2}M$ of length $6$; cf. Theorem 3.7 which in
fact proves more.

Case 2: Suppose $c=3$ and $\operatorname{depth}_{J}B_{t}\geq 5$. Recall that
the Eagon-Northcott resolution of $I_{t}$ yields

(4.9) $(P_{\bullet}):\quad 0\longrightarrow S_{2}F(-\ell)\longrightarrow
G\otimes F(-\ell)\longrightarrow\bigwedge^{2}G(-\ell)\longrightarrow
I_{t}\longrightarrow 0$

with $\ell:=\ell_{3}=\sum_{j=1}^{t+2}a_{j}-\sum_{i=1}^{t}b_{i}$, as
$\bigwedge^{t+i}G\otimes\bigwedge^{t}F\cong\bigwedge^{2-i}G(-\ell)\ $.
Firstly, we describe a minimal free $R$-resolution of the normal module
$N_{B_{t}}:=\operatorname{Hom}_{R}(I_{t},B_{t})\cong\operatorname{Ext}^{1}_{R}(I_{t},I_{t})$.
As a main tool we will use again the generalization of the mapping cone
process. We have:

######  Theorem 4.7.

Let $\varphi:F:=\oplus_{i=1}^{t}R(b_{i})\longrightarrow
G:=\oplus_{j=1}^{t+2}R(a_{j})$ be a graded minimal $R$-morphism between two
free $R$-modules of rank $t$ and $t+2$, respectively. Set
$\ell:=\ell_{3}=\sum_{j=1}^{t+2}a_{j}-\sum_{i=1}^{t}b_{i}$ and assume
$\operatorname{depth}_{J}B_{t}\geq 4$ where $J=I_{t-1}(\varphi)$. Then the
normal module
$N_{B_{t}}:=\operatorname{Hom}_{R}(I_{t},B_{t})\cong\operatorname{Ext}^{1}_{R}(I_{t},I_{t})$
has a minimal free $R$-resolution of the following type:

$(N_{\bullet}):\quad
0\longrightarrow\bigwedge^{2}F(-\ell)\longrightarrow(G\otimes
F(-\ell))\longrightarrow(F\otimes G^{*})\oplus S_{2}G(-\ell)$
$\longrightarrow((F\otimes F^{*})\oplus(G\otimes G^{*}))/R\longrightarrow
G\otimes F^{*}\longrightarrow N_{B_{t}}\longrightarrow 0.$

###### Proof.

Since $\operatorname{depth}_{J}B_{t}\geq 4$, using [9]; Corollary 4.9 and
Theorem 5.2, we get $N_{B_{t}}\cong\operatorname{Ext}^{1}_{R}(M,M)$. Then,
[9]; Theorem 3.1 implies
$\operatorname{depth}N_{B_{t}}=\operatorname{depth}\operatorname{Ext}^{1}_{R}(M,M)\geq\dim
B_{t}-1$ and its proof implies that the sequence

(4.10) $0\longrightarrow B_{t}\longrightarrow F\otimes M\longrightarrow
G\otimes M\longrightarrow N_{B_{t}}\longrightarrow 0$

is exact. This last exact sequence together with Lemma 4.5 will allow us to
construct a minimal free $R$-resolution of the normal module $N_{B_{t}}$.
Indeed, the Buchsbaum-Rim resolution (see Proposition 2.2):

$0\longrightarrow F(-\ell)\cong\bigwedge^{t+2}G^{*}\otimes
F\otimes\bigwedge^{t}F\longrightarrow\bigwedge^{t+1}G^{*}\otimes\bigwedge^{t}F\cong
G(-\ell)\longrightarrow G^{*}\longrightarrow F^{*}\longrightarrow
M\longrightarrow 0$

together with the resolution $(P_{\bullet})$ of $B_{t}$ yields the diagram

---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{2}F(-\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{2}}$$\textstyle{F\otimes
F(-\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\otimes
F(-\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\otimes
F(-\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{1}}$$\scriptstyle{\quad\quad\ell_{1}}$$\textstyle{F\otimes
G(-\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\otimes
G(-\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigwedge^{2}G(-\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{0}}$$\scriptstyle{\quad\quad\ell_{0}}$$\textstyle{F\otimes
G^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\otimes
G^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{-1}}$$\scriptstyle{\quad\quad\ell_{-1}}$$\textstyle{F\otimes
F^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\otimes
F^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F\otimes
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\otimes
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{B_{t}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

where $\ell_{1}$, $\ell_{0}$, $\ell_{-1}$, $\tau_{2}$, $\tau_{1}$ and
$\tau_{-1}$ are the natural injections. Since
$\operatorname{coker}(\tau_{2})\cong\bigwedge^{2}F(-\ell)$,
$\operatorname{coker}(\ell_{0})\cong S_{2}G(-\ell)$ and
$\tau_{1}=\operatorname{id}$, the mapping cone construction (see Lemma 4.5)
gives us the minimal free $R$-resolution of $N_{B_{t}}$ (it is minimal as all
vertical and rightmost horizontal maps in the above diagram are minimal):

$(N_{\bullet}):\quad 0\longrightarrow\bigwedge^{2}F(-\ell)\longrightarrow
G\otimes F(-\ell)\longrightarrow(F\otimes G^{*})\oplus S_{2}G(-\ell)$
$\longrightarrow((F\otimes F^{*})\oplus(G\otimes G^{*}))/R\longrightarrow
G\otimes F^{*}\longrightarrow N_{B_{t}}\longrightarrow 0$

which proves what we want. ∎

Let us now compute a free $R$-resolution of $S_{2}M\otimes I_{t}$. We split
$(P_{\bullet})$ into two short exact sequences:

(4.11) $0\longrightarrow P_{-3}:=S_{2}F(-\ell)\longrightarrow P_{-2}:=G\otimes
F(-\ell)\longrightarrow E\longrightarrow 0$ (4.12) $0\longrightarrow
E\longrightarrow P_{-1}:=\bigwedge^{2}G(-\ell)\longrightarrow
I_{t}\longrightarrow 0$

and we use Lemma 4.3 which implies $S_{2}M(\ell)\otimes
I_{t}\cong\operatorname{Ext}^{2}_{R}(I_{t},I_{t})$. By (4.12) we have
$\operatorname{Ext}^{2}_{R}(I_{t},I_{t})\cong\operatorname{Ext}^{1}_{R}(E,I_{t})$,
and

(4.13)
$\begin{array}[]{ccccccccccc}0\longrightarrow&\operatorname{Hom}(I_{t},I_{t})&\longrightarrow&\operatorname{Hom}(P_{-1},I_{t})&\longrightarrow&\operatorname{Hom}(E,I_{t})&\longrightarrow&\operatorname{Ext}^{1}_{R}(I_{t},I_{t})&\longrightarrow
0\\\ &\cong\uparrow&&\cong\uparrow&&\parallel&&\cong\uparrow\\\
&R&\longrightarrow&P_{-1}^{*}\otimes
I_{t}&\longrightarrow&\operatorname{Hom}(E,I_{t})&\longrightarrow&N_{B_{t}}&\longrightarrow
0\\\ &&&\uparrow&&&&\uparrow\\\ &&&P_{-1}^{*}\otimes P_{-1}&&&&G\otimes
F^{*}\\\ &&&\uparrow&&&&\uparrow\\\ &&&P_{-1}^{*}\otimes P_{-2}&&&&((F\otimes
F^{*})\oplus(G\otimes G^{*}))/R\\\ &&&\uparrow&&&&\uparrow\\\
&&&P_{-1}^{*}\otimes P_{-3}&&&&(F\otimes G^{*})\oplus S_{2}G(-\ell)\\\
&&&\uparrow&&&&\uparrow\\\ &&&0&&&&G\otimes F(-\ell)\\\ &&&&&&&\uparrow\\\
&&&&&&&\bigwedge^{2}F(-\ell)\\\ &&&&&&&\uparrow\\\ &&&&&&&0\end{array}$

whence a free resolution of $\operatorname{Hom}(E,I_{t})$ is:

$0\longrightarrow\bigwedge^{2}F(-\ell)\longrightarrow G\otimes
F(-\ell)\longrightarrow(F\otimes G^{*})\oplus
S_{2}G(-\ell)\oplus(P_{-1}^{*}\otimes P_{-3})\longrightarrow$ $((F\otimes
F^{*})\oplus(G\otimes G^{*}))/R\oplus(P_{-1}^{*}\otimes
P_{-2})\longrightarrow(G\otimes F^{*})\oplus(P_{-1}^{*}\otimes
P_{-1}/R)\longrightarrow\operatorname{Hom}(E,I_{t})\longrightarrow 0.$

Now, we apply the functor $\operatorname{Hom}(-,I_{t})$ to the exact sequence
(4.11) and we get

$\begin{array}[]{ccccccccccc}0\longrightarrow\operatorname{Hom}(E,I_{t})&\longrightarrow&\operatorname{Hom}(P_{-2},I_{t})&\longrightarrow&\operatorname{Hom}(P_{-3},I_{t})&\longrightarrow&\operatorname{Ext}^{1}_{R}(E,I_{t})\longrightarrow
0.\\\ \parallel&&\cong\uparrow&&\cong\uparrow&&\cong\uparrow\\\
0\longrightarrow\operatorname{Hom}(E,I_{t})&\longrightarrow&P_{-2}^{*}\otimes
I_{t}&\longrightarrow&P_{-3}^{*}\otimes
I_{t}&\longrightarrow&\operatorname{Ext}^{2}_{R}(I_{t},I_{t})\\\
\uparrow&&\uparrow&&\uparrow\\\ (G\otimes F^{*})\oplus(P_{-1}^{*}\otimes
P_{-1}/R)&\longrightarrow&P_{-2}^{*}\otimes
P_{-1}&\longrightarrow&P_{-3}^{*}\otimes P_{-1}\\\
\uparrow&&\uparrow&&\uparrow\\\ ((F\otimes F^{*})\oplus(G\otimes
G^{*}))/R\oplus(P_{-1}^{*}\otimes P_{-2})&\longrightarrow&P_{-2}^{*}\otimes
P_{-2}&\longrightarrow&P_{-3}^{*}\otimes P_{-2}\\\
\uparrow&&\uparrow&&\uparrow\\\ (F\otimes G^{*})\oplus
S_{2}G(-\ell)\oplus(P_{-1}^{*}\otimes
P_{-3})&\longrightarrow&P_{-2}^{*}\otimes
P_{-3}&\longrightarrow&P_{-3}^{*}\otimes P_{-3}\\\
\uparrow&&\uparrow&&\uparrow&&&\\\ G\otimes F(-\ell)&&0&&0&&\\\
\uparrow&&&&&\\\ \bigwedge^{2}F(-\ell)&&&&&&\\\ \uparrow&&&&&&\\\
0&&&&&&\end{array}$

Finally, applying the mapping cone process we obtain the following free
$R$-resolution of $\operatorname{Ext}^{2}(I_{t},I_{t})(-\ell)\cong
S_{2}M\otimes I_{t}$:

$0\longrightarrow\bigwedge^{2}F(-2\ell)\longrightarrow G\otimes
F(-2\ell)\longrightarrow(F\otimes G^{*})(-\ell)\oplus
S_{2}G(-2\ell)\oplus(P_{-1}^{*}\otimes P_{-3})(-\ell)\longrightarrow$
$((F\otimes F^{*})\oplus(G\otimes G^{*}))/R\oplus(P_{-1}^{*}\otimes
P_{-2})(-\ell)\oplus(P_{-2}^{*}\otimes P_{-3})(-\ell)\longrightarrow$
$(G\otimes F^{*})\oplus((P_{-1}^{*}\otimes P_{-1})\oplus(P_{-2}^{*}\otimes
P_{-2})\oplus(P_{-3}^{*}\otimes P_{-3}))/R(-\ell)\longrightarrow$
$\begin{array}[]{ccccc}(P_{-2}^{*}\otimes
P_{-1})(-\ell)\oplus(P_{-3}^{*}\otimes
P_{-2})(-\ell)&\longrightarrow&(P_{-3}^{*}\otimes
P_{-1})(-\ell)&\longrightarrow S_{2}M\otimes I_{t}\longrightarrow 0.\\\
\parallel&&\parallel\\\ (G^{*}\otimes
F^{*}\otimes\bigwedge^{2}G)(-\ell)\oplus(S_{2}F^{*}\otimes G\otimes
F)(-\ell)&\longrightarrow&S_{2}F^{*}\otimes\bigwedge^{2}G(-\ell)\end{array}$

Once we have a free $R$-resolution of $S_{2}M\otimes I_{t}$ arguing as in Case
1, see also the proof of Theorem 4.6, we get a free $R$-resolution of
$\bigwedge^{2}M$ of length $8$.

######  Remark 4.8.

Using Macaulay2 we have checked that the above free $R$-resolution of
$S_{2}M\otimes I_{t}$ can contain redundant summands.

##  5\. Final comments

Based on the previous results and on a series of examples computed with
Macaulay2, we state the following guesses:

######  Conjecture 5.1.

Set $R=k[x_{0},\cdots,x_{n}]$. Let $\varphi^{*}:G^{*}\longrightarrow F^{*}$ be
a graded morphism between free $R$-modules of rank $t+c-1$ and $t$,
respectively. Assume that $\operatorname{depth}_{I_{t+1-p}}R=p(c+p-1)$ and
$c\geq 2$. Then for any $1\leq i\leq c+p-1$, $S_{i}(\bigwedge^{p}(M))$ is a
maximal Cohen-Macaulay $R/I_{t+1-p}$-module.

######  Remark 5.2.

(1) By [15], Proposition 4, we know that Conjecture 5.1 is true for $i=1$ and
any $t$, $c$ and $p$.

(2) Using Macaulay2 we have checked Conjecture 5.1 for any $1\leq i\leq c+p-1$
and any $p$ provided $c\leq 4$ and $t\leq 5$ and found it to be true.

######  Conjecture 5.3.

Set $R=k[x_{0},\cdots,x_{n}]$ and let $B_{t}:=R/I_{t}(\varphi)$ be standard
determinantal with $\operatorname{depth}_{J}B_{t}\geq 2$,
$J:=/I_{t-1}(\varphi)$ where $\varphi:F\to G$ is a graded morphism between
free $R$-modules of ranks $t$ and $t+c-1$ respectively. Let $j,r,s$ and $q$ be
integers and assume $j>\max(0,\frac{c+3-\operatorname{depth}_{J}B_{t}}{2})$
and $c\geq 2$. Then for $r,s,r-q,s+q\in[-1,c]$, it holds

(5.1)
$\operatorname{Tor}^{R}_{j}(S_{r}M,S_{s}M)\cong\operatorname{Tor}^{R}_{j}(S_{r-q}M,S_{s+q}M).$

######  Remark 5.4.

Using Macaulay2 we have checked that the hypothesis
$j>\max(0,\frac{c+3-\operatorname{depth}_{J}B_{t}}{2})$ in Conjecture 5.3
cannot be dropped, at least not when $\operatorname{depth}_{J}B_{t}=\dim
B_{t}<c+2$. Indeed,

(1) We consider the polynomial ring $R=\mathbb{Q}[a,b,c,d,e,f,g]$ and
$\mathcal{A}$ a $3\times 5$ matrix with entries general linear forms in $R$.
Associated to $\mathcal{A}$ we have a graded morphism $\varphi$ between free
$R$-modules of rank $t=3$ and $t+c-1=5$. We have got that
$\operatorname{Tor}^{R}_{2}(M,M)$ and
$\operatorname{Tor}^{R}_{2}(S_{2}M,B_{t})$ have the same graded Betti numbers
and thus are likely isomorphic. Moreover we got
$\operatorname{pd}\operatorname{Tor}^{R}_{1}(S_{2}M,B_{t})=6$, thus
$\operatorname{depth}\operatorname{Tor}^{R}_{1}(S_{2}M,B_{t})=1$ by Auslander-
Buchsbaum’s formula. As
$\operatorname{depth}\operatorname{Tor}^{R}_{1}(M,M)\geq 2$ by Remark 4.1,
$\operatorname{Tor}^{R}_{1}(S_{2}M,B_{t})$ cannot be isomorphic to
$\operatorname{Tor}^{R}_{1}(M,M)$. Note $\operatorname{depth}_{J}B_{t}=\dim
B_{t}=4$.

(2) Analogous result replacing $R$ by $\mathbb{Q}[a,b,c,d,e,f]$, but now
$\operatorname{pd}\operatorname{Tor}^{R}_{1}(S_{2}M,B_{t})=6$ yields
$\operatorname{depth}\operatorname{Tor}^{R}_{1}(S_{2}M,B_{t})=0$ and thus
$\operatorname{Tor}^{R}_{1}(S_{2}M,B_{t})\not\cong\operatorname{Tor}^{R}_{1}(M,M)$
while $\operatorname{depth}_{J}B_{t}=\dim B_{t}=3$.

(3) Taking $R=\mathbb{Q}[a,b,c,d,e]$ and $\mathcal{A}$ a $3\times 5$ matrix
with entries general linear forms in $R$, we got that
$\operatorname{Tor}^{R}_{j}(M,M)$ and
$\operatorname{Tor}^{R}_{j}(S_{2}M,B_{t})$ have different graded Betti numbers
for $j=1$ and for $j=2$. Thus
$\operatorname{Tor}^{R}_{j}(S_{2}M,B_{t})\not\cong\operatorname{Tor}^{R}_{j}(M,M)$
for $j=1$ and for $j=2$. Note $\operatorname{depth}_{J}B_{t}=\dim B_{t}=2$.

(4) Finally taking $R=\mathbb{Q}[a,b,c,d]$ and $\mathcal{A}$ as above, we got
that $\operatorname{Hom}(S_{-1}M,B_{t})$ and $M$ have different graded Betti
numbers. As
$\operatorname{Tor}^{R}_{3}(S_{3}M,B_{t})\cong\operatorname{Hom}(S_{-1}M,B_{t})$
and $\operatorname{Tor}^{R}_{3}(S_{2}M,M)\cong\operatorname{Hom}(B_{t},M)\cong
M$ by (5.2) below, the hypothesis $\operatorname{depth}_{J}B_{t}\geq 2$ cannot
in general be dropped.

The conjecture may even hold outside $[-1,c]$, but inside $[-1,c]$ we may use

(5.2)
$\operatorname{Ext}^{c-j}_{R}(S_{c-1-r}M,S_{s}M)\cong\operatorname{Tor}^{R}_{j}(S_{r}M,S_{s}M)(\ell)\,,\
\ \ \ {\rm for}\ -1\leq r,s\leq c$

of Remark 3.22(ii) of [11] (see the latest version in arXiv!) and the
corresponding isomorphism with $(r-q,s+q)$ in place of $(r,s)$ in (5.2), to
partially show Conjecture 5.3. Indeed we proved in Corollary 3.17 (in the
latest version in arXiv) that for $s^{\prime}\geq-1$, $r^{\prime}-1\leq
s^{\prime}\leq c$,

(5.3)
$\operatorname{Ext}^{i}_{R}(S_{r^{\prime}}M,S_{s^{\prime}}M)\cong\operatorname{Ext}^{i}_{R}(B_{t},S_{s^{\prime}-r^{\prime}}M)\
\ \ {\rm for\ every\ }i\leq\min\\{a,r^{\prime}-s^{\prime}+c\\}$

where a is an integer satisfying $\operatorname{depth}_{J}B_{t}\geq 2a+2$ (or
$\operatorname{depth}_{J}B_{t}\geq 2a+1$ if
$s^{\prime}-r^{\prime}\in\\{-1,0,c-2\\}$ and $a>1$) with $J=I_{t-1}(\varphi)$.
Let’s use (5.3) twice, both to $(r^{\prime},s^{\prime})=(c-1-r,s)$ and to
$(r^{\prime},s^{\prime})=(c-1-r+q,s+q)$, noticing that
$s^{\prime}-r^{\prime}=r+s+1-c$ in both cases. Also $r^{\prime}-1\leq
s^{\prime}$ yields $c-2\leq r+s$ and $r^{\prime}-s^{\prime}+c$ equals
$2c-1-r-s$ in both cases. So by (5.3) we get

$\operatorname{Ext}^{i}_{R}(S_{c-1-r}M,S_{s}M)\cong\operatorname{Ext}^{i}_{R}(S_{c-1-r+q}M,S_{s+q}M)\
\ {\rm for\ every\ }i\leq\min\\{a,2c-1-r-s\\}$

under the assumption $c-2\leq r+s$ and $-1\leq s,s+q\leq c$. This show that
(5.1) of Conjecture 5.3 holds when

(5.4)
$j\geq\max\\{r+s-c+1,c+1-\frac{\operatorname{depth}_{J}B_{t}}{2}\\}\quad{\rm
and}\quad r+s\geq c-2.$

The “resp.-assumptions” above gives slightly more when
$r+s+1-c\in\\{-1,0,c-2\\}$, namely that

(5.5) $c-2\geq
j\geq\max\\{r+s-c+1,c+\frac{1-\operatorname{depth}_{J}B_{t}}{2}\\}\quad{\rm
and}\quad r+s\geq c-2$

imply (5.1); note $c-2\geq j$ is included here to cover the assumption $a>1$.
For example if $(c,r,s)=(3,1,1)$, we get (5.1) for $j=1$ (and $-2\leq q\leq
2$) by (5.5) which imply what we separately prove in the first part of Lemme
4.3(ii).

Note that Conjecture 5.3 holds for $j\geq c$. Indeed $S_{i}M$ for
$i\in\\{s,s+q\\}$ is maximally Cohen-Macaulay, thus has projective dimension
$c$. So both $\operatorname{Tor}_{j}$-groups in (5.1) vanish for $j>c$. For
$j=c$ these $\operatorname{Tor}_{j}$-groups are isomorphic by (5.2) as
$\operatorname{depth}_{J}B_{t}\geq 2$ yields
$\operatorname{depth}_{J}\operatorname{Hom}_{R}(S_{k}M,S_{i}M)\geq 2$ and thus
$\operatorname{Hom}(S_{k}M,S_{i}M)\cong\operatorname{H}_{*}^{0}(U,\widetilde{\operatorname{Hom}(S_{k}M,S_{i}M)})$,
$U:=\operatorname{Proj}B_{t}\setminus V(J)$ for any $k\geq-1$. As
$\widetilde{S_{k}M}$ is invertible in $U$,
$\widetilde{\operatorname{Hom}(S_{r^{\prime}}M,S_{s}M)}\cong\widetilde{\operatorname{Hom}(S_{r^{\prime}+q}M,S_{s+q}M)}$
over $U$. Letting $r^{\prime}=c-1-r$ and using (5.2) we get that (5.1) holds.

Also notice, as explained in Remark 4.4 that for generic determinantal rings
we have used Macaulay2 over the field $\mathbb{Q}$ to get
$\operatorname{Tor}^{R}_{1}(S_{2}M,B_{t})\cong\operatorname{Tor}^{R}_{1}(M,M)$
for $2\leq c\leq 5$ and $t=3$ and also for $2\leq c\leq 9$ and $t=2$, giving
further evidence to an important case of (5.1) of Conjecture 5.3.

An interesting consequence of the conjecture is that if
$\max\\{0,\frac{c+3-\operatorname{depth}_{J}B_{t}}{2},r+s-c\\}<j\leq r+s+1$
and $\operatorname{depth}_{J}B_{t}\geq 2$, then each
$\operatorname{Tor}_{j}$-group of the conjecture has $\operatorname{depth}$ at
least $2$ at $J$. So if $U=\operatorname{Proj}B_{t}\setminus V(J)$, then

(5.6)
$\operatorname{Tor}^{R}_{j}(S_{r}M,S_{s}M)\cong\operatorname{H}_{*}^{0}(U,\widetilde{\operatorname{Tor}^{R}_{j}(S_{r}M,S_{s}M)}).$

This is because the conjecture implies
$\operatorname{Tor}^{R}_{j}(S_{r}M,S_{s}M)\cong\operatorname{Tor}^{R}_{j}(S_{j}M,S_{s+r-j}M)$
where the latter $\operatorname{Tor}_{j}$-group is the $j$-th homology group
of the $R$-free resolution of $S_{j}M$ in (2.1):

$\to
S_{0}(F)\otimes\bigwedge^{t}F\stackrel{{\scriptstyle\epsilon_{j}}}{{\longrightarrow}}\bigwedge^{j}G^{*}\otimes
S_{0}(F^{*})\rightarrow\bigwedge^{j-1}G^{*}\otimes
S_{1}(F^{*})\rightarrow\ldots\rightarrow\bigwedge^{0}G^{*}\otimes
S_{j}(F^{*})\to S_{j}M\rightarrow 0,$

tensored by $N:=S_{s+r-j}M$. Since ${\epsilon_{j}}\otimes id_{N}=0$ it follows
that $\operatorname{Tor}^{R}_{j}(S_{j}M,S_{s+r-j}M)$ is the kernel of
$G^{*}\otimes S_{0}(F^{*})\otimes N\rightarrow\bigwedge^{j-1}G^{*}\otimes
S_{1}(F^{*})\otimes N$ and since $N=S_{s+r-j}M$ is maximally Cohen-Macaulay,
due to $r+s-c\leq j\leq r+s+1$, we get $\operatorname{depth}_{J}N\geq 2$ and
hence (5.6).

It is also of interest to ask if (5.1) holds when $c=1$. In fact if
$\operatorname{depth}_{J}B_{t}\geq 2$ the answer is yes. Indeed using that
$\operatorname{Tor}^{R}_{i}(N_{1},N_{2})\cong\operatorname{Tor}^{R}_{i}(N_{2},N_{1})$
where $N_{j}$, $j=1,2$ are finitely generated $R$-modules, there’s only one
non-trivial case to consider, namely to show

(5.7)
$\operatorname{Tor}^{R}_{1}(B_{t},B_{t})\cong\operatorname{Tor}^{R}_{1}(M,S_{-1}M)\,.$

Let’s show (5.7) for any $c\geq 1$ assuming $\operatorname{depth}_{J}B_{t}\geq
c+1$, and that all entries of $\mathcal{A}$ are general of positive degree if
$c\geq 2$. Firstly $\operatorname{Tor}^{R}_{1}(B_{t},B_{t})\cong B_{t}\otimes
I_{t}\cong I_{t}/I_{t}^{2}$ has $\operatorname{depth}_{J}\geq 2$ by [11,
Proposition 4.1] if $c\geq 2$ (and $B_{t}$ is a complete intersection if
$c=1$). Next
$\operatorname{depth}_{J}\operatorname{Tor}^{R}_{1}(M,S_{-1}M)\geq 2$ by
Remark 4.1. Thus (5.6) applies to both $\operatorname{Tor}_{1}$-groups of
(5.7), and since, locally on $U$,

$\widetilde{\operatorname{Tor}_{1}(M,S_{-1}M)}\cong\widetilde{S_{-1}M\otimes\operatorname{Tor}^{R}_{1}(M,B_{t})}\cong\widetilde{S_{-1}M\otimes
M\otimes I_{t}}\cong\widetilde{I_{t}/I_{t}^{2}}$

by arguing as in the proof of Lemma 4.2 with $S_{-1}M$ in place of $M$, both
$\operatorname{Tor}_{1}$-groups are isomorphic, as desired, (cf. Remarks
3.22(v) of the latest version of [11] on the arXiv).

######  Remark 5.5.

Note that by (5.2), Conjecture 5.3 is equivalent to:

Let $i<\min(c,\frac{c-3+\operatorname{depth}_{J}B_{t}}{2}),r,s$ and $q$ be
integers and assume $\operatorname{depth}_{J}B_{t}\geq 2$ and $c\geq 2$. It
holds

(5.8) $\operatorname{Ext}^{i}_{R}(S_{r}M,S_{s}M)\cong\
\operatorname{Ext}^{i}_{R}(S_{r+q}M,S_{s+q}M)\ \ {\rm for}\ \
r,s,r+q,s+q\in[-1,c]\,,$

see [11, Proposition 3.9] and its correction (Corollary 3.17 and Remarks 3.18
and 3.22) appearing in the latest arXiv version of [11]. In Remark 3.18 we say
“we expect Corollary 3.17 to be true for $i\leq c-1$ and $X$ general” (with
$X:=\operatorname{Proj}B_{t}$), but Remark 5.4(i) and (ii) of this paper
provide examples with $c-1=2$ where
$\operatorname{Tor}^{R}_{1}(S_{2}M,B_{t})\not\cong\operatorname{Tor}^{R}_{1}(M,M)$.
Thus by (5.2),

(5.9) $\operatorname{Ext}^{i}_{R}(M,M)\not\cong\
\operatorname{Ext}^{i}_{R}(B_{t},B_{t})$

for $i=2$, and Corollary 3.17 (cf. (5.3) with $r^{\prime}=s^{\prime}=1$) is in
general not true for $i=c-1$. In Remark 5.4(iii) we even have an example where
also the corresponding $\operatorname{Tor}_{2}$-groups are non-isomorphic.
Thus (5.2) imply that (5.9) holds also for $i=1$ (and $i=2$) showing that
Corollary 3.17 doesn’t extend to cover the case $i=c-2$ either. In this latter
example, $i\leq c/2$ and the first two sentences in Remark 3.18 state that
Corollary 3.17 should be true in this case as a general determinantal $B_{t}$
satisfy $\operatorname{depth}_{J}B_{t}=c+2$. However, the latter statement
requires “enough room”, i.e. $\dim B_{t}\geq c+2$, which isn’t the case in
Remark 5.4(iii).

Note the “counterexamples above to extend Corollary 3.17” don’t satisfy
$i<\min(c,\frac{c-3+\operatorname{depth}_{J}B_{t}}{2})$ of (5.8) which is
included as an assumption to (5.8). Moreover we remark that the weak depth
condition $\operatorname{depth}_{J}B_{t}\geq 2$ is assumed in Conjecture 5.3
also because we often use $\operatorname{depth}_{J}B_{t}\geq 2$ when we
partially prove (5.1) of the conjecture, or equivalently (5.8).

Finally if we in (5.4) and (5.5) replace $j$ by $c-i$ and $r$ by $c-1-r$ we
get conditions under which (5.8) holds.

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