# Buchsbaum-Eisenbud complexes of OI-modules

Nathan Fieldsteel  Department of Mathematics, University of Kentucky,
Lexington, KY 40506 USA<EMAIL_ADDRESS>http://nathanfieldsteel.github.io and  Uwe Nagel  Department of Mathematics,
University of Kentucky, Lexington, KY 40506 USA<EMAIL_ADDRESS>http://www.ms.uky.edu/~uwenagel/

###### Abstract.

We extend the theory of Koszul and Buchsbaum-Eisenbud complexes to modules
over commutative OI-algebras and show that they still have the familiar
properties of the classical complexes. In particular, the OI-complexes are
generically acyclic and often provide width-wise minimal free resolutions.

###### 2020 Mathematics Subject Classification:

13D02, 16W22

The second author was partially supported by Simons Foundation grant #636513.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Classical Complexes Constructed via Multilinear Algebra
  4. 4 Exterior and Symmetric Powers and Duals of Free $\operatorname{OI}$-modules
  5. 5 The OI Koszul Complex
  6. 6 OI Buchsbaum-Eisenbud Complexes

##  1\. Introduction

The desire to study spaces of different dimensions simultaneously has
motivated many concepts and results. For example, the theory of
$\operatorname{FI}$-modules (over a fixed ring) allows one to express and
establish results in representation stability, where $\operatorname{FI}$ is
the category of finite sets with injections as morphisms. This theory has led
to many further exciting developments (see, e.g., [4, 5, 14]). In order to
incorporate the study of Markov bases in algebraic statistics (see, e.g.,
[9]), $\operatorname{FI}$-modules over an $\operatorname{FI}$-algebra
$\mathbf{A}$ were introduced in [11]. Establishing finiteness results required
even further generality by using the category $\operatorname{OI}$ of ordered
finite sets with order-preserving injections as morphisms. Gröbner bases for
$\operatorname{OI}$-modules over a polynomial $\operatorname{OI}$-algebra
$\mathbf{A}$ were introduced in [11]. A theory of Hilbert functions of graded
$\operatorname{OI}$-modules has been initiated in [10, 12] (see also [13]).
Over a noetherian polynomial $\operatorname{OI}$-algebra, the existence of
resolutions by finitely generated free $\mathbf{A}$-modules was established in
[11]. However, very few examples of such resolutions or general constructions
are known. This is not too surprising because even in the classical setting of
commutative algebra rather few widely applicable methods exist. Arguably, the
most famous complex in commutative algebra is the Koszul complex associated to
an $R$-module homomorphism $F\to R$. More generally, there is a family of
complexes associated to an $R$-module homomorphism $F\to G$ where both $F$ and
$G$ are finitely generated free $R$-modules. We call these complexes
Buchsbaum-Eisenbud complexes because of their appearance in [3]. They include
the Eagon-Northcott complex [6].

In this paper, we extend both constructions to the category of
$\operatorname{OI}$-modules.

######  Theorem 1.1.

If $\mathbf{A}$ is a commutative $\operatorname{OI}$-algebra and
$\varphi\colon\mathbf{F}\rightarrow\mathbf{A}$ is a morphism from a finitely
generated free $\mathbf{A}$-module to $\mathbf{A}$, then there exists a
complex $K_{\bullet}(\varphi)$ of finitely generated free $\mathbf{A}$-modules
for which the module in homological degree $d$ is the $d^{\text{th}}$ exterior
power of $\mathbf{F}$, and in each width $w$ the complex
$K_{\bullet}(\varphi)(w)$ is the ordinary Koszul complex on
$\varphi(w)\colon\mathbf{F}(w)\rightarrow\mathbf{A}(w)$. Furthermore, the
complex $K_{\bullet}(\varphi)$ is generically acyclic.

We use the term “generically acyclic” to indicate that there are circumstances
in which $K_{\bullet}(\varphi)$ is acyclic.

######  Theorem 1.2.

Let $\mathbf{A}$ be a commutative $\operatorname{OI}$-algebra and let
$\varphi:\mathbf{F}\rightarrow\mathbf{G}$ be a morphism of finitely-generated
free $\mathbf{A}$-modules where $\mathbf{G}$ is generated in width $0$. For
each $i\geq 0$, there exists a complex
$\operatorname{BE}^{i}_{\bullet}(\varphi)$ of finitely generated free
$\mathbf{A}$-modules that in each width $w$ is a Buchsbaum-Eisenbud complex on
$\varphi(w)$. Furthermore, $\operatorname{BE}^{i}_{\bullet}(\varphi)$ is
generically acyclic.

Both of these constructions produce complexes of graded modules if $\varphi$
is a morphism of graded $\mathbf{A}$-modules.

Minimality of graded free resolutions over a polynomial
$\operatorname{OI}$-algebra was investigated in [8]. In particular, it was
shown that, if it exists, a minimal graded free resolution of an $\mathbf{A}$
module is unique up to isomorphism and that, if $\mathbf{A}$ is noetherian,
any $\mathbf{A}$-module has such a minimal resolution.

Note that any free resolution $\mathbf{F}_{\bullet}$ of an $\mathbf{A}$-module
$\mathbf{M}$ provides in each width $w$, a free resolution of $\mathbf{M}(w)$
as an $\mathbf{A}(w)$-module, namely its width $w$ component
$\mathbf{F}_{\bullet}(w)$. If $\mathbf{F}_{\bullet}(w)$ is a minimal free
resolution for each $w$, the resolution $\mathbf{F}_{\bullet}$ is called
_width-wise minimal_. Such a resolution is necessarily minimal, but not every
minimal resolution is width-wise minimal (see [8]). Using cellular
resolutions, width-wise minimal free resolutions of some monomial ideals were
constructed in [8]. The complexes above provide many more such examples. If
$\varphi$ is graded and $\operatorname{BE}^{i}_{\bullet}(\varphi)$ is acyclic,
it gives a width-wise minimal free resolution. It resolves the $i^{\text{th}}$
symmetric power of $\operatorname{coker}\varphi$ if $i\geq 1$.

######  Example 1.3.

Let $\mathbf{A}$ be a polynomial $\operatorname{OI}$-algebra with three width
$1$ variables, so that in width $w$ we have $\mathbf{A}(w)=k[x_{i,j}\mid 1\leq
i\leq 3,~{}1\leq j\leq w]$. Let $\mathbf{F}$ and $\mathbf{G}=\mathbf{A}^{3}$
be free $\mathbf{A}$-modules, where $\mathbf{F}$ has rank $1$ and is generated
in width $1$, and consider the morphism
$\varphi\colon\mathbf{F}\rightarrow\mathbf{G}$ determined by sending the
generator of $\mathbf{F}$ to $\begin{bmatrix}x_{1,1}\\\ x_{2,1}\\\
x_{3,1}\end{bmatrix}\in\mathbf{G}(1)$. Thus, in each width $w$, the map
$\varphi(w)$ is represented by a generic $3\times w$ matrix containing the
variables of $\mathbf{A}(w)$. If we let $\mathbf{I}$ denote the ideal in
$\mathbf{A}$ generated in width $3$ by the determinant of $\varphi(3)$, then
$\mathbf{I}(w)$ is generated by the maximal minors of $\varphi(w)$ for all
$w$. In this situation, $\operatorname{BE}^{0}_{\bullet}(\varphi)$ gives a
width-wise minimal graded resolution of $\mathbf{I}$ by free
$\mathbf{A}$-modules. Any width $w\geq 3$ component
$\operatorname{BE}^{0}_{\bullet}(\varphi)(w)$ is the classical Eagon-Northcott
complex to $\varphi(w)$.

Our constructions of the above complexes mirror the classical constructions,
interpreted suitably. They use multilinear algebra and duals of modules.
Analogous complexes can be constructed in the category of
$\operatorname{FI}$-modules. However, we are not aware of any (non-trivial)
instance where these complexes provide width-wise minimal free resolutions.
Thus, we focus on $\operatorname{OI}$-modules in this paper.

In Section 2, we recall basic definitions, in particular, of polynomial
$\operatorname{OI}$-algebras and free modules. We also discuss basic
properties of tensor products and symmetric and exterior powers of modules
over any commutative $\operatorname{OI}$-algebra. Furthermore, we introduce an
internal hom functor that has the adjunction property (see Definition 2.6). In
Section 3, we review the classical complexes. In particular, we write the
classical Buchsbaum-Eisenbud complexes in a form so that they become amenable
to extensions for $\mathbf{A}$-modules. To this end, we describe the
differentials very explicitly. We conclude the preparations in Section 4,
where we show that tensor products and exterior and symmetric powers of
finitely generated free $\mathbf{A}$-modules are free and finitely generated.
Note that these statements witness identities involving binomial coefficients
(see, e.g, Example 4.3). We also characterize when a finitely generated free
$\mathbf{A}$-module is isomorphic to its dual. Finally, our extensions of
Koszul and Buchsbaum-Eisenbud complexes for $\operatorname{OI}$-modules are
presented in Sections 5 and 6. We illustrate them with several examples.

##  2\. Preliminaries

We are concerned with $\operatorname{OI}$-algebras over a commutative ring $k$
and modules over these algebras. For the sake of completeness, we briefly
recall the relevant definitions from [11], following [8].
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citations here We also introduce a suitable $\operatorname{Hom}$ functor and
review multilinear constructions in the category of
$\operatorname{OI}$-modules.

######  Definition 2.1.

Let $\operatorname{OI}$ be the category whose objects are totally-ordered
finite sets and whose maps are order-preserving injective functions. An
$\operatorname{OI}$-module (over $k$) is a functor from $\operatorname{OI}$ to
the category of $k$-modules.

The skeleton of $\operatorname{OI}$ is the subcategory of $\operatorname{OI}$
whose unique object of cardinality $w\in\mathbb{N}_{0}$ is
$\\{1,2,\ldots,w\\}$. By abuse of notation we use $w$ to refer to this object.
We use the term width to refer to this parameter. So for an
$\operatorname{OI}$-module $\mathbf{M}$, we call $\textbf{M}(w)$ the width $w$
component of M. By an element of $\mathbf{M}$ we mean an element $m$ in some
$\textbf{M}(w)$, which we call a width $w$ element of M.
$\operatorname{OI}$-modules form a functor category, where the morphisms
between two $\operatorname{OI}$ modules $\mathbf{M}$ and $\mathbf{N}$ are the
natural transformations $\textbf{M}\to\textbf{N}$. The set of these is denoted
$\operatorname{Nat}(\mathbf{M},\mathbf{N})$.

A submodule of an $\operatorname{OI}$-module $\mathbf{M}$ is just a subfunctor
of $\mathbf{M}$ that is an $\operatorname{OI}$-module itself. Given a set $G$
of elements of $\mathbf{M}$, the submodule generated by $G$ is the smallest
submodule that contains $G$.

We are interested in a more general class of objects where, instead of looking
at a functor from $\operatorname{OI}$ to $k$-mod, we consider families of
modules over different rings where both the modules and the rings are
parametrized by $\operatorname{OI}$.

######  Definition 2.2.

An $\operatorname{OI}$-algebra is a covariant functor from $\operatorname{OI}$
to the category of $k$-algebras. An $\operatorname{OI}$-algebra is called
commutative, graded or graded-commutative if it is a functor to the category
of $k$-algebras with the corresponding property. In this paper we will work
with commutative $\operatorname{OI}$-algebras unless otherwise stated, though
we will typically drop the adjective.

For an $\operatorname{OI}$-algebra $\mathbf{A}$, an $\mathbf{A}$-module is an
$\operatorname{OI}$-module $\mathbf{M}$ together with an action
$\mathbf{A}\otimes\mathbf{M}\stackrel{{\scriptstyle\mu}}{{\longrightarrow}}\mathbf{M}$
that satisfies the usual commuting diagrams required of an action map (see
[11, Definition 3.1]).

Equivalently, an $\operatorname{OI}$-algebra $\mathbf{A}$ is a commutative
monoid in the category of $\operatorname{OI}$-modules over $k$, and an
$\mathbf{A}$-module is a module in the category-theoretic sense. We will not
make use of this perspective in this paper.

Note that the constant functor from $\operatorname{OI}$ with image $k$ defines
not only an $\operatorname{OI}$-module, but also an
$\operatorname{OI}$-algebra. Modules over this $\operatorname{OI}$-algebra are
exactly the ordinary $\operatorname{OI}$-modules defined above. So the theory
of modules over an $\operatorname{OI}$-algebra is a generalization of the
theory of $\operatorname{OI}$-modules. From now on we will only consider
modules over an $\operatorname{OI}$-algebra.

######  Example 2.3.

Consider the functor $\mathbf{A}\colon\operatorname{OI}\rightarrow
k\text{-alg}$ for which, in any width $w$, $\mathbf{A}(w)$ is the coordinate
ring of the space of $3\times w$ matrices over $k$, that is,

$\mathbf{A}(w)=k\begin{bmatrix}x_{11}&x_{12}&&x_{1w}\\\
x_{21}&x_{22}&\cdots&x_{2w}\\\ x_{31}&x_{32}&&x_{3w}\end{bmatrix},$

and where any $\operatorname{OI}$-morphism $\varepsilon\colon u\rightarrow v$
induces an algebra map
$\varepsilon_{*}=\mathbf{A}(\varepsilon)\colon\mathbf{A}(u)\rightarrow\mathbf{A}(v)$
defined by $x_{ij}\mapsto x_{i\varepsilon(j)}$. In the language of [11], this
is the polynomial $\operatorname{OI}$-algebra
$\left(\mathbf{X}^{\operatorname{OI},1}\right)^{\otimes 3}$

Define a functor $\mathbf{I}\colon\operatorname{OI}\rightarrow k$-mod so that
$\mathbf{I}(w)\subseteq\mathbf{A}(w)$ is the ideal generated by the $3\times
3$ minors of the above $3\times w$ matrix. The maps
$\mathbf{A}(u)\rightarrow\mathbf{A}(v)$ induce maps
$\mathbf{I}(u)\rightarrow\mathbf{I}(v)$, and with these maps $\mathbf{I}$ is
an $\mathbf{A}$-module. As it is a submodule of $\mathbf{A}$, we call
$\mathbf{I}$ an ideal. It is generated as an $\mathbf{A}$-module by a single
element in width $3$, namely the determinant of the generic $3\times 3$
matrix, as any $3\times 3$ minor in any $\mathbf{I}(w)$ with $w\geq 3$ is the
image of this determinant under an appropriately chosen
$\operatorname{OI}$-morphism.

The most important $\mathbf{A}$-modules for our purposes are the free
$\mathbf{A}$-modules, which we define now.

######  Definition 2.4.

For any integer $n\geq 0$, define an $\mathbf{A}$-module
$\mathbf{F}_{\mathbf{A}}^{\operatorname{OI},n}$ as follows. For any
$w\in\mathbb{N}_{0}$, its width $w$ component
$\mathbf{F}_{\mathbf{A}}^{\operatorname{OI},n}(w)$ is the free
$\mathbf{A}(w)$-module with basis
$\left\\{e_{\nu}\mid\nu\in\operatorname{Hom}_{\operatorname{OI}}(n,w)\right\\}$,
and any $\operatorname{OI}$-morphism $\varepsilon\colon w\rightarrow
w^{\prime}$ acts on an element of
$\mathbf{F}_{\mathbf{A}}^{\operatorname{OI},n}(w)$ by

$\sum\limits_{\nu\in\operatorname{Hom}_{\operatorname{OI}}(n,w)}a_{\nu}e_{\nu}\mapsto\sum\limits_{\nu\in\operatorname{Hom}_{\operatorname{OI}}(n,w)}\varepsilon_{*}(a_{\nu})e_{\varepsilon\circ\nu}\in{\mathbf{F}}^{\operatorname{OI},n}_{\mathbf{A}}(w^{\prime}).$

$\mathbf{F}_{\mathbf{A}}^{\operatorname{OI},n}$ is generated as an
$\mathbf{A}$-module by the element $e_{\text{id}}$ in width $n$.

A finitely generated free $\mathbf{A}$-module of rank $r$ is an
$\mathbf{A}$-module that is isomorphic to a direct sum of the form

$\mathbf{F}_{\mathbf{A}}^{\operatorname{OI},n_{1}}\oplus\mathbf{F}_{\mathbf{A}}^{\operatorname{OI},n_{2}}\oplus\cdots\oplus\mathbf{F}_{\mathbf{A}}^{\operatorname{OI},n_{r}}.$

If $\mathbf{A}$ is clear from context, we often simply write
${\mathbf{F}}^{\operatorname{OI},w}$ instead of
${\mathbf{F}}^{\operatorname{OI},w}_{\mathbf{A}}$. By [8, Lemma 3.6], if
$\mathbf{F}$ is a finitely generated free $\mathbf{A}$-module, the number of
elements in any basis and their widths are unique. In particular, the number
of basis elements is equal to the rank of $\mathbf{F}$.

A morphism of $\mathbf{A}$-modules is a natural transformation
$\varphi\colon\mathbf{M}\rightarrow\mathbf{N}$ such that every map
$\varphi(w)\colon\mathbf{M}(w)\rightarrow\mathbf{N}(w)$ is a homomorphism of
$\mathbf{A}(w)$-modules. Note that any morphism out of a free
$\mathbf{A}$-module $\mathbf{F}$ is uniquely determined by the images of the
generators of $\mathbf{F}$. This justifies the name “free.”

######  Example 2.5.

The map $\varphi$ in Example 1.3 is an example of a map between free
$\mathbf{A}$-modules, where $\mathbf{A}$ is the $\operatorname{OI}$-algebra
considered in Example 2.3

For any $\operatorname{OI}$-algebra $\mathbf{A}$, the category of
$\mathbf{A}$-modules has a pointwise direct sum and a pointwise tensor product
defined by
$\left(\mathbf{M}\oplus\mathbf{N}\right)(w)=\mathbf{M}(w)\oplus\mathbf{N}(w)$
and
$\left(\mathbf{M}\otimes_{\mathbf{A}}\mathbf{N}\right)(w)=\mathbf{M}(w)\otimes_{\mathbf{A}(w)}\mathbf{N}(w)$,
respectively. One checks that this width-wise definition of the tensor product
agrees with the categorical definition of the tensor product as the
coequalizer of the diagram

${\mathbf{M}\otimes\mathbf{A}\otimes\mathbf{N}}$${\mathbf{M}\otimes\mathbf{N}.}$

Later, we want to consider the dual of an $\mathbf{A}$-module. This requires
some care and preparation. We use
$\operatorname{Nat}_{\mathbf{A}}(\mathbf{M},\mathbf{N})$ to denote the set of
$\mathbf{A}$-module morphisms $\mathbf{M}\to\mathbf{N}$. We note that while
this set is a $k$-module, it does not have an $\mathbf{A}$-module structure or
even an $\operatorname{OI}$-module structure. So it is not the internal
$\operatorname{Hom}$ in the category of $\mathbf{A}$-modules. To define the
latter, we use a different approach.

By the adjoint functor theorem, the functor
${-}\otimes_{\mathbf{A}}\mathbf{N}$ has a right adjoint, which we will denote
by $\operatorname{Hom}_{\mathbf{A}}(\mathbf{N},{-})$, a functor from
$\mathbf{A}$-mod to $\mathbf{A}$-mod. The enriched Yoneda lemma implies that
in every width $w$ we have

$\operatorname{Hom}_{\mathbf{A}}(\mathbf{N},{-})(w)=\operatorname{Nat}_{\mathbf{A}}(\mathbf{F}^{\operatorname{OI},w}_{\mathbf{A}}\otimes_{\mathbf{A}}\mathbf{N},{-}).$

This suggests the following definition.

######  Definition 2.6.

For $\mathbf{A}$-modules $\mathbf{M}$ and $\mathbf{N}$, define a functor
$\operatorname{Hom}_{\mathbf{A}}(\mathbf{M},\mathbf{N})\colon\operatorname{OI}\to
k$-mod: For any $w\in\mathbb{N}_{0}$, set

$\operatorname{Hom}_{\mathbf{A}}(\mathbf{M},\mathbf{N})(w)=\operatorname{Nat}_{\mathbf{A}}(\mathbf{F}^{\operatorname{OI},w}_{\mathbf{A}}\otimes_{\mathbf{A}}\mathbf{M},\mathbf{N}),$

and, for any $\operatorname{OI}$-morphism $\varepsilon\colon w\rightarrow
w^{\prime}$ and any $\mathbf{A}$-module morphism
$\varphi\colon\mathbf{F}^{\operatorname{OI},w}_{\mathbf{A}}\otimes_{\mathbf{A}}\mathbf{M}\rightarrow\mathbf{N}$,
define a $k$-module homomorphism
$\varepsilon_{*}\colon\operatorname{Nat}_{\mathbf{A}}(\mathbf{F}^{\operatorname{OI},w}_{\mathbf{A}}\otimes_{\mathbf{A}}\mathbf{M},\mathbf{N})\to\operatorname{Nat}_{\mathbf{A}}(\mathbf{F}^{\operatorname{OI},w^{\prime}}_{\mathbf{A}}\otimes_{\mathbf{A}}\mathbf{M},\mathbf{N})$
by putting

$\big{(}\varepsilon_{*}(\varphi)(v)\big{)}\left(e_{\eta}\otimes
m\right)=\varphi(e_{\eta\circ\varepsilon}\otimes m)\in\mathbf{N}(v),$

where $v\in\mathbb{N}_{0}$,
$\eta\in\operatorname{Hom}_{\operatorname{OI}}(w^{\prime},v)$ and
$m\in\mathbf{M}(v)$.

######  Proposition 2.7.

Consider $\mathbf{A}$-modules $\mathbf{M}$ and $\mathbf{N}$. For any integer
$w\geq 0$, any $a\in\mathbf{A}(w)$ and any $\mathbf{A}$-module morphism
$\varphi\colon{\mathbf{F}}^{\operatorname{OI},w}\otimes_{\mathbf{A}}\mathbf{M}\to\mathbf{N}$,
define an $\mathbf{A}$-module morphism
$a\varphi=a\cdot\varphi\colon{\mathbf{F}}^{\operatorname{OI},w}\otimes_{\mathbf{A}}\mathbf{M}\to\mathbf{N}$
by mapping $e_{\mu}\otimes
m\in{\mathbf{F}}^{\operatorname{OI},w}(v)\otimes_{\mathbf{A}(v)}\mathbf{M}(v)$
with $v\in\mathbb{N}_{0}$ (and so
$\mu\in\operatorname{Hom}_{\operatorname{OI}}(w,v)$) onto
$\varphi\big{(}\mu_{*}(a)e_{\mu}\otimes m\big{)}\in\mathbf{N}(v)$. This gives
$\operatorname{Nat}_{\mathbf{A}}(\mathbf{F}^{\operatorname{OI},w}_{\mathbf{A}}\otimes_{\mathbf{A}}\mathbf{M},\mathbf{N})$
the structure of an $\mathbf{A}(w)$-module and endows
$\operatorname{Hom}_{\mathbf{A}}(\mathbf{M},\mathbf{N})$ with the structure of
an $\mathbf{A}$-module.

###### Proof.

It is routine to check that the width-wise actions described above make
$\operatorname{Hom}_{\mathbf{A}}(\mathbf{M},\mathbf{N})(w)$ an
$\mathbf{A}(w)$-module. To verify that
$\operatorname{Hom}_{\mathbf{A}}(\mathbf{M},\mathbf{N})$ is an
$\mathbf{A}$-module, we also need to show that for any
$\operatorname{OI}$-morphism $\eta\colon v\rightarrow v^{\prime}$, the diagram

${\mathbf{A}(v)\otimes\operatorname{Hom}_{\mathbf{A}}(\mathbf{M},\mathbf{N})(v)}$${\operatorname{Hom}_{\mathbf{A}}(\mathbf{M},\mathbf{N})(v)}$${\mathbf{A}(v^{\prime})\otimes\operatorname{Hom}_{\mathbf{A}}(\mathbf{M},\mathbf{N})(v^{\prime})}$${\operatorname{Hom}_{\mathbf{A}}(\mathbf{M},\mathbf{N})(v^{\prime})}$

determined by $\eta$ commutes. This means, for any $\mathbf{A}$-module
morphism
$\varphi\colon{\mathbf{F}}^{\operatorname{OI},v}\otimes_{\mathbf{A}}\mathbf{M}\to\mathbf{N}$
and any $m\in\mathbf{A}(v)$, one has
$\eta_{*}(a\cdot\varphi)=\eta_{*}(a)\cdot\eta_{*}(\varphi)$. To check this is
in any width $w$, consider any
$\mu\in\operatorname{Hom}_{\operatorname{OI}}(v^{\prime},w)$ and any
$m\in\mathbf{M}(w)$. One computes

$\displaystyle\eta_{*}(a\cdot\varphi)(e_{\mu}\otimes m)$
$\displaystyle=\big{(}a\cdot\varphi\big{)}(e_{\mu\circ\eta}\otimes m)$
$\displaystyle=\varphi\big{(}(\mu\circ\eta)_{*}(a)\cdot
e_{\mu\circ\eta}\otimes m\big{)}$
$\displaystyle=\varphi\big{(}\mu_{*}(\eta_{*}(a))\cdot e_{\mu\circ\eta}\otimes
m\big{)}$ $\displaystyle=\eta_{*}(\varphi)\big{(}\mu_{*}(\eta_{*}(a))\cdot
e_{\mu}\otimes m\big{)}$
$\displaystyle=\big{(}\eta_{*}(a)\cdot\eta_{*}(\varphi)\big{)}(e_{\mu}\otimes
m),$

as desired. ∎

We always consider the $\mathbf{A}$-module structure on
$\operatorname{Hom}_{\mathbf{A}}(\mathbf{M},\mathbf{N})$ as defined above and
refer to it as the internal hom in the category of $\mathbf{A}$-modules. By
its definition, it has the adjunction property

$\operatorname{Nat}_{\mathbf{A}}(\mathbf{M}\otimes_{\mathbf{A}}\mathbf{N},\mathbf{P})=\operatorname{Nat}_{\mathbf{A}}(\mathbf{M},\operatorname{Hom}_{\mathbf{A}}(\mathbf{M},\mathbf{P})).$

We also need analogs of multilinear constructions in the category of
$\mathbf{A}$-modules.

######  Definition 2.8.

Let $\mathbf{A}$ be a commutative OI-algebra over $k$, and let $\mathbf{M}$ be
an $\mathbf{A}$-module. Define the tensor algebra of $\mathbf{M}$ over
$\mathbf{A}$ as

$\mathbf{T}(\mathbf{M})=\mathbf{A}\oplus(\mathbf{M})\oplus(\mathbf{M}\otimes_{\mathbf{A}}\mathbf{M})\oplus(\mathbf{M}\otimes_{\mathbf{A}}\mathbf{M}\otimes_{\mathbf{A}}\mathbf{M})\oplus\cdots.$

Note that in any width, $\big{(}\mathbf{T}(\mathbf{M})\big{)}(w)$ is simply
the tensor algebra of $\mathbf{M}(w)$ as an $\mathbf{A}(w)$-module. This means
that $\mathbf{T}(\mathbf{M})$ is itself an $\mathbf{A}$-module, though it also
comes equipped with an $\mathbf{A}$-linear multiplication map
$\mathbf{T}(\mathbf{M})\otimes_{\mathbf{A}}\mathbf{T}(\mathbf{M})\longrightarrow\mathbf{T}(\mathbf{M})$
determined by the width-wise algebra structures.

######  Definition 2.9.

For any $\mathbf{A}$-module $\mathbf{M}$, consider the two-sided ideal of the
tensor algebra $\mathbf{T}(\mathbf{M})$,

$\mathbf{I}=\left\langle m\otimes n-n\otimes m\mid
n,m~{}\text{in}~{}\mathbf{M}(w)~{}\text{for some
width}~{}w\in\mathbb{N}_{0}\right\rangle.$

The quotient of $\mathbf{T}(\mathbf{M})$ by $\mathbf{I}$ is called the
symmetric algebra of $\mathbf{M}$, denoted
$\operatorname{Sym}_{\bullet}(\mathbf{M})$.

The symmetric algebra of $\mathbf{M}$ is graded, and we denote the components
of its decomposition using the usual notation

$\operatorname{Sym}_{\bullet}(\mathbf{M})=\bigoplus\limits_{i=0}^{\infty}\operatorname{Sym}_{i}(\mathbf{M}),$

where each $\operatorname{Sym}_{i}(\mathbf{M})$ is an $\mathbf{A}$-module.

######  Definition 2.10.

For any $\mathbf{A}$-module $\mathbf{M}$, consider the two-sided ideal of the
tensor algebra $\mathbf{T}(\mathbf{M})$,

$\mathbf{J}=\left\langle m\otimes m\mid
m~{}\text{in}~{}\mathbf{M}(w)~{}\text{for some width}~{}w\right\rangle.$

The quotient of $\mathbf{T}(\mathbf{M})$ by $\mathbf{J}$ is called the
exterior algebra of $\mathbf{M}$, denoted
$\bigwedge\nolimits^{\bullet}(\mathbf{M})$.

The exterior algebra of $\mathbf{M}$ is also graded, and we denote the graded
components using the usual notation

$\bigwedge\nolimits^{\bullet}(\mathbf{M})=\bigoplus\limits_{i=0}^{\infty}\bigwedge\nolimits^{i}(\mathbf{M}),$

where each $\bigwedge\nolimits^{i}(\mathbf{M})$ is an $\mathbf{A}$-module.

The exterior and the symmetric algebra of $\mathbf{M}$ could be defined point-
wise. In fact, one checks that the width $w$ component of the exterior algebra
of $\mathbf{M}$ over $\mathbf{A}$ is simply the exterior algebra of
$\mathbf{M}(w)$ over $\mathbf{A}(w)$, that is,

$\left(\bigwedge\nolimits^{\bullet}(\mathbf{M})\right)(w)=\bigwedge\nolimits^{\bullet}\left(\mathbf{M}(w)\right).$

Similarly, one has

$\left(\operatorname{Sym}_{\bullet}(\mathbf{M})\right)(w)=\bigoplus\limits_{i=0}^{\infty}\operatorname{Sym}_{i}(\mathbf{M}(w)).$

##  3\. Classical Complexes Constructed via Multilinear Algebra

In this section we review classical complexes constructed by multilinear
algebra, which we will generalize to the $\operatorname{OI}$ setting in
Sections 5 and 6. In particular, we will discuss the Koszul complex and the
Eagon-Northcott complex. The latter is part of a family of complexes we dub
Buchsbaum-Eisenbud complexes because of their appearance in [3]. It is
important to note that our treatment of these complexes differs slightly from
their presentations in other sources (see, e.g., [2, 7]). The differences in
our presentation do not meaningfully change the complexes in the classical
setting, but allow us to obtain analogs in the category of
$\operatorname{OI}$-modules in subsequent sections. With this goal in mind, we
describe all chain maps in detail.

We begin with the Koszul complex, which is a complex of free $R$-modules
determined by the data of a map from a free $R$-module to $R$.

######  Definition 3.1.

Let $R$ be a commutative ring, let $F$ be a finitely generated free $R$-module
of rank $n$, and let $\varphi:F\rightarrow R$ be an $R$-module map. We define
_the Koszul complex_ $K_{\bullet}(\varphi)$ to be the chain complex with
modules $K_{d}(\varphi)=\bigwedge\nolimits^{d}(F)$ for $d\geq 0$, where for
$d\geq 1$ the map $K_{d}(\varphi)\rightarrow K_{d-1}(\varphi)$ is defined on
any wedge of $d$ elements of $F$ by

$e_{1}\wedge e_{2}\wedge\cdots\wedge
e_{d}\mapsto\sum\limits_{j=1}^{d}(-1)^{j+1}\varphi(e_{j})e_{1}\wedge
e_{2}\wedge\cdots\wedge\cancel{e_{j}}\wedge\cdots\wedge e_{n}.$

The Koszul complex is exact if and only if the elements
$\varphi(e_{1}),\ldots,\varphi(e_{n})$ form a regular sequence. In this case,
if $R$ is a standard graded algebra over a field it provides a graded minimal
free resolution of the ideal generated by the image of $\varphi$. For example,
if $R$ is a noetherian polynomial ring, and $\varphi:F\rightarrow R$ sends the
basis of $F$ to the variables of $R$, the Koszul complex
$K_{\bullet}(\varphi)$ provides a minimal free resolution of the graded
irrelevant ideal of $R$.

The family of Buchsbaum-Eisenbud complexes provides a significant
generalization of the Koszul complex. Their input is a homomorphism of free
$R$-modules.

######  Definition 3.2.

Let $F$ and $G$ be free $R$-modules with bases $\\{f_{1},\ldots,f_{n}\\}$ and
$\\{g_{1},\ldots,g_{r}\\}$, and let $\varphi\colon F\rightarrow G$ be an
$R$-module map, represented by a matrix $A=\left(a_{i,j}\right)\in R^{r\times
n}$ with respect to these bases. Let $({-})^{*}$ be the usual dualizing
functor $\operatorname{Hom}_{R}({-},R)$ and let
$\\{f_{1}^{*},\ldots,f_{n}^{*}\\}$ be the usual dual basis for $F^{*}$. For
any integer $i\geq 0$, we define the $i^{\text{th}}$ Buchsbaum-Eisenbud
complex to $\varphi$ as the chain complex $\text{BE}^{i}_{\bullet}(\varphi)$,

$\displaystyle 0$ $\displaystyle\to\bigwedge\nolimits^{r}(G^{*})\otimes
S_{n-r-i}(G^{*})\otimes\bigwedge\nolimits^{n}F\to\bigwedge\nolimits^{r}(G^{*})\otimes
S_{n-r-i-1}(G^{*})\otimes\bigwedge\nolimits^{n-1}F\to\cdots$
$\displaystyle\to\bigwedge\nolimits^{r}(G^{*})\otimes
S_{0}(G^{*})\otimes\bigwedge\nolimits^{r+i}F\stackrel{{\scriptstyle\alpha_{i}}}{{\longrightarrow}}\bigwedge\nolimits^{i}F\otimes
S_{0}(G)\to\bigwedge\nolimits^{i-1}F\otimes S_{1}(G)\to\cdots$
$\displaystyle\to\bigwedge\nolimits^{1}F\otimes
S_{i-1}(G)\to\bigwedge\nolimits^{0}F\otimes S_{i}(G)\to 0,$

where $S_{i}$ is short for $\operatorname{Sym}_{i}$ and the map
$\alpha_{i}\colon\bigwedge\nolimits^{r}(G^{*})\otimes
S_{0}(G^{*})\otimes\bigwedge\nolimits^{r+i}F\rightarrow\bigwedge\nolimits^{i}F\otimes
S_{0}(G)$ is defined by

$g_{1}^{*}\wedge\cdots\wedge g_{r}^{*}\otimes f_{j_{1}}\wedge\cdots\wedge
f_{j_{r+i}}\mapsto\sum\limits_{\begin{subarray}{c}I\subseteq J\\\
|I|=r\end{subarray}}\operatorname{sgn}(I\subseteq
J)\cdot\operatorname{det}(A_{I})f_{J\setminus I}.$

Here, $J=\\{j_{1}<\ldots<j_{r+i}\\}$, $f_{J}=f_{j_{1}}\wedge\cdots\wedge
f_{j_{r+i}}$, $f_{J\setminus I}$ is the wedge of elements with indices in
$J\setminus I$, $A_{I}$ is the $r$-minor of $A$ with column indices in $I$ and
$\operatorname{sgn}(I\subseteq J)$ is the sign of the permutation
$\begin{pmatrix}j_{1}&\ldots&j_{r}&j_{r+1}&\ldots&j_{r+i}\\\
i_{1}&\ldots&i_{r}&k_{1}&\ldots&k_{i}\end{pmatrix}$ with
$I=\\{i_{1}<\cdots<i_{r}\\}$ and $J\setminus I=\\{k_{1}<\cdots<k_{i}\\}$. By
definition, if $i+r>n$ the length of
$\operatorname{BE}^{i}_{\bullet}(\varphi)$ is $\min\\{i,n\\}$.

The part of $\operatorname{BE}^{i}_{\bullet}(\varphi)$ to the right of
$\alpha_{i}$ is a part of the Koszul complex determined by the $S(G)$-linear
map $F\otimes S(G)\to S(G)$ induced by $\varphi$. Concretely, the map
$\bigwedge\nolimits^{p}(F)\otimes
S_{q}(G)\rightarrow\bigwedge\nolimits^{p-1}(F)\otimes S_{q+1}(G)$ is defined
by

$f_{i_{1}}\wedge\cdots\wedge f_{i_{p}}\otimes g_{j_{1}}\cdots
g_{j_{q}}\mapsto\sum\limits_{\ell=1}^{p}\left(f_{i_{1}}\wedge\cdots\wedge\cancel{f_{i_{\ell}}}\wedge\cdots\wedge
f_{i_{p}}\otimes\varphi(f_{i_{\ell}})g_{j_{1}}\cdots g_{j_{q}}\right).$

Any map $\bigwedge\nolimits^{r}(G^{*})\otimes
S_{p}(G^{*})\otimes\bigwedge\nolimits^{q}F\rightarrow\bigwedge\nolimits^{r}(G^{*})\otimes
S_{p-1}(G^{*})\otimes\bigwedge\nolimits^{q-1}F$ to the left of $\alpha_{i}$ is
defined by

$g_{i_{1}}^{*}\cdots g_{i_{p}}^{*}\otimes f_{j_{1}}\wedge\cdots\wedge
f_{j_{q}}\mapsto\sum_{i\in\\{i_{1},\ldots,i_{p}\\}}\Big{(}g_{i}(g_{i_{1}}^{*}\cdots
g_{i_{p}}^{*})\otimes\sum\limits_{\ell=1}^{q}(-1)^{\ell+1}a_{i,j_{\ell}}f_{j_{1}}\wedge\cdots\wedge\cancel{f_{j_{\ell}}}\wedge\cdots\wedge
f_{j_{q}}\Big{)}.$

Here, $g_{i}(g_{i_{1}}^{*}\cdots g_{i_{p}}^{*})$ denotes the divided power
action of $S_{\bullet}(G)$ on $S_{\bullet}(G^{*}).$

Often, the part of $\operatorname{BE}^{i}_{\bullet}(\varphi)$ to the left of
the map $\alpha_{i}$ is given as the dual of a Koszul complex used for the
beginning of $\text{BE}^{i}_{\bullet}(\varphi)$ (see, e.g., [2]). Applying the
explicit isomorphisms given in [1, Section 1.6], one checks that one obtains
the maps as given above. The description of the above maps is not coordinate
free, but turns out useful in what follows.

The factor $\bigwedge\nolimits^{r}(G^{*})$ appearing in terms to the left of
$\alpha_{i}$ in $\operatorname{BE}^{i}_{\bullet}(\varphi)$ is often omitted.
If $R$ is a standard graded algebra, we assume that $\varphi$ is a map of
graded $R$-modules, i.e., it has degree zero. In this case, the factor
$\bigwedge\nolimits^{r}(G^{*})$ is needed to turn
$\operatorname{BE}^{i}_{\bullet}(\varphi)$ into a complex of graded free
$R$-modules.

It is well-known that the complexes $\text{BE}^{i}_{\bullet}(\varphi)$ are
generically exact (see, e.g., [7, Theorem A2.10]). Note that the arguments
also work if $i>n-r+1$ or if $n<r$.

######  Example 3.3.

For a generic $3\times w$ matrix $B$, denote by $I_{3}$ the ideal generated by
the $3$-minors of $B$. Using the notation of Example 1.3, if $w\geq 3$ then
the quotient $\mathbf{A}(w)/\mathbf{I}_{3}(A)$ is resolved as an
$\mathbf{A}(w)$-module by $\text{BE}^{0}_{\bullet}(\varphi)$, where
$\varphi\colon(\mathbf{A}(w))^{w}\to(\mathbf{A}(w))^{3}$ is the map given by
$B$. This was shown first by Eagon and Northcott. Often, the complex
$\text{BE}^{0}_{\bullet}(\varphi)$ is called an Eagon-Northcott complex
because of [6].

##  4\. Exterior and Symmetric Powers and Duals of Free
$\operatorname{OI}$-modules

Our goal is to generalize the complexes from the previous section to the
setting of modules over $\operatorname{OI}$-algebras. In this section, we
establish some needed preparations. Most of the following statements are
familiar classical results with commutative rings replaced by commutative
$\operatorname{OI}$-algebras. However, these similarities can mask important
differences. For example, the tensor product of two free $\mathbf{A}$-modules
of rank $1$ can be very complicated. Furthermore, duality behaves poorly in
many cases.

We begin with a criterion for checking whether an $\mathbf{A}$-module is free.

######  Lemma 4.1.

Let $\mathbf{A}$ be a commutative OI-algebra, and let $\mathbf{M}$ be a
finitely-generated $\mathbf{A}$-module. $\mathbf{M}$ is a free
$\mathbf{A}$-module if and only if $\mathbf{M}$ has a generating set
$G=\\{m_{1},\ldots,m_{n}\\}$ satisfying the following two conditions:

  1. (i)

For each $m_{i}$ in $G$, the $\mathbf{A}$-submodule of $\mathbf{M}$ generated
by $m_{i}$ is a free $\mathbf{A}$-module of rank 1.

  2. (ii)

For each $m_{i}$ in $G$, the intersection of the $\mathbf{A}$-submodule of
$\mathbf{M}$ generated by $m_{i}$ and the $\mathbf{A}$-submodule of
$\mathbf{M}$ generated by $G\setminus\\{m_{i}\\}$ is $0$.

###### Proof.

It is immediate that for a free $\mathbf{A}$ module $\mathbf{F}$, any basis
will satisfy these two conditions. Conversely, let $\mathbf{M}$ be a finitely
generated $\mathbf{A}$-module, and let $G$ be a generating set satisfying both
conditions. We will prove that $\mathbf{M}$ is free by induction on $|G|$.

If $|G|=1$, condition (i) implies that $\mathbf{M}$ is free. If $|G|>1$, then
let $\mathbf{M}^{\prime}$ be the free submodule generated by $m_{1}$, and let
$\mathbf{M}^{\prime\prime}$ be the submodule generated by
$G\setminus\\{m_{1}\\}$. The set $G\setminus\\{m_{1}\\}$ satisfies the two
conditions from the lemma, so by induction $\mathbf{M}^{\prime\prime}$ is
free. We can combine the submodule inclusion maps
$\mathbf{M}^{\prime}\hookrightarrow\mathbf{M}$ and
$\mathbf{M}^{\prime\prime}\hookrightarrow\mathbf{M}$ into a map on the direct
sum
$\mathbf{M}^{\prime}\oplus\mathbf{M}^{\prime\prime}\longrightarrow\mathbf{M}$,
yielding a map which is surjective since $G$ is a generating set for
$\mathbf{M}$, and injective because
$\mathbf{M}^{\prime}\cap\mathbf{M}^{\prime\prime}$ is zero. This completes the
proof. ∎

The first use we find for this lemma is to prove that the tensor product of
free $\mathbf{A}$-modules is free.

######  Proposition 4.2.

Let $\mathbf{A}$ be a commutative $\operatorname{OI}$-algebra. Given two
finitely generated free $\mathbf{A}$-modules $\mathbf{F}$ and
$\mathbf{F}^{\prime}$, their tensor product
$\mathbf{F}\otimes_{\mathbf{A}}\mathbf{F}^{\prime}$ is also free.

###### Proof.

Note that any tensor product of free $\mathbf{A}$-modules of finite rank can
be written as a direct sum of tensor products of rank 1 $\mathbf{A}$-modules.
It thus suffices to prove the assertion in the case where $\mathbf{F}$ and
$\mathbf{F}^{\prime}$ have rank 1.

To prove the proposition in the rank $1$ case, we explicitly describe a
generating set for
$\mathbf{M}=\mathbf{F}^{\text{OI},u}\otimes_{\mathbf{A}}\mathbf{F}^{\text{OI},v}$
in terms of $u$ and $v$ and show that it satisfies the two conditions in Lemma
4.1. In a fixed width $w$, the module
$\left(\mathbf{F}^{\text{OI},u}\otimes_{\mathbf{A}}\mathbf{F}^{\text{OI},v}\right)(w)$
is a free $\mathbf{A}(w)$-module with an $\mathbf{A}(w)$-module basis given by
the set of pure tensors of basis elements of $\mathbf{F}^{\text{OI,u}}(w)$ and
$\mathbf{F}^{\text{OI,v}}(w)$, i.e. it has a basis of the form

$\left\\{e_{\mu}\otimes
e_{\nu}\mid\mu\in\operatorname{Hom}_{\text{OI}}(u,w)~{}\text{and}~{}\nu\in\operatorname{Hom}_{\text{OI}}(v,w)\right\\}.$

For a particular pure tensor $e_{\mu}\otimes e_{\nu}$ in this set, we will use
$I_{\mu,\nu}$ to denote the union of the images of the indexing maps

$I_{\mu,\nu}=\operatorname{im}(\mu)\cup\operatorname{im}(\nu),$

which we think of as a subset of $[w]=\\{1,2,\ldots,w\\}$. Let $G$ be the set
of pure tensors in $\mathbf{M}$ for which the union of these images is the
full set $[w]$, i.e.

$G=\\{e_{\mu}\otimes e_{\nu}\in\mathbf{M}(w)\mid w\in\mathbb{N}_{0}\text{ and
}\ I_{\mu,\nu}=[w]\\}.$

We claim that $G$ satisfies the conditions of Lemma 4.1. To prove that $G$ is
a generating set, let $e_{\mu}\otimes e_{\nu}$ be an arbitrary pure tensor in
width $w$, with $|I_{\mu,\nu}|=w^{\prime}\leq w$. There is a unique OI
morphism $\varepsilon\in\operatorname{Hom}_{\text{OI}}(w^{\prime},w)$ whose
image is $I_{\mu,\nu}$, namely the inclusion from $[w^{\prime}]$ to
$I_{\mu,\nu}$. If we define
$\mu^{\prime}\in\operatorname{Hom}_{\text{OI}}(u,w^{\prime})$ by
$\mu^{\prime}(i)=\varepsilon^{-1}(\mu(i))$, and
$\nu^{\prime}\in\operatorname{Hom}_{\text{OI}}(v,w^{\prime})$ by
$\nu^{\prime}(i)=\varepsilon^{-1}(\nu(i))$, we see that
$e_{\mu^{\prime}}\otimes e_{\nu^{\prime}}$ is in $G$, and $e_{\mu}\otimes
e_{\nu}$ is in the $\mathbf{A}$-submodule generated by
$e_{\mu^{\prime}}\otimes e_{\nu^{\prime}}$.

To verify condition (i), let $e_{\mu}\otimes e_{\nu}$ be any width $w$ element
of $G$. Define a morphism $\mathbf{F}^{\text{OI},w}\to\langle e_{\mu}\otimes
e_{\nu}\rangle$ by mapping
$e_{\varepsilon}\in\mathbf{F}^{\text{OI},w}(w^{\prime})$ onto
$e_{\varepsilon\circ\mu}\otimes e_{\varepsilon\circ\nu}$. One checks that this
map is an isomorphism of $\mathbf{A}$-modules.

To prove that condition (ii) holds, it is enough to show, for any two distinct
elements $e_{\mu}\otimes e_{\nu}\in\mathbf{M}(w)$ and $e_{\mu^{\prime}}\otimes
e_{\nu^{\prime}}\in\mathbf{M}(w^{\prime})$ in $G$ and any two maps
$\varepsilon\in\operatorname{Hom}_{\operatorname{OI}}(w,\tilde{w})$,
$\varepsilon^{\prime}\in\operatorname{Hom}_{\operatorname{OI}}(w^{\prime},\tilde{w})$,
one has
$(\varepsilon\circ\mu,\varepsilon\circ\nu)\neq(\varepsilon^{\prime}\circ\mu^{\prime},\varepsilon^{\prime}\circ\nu^{\prime})$.
Indeed, using that
$w=|I_{\mu,\nu}|=|I_{\varepsilon\circ\mu,\varepsilon\circ\nu}|$ and the
analogous fact for the other element, this is not too difficult to show. ∎

Observe that the generators of
$\mathbf{F}^{\text{OI},u}\otimes_{\mathbf{A}}\mathbf{F}^{\text{OI},v}$ have
any width $w$ with $\max\\{u,v\\}\leq w\leq u+v$.

######  Example 4.3.

Consider the free rank 1 $\mathbf{A}$-modules
$=\mathbf{F}^{\operatorname{OI},2}_{\mathbf{A}}$ and
$\mathbf{F}^{\operatorname{OI},3}_{\mathbf{A}}$. Their tensor product has a
basis with 3, 12 and 10 elements of width 3, 4 and 5, respectively. Thus,
there is an isomorphism

$\mathbf{F}^{\operatorname{OI},2}_{\mathbf{A}}\otimes_{\mathbf{A}}\mathbf{F}^{\operatorname{OI},3}_{\mathbf{A}}\cong({\mathbf{F}}^{\operatorname{OI},3})^{3}\oplus({\mathbf{F}}^{\operatorname{OI},4})^{12}\oplus({\mathbf{F}}^{\operatorname{OI},5})^{10}.$

Comparing ranks in width $w$, we obtain for any $w\in\mathbb{N}$ the identity,

$\binom{w}{2}\cdot\binom{w}{3}=3\binom{w}{3}+12\binom{w}{4}+10\binom{w}{5}.$

Turning to exterior powers, we show that a familiar formula in the classical
setting remains true for $\mathbf{A}$-modules.

######  Lemma 4.4.

If $\mathbf{M}$ and $\mathbf{N}$ are $\mathbf{A}$-modules, then one has for
every integer $i\geq 0$,

$\bigwedge\nolimits^{i}\left(\mathbf{M}\oplus\mathbf{N}\right)\cong\bigoplus\limits_{j=0}^{i}\left(\bigwedge\nolimits^{i-j}\mathbf{M}\right)\otimes_{\mathbf{A}}\left(\bigwedge\nolimits^{j}\mathbf{N}\right).$

###### Proof.

By Equation (2), we have in each width $w$ a classical isomorphism of
$\mathbf{A}(w)$-modules

$\displaystyle\left(\bigwedge\nolimits^{i}\left(\mathbf{M}\oplus\mathbf{N}\right)\right)(w)=\bigwedge\nolimits^{i}\big{(}\mathbf{M}(w)\oplus\mathbf{N}(w)\big{)}\stackrel{{\scriptstyle\varphi}}{{\rightarrow}}$
$\displaystyle\bigoplus\limits_{j=0}^{i}\bigwedge\nolimits^{i-j}(\mathbf{M}(w))\otimes_{\mathbf{A}(w)}\bigwedge\nolimits^{j}(\mathbf{N}(w))=$
$\displaystyle\Big{(}\bigoplus\limits_{j=0}^{i}\big{(}\bigwedge\nolimits^{i-j}\mathbf{M}\big{)}\otimes_{\mathbf{A}}\big{(}\bigwedge\nolimits^{j}\mathbf{N}\big{)}\Big{)}(w).$

Thus, it is enough to verify that for any $\operatorname{OI}$-morphism
$\varepsilon\colon w\to w^{\prime}$ the following diagram is commutative:

${\left(\bigwedge\nolimits^{i}(\mathbf{M}\oplus\mathbf{N})\right)(w)}$${\Big{(}\bigoplus\limits_{j=0}^{i}\big{(}\bigwedge\nolimits^{i-j}\mathbf{M}\big{)}\otimes_{\mathbf{A}}\big{(}\bigwedge\nolimits^{j}\mathbf{N}\big{)}\Big{)}(w)}$${\left(\bigwedge\nolimits^{i}(\mathbf{M}\oplus\mathbf{N})\right)(w^{\prime})}$${\Big{(}\bigoplus\limits_{j=0}^{i}\big{(}\bigwedge\nolimits^{i-j}\mathbf{M}\big{)}\otimes_{\mathbf{A}}\big{(}\bigwedge\nolimits^{j}\mathbf{N}\big{)}\Big{)}(w^{\prime})}$$\scriptstyle{\varphi}$$\scriptstyle{\varepsilon_{*}}$$\scriptstyle{\varepsilon_{*}}$$\scriptstyle{\varphi}$

This is straightforward. ∎

Next, we show that the exterior power of a free $\mathbf{A}$-module is still
free.

######  Proposition 4.5.

Let $\mathbf{A}$ be a commutative $\operatorname{OI}$-algebra, and let
$\mathbf{F}$ be a finitely-generated free $\mathbf{A}$-module. Then for every
integer $i\geq 0$, the exterior power $\bigwedge\nolimits^{i}\mathbf{F}$ is
also a finitely-generated free $\mathbf{A}$-module.

###### Proof.

Because of Lemma 4.4, it suffices to prove the statement in the case where
$\mathbf{F}=\mathbf{F}^{\text{OI},d}$ is a free $\mathbf{A}$-module of rank
$1$. As for Proposition 4.2, the proof proceeds by giving an explicit
description of a generating set $G$ for
$\bigwedge\nolimits^{i}\mathbf{F}^{\text{OI},d}$ which satisfies the
conditions in Lemma 4.1.

In any given width $w$, using the standard lexicographic ordering on the set
of maps $\operatorname{Hom}_{\text{OI}}(d,w)$, we have a basis $B(w)$ for the
free $\mathbf{A}(w)$-module
$\left(\bigwedge\nolimits^{i}\mathbf{F}^{\text{OI},d}\right)(w)$ given by

$B(w)=\left\\{e_{\varepsilon_{1}}\wedge\ldots\wedge
e_{\varepsilon_{i}}~{}|~{}\varepsilon_{1},\ldots,\varepsilon_{i}\in\operatorname{Hom}_{\text{OI}}(d,w)~{}\text{and}~{}\varepsilon_{1}<\ldots<\varepsilon_{i}\right\\}.$

For such a basis element $e_{\varepsilon_{1}}\wedge\ldots\wedge
e_{\varepsilon_{i}}$, consider the union of the images of these indexing maps,
which we denote by $I_{\varepsilon_{1},\ldots,\varepsilon_{i}}$. Let $G$ be
the following subset of the union of the bases $B(w)$:

$G=\\{e_{\varepsilon_{1}}\wedge\ldots\wedge e_{\varepsilon_{i}}\mid
e_{\varepsilon_{1}}\wedge\ldots\wedge e_{\varepsilon_{i}}\in B(w)~{}\text{for
some}~{}w\in\mathbb{N}_{0}~{}\text{and}~{}I_{\varepsilon_{1},\ldots,\varepsilon_{i}}=[w]\\}.$

To show that $G$ generates $\bigwedge\nolimits^{i}\mathbf{F}$, it is enough to
establish that any $e_{1}\wedge\ldots\wedge e_{i}$ in any $B(w)$ is in the
$\mathbf{A}$-submodule of $\bigwedge\nolimits^{i}\mathbf{F}$ generated by $G.$
If $|I_{\varepsilon_{1},\ldots,\varepsilon_{i}}|=w$ the element is in $G$, and
we are done. If $|I_{\varepsilon_{1},\ldots,\varepsilon_{i}}|=w^{\prime}<w$,
let $\varepsilon$ be the unique $\operatorname{OI}$-morphism $\varepsilon$
from $[w^{\prime}]$ to $[w]$ whose image is
$I_{\varepsilon_{1},\ldots,\varepsilon_{i}}$. Defining
$\varepsilon_{j}^{\prime}\in\operatorname{Hom}_{\operatorname{OI}}(d,w^{\prime})$
on any $k\in[d]$ by $\varepsilon^{-1}(\varepsilon_{j}(k))$, we see that the
element $e_{\varepsilon_{1}^{\prime}}\wedge\ldots\wedge
e_{\varepsilon_{i}^{\prime}}$ is in $G$ and that the induced map
$\varepsilon\colon\bigwedge\nolimits^{i}\mathbf{F}(w^{\prime})\rightarrow\bigwedge\nolimits^{i}\mathbf{F}(w)$
sends this element to $e_{\varepsilon_{1}}\wedge\ldots\wedge
e_{\varepsilon_{i}}$. So $G$ is a generating set for
$\bigwedge\nolimits^{i}\mathbf{F}$ as an $\mathbf{A}$-module.

To verify that $G$ satisfies Condition (i) of Lemma 4.1, consider any element
$e_{\varepsilon_{1}}\wedge\ldots\wedge e_{\varepsilon_{i}}\in B(w)$ of $G$.
Define a morphism ${\mathbf{F}}^{\operatorname{OI},w}\to\langle
e_{\varepsilon_{1}}\wedge\ldots\wedge e_{\varepsilon_{i}}\rangle$ by mapping
$e_{\varepsilon}\in\mathbf{F}^{\text{OI},w}(w^{\prime})$ onto
$e_{\varepsilon\circ\varepsilon_{1}}\wedge\ldots\wedge
e_{\varepsilon\circ\varepsilon_{i}}$. One checks that this map is an
isomorphism of $\mathbf{A}$-modules.

To prove that $G$ satisfies Condition (ii) of Lemma 4.1, it is enough to show
for any two distinct elements $e_{\varepsilon_{1}}\wedge\ldots\wedge
e_{\varepsilon_{i}}\in B(w)$ and
$e_{\varepsilon^{\prime}_{1}}\wedge\ldots\wedge
e_{\varepsilon^{\prime}_{i}}\in B(w^{\prime})$ of $G$ and any two maps
$\varepsilon\in\operatorname{Hom}_{\operatorname{OI}}(w,\tilde{w})$,
$\varepsilon^{\prime}\in\operatorname{Hom}_{\operatorname{OI}}(w^{\prime},\tilde{w})$,
one has
$(\varepsilon\circ\varepsilon_{1},\ldots,\varepsilon\circ\varepsilon_{i})\neq(\varepsilon^{\prime}\circ\varepsilon^{\prime}_{1},\ldots,\varepsilon^{\prime}\circ\varepsilon^{\prime}_{i})$.
As in the proof of Proposition 4.2, this follows by using
$|I_{\varepsilon\circ\varepsilon_{1},\ldots,\varepsilon\circ\varepsilon_{i}}|=|I_{\varepsilon_{1},\ldots,\varepsilon_{i}}|$.
∎

Observe that the generators of
$\bigwedge\nolimits^{i}{\mathbf{F}}^{\operatorname{OI},d}$ have any width $w$
with $w_{0}\leq w\leq di$, where
$w_{0}=\min\\{w\in\mathbb{N}_{0}\mid\binom{w}{d}\geq i\\}$.

######  Example 4.6.

The $\mathbf{A}$-module $\bigwedge^{2}{\mathbf{F}}^{\operatorname{OI},3}$ has
rank 31. It has a basis with 6, 15 and 10 elements of width 4, 5 and 6,
respectively. Thus, there is an isomorphism

$\bigwedge^{2}{\mathbf{F}}^{\operatorname{OI},3}\cong({\mathbf{F}}^{\operatorname{OI},4})^{6}\oplus({\mathbf{F}}^{\operatorname{OI},5})^{15}\oplus({\mathbf{F}}^{\operatorname{OI},6})^{10}.$

Comparing ranks in width $w$, we obtain for any $w\in\mathbb{N}$ the identity

$\binom{\binom{w}{3}}{2}=6\binom{w}{4}+15\binom{w}{5}+10\binom{w}{6}.$

Now we establish the analogous result for the symmetric power of a free
$\mathbf{A}$-module.

######  Proposition 4.7.

Let $\mathbf{A}$ be a commutative $\operatorname{OI}$-algebra, and let
$\mathbf{F}$ be a finitely-generated free $\mathbf{A}$-module. Then, for every
integer $q\geq 0$, the symmetric power $\operatorname{Sym}_{q}(\mathbf{F})$ is
also a finitely-generated free $\mathbf{A}$-module.

###### Proof.

For any $\mathbf{A}$-modules $\mathbf{F}$ and $\mathbf{G}$, one has a
decomposition

$\operatorname{Sym}_{q}(\mathbf{F}\oplus\mathbf{G})=\bigoplus\limits_{i+j=q}\operatorname{Sym}_{i}(\mathbf{F})\otimes\operatorname{Sym}_{j}(\mathbf{G})$

This follows as in the proof of Lemma 4.4. Thus, it suffices to show the
statement in the case where $\mathbf{F}=\mathbf{F}^{\operatorname{OI},d}$. Now
one argues as in the proof of Proposition 4.5. ∎

To generalize the Buchsbaum-Eisenbud complexes of Section 3 to the
$\mathbf{A}$-module setting, we need to define the dual of a free module. If
$\mathbf{M}$ is an $\mathbf{A}$-module, we have already seen that
$\operatorname{Nat}_{\mathbf{A}}(\mathbf{M},\mathbf{A})$ is not a suitable
dual, because this is in general not an $\mathbf{A}$-module. Taking the width-
wise dual of $\mathbf{M}$ would produce a contravariant functor from
$\operatorname{OI}$ to $k$-mod, which is not even an
$\operatorname{OI}$-module. The natural candidate for the dual of an
$\mathbf{A}$-module is obtained by using the internal hom defined in
Definition 2.6.

######  Definition 4.8.

For an $\mathbf{A}$-module $\mathbf{M}$, we define the dual of $\mathbf{M}$ to
be $\operatorname{Hom}_{\mathbf{A}}(\mathbf{M},\mathbf{A})$, and we denote it
by $\mathbf{M}^{*}$.

Thus, in width $w$, we have
$\mathbf{M}^{*}(w)=\operatorname{Nat}_{\mathbf{A}}(\mathbf{F}^{\operatorname{OI},w}_{\mathbf{A}}\otimes_{\mathbf{A}}\mathbf{M},\mathbf{A})$.

Unfortunately, many statements about the dual in the category of modules over
a commutative ring $R$ do not generalize to the setting of modules over an
$\operatorname{OI}$-algebra $\mathbf{A}$. For example, if $d\geq 1$ and
$\mathbf{F}={\mathbf{F}}^{\operatorname{OI},d}$, then $\mathbf{F}(0)$ is zero,
but
$\mathbf{F}^{*}(0)=\operatorname{Nat}_{\mathbf{A}}({\mathbf{F}}^{\operatorname{OI},d},\mathbf{A})$
is not zero, and so ${\mathbf{F}}^{\operatorname{OI},d}$ is not isomorphic to
its dual. However, free modules generated in width zero are isomorphic to
their duals.

######  Proposition 4.9.

Let $\mathbf{F}$ be a finitely generated free $\mathbf{A}$-module generated in
width $0$, that is, $\mathbf{F}\cong({\mathbf{F}}^{\operatorname{OI},0})^{n}$
for some $n\in\mathbb{N}$. Then there is an isomorphism
$\mathbf{F}\rightarrow\mathbf{F}^{*}$ of $\mathbf{A}$-modules.

###### Proof.

Note that $\mathbf{F}^{\operatorname{OI},0}_{\mathbf{A}}$ is isomorphic to
$\mathbf{A}$ as an $\mathbf{A}$-module. Assume
$\mathbf{F}\cong({\mathbf{F}}^{\operatorname{OI},0})^{n}$ and let
$\\{f_{1},f_{2},\ldots,f_{n}\\}\subset\mathbf{F}(0)$ be a basis for
$\mathbf{F}$. If we let $e_{\text{id}_{w}}$ be the generator of
$\mathbf{F}^{\operatorname{OI},w}_{\mathbf{A}}$, then
$\mathbf{F}^{\operatorname{OI},w}_{\mathbf{A}}\otimes_{\mathbf{A}}\mathbf{F}$
is a free $\mathbf{A}$-module of rank $n$. Due to Proposition 4.2, it has a
basis consisting of width $w$ elements

$\\{e_{\text{id}_{w}}\otimes\eta_{*}(f_{1}),e_{\text{id}_{w}}\otimes\eta_{*}(f_{2}),\ldots,e_{\text{id}_{w}}\otimes\eta_{*}(f_{n})\\},$

where $\eta$ is the unique $\operatorname{OI}$ morphism in
$\operatorname{Hom}_{\operatorname{OI}}(0,w)$. Hence, any $\mathbf{A}$-module
morphism
$\varphi\colon{\mathbf{F}}^{\operatorname{OI},w}\otimes_{\mathbf{A}}\mathbf{F}\to\mathbf{A}$
is determined by the images of $e_{\text{id}_{w}}\otimes\eta_{*}(f_{j})$ in
$\mathbf{A}(w)$, say $a_{j}$.

For $i\in[n]$, define an $\mathbf{A}$-module morphism
$f_{i}^{*}\colon{\mathbf{F}}^{\operatorname{OI},0}\otimes_{\mathbf{A}}\mathbf{F}\to\mathbf{A}$
by mapping $e_{\text{id}_{0}}\otimes f_{j}$ onto $1$ if $j=i$ and onto $0$ if
$j\neq i$. Using $\varepsilon\in\operatorname{Hom}_{\operatorname{OI}}(0,w)$
and the definition of the $\mathbf{A}$-module structure of
$\mathbf{F}^{*}=\operatorname{Hom}_{\mathbf{A}}(\mathbf{F},\mathbf{A})$, a
computation shows for the above morphism $\varphi$ that
$\varphi=a_{1}\mathbf{F}^{*}(\varepsilon)(f_{1}^{*})+\cdots+a_{n}\mathbf{F}^{*}(\varepsilon)(f_{n}^{*})$.
It follows that the $\mathbf{A}$-module morphism $\mathbf{F}\to\mathbf{F}^{*}$
defined by $f_{i}\mapsto f_{i}^{*}$ is an isomorphism. ∎

##  5\. The OI Koszul Complex

inline,color=gray!10,linecolor=gray!40!white,bordercolor=gray!30!white,size=inline,color=gray!10,linecolor=gray!40!white,bordercolor=gray!30!white,size=todo:
inline,color=gray!10,linecolor=gray!40!white,bordercolor=gray!30!white,size=Put
Thm 1.1 here

If $R$ is a commutative ring and $F$ is a finitely generated free $R$-module,
then the Koszul complex to an $R$-module homomorphism $\varphi\colon F\to R$
is a bounded complex of finitely generated free $R$-modules that is
generically acyclic and, if it is acyclic, resolves
$\operatorname{coker}\varphi$. After the preceding preparations, we establish
in this section an analogous result for modules over a commutative
$\operatorname{OI}$-algebra $\mathbf{A}$.

######  Theorem 5.1.

Let $\mathbf{A}$ be a commutative $\operatorname{OI}$-algebra and consider a
morphism of $\mathbf{A}$-modules $\varphi\colon\mathbf{F}\to\mathbf{A}$, where
$\mathbf{F}$ is a finitely generated free $\mathbf{A}$-module. Then there is a
complex of finitely generated free $\mathbf{A}$-modules

$\cdots\longrightarrow\bigwedge\nolimits^{3}\mathbf{F}\longrightarrow\bigwedge\nolimits^{2}\mathbf{F}\longrightarrow\bigwedge\nolimits^{1}\mathbf{F}\longrightarrow\bigwedge\nolimits^{0}\mathbf{F}=\mathbf{A}\longrightarrow
0,$

where each differential
$\partial_{i}\colon\bigwedge\nolimits^{i}\mathbf{F}\longrightarrow\bigwedge\nolimits^{i-1}\mathbf{F}$
is defined width-wise by letting
$\partial_{i}(w)\colon\bigwedge\nolimits^{i}(\mathbf{F}(w))\longrightarrow\bigwedge\nolimits^{i-1}(\mathbf{F}(w))$
be the degree $i$ map of the classical Koszul complex to the
$\mathbf{A}(w)$-module homomorphism
$\varphi(w)\colon\mathbf{F}(w)\rightarrow\mathbf{A}(w)$. This complex is
generically acyclic and, if it is acyclic, resolves
$\operatorname{coker}\varphi$.

Furthermore, if $\varphi$ is a morphism of graded $\mathbf{A}$-modules, then
the above complex is a graded complex.

We refer to the above complex as the $\operatorname{OI}$ Koszul complex to
$\varphi$ and denote it by $K_{\bullet}(\varphi)$. Note that the above result
covers Theorem 1.1 of the Introduction.

###### Proof of Theorem 5.1.

By Proposition 4.5, we know that any module $\bigwedge^{i}\mathbf{F}$ is
finitely generated and free and that, for any $w\in\mathbb{N}_{0}$, one has
$\big{(}\bigwedge^{i}\mathbf{F}\big{)}(w)=\bigwedge^{i}(\mathbf{F}(w))$.
Moreover, since $\partial_{i}(w)$ is the differential of a classical Koszul
complex, the stated sequence is indeed a complex. Analogously, it follows that
$K_{\bullet}(\varphi)$ is graded if $\varphi$ is graded.

It remains to show that the family of $\mathbf{A}(w)$-module homomorphisms
$\partial_{i}(w)$ can be used to define a morphism of $\mathbf{A}$-modules
$\partial_{i}\colon\bigwedge\nolimits^{i}\mathbf{F}\longrightarrow\bigwedge\nolimits^{i-1}\mathbf{F}$,
that is, for any
$\varepsilon\in\operatorname{Hom}_{\operatorname{OI}}(w,w^{\prime})$, the
diagram

${\big{(}\bigwedge^{i}\mathbf{F}\big{)}(w)}$${\big{(}\bigwedge^{i-1}\mathbf{F}\big{)}(w)}$${\big{(}\bigwedge^{i}\mathbf{F}\big{)}(w^{\prime})}$${\big{(}\bigwedge^{i-1}\mathbf{F}\big{)}(w^{\prime})}$$\scriptstyle{\partial_{i}(w)}$$\scriptstyle{(\bigwedge^{i}\mathbf{F})(\varepsilon)}$$\scriptstyle{(\bigwedge^{i-1}\mathbf{F})(\varepsilon)}$$\scriptstyle{\partial_{i}(w^{\prime})}$

commutes. This is a straightforward computation. We leave the details to the
reader. ∎

Since the modules $\bigwedge^{i}\mathbf{F}$ are free, there is an alternative
way to describe the above $\operatorname{OI}$ Koszul complex. One can define
the differential $\partial_{i}$ directly as a morphism of $\mathbf{A}$-modules
by specifying the images of the basis elements of $\bigwedge^{i}\mathbf{F}$
and then arguing that in every width $w$, the map $\partial_{i}(w)$ is the
differential in a classical complex of $\mathbf{A}(w)$-modules. Note that
$\operatorname{OI}$ Koszul complexes are infinite, in contrast to the
classical Koszul complex.

Special cases of the above Koszul complex have appeared previously in the
literature.

######  Example 5.2.

The first $\operatorname{OI}$ Koszul complex was constructed in [11]. Consider
any $\mathbf{A}$-module morphism
$\varphi\colon{\mathbf{F}}^{\operatorname{OI},1}_{\mathbf{A}}\to\mathbf{A}$.
It is determined by $a=\varphi(e_{\operatorname{id}})\in\mathbf{A}(1)$.
Moreover, one computes
$\bigwedge^{i}{\mathbf{F}}^{\operatorname{OI},1}_{\mathbf{A}}\cong{\mathbf{F}}^{\operatorname{OI},i}_{\mathbf{A}}$,
and so the complex $K_{\bullet}(\varphi)$ has a particular simple form

$\cdots\longrightarrow{\mathbf{F}}^{\operatorname{OI},3}_{\mathbf{A}}\longrightarrow{\mathbf{F}}^{\operatorname{OI},2}_{\mathbf{A}}\longrightarrow{\mathbf{F}}^{\operatorname{OI},1}_{\mathbf{A}}\longrightarrow\mathbf{A}\longrightarrow
0.$

This is the version given in [11, Lemma 3.8]. In the case where, using
notation from [11], $\mathbf{A}$ is the polynomial $\operatorname{OI}$-algebra
$\mathbf{A}=\mathbf{X}^{\operatorname{OI},1}\cong\operatorname{Sym}_{\bullet}({\mathbf{F}}^{\operatorname{OI},1})$,
and so $\mathbf{A}(w)=k[x_{j}\mid j\in[w]]$ is a polynomial ring in $w$
variables, the Koszul complex to
$\varphi\colon{\mathbf{F}}^{\operatorname{OI},1}_{\mathbf{A}}\to\mathbf{A}$
with $\varphi(e_{\operatorname{id}})=x_{1}$ is acyclic and has the structure
of a cellular resolution by [8, Theorem 4.10]. It resolves
$\operatorname{coker}\varphi=\mathbf{A}/\langle x_{1}\rangle$, and in fact
gives a minimal graded free resolution of $\operatorname{coker}\varphi$. The
existence of minimal resolutions in certain cases has been established in [8,
Theorem 3.10]. They are unique up to isomorphism.

Moreover, $K_{\bullet}(\varphi)$ is a width-wise minimal free resolution in
the sense of [8], that is, in each width $w$, the complex
$(K_{\bullet}(\varphi))(w)$ is a minimal graded free resolution of
$(\operatorname{coker}\varphi)(w)$, which is $k[x_{1},\ldots,x_{w}]/\langle
x_{1},\ldots,x_{w}\rangle\cong k$.

It is worth mentioning the following generalization of the previous example.

######  Example 5.3.

(i) For any integer $d\geq 1$, consider the Koszul complex to the
$\operatorname{OI}$-morphism
$\varphi\colon{\mathbf{F}}^{\operatorname{OI},d}_{\mathbf{A}}\to\mathbf{A}$.
It is determined by $a=\varphi(e_{\operatorname{id}_{d}})\in\mathbf{A}(d)$.
Again, there are cases in which $K_{\bullet}(\varphi)$ is acyclic. For
example, using notation from [11], let $\mathbf{A}$ be the polynomial
$\operatorname{OI}$-algebra
$\mathbf{A}=\mathbf{X}^{\operatorname{OI},d}\cong\operatorname{Sym}_{\bullet}({\mathbf{F}}^{\operatorname{OI},d})$,
and so
$\mathbf{A}(w)=k[x_{\pi}\mid\pi\in\operatorname{Hom}_{\operatorname{OI}}(d,w)]$
is a polynomial ring in $\binom{w}{d}$ variables. Then the complex
$K_{\bullet}(\varphi)$ to
$\varphi\colon{\mathbf{F}}^{\operatorname{OI},d}_{\mathbf{A}}\to\mathbf{A}$
with $\varphi(e_{\operatorname{id}_{d}})=x_{\operatorname{id}_{w}}$ is
acyclic. In fact, it gives a minimal and width-wise minimal graded free
resolution of $\operatorname{coker}\varphi$. Note that
$\mathbf{A}=\mathbf{X}^{\operatorname{OI},d}$ is not a noetherian
$\operatorname{OI}$-algebra if $d\geq 2$ (see [11, Proposition 4.8]), and so
the existence of free resolutions with finitely generated $\mathbf{A}$-modules
is not guaranteed in general.

(ii) Analogous results are true over any polynomial
$\operatorname{OI}$-algebra. For example, consider
$A=\mathbf{X}^{\operatorname{OI},2}\otimes_{k}\mathbf{X}^{\operatorname{OI},3}$
and so
$\mathbf{A}(w)=k[x_{\pi},y_{\sigma}\mid\pi\in\operatorname{Hom}_{\operatorname{OI}}(2,w),\sigma\in\operatorname{Hom}_{\operatorname{OI}}(3,w)]$
is a poynomial ring in $\binom{w}{2}+\binom{w}{3}$ variables. Then the complex
$K_{\bullet}(\varphi)$ to
$\varphi\colon{\mathbf{F}}^{\operatorname{OI},2}_{\mathbf{A}}\oplus{\mathbf{F}}^{\operatorname{OI},3}_{\mathbf{A}}\to\mathbf{A}$
with $\varphi(e_{\operatorname{id}_{2}})=x_{\operatorname{id}_{2}}$ and
$\varphi(e_{\operatorname{id}_{3}})=y_{\operatorname{id}_{3}}$ is acyclic and
gives a minimal and width-wise minimal free resolution of
$\operatorname{coker}\varphi$.

######  Example 5.4.

As in the classical case, acyclicity of $K_{\bullet}(\varphi)$ depends on
$\mathbf{A}$ and the choice of the morphism
$\varphi\colon\mathbf{F}\to\mathbf{A}$. For example, let
$\mathbf{A}=\mathbf{X}^{\operatorname{OI},1}\cong\operatorname{Sym}_{\bullet}({\mathbf{F}}^{\operatorname{OI},1})$
as in Example 5.2 and consider the morphism
$\varphi\colon{\mathbf{F}}^{\operatorname{OI},2}_{\mathbf{A}}\to\mathbf{A}$
with $\varphi(e_{\operatorname{id}})=x_{2}\in\mathbf{A}(2)$. Then the Koszul
complex $K_{\bullet}(\varphi)$ is not acyclic. For example, its width 3
component is not acylic because it is the Koszul complex on the sequence
$(x_{2},x_{3},x_{3})$.

##  6\. OI Buchsbaum-Eisenbud Complexes

Over a commutative ring $R$, the classical Buchsbaum-Eisenbud complexes are
determined by an $R$-module homomorphism $\varphi\colon F\to G$ of finitely
generated free $R$-modules (see Definition 3.2). Similar to the Koszul complex
in the previous section, we now generalize the Buchsbaum-Eisenbud complexes to
the $\operatorname{OI}$ setting.

######  Theorem 6.1.

Let $\mathbf{A}$ be a commutative $\operatorname{OI}$-algebra and consider a
morphism of finitely generated free $\mathbf{A}$-modules
$\varphi\colon\mathbf{F}\to\mathbf{G}$, where $\mathbf{G}$ is generated in
width zero and has rank $r$. Then, for every integer $i\geq 0$, there is a
complex of finitely generated free $\mathbf{A}$-modules

$\displaystyle\cdots$
$\displaystyle\to\bigwedge\nolimits^{r}(\mathbf{G}^{*})\otimes
S_{p+1}(\mathbf{G}^{*})\otimes\bigwedge\nolimits^{r+i+p+1}\mathbf{F}\to\bigwedge\nolimits^{r}(\mathbf{G}^{*})\otimes
S_{p}(\mathbf{G}^{*})\otimes\bigwedge\nolimits^{r+i+p}\mathbf{F}\to\cdots$
$\displaystyle\to\bigwedge\nolimits^{r}(\mathbf{G}^{*})\otimes
S_{0}(\mathbf{G}^{*})\otimes\bigwedge\nolimits^{r+i}\mathbf{F}\stackrel{{\scriptstyle\alpha_{i}}}{{\longrightarrow}}\bigwedge\nolimits^{i}\mathbf{F}\otimes
S_{0}(\mathbf{G})\to\bigwedge\nolimits^{i-1}\mathbf{F}\otimes
S_{1}(\mathbf{G})\to\cdots$
$\displaystyle\to\bigwedge\nolimits^{1}\mathbf{F}\otimes
S_{i-1}(\mathbf{G})\to\bigwedge\nolimits^{0}\mathbf{F}\otimes
S_{i}(\mathbf{G})\to 0,$

where $S_{i}$ is short for $\operatorname{Sym}_{i}$ and each differential
$\partial_{j}$ is defined width-wise by letting $\partial_{j}(w)$ be the
degree $j$ map of the classical Buchsbaum-Eisenbud complex
$\operatorname{BE}^{i}_{\bullet}(\varphi(w))$ to the $\mathbf{A}(w)$-module
homomorphism $\varphi(w)\colon\mathbf{\mathbf{F}}(w)\rightarrow\mathbf{G}(w)$.
This complex is generically acyclic and, if it is acyclic, resolves
$\operatorname{coker}\partial_{1}$.

Furthermore, if $\varphi$ is a morphism of graded $\mathbf{A}$-modules, then
the above complex is a graded complex.

We refer to the above complex as the $\operatorname{OI}$ Buchsbaum-Eisenbud
complex to $\varphi$ and denote it by
$\operatorname{BE}^{i}_{\bullet}(\varphi)$. Note that the above result covers
Theorem 1.2 of the Introduction. The assumption that $\mathbf{G}$ is generated
in width zero guarantees that the rank of $\mathbf{G}(w)$ as an
$\mathbf{A}(w)$-module is equal to the rank of $\mathbf{G}$ for every width
$w$.

###### Proof of Theorem 6.1.

By Propositions 4.2, 4.5 4.7, we know that each of the modules
$S_{p}(\mathbf{G}^{*})\otimes_{\mathbf{A}}\bigwedge\nolimits^{q}\mathbf{F}$ is
finitely generated and free and satisfies

$\big{(}S_{p}(\mathbf{G}^{*})\otimes_{\mathbf{A}}\bigwedge\nolimits^{q}\mathbf{F})(w)=S_{p}(\mathbf{G}^{*}(w))\otimes_{\mathbf{A}(w)}\bigwedge\nolimits^{q}(\mathbf{F}(w))$

in every width $w$. These properties remain true if one tensors with
$\bigwedge^{r}(\mathbf{G}^{*})$. Moreover, since $\partial_{j}(w)$ is the
differential of a classical Buchsbaum-Eisenbud complex
$\operatorname{BE}^{i}_{\bullet}(\varphi(w))$, the stated sequence is indeed a
complex. Analogously, it follows that
$\operatorname{BE}^{i}_{\bullet}(\varphi)$ is graded if $\varphi$ is graded.

It remains to show that the family of $\mathbf{A}(w)$-module homomorphisms
$\partial_{j}(w)$ can be used to define a morphism of $\mathbf{A}$-modules
$\partial_{j}$, which requires showing that certain diagrams commute. In
$\operatorname{BE}^{i}_{\bullet}(\varphi)$, there are three types of
differentials $\partial_{j}$ depending on $j=i+1$, $j\leq i$ or $j\geq i+2$.
We discuss the needed commutativity in the most complex case, where $j=i+1$,
that is, if the differential is the splicing map $\alpha_{i}$, and leave the
other cases to the interested reader.

We have to show for any map
$\varepsilon\in\operatorname{Hom}_{\operatorname{OI}}(w,w^{\prime})$ that the
following diagram commutes:

${\left(\bigwedge\nolimits^{r}(\mathbf{G}^{*})\otimes
S_{0}(\mathbf{G}^{*})\otimes\bigwedge\nolimits^{r+i}(\mathbf{F})\right)(w)}$${\left(\bigwedge\nolimits^{i}(\mathbf{F})\otimes
S_{0}(\mathbf{G})\right)(w)}$${\left(\bigwedge\nolimits^{r}(\mathbf{G}^{*})\otimes
S_{0}(\mathbf{G}^{*})\otimes\bigwedge\nolimits^{r+i}(\mathbf{F})\right)(w^{\prime})}$${\left(\bigwedge\nolimits^{i}(\mathbf{F})\otimes
S_{0}(\mathbf{G})\right)(w^{\prime})}$$\scriptstyle{\alpha_{i}(w)}$$\scriptstyle{\varepsilon_{*}}$$\scriptstyle{\varepsilon_{*}}$$\scriptstyle{\alpha_{i}(w^{\prime})}$

To this end fix bases $\\{g_{1},\ldots,g_{r}\\}$ and
$\\{f_{1},\ldots,f_{s}\\}$ of $\mathbf{G}$ and $\mathbf{F}$, respectively.
Thus, for each $v\in\mathbb{N}_{0}$, the module $\mathbf{G}(v)$ has a basis
$\\{\tau_{*}(g_{1}),\ldots,\tau_{*}(g_{r})\\}$, where $\tau$ is the unique
$\operatorname{OI}$ morphism from width $0$ to width $v$. To simplify
notation, we still use $g_{j}$ to denote $\tau_{*}(g_{j})$. Similarly, we
write $g_{j}^{*}$ for the basis elements of $\mathbf{G}^{*}$ and
$\mathbf{G}^{*}(v)$ as introduced in the proof of Proposition 4.9. The
elements $\pi_{*}(f_{k})$ with $k\in[s]$ and
$\pi\in\operatorname{Hom}_{\operatorname{OI}}(w_{k},v)$, where $w_{k}$ is the
width of $f_{k}$, form a basis of $\mathbf{F}(v)$. We order these elements by
using the following order of the set $\\{(\pi,k)\mid
k\in[s],\pi\in\operatorname{Hom}_{\operatorname{OI}}(w_{k},v)\\}$: Put
$(\pi,j)>(\tau,k)$ if $j<k$ or if $j=k$ and
$\operatorname{im}\pi>\operatorname{im}\tau$ in the lexicographic order of the
set $\mathbb{N}^{w_{k}}$. Observe that $(\pi,j)>(\tau,k)$ implies
$(\rho\circ\pi,j)>(\rho\circ\tau,k)$ for every
$\rho\in\operatorname{Hom}_{\operatorname{OI}}(v,v^{\prime})$.

Using this order, lets us rewrite the basis of $\mathbf{F}(w)$ as
$\\{f_{1},\ldots,f_{s^{\prime}}\\}$ where $s^{\prime}$ is the rank of
$\mathbf{F}(w)$. For a subset $J=\\{j_{1},\ldots,j_{i+r}\\}$ of $[s^{\prime}]$
with cardinality $i+r$, set $f_{J}=f_{j_{1}}\wedge\cdots\wedge f_{j_{i+r}}$.
Denote by $\varepsilon(J)$ the set $\\{k_{1},\ldots,k_{i+r}\\}$, where $k_{m}$
is the index of $\varepsilon_{*}(f_{j_{m}})$ in the ordered basis
$\\{f^{\prime}_{1},\ldots,f^{\prime}_{s^{\prime\prime}}\\}$ of
$\mathbf{F}(w^{\prime})$. Thus,
$\varepsilon_{*}(f_{J})=f^{\prime}_{\varepsilon(J)}$. Abusing notation, write
$\varphi(v)$ for the coordinate matrix of $\varphi(v)$ with respect to the
above ordered bases, and let $\varphi(v)_{I}$ be the submatrix formed by the
columns with indices in $I$. Now we compute, where we use the facts on the
order of basis elements mentioned above and the sign
$\operatorname{sgn}(I\subseteq J)$ as introduced in Definition 3.2:

$\displaystyle\big{(}\varepsilon_{*}\circ\alpha_{i}\big{)}\left(g_{1}^{*}\wedge\ldots\wedge
g_{r}^{*}\otimes f_{J}\right)$
$\displaystyle=\varepsilon_{*}\Big{(}\sum\limits_{\begin{subarray}{c}I\subseteq
J\\\ |I|=r\end{subarray}}\operatorname{sgn}(I\subseteq
J_{w})\cdot\operatorname{det}(\varphi(w)_{I})f_{J\setminus I}\Big{)}$
$\displaystyle=\sum\limits_{\begin{subarray}{c}I\subseteq J\\\
|I|=r\end{subarray}}\operatorname{sgn}(\varepsilon(I)\subseteq\varepsilon(J))\cdot\operatorname{det}\left(\varphi(w^{\prime})_{\varepsilon(I)}\right)f^{\prime}_{\varepsilon(J)\setminus\varepsilon(I)})$
$\displaystyle=\alpha_{i}(g_{1}^{*}\wedge\ldots\wedge g_{r}^{*}\otimes
f^{\prime}_{\varepsilon(J)})$
$\displaystyle=(\alpha_{i}\circ\varepsilon_{*})\left(g_{1}^{*}\wedge\ldots\wedge
g_{r}^{*}\otimes f_{J}\right).$

This proves the desired commutativity of the above diagram. ∎

We illustrate Theorem 6.1 with a few examples, in which we focus on acyclic
complexes that provide minimal free resolutions in every width $w$, that is,
$\operatorname{BE}^{i}_{\bullet}(\varphi)$ gives a width-wise minimal free
resolution over the algebra $\mathbf{A}$.

######  Example 6.2.

For any integer $c\geq 1$, let $\mathbf{A}$ be the graded polynomial
$\operatorname{OI}$-algebra $(\mathbf{X}^{\operatorname{OI},1})^{\otimes
c}\cong\operatorname{Sym}_{\bullet}(({\mathbf{F}}^{\operatorname{OI},1})^{c})$,
and so $\mathbf{A}(w)=k[x_{i,j}\mid i\in[c],j\in[w]]$ is a standard-graded
polynomial ring in $c\cdot w$ variables. Consider the graded
$\mathbf{A}$-module morphism
$\varphi\colon\mathbf{F}={\mathbf{F}}^{\operatorname{OI},1}_{\mathbf{A}}(-1)\to\mathbf{A}^{c}$
determined by $\varphi(e_{\operatorname{id}})=\begin{bmatrix}x_{1,1}\\\
\vdots\\\ x_{c,1}\end{bmatrix}$. Thus, using the above standard bases, the
coordinate matrix of $\varphi(w)$ is a generic $c\times w$ matrix $(x_{i,j})$.
For every $i\geq 0$, the complex $\operatorname{BE}^{i}_{\bullet}(\varphi)$ is
acyclic and gives a width-wise minimal free resolution of $\partial_{1}$. For
example, $\operatorname{BE}^{0}_{\bullet}(\varphi)$ is in every witdth $w\geq
c$ the classical Eagon-Northcott complex that resolves
$\mathbf{A}(w)/\mathbf{I}(w)$, where $\mathbf{I}(w)$ is the ideal generated by
the maximal minors of the generic $c\times w$ matrix. If $0\leq w<c$ then
$\operatorname{BE}^{0}_{\bullet}(\varphi)(w)$ has length zero, and the
cokernel of $\partial_{1}=\alpha_{0}$ is $\mathbf{A}(w)$.

Note that for $c=3$, this covers the resolution discussed in Example 1.3.

A similar construction gives width-wise minimal free resolutions over non-
noetherian polynomial algebras. We use the notation of Example 5.3

######  Example 6.3.

(i) For any integers $c,d\geq 1$, consider the Buchsbaum-Eisenbud complexes to
a graded $\operatorname{OI}$-morphism
$\varphi\colon{\mathbf{F}}^{\operatorname{OI},d}(-1)\to\mathbf{A}^{c}$, where
$\mathbf{A}=(\mathbf{X}^{\operatorname{OI},d})^{\otimes
c}\cong\operatorname{Sym}_{\bullet}(({\mathbf{F}}^{\operatorname{OI},d})^{c})$
and $\varphi$ is determined by
$\varphi(e_{\operatorname{id}_{d}})=\begin{bmatrix}x_{1,\operatorname{id}_{d}}\\\
\vdots\\\ x_{c,\operatorname{id}_{d}}\end{bmatrix}$. Thus, a coordinate matrix
of $\varphi(w)$ is given by the generic $c\times\binom{w}{d}$ matrix
$(x_{i,\pi})$ with $i\in[c]$ and
$\pi\in\operatorname{Hom}_{\operatorname{OI}}(d,w)$. Let $\mathbf{I}$ the
ideal of $\mathbf{A}$ such that $\mathbf{I}(w)\subset\mathbf{A}(w)$ is the
ideal generated by the maximal minors of $\varphi(w)$. Considering
$\operatorname{BE}^{0}_{\bullet}(\varphi)$, note that
$\mathbf{I}=\operatorname{im}\alpha_{0}$ and that
$\operatorname{BE}^{0}_{\bullet}(\varphi)$ is a width-wise minimal graded free
resolution of $\mathbf{I}$. If $d=1$, then $\mathbf{I}$ is the ideal generated
in width $c$ by the determinant of $\varphi(c)$. However, if $d\geq 2$ then
the minimal generators of $\mathbf{I}$ have different widths. In fact, their
widths are given by the widths of the basis elements of
$\bigwedge^{c}({\mathbf{F}}^{\operatorname{OI},d}_{\mathbf{A}})$. By
Proposition 4.5, it follows that $\mathbf{I}$ has minimal generators in any
width $w$ with $w_{0}\leq w\leq dc$, where
$w_{0}=\min\\{w\in\mathbb{N}_{0}\mid\binom{w}{d}\geq c\\}$. Note, that in this
case $\operatorname{BE}^{i}_{\bullet}(\varphi)$ gives a width-wise minimal
graded free resolution of $\operatorname{coker}\partial_{1}$ for every $i\geq
0$.

(ii) Similar results are true over any polynomial $\operatorname{OI}$-algebra.
For example, consider the graded algebra
$A=\big{(}\mathbf{X}^{\operatorname{OI},2}\otimes_{k}\mathbf{X}^{\operatorname{OI},3}\big{)}$.
Then, for every $i\geq 0$, the Buchsbaum-Eisenbud complex
$\operatorname{BE}^{i}_{\bullet}(\varphi)$ to
$\varphi\colon{\mathbf{F}}^{\operatorname{OI},2}_{\mathbf{A}}(1)\oplus{\mathbf{F}}^{\operatorname{OI},3}_{\mathbf{A}}(-1)\to\mathbf{A}^{4}$
with

$\varphi(e_{\operatorname{id}_{2}})=\begin{bmatrix}x_{1,\operatorname{id}_{2}}\\\
\vdots\\\ x_{4,\operatorname{id}_{2}}\end{bmatrix}\text{ and
}~{}\varphi(e_{\operatorname{id}_{3}})=\begin{bmatrix}y_{1,\operatorname{id}_{3}}\\\
\vdots\\\ y_{4,\operatorname{id}_{3}}\end{bmatrix}$

is acyclic and gives a minimal and width-wise minimal graded free resolution
of $\operatorname{coker}\partial_{1}$. In particular,
$\operatorname{BE}^{1}_{\bullet}(\varphi)$ resolves
$\operatorname{coker}\varphi$.

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talk about regular sequences and depth as a “future directions” remark

## References

  * [1] W. Bruns and J. Herzog, _Cohen-Macaulay rings_ , Revised edition, Cambridge Studies Adv. Math. 39, University Press, Cambridge, 1998.
  * [2] W. Bruns and U. Vetter, _Determinantal rings_ , Lecture Notes in Mathematics 1327, Springer-Verlag, Berlin, Heidelberg, 1988.
  * [3] D. A. Buchsbaum and D. Eisenbud, _Generic free resolutions and a family of generically perfect ideals_ , Adv. Math 18 (1975), 245–301.
  * [4] T. Church, J. S. Ellenberg, and B. Farb, _$\operatorname{FI}$ -modules and stability for representations of symmetric groups_, Duke Math. J. 164 (2015), 1833–1910.
  * [5] J. Draisma, R.H. Eggermont, R. Krone, and A. Leykin, _Noetherianity for infinite-dimensional toric varieties_ , Algebra Number Theory 9 (2015), 1857–1880.
  * [6] J.A. Eagon and D.G.Northcott, _Ideals defined by matrices and a certain complex associated with them_ , Proc. Roy. Soc. London Ser. A 269 (1962), 188-204.
  * [7] D. Eisenbud, _Commutative algebra_ , Graduate Texts in Mathematics 150, Springer-Verlag, New York,1995.
  * [8] N. Fieldsteel and U. Nagel, _Minimal and cellular free resolutions over polynomial OI-algebras_ , Preprint, 2021; available at arXiv:2105.08603.
  * [9] C.J. Hillar and S. Sullivant, _Finite Gröbner bases in infinite dimensional polynomial rings and applications_ , Adv. Math. 229 (2012), 1–25.
  * [10] U. Nagel and T. Römer, _Equivariant Hilbert series in non-Noetherian Polynomial Rings_ , J. Algebra 486 (2017), 204–245.
  * [11] U. Nagel and T. Römer, _$\operatorname{FI}$ \- and $\operatorname{OI}$-modules with varying coefficients_, J. Algebra 535 (2019), 286–322.
  * [12] U. Nagel, _Rationality of Equivariant Hilbert Series and Asymptotic Properties_ , Trans. Amer. Math. Soc. 374 (2021), 7313–7357.
  * [13] D.V. Le, U. Nagel, H.D. Nguyen, and T. Römer, _Castelnuovo-Mumford regularity up to symmetry_ , Int. Math. Res. Not. 2021, no. 14 (2021), 11010–11049.
  * [14] S. V. Sam and A. Snowden, _Gröbner methods for representations of combinatorial categories_ , J. Amer. Math. Soc. 30 (2017), 159–203.