# Boij-Söderberg Conjectures for Differential Modules

Maya Banks

###### Abstract.

Boij-Söderberg theory gives a combinatorial description of the set of Betti
tables belonging to finite length modules over the polynomial ring
$S=\mathbbm{k}[x_{1},\ldots,x_{n}]$. We posit that a similar combinatorial
description can be given for analogous numerical invariants of _graded
differential $S$-modules_, which are natural generalizations of chain
complexes. We prove several results that lend evidence in support of this
conjecture, including a categorical pairing between the derived categories of
graded differential $S$-modules and coherent sheaves on $\mathbbm{P}^{n-1}$
and a proof of the conjecture in the case where $S=\mathbbm{k}[t]$.

##  1\. Introduction

_Differential modules_ , modules equipped with an endomorphism squaring to
zero, are natural generalizations of chain complexes. One enlightening way to
picture a differential module is as a complex absent the structure of a
homological grading. Initially appearing in more of a book-keeping capacity in
[CE16], differential modules have increasingly become objects of interest in
commutative algebra, algebraic geometry, and algebraic topology in particular
as useful tools for generalizing classical results and conjectures [ŞÜ19,
IW18, BE21b, ABI07, BE21a, BD10]. An overarching question in recent years can
be distilled down to the following.

######  Question 1.1.

When can results about familiar objects be generalized to results about
differential modules, and what insight does the ability or failure to
generalize yield about the original results?

Avramov-Buchweitz-Iyengar [ABI07] investigate this question for a special
class of differential modules— _flag differential modules_ —and find
generalizations of several results in commutative algebra, including a
generalization of the New Intersection Theorem of Hochster, Peskine, Roberts,
and Szpiro [Hoc74, PS73, Rob89]. Brown and Erman give a generalization of the
notion of a minimal free resolution [BE21a], and Banks and VandeBogert
construct a differential module generalization of the Koszul complex [BV22].
On the other hand, [BV22] also provide examples of the failure of certain
notions to generalize nicely to the realm of differential modules, while
Iyengar and Walker similarly show that bounds on the Betti numbers of a free
resolution do not generalize to differential modules even in the case of free
flags [IW18].

Our goal in this paper is to explore Question 1.1 for Boij-Söderberg theory,
which concerns the combinatorial structure of the set of all possible Betti
tables of modules over a given ring. Initially conjectured by Boij and
Söderberg [BS08] for modules over the polynomial ring, this theory brought a
new perspective to the study of Betti tables. The Boij-Söderberg conjectures
were proven in the finite length case in 2008 by Eisenbud and Schreyer via a
duality between free resolutions of graded modules over the polynomial ring
and vector bundles on projective space [ES]. Their results describe the
positive rational cone spanned by Betti tables belonging to modules over the
polynomial ring and give a classification, up to rational multiple, of all
such tables.

######  Remark 1.2.

The original theory was developed in the finite length case and subsequently
extended to other codimensions. For simplicity, we restrict ourselves here to
the finite length case as well.

The results of [ES, BS12] were soon generalized and extended in various
directions, such as [ES09, Erm09, Flø10, FJK11, BEKS11, BBEG12, BS15, ES16,
FLS18, FL18, SS21] as well as most recently in [IMW22]. In a particularly
interesting extension of the original theory, Eisenbud and Erman [EE17] showed
that the duality in [ES] does not actually depend on the structure of a free
resolution—the duality holds for minimal free complexes with finite length
homology more generally (a theme further explored by [IMW22]). This allowed
the original Boij-Söderberg theory to be extended from free resolutions to the
more general setting of free complexes, but it also showed that the
foundations of the theory rested on a different level of structure than what
was originally assumed and conjectured. In particular, one interpretation of
this generalization is that we should think of Boij-Söderberg theory as
_actually_ being a theory about the numerics of minimal finite free complexes,
rather than a theory about invariants of modules. This leads us to question
what the “correct” level of generality for this theory really is. That is, how
far can the original Boij-Söderberg theory extend, and what is the key
structure on which the theory truly relies? Can Boij-Söderberg theory be
further generalized beyond free complexes, and what new insight might such a
generalization give us?

We posit a generalization of the Boij-Söderberg conjectures for differential
modules and build evidence in support of this conjecture, including some
partial results that shed new light on some of the key structures underlying
the original theory, while simultaneously giving insight on what makes
differential modules similar to and different from complexes and free
resolutions.

Let $S=\mathbbm{k}[x_{1},\ldots,x_{n}]$ for any field $\mathbbm{k}$ and denote
by $\operatorname{DM}(S,a)$ the category of differential modules of degree $a$
over $S$, that is the graded modules $D$ over $S$ with a square 0 endomorphism
$D\to D(a)$. For a differential module $D\in\operatorname{DM}(S,a)$, the
_Betti vector_ of $D$, denoted $\beta^{\text{DM}}(D)$ is the vector whose
$i{{}^{\text{th}}}$ entry counts the number of degree $i$ generators in a
minimal free resolution of $D$ (see Definition 2.7).

Let $\mathbbm{V}=\bigoplus_{i,j\in\mathbbm{Z}}\mathbbm{Q}$ and
$\mathbbm{B}=\bigoplus_{j\in\mathbbm{Z}}\mathbbm{Q}$, so that Betti tables and
Betti vectors are naturally elements of $\mathbbm{V}$ and $\mathbbm{B}$,
respectively. We denote by $BS_{\text{mod}}(S)\subset\mathbbm{V}$ and
$BS_{\text{DM}}(S,a)\subset\mathbbm{B}$ the rational cones of Betti tables of
graded, finite length $S$-modules and Betti vectors of degree $a$ differential
$S$-modules with finite length homology.

The Betti vector of a differential module is an analogous numerical object to
the Betti table of a minimal free resolution, but it represents a certain
coarsening of the data of a Betti table. More precisely, for any
$a\in\mathbbm{Z}$ we may regard any $S$-complex as a differential module in
$\operatorname{DM}(S,a)$ via what [BE21a] call the _folding functor_ , which
we define in more detail in Section 2. As an operation on Betti tables, the
folding functor “flattens” the Betti table of a minimal free resolution by
forgetting all homological data and retaining only the internal grading. In
other words, this flattening is a map $\mathbbm{V}\to\mathbbm{B}$ that maps
Betti tables of minimal free resolutions of $S$-modules into
$BS_{\text{DM}}(S,a)$ for any $a$. We propose the following:

######  Conjecture 1.3.

This “flattening” gives a surjection of cones $BS_{\text{mod}}(S)\to
BS_{\text{DM}}(S,a)$ for every $a$. More precisely Any differential module
with finite length homology has a Betti vector that can be written as a
positive rational combination of Betti vectors of differential modules whose
homology have Betti tables that are extremal in $BS_{\text{mod}}(S)$.

Before moving on to our results in support of this conjecture, we first
consider a small example

######  Example 1.4.

Consider the two variable case when $S=\mathbbm{k}[x,y]$ and $(D,{\partial})$
is the degree 0 differential module where

$\displaystyle D=S\oplus S(-1)^{2}\oplus S(-2)\text{ and
}{\partial}=\begin{bmatrix}0&x&y&0\\\ 0&0&0&-y\\\ 0&0&0&x\\\
0&0&0&0\end{bmatrix}$

This differential module has homology $\mathbbm{k}$ and Betti vector
$\beta^{\text{DM}}(D)=(1^{\circ},2,1)$ where $1^{\circ}$ indicates that the 1
is in the degree 0 position in the vector. In fact, by [BV22], every degree 0
differential module over $S=\mathbbm{k}[x,y]$ with homology $\mathbbm{k}$ has
this Betti vector.

On the other hand, suppose we instead let $(D,{\partial})$ be a degree 2
differential module over $S$, with

$\displaystyle D=S\oplus S(1)^{2}\oplus S(2)\text{ and
}{\partial}=\begin{bmatrix}0&x&y&0\\\ 0&0&0&-y\\\ 0&0&0&x\\\
0&0&0&0\end{bmatrix}$

Again, $D$ has homology $\mathbbm{k}$ and now its Betti vector is
$\beta^{\text{DM}}(D)=(1,2,1^{\circ})$. On the other hand, let
$(D^{\prime},{\partial}^{\prime})$ be the degree 2 differential module where

$\displaystyle D^{\prime}=S\oplus S(1)^{2}\oplus S(2)\text{ and
}{\partial}=\begin{bmatrix}0&x&y&1\\\ 0&0&0&-y\\\ 0&0&0&x\\\
0&0&0&0\end{bmatrix}$

This differential module also has homology $\mathbbm{k}$ but, unlike
$(D,{\partial})$, $(D^{\prime},{\partial}^{\prime})$ is not minimal. It has
minimal free resolution

$\begin{bmatrix}xy&x^{2}\\\ -y^{2}&-xy\end{bmatrix}$

and Betti vector $\beta^{\text{DM}}(D^{\prime})=(0,2,0^{\circ})$.

We draw attention to three particular features of this example and the
implications that they highlight. First, observe that
$\beta^{\text{DM}}(D^{\prime})$ cannot be the Betti vector of a fold of
minimal free resolution of a finite length, graded $S$-module since any such
module must have total Betti numbers at least 4. In other words, while we
conjecture that flattening gives a surjection on _cones_ , this example shows
that it certainly cannot be a surjection onto the actual semigroup of Betti
vectors. The second observation is that we have two differential modules whose
homologies are isomorphic, but whose minimal free resolutions–and thus their
Betti vectors–are genuinely different. This behavior represents a stark
departure from what we are used to in the realm of finite length, graded
$S$-modules. Finally, we point out the differing behavior exhibited by the
differential modules in $\operatorname{DM}(S,a)$ as $a$ varies from 0 to 2.

This third point is actually a more general phenomenon that will continue to
appear. When dealing with graded complexes, we are always free to twist the
individual modules in the complex so that the differential is of degree 0. For
differential modules however, the source and target of our differential are
the same, so we can not twist one without twisting the other. This means that
we are unable to control the degree of the differential in the way that we are
used to, and must contend with differentials of nonzero degree. While a
differential of nonzero degree may seem innocuous at first glance, we will see
that this variance can actually have drastic impacts on the numerical
invariants at hand. This phenomenon was observed as well in [BE21a, Example
5.4] and [BV22, Corollary 3.7, Proposition 3.8], and turns out to be a theme
underpinning our results here as well. In general, it seems that while for
degree 0 we may expect differential modules to behave more or less similarly
to complexes in many regards, the same is not the case in more general degree.
For Boij-Söderberg theory specifically, this behavior seems to support the
idea that the ability to force the differential to be degree 0 is actually an
indispensable piece of the structure on which the theory rests. More broadly,
it seems that one answer to the question of what makes differential modules
similar to or different from complexes lies in their degree.

###  1.1. Results

We split our results into three cases depending on whether the degree of the
differential module is positive, negative, or zero. The positive degree case
turns out to be much more straightforward—as well as much less reminiscent of
the standard Boij-Söderberg theory—and we are able prove the following in full
generality via quite elementary methods.

######  Theorem 1.5.

Conjecture 1.3 is true for differential modules of degree $a>0$.

We are able to prove this result by explicitly constructing all of the
extremal rays of the Betti cone. The proof relies on a certain level of
“cancellation” that can appear when passing from a differential module to its
minimal free resolution–as demonstrated in Example 1.4. Our next result says
essentially that when the degree of the differential module is not positive,
there is a limit to this cancellation.

We prove the following which we show arises as a consequence of Avramov,
Buchweitz, and Iyengar’s Class Inequality [ABI07, Theorem 4.1].

######  Proposition 1.6.

Let $F\in\operatorname{DM}(S,a)$ be a free differential module with finite
length homology, and assume that $a\leq 0$. Then the Betti vector
$\beta^{\text{DM}}(F)$ has at least $n+1$ nonzero entries.

Recall that in standard Boij-Söderberg theory the extremal rays of the Betti
cone come from the Betti tables of modules with _pure resolutions_ , those
with the fewest possible nonzero entries. Over
$S=\mathbbm{k}[x_{1},\ldots,x_{n}]$, Betti tables of pure resolutions have
exactly $n+1$ nonzero entries. Since folds of pure resolutions in degree
$a\leq 0$ have Betti vectors with exactly $n+1$ nonzero entries, Proposition
1.6 gives a hint that the extremal rays of $BS_{\text{mod}}(S)$ might remain
extremal after flattening. This would mean that the extremal rays in
$BS_{\text{DM}}(S,a)$ come from analogous objects as in the standard theory.
This provides some motivation for Conjecture 1.3 in the $a\leq 0$ case as well
as some hope that familiar methods may be applicable.

Indeed, when we restrict to the case $a=0$, we are able to make use of
familiar methods by generalizing the categorical pairing of Eisenbud and Erman
[EE17]. The categorified duality in [EE17] clarified the pairing between Betti
tables of finite length $S$-modules and cohomology tables of vector bundles on
$\mathbbm{P}^{n-1}$ which played a crucial role in the proof of the original
Boij-Söderberg conjectures [ES]. In the original proof, Eisenbud and Schreyer
show a sort of duality between $BS_{\text{mod}}(S)$ and the cone
$C_{\text{vb}}(\mathbbm{P}^{n-1})$ of cohomology tables of vector bundles on
$\mathbbm{P}^{n-1}$, which they then use to prove that the facet equations of
the cone spanned by the pure Betti tables are nonnegative on the Betti table
of any finite length $S$-module. Eisenbud and Erman then showed that there was
in fact a pairing

$BS_{\text{mod}}(S)\times C_{\text{vb}}(\mathbbm{P}^{n-1})\to
BS_{\text{mod}}(\mathbbm{k}[t])$

which captured this duality.

We generalize this pairing to the setting of degree 0 differential modules and
prove analogs of two key results from [EE17]. In particular, we construct a
functor

$\Phi\colon\operatorname{D^{b}_{DM}}(S,0)\times\operatorname{D^{b}}(\mathbbm{P}^{n-1})\to\operatorname{D^{b}_{DM}}(A,0)$

where $A=\mathbbm{k}[t]$, $\operatorname{D^{b}_{DM}}(S,0)$ and
$\operatorname{D^{b}_{DM}}(A,0)$ denote the bounded derived categories of
degree 0 differential modules over $S$ and $A$, and
$\operatorname{D^{b}}(\mathbbm{P}^{n-1})$ denotes the bounded derived category
of coherent sheaves on $\mathbbm{P}^{n-1}$. We show that $\Phi$ satisfies two
crucial properties.

######  Theorem 1.7.

Let $F\in\operatorname{D^{b}_{DM}}(S,0)$ be free and
${\mathcal{E}}\in\operatorname{D^{b}}(\mathbbm{P}^{n-1})$. Then

  1. (i)

The Betti vector of $\Phi(F,{\mathcal{E}})$ depends only on the Betti vector
of $F$ and the absolute Hilbert function (see Definition 4.5) of
${\mathcal{E}}$.

  2. (ii)

If $\tilde{F}\otimes{\mathcal{E}}$ is exact, then $\Phi(F,{\mathcal{E}})$ has
finite length homology.

Thus our $\Phi$ also induces a pairing

$BS_{\text{DM}}(S,0)\times C_{\text{vb}}(\mathbbm{P}^{n-1})\to
BS_{\text{DM}}(A,0).$

The existence of this pairing leads us to conjecture the following analog of
the duality displayed in [EE17, Theorem 0.2].

######  Conjecture 1.8.

Let $\mathbf{b}\in\mathbbm{B}$. Then the following are equivalent:

  1. (i)

$\mathbf{b}=\beta^{\text{DM}}(F)$ for a degree 0 differential module $F$ over
$S$ with finite length homology.

  2. (ii)

For every vector bundle ${\mathcal{E}}$ on $\mathbbm{P}^{n-1}$, $\mathbf{b}$
pairs with the absolute Hilbert function of ${\mathcal{E}}$ to give the Betti
vector of a differential module in $\operatorname{DM}(A,0)$ with finite length
homology.

Finally, we prove that Conjecture 1.3 holds for degree 0 differential modules
over $A$, explicitly:

######  Theorem 1.9.

Every degree 0 differential modules over $A=\mathbbm{k}[t]$ with finite length
homology has a Betti vector that can be expressed as a positive rational
combination of Betti vectors of folds of pure resolutions whose degree
sequences form a chain.

This result is part of what gives the categorical pairing its power. With a
complete description of the extremal rays and facets of the Betti cone over
$A$, the pairing $\Phi$ gives a machine for proving that a certain class of
linear functionals are nonnegative on every Betti vector in
$BS_{\text{DM}}(S,0)$ by reducing to $BS_{\text{DM}}(A,0)$ as a sort of “base
case”. Proving this nonnegativity is one of the main steps in a proof of
Conjecture 3.7 in the general case. What ultimately remains standing in the
way of a full proof of Conjecture 1.3 for degree 0 is a combinatorial
description of the cone spanned by the pure Betti vectors. The authors of [ES]
describe a simplicial fan structure which gives a facet description of the
cone for finite length $S$-modules. While we do this for our base case of
$BS_{\text{DM}}(A,0)$, we have not yet been able to replicate this for
$BS_{\text{DM}}(S,0)$ more generally, though computational data enables us to
conjecture that the extremal facets arise similarly as in [EE17]. Once such a
description is proven, we can use the categorical pairing to prove that the
equations of these facets are actually nonnegative on all Betti vectors of
degree 0 differential modules with finite length homology.

In the case where $a<0$, we are forced to abandon some of the methods that
generalized nicely from complexes to degree 0 differential modules, such as
the categorical pairing. However, the positive results we have obtained in the
degree $\geq 0$ cases lead us to believe that it is natural to extend
Conjecture 1.3 to the negative degree case as well. That being said, in this
setting, new methods appear to be needed to tackle the conjecture and
computational data has thus far proved perplexing.

### Acknowledgements

The author would like to thank Daniel Erman for invaluable guidance,
suggestions, and feedback throughout this work, and Michael Brown for helpful
conversations contributing to this project. Additionally, the computer algebra
system Macaulay2 [GS] provided crucial insight and examples.

##  2\. Background

For the rest of the paper, our convention will be to assume that $R$ is
graded-local with maximal ideal ${\mathfrak{m}}$ and residue field
$\mathbbm{k}=R/{\mathfrak{m}}$.

######  Definition 2.1.

For a ring $R$, a _differential $R$-module_ is a pair $(D,{\partial})$ where
$D$ is a module over $R$ and ${\partial}\colon D\to D$ is an $R$-module
endomorphism squaring to 0. In the case where $D$ is graded, we may define,
for any $a\in\mathbbm{Z}$, a _degree $a$ differential module_ over $R$ to be a
differential module $(D,{\partial})$ where ${\partial}\colon D\to D(a)$ is
homogeneous. We denote the category of degree $a$ differential modules over
$R$ by $\operatorname{DM}(R,a)$. A _morphism_ of differential modules is a
module map that respects the differential, that is a map
$f\colon(D,{\partial})\to(D^{\prime},{\partial}^{\prime})$ satisfying
$f{\partial}={\partial}^{\prime}f$.

Note that the data of a differential $R$-module $(D,{\partial})$ is equivalent
to the data of a graded module over the ring
$R[\varepsilon]/(\varepsilon^{2})$ with $\deg(\varepsilon)=a$, where the
action of $\varepsilon$ corresponds to the action of the differential
${\partial}$. This implies that $\operatorname{DM}(R,a)$ is an abelian
category.

######  Definition 2.2.

For a differential module $(D,{\partial})\in\operatorname{DM}(R,a)$, the
_homology_ of $D$ is

$H(D)=\ker({\partial})/\operatorname{Im}({\partial})(-a)$

. A morphism of differential modules is a _quasi-isomorphism_ if it induces an
isomorphism on homology.

Differential modules are in a sense a generalization of chain complexes. In
particular, any chain complex may be considered as a differential module where
the underlying module is the direct sum of the modules in the complex and the
differential is the differential from the complex. More precisely, we have a
functor

$\displaystyle\operatorname{Fold}_{a}\colon$
$\displaystyle\operatorname{Ch}(R)\to\operatorname{DM}(R,a)$
$\displaystyle(C_{\bullet},{\partial})\mapsto\left(\bigoplus
C_{i}(ia),{\partial}\right).$

In general, there is no way to consistently define a tensor product of two
differential modules. However, we can define the tensor product of a
differential module with a _complex_ as follows. Given a differential module
$D\in\operatorname{DM}(R,a)$ and a graded degree $a$ complex $C_{\bullet}$ of
$R,R^{\prime}$-bimodules, the tensor product
$D\otimes^{DM}_{R}C_{\bullet}\in\operatorname{DM}(R^{\prime},a)$ is the
differential module whose underlying module is
$\operatorname{Fold}_{a}(D\otimes_{R}C_{\bullet})$ and whose differential is
given by

$d\otimes c\mapsto
d\otimes{\partial}^{C}(c)+(-1)^{\deg(c)}c\otimes{\partial}^{D}(d).$

Since we may regard any $R$-module as a complex concentrated in degree 0, this
defines a tensor product $D\otimes^{DM}_{R}N$ for any $R$-module $N$.

While a differential module is a generalization of a chain complex, passing
from complexes to more general differential modules destroys the homological
grading which is key to proving many results for complexes. However, we are
able to regain some of the benefits of a homological grading for certain
differential modules defined below.

######  Definition 2.3.

A _flag_ is a differential module $(F,{\partial})$ of the form $F=\bigoplus
F_{i}$ where

${\partial}(F_{i})\subset\bigoplus_{j<i}F_{j}$

. A flag $F$ is _free/projective_ if each of the $F_{i}$ is free/projective.
For a differential module $F$, the _free/projective class_ of $F$ is

$\min\left\\{n\in\mathbbm{N}\text{ such that }F\text{ admits a free/projective
flag structure }F=\bigoplus_{i=0}^{n}F_{i}\right\\}.$

The free class of $F$ over a ring $R$ is denoted
$\operatorname{freeclass}_{R}{F}$. If no flag structure on $F$ exists, then we
say $\operatorname{freeclass}_{R}{F}=\infty$

######  Theorem 2.4 ([ABI07] Class Inequality).

Let $D$ be a finitely generated differential module over the polynomial ring
$S=\mathbbm{k}[x_{1},\ldots,x_{n}]$. Then

$\operatorname{freeclass}_{S}{D}\geq 0pt(\operatorname{Ann}H(D)).$

Given a (free) flag $F$, we can always extract certain complexes from $F$.
First observe that a free flag $F$ over $R$ consists of the data of a family
of free $R$-modules $\\{F_{i}\\}_{i\in\mathbbm{N}}$ and a family of morphisms
$\\{A_{i,j}\colon F_{i}\to F_{j}\\}_{i>j}$ such that

$\sum_{i>j>k}A_{j,k}A_{i,j}=0.$

In particular, this requires that $A_{i,i-1}A_{i+1,i}=0$ for all $i\geq 1$, so
we there is a free complex

$\mathbf{F}=\quad
F_{0}\xleftarrow{A_{1,0}}F_{1}\xleftarrow{A_{2,1}}F_{2}\leftarrow\cdots$

######  Definition 2.5.

In the setting detailed above, we say that the flag $\left(\bigoplus_{i\geq
0}F_{i},\\{A_{i,j}\\}\right)$ is _anchored by_ the complex $\mathbf{F}$. This
definition was given in [BV22].

One main goal in the study of differential modules has been to generalize
constructions and results for complexes to the differential module setting.
Key progress in this endeavor was made by Brown and Erman [BE21a] in their
definition of free flag and minimal free resolutions of differential modules,
which we reiterate here.

######  Definition 2.6.

Let $D$ be a differential module over a ring $R$. A _free flag resolution_ of
$D$ is a quasi-isomorphism $F\to D$ where $F$ is a free flag.

A result of [BE21a] highlights an important link between a differential module
$D$ and the minimal free resolution of its homology. Their theorem says that
_every_ differential module has a free flag resolution anchored on the minimal
free resolution of its homology. Following the terminology of [BV22], we call
these free flag resolutions _anchored free flags_.

######  Definition 2.7.

We say a differential $R$-module $(M,{\partial})$ is _minimal_ if
${\partial}(M)\subseteq{\mathfrak{m}}M$. For a differential $R$-module $D$,
_minimal free resolution_ of $D$ is a quasi-isomorphism $M\to D$ that factors
through a splitting map $M\to F$ where $F$ is a free flag resolution of $D$
and $M$ is minimal.

It is a further result of [BE21a] that for differential modules admitting a
finitely generated free flag resolution, minimal free resolutions exist, are
finitely generated, and are unique up to differential module isomorphism.

We can define certain numerical invariants associated to a differential
module. In particular, we have a parallel notion of the Betti numbers of
graded differential module:

######  Definition 2.8.

Let $D$ be a differential $R$-module with minimal free resolution $F$ and let
$N$ be an $R$-module. Then we define

$\operatorname{Tor}_{DM}^{R}(D,N):=H(F\otimes_{R}^{DM}N).$

The _Betti numbers_ of $D$ are then defined as

$\beta^{\text{DM}}_{j}(D)=\dim_{\mathbbm{k}}\operatorname{Tor}_{DM}^{R}(D,\mathbbm{k})_{j}$

######  Remark 2.9.

One important observation is that, if $M$ is a minimal free resolution of $D$,
then $\beta^{\text{DM}}_{j}(D)$ is equal to the number of generators of the
degree $j$ part of $M$ as a $\mathbbm{k}$-vector space (see [BE21a, Remark
1.3(4)]).

Recall that a Betti diagram is called _pure_ if it has at most one nonzero
entry in each column, and that a resolution with a pure Betti diagram is
called a _pure resolution_. We say that a Betti vector is pure if it is the
Betti vector of the fold of a pure resolution.

Each pure Betti vector has an associated _degree sequence_ , an increasing
sequence of integers corresponding to the degrees of the nonzero entries in
the Betti vector. We put a partial order on the set of degree sequences of
length $\ell$ by saying that
$\mathbf{a}=(a_{0}<a_{1}<\cdots<a_{\ell})\leq\mathbf{b}=(b_{0}<b_{1}<\cdots<b_{\ell})$
if $a_{i}\leq b_{i}$ for every $i$.

##  3\. From Free Complexes to Differential Modules

In this section, we compare the theory of Betti tables of free resolutions and
free complexes to that of Betti vectors of differential modules and propose a
generalization of the Boij-Söderberg conjectures for differential modules.

Let $S=\mathbbm{k}[x_{1},\ldots,x_{n}]$ for any fixed field $\mathbbm{k}$. We
denote the space of _rational Betti vectors_ by
$\mathbbm{B}=\bigoplus_{j\in\mathbbm{Z}}\mathbbm{Q}$. As in the case of the
original Boij-Söderberg theory, our goal is to describe the rational cone
spanned by the vectors arising as the Betti vectors of differential modules
with finite length homology.

We begin by exploring how Betti vectors of differential modules over $S$ are
related to Betti tables of free $S$-complexes. First note that by Remark 2.9
it is enough to consider minimal free resolutions of differential modules, and
by [BE21a, Theorem 3.2, Theorem 4.2] we may assume that up to differential
module isomorphism any minimal free resolution of a differential module $D$ is
a summand of an anchored free flag resolution $F\to D$ where $F$ is the fold
of the minimal free resolution of the homology $H(D)$. We thus have that
$\beta^{\text{DM}}_{j}(D)$ is at most the number of degree $j$ generators of
$F$, which is in turn the sum $\sum_{i}\beta_{i,j}(H(D))$.

######  Definition 3.1.

Let $(\beta_{i,j})$ be the Betti table of a module $M$ over $S$. The _degree
$a$ flattening_ of $(\beta_{i,j})$ is the vector
$\mathbf{b}\in\bigoplus_{j\in\mathbbm{Z}}\mathbbm{Q}$ defined by

$\mathbf{b}_{j}=\sum_{i}\beta_{i,ai+j}$

######  Remark 3.2.

Combining the definitions of flattening and folding, we get that for
$\mathbf{F}$ a minimal free resolution of $M$, the Betti vector of
$~{}\operatorname{Fold}_{a}(\mathbf{F})$ is the degree $a$ flattening of the
Betti table $(\beta_{i,j})$ of $M$.

Remark 3.2 says that for certain differential modules $D$, the Betti vector is
equal to the flattening of the Betti table of the $H(D)$. While this is always
the case when $D$ is quasi-isomorphic to its homology, it is not true for all
differential modules. We saw this in Example 1.4 with a degree 2 differential
module, but such behavior also occurs in other degrees, as we see with the
following.

######  Example 3.3.

Let $R=\mathbbm{k}[x,y]$ and let $D\in\operatorname{DM}(R,0)$ have underlying
module

$R\oplus R(-2)\oplus R(-1)^{2}\oplus R(-3)^{2}\oplus R(-2)\oplus R(-4)$

and differential given by the matrix

$\begin{bmatrix}0&0&x&y&0&0&0&0\\\ 0&0&0&0&x&y&1&0\\\ 0&0&0&0&0&0&-y&0\\\
0&0&0&0&0&0&x&0\\\ 0&0&0&0&0&0&0&-y\\\ 0&0&0&0&0&0&0&x\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\end{bmatrix}.$

The homology $H(D)$ is isomorphic to $\mathbbm{k}\oplus\mathbbm{k}(-2)$ which
has Betti table

$\begin{matrix}&0&1&2\\\ \text{0:}&1&2&1\\\
\text{1:}&\text{-}&\text{-}&\text{-}\\\ \text{2:}&1&2&1\\\ \end{matrix}$

This Betti table has degree 0 flattening $(1,2,2,2,1,0,\ldots)$, but the
differential module $D$ is not minimal, so this is not the Betti vector of
$D$. Rather, $D$ has Betti vector $(1,2,0,2,1)$. This reflects some
“cancellation” of the $R(-2)$ summands that occurs when passing from $D$ to
its minimal free resolution.

######  Remark 3.4.

In fact, a careful analysis (which we omit here) shows that there is no finite
length graded module over $\mathbbm{k}[x,y]$ whose Betti table flattens to
$(1,2,0,2,1)$. However, there is a free _complex_ with finite length homology
that gives this Betti vector. The complex in question has homology
$\mathbbm{k}$ in degree 0 and $\mathbbm{k}(-2)$ in degree 1. Furthermore, the
fold of this complex is isomorphic to the minimal free resolution of the
differential module in the example. We note also that there _is_ a module
whose Betti table flattens to a rational multiple of the vector $(1,2,0,2,1)$,
so this vector lives on a ray contained in the Betti cone for finite length
$S$-modules.

Example 3.3 shows that there can be some difference between the Betti vector
of a differential module and the Betti vector of the fold of the minimal free
resolution of its homology. To understand the Betti cone of differential
modules, we want to know how big this difference can be, that is, how much
“cancellation” can occur. It turns out that the answer to this question
depends substantially on the degree of the differential module, with the cases
of positive degree and nonpositive degree exhibiting quite different
properties. We thus divide the remainder of our discussion into those two
cases.

###  3.1. Differential modules of degree $a\leq 0$

We begin with the nonpositive case, which we will see is the case that more
closely resembles the behavior of complexes and resolutions. The following
result says, in essence, that there is a limit to how much cancellation can
occur in the Betti vector so long as the degree of the differential module is
not positive.

######  Proposition 3.5.

Let $F\in\operatorname{DM}(S,a)$ be a free differential module with finite
length homology for some $a\leq 0$. Then $F$ is generated in at least $n+1$
distinct degrees.

###### Proof.

Let $(F,{\partial})$ be a minimal, free, graded differential module generated
in degrees $d_{0},\ldots,d_{\ell}$ and let $H=H(F)$, the homology of $F$. Then
the grading gives a natural flag structure $F=\bigoplus_{i}F_{i}$ where
$F_{i}$ is the piece of $F$ generated in degree at least $d_{i}$. This defines
a flag structure on $F$, since minimality and nonpositivity of $a$ ensures
that ${\partial}(F_{i})\subset\bigoplus_{i<j}F_{j}$. Applying the Class
Inequality (Theorem 2.4), $\ell\geq 0pt\operatorname{Ann}(H)$ which is equal
to $n$ by the finite length assumption. So $F$ must be generated in at least
$n+1$ distinct degrees. ∎

Note that in the case of finite length modules over $S$, any Betti table with
exactly $n+1$ nonzero entries is the Betti table of a module with a pure
resolution and, furthermore, Betti table is determined up to scalar multiple
by the choice of nonzero entries. What we have shown for differential modules
is weaker than this, but we conjecture that a similar phenomenon holds. That
is, based on experimentation, we guess that the choice of nonzero entries
determines the Betti vector up to scalar multiple, and that any Betti vector
with exactly $n+1$ nonzero entries belongs to a differential module with pure
homology. The reason this is nontrivial is that, due to the type of
cancellation we have observed in Betti vectors, one could imagine a
differential module $D$ with non-pure homology $H$ where some of the nonzero
entries in the Betti table for $H$ cancel out, causing the number of nonzero
entries in $\beta^{\text{DM}}(D)$ to be exactly $n+1$.

######  Remark 3.6.

Proposition 3.5 relies very heavily on the degree $\leq 0$ condition. We have
already seen that for differential modules in degree $a>0$ it may fail
spectacularly. We will return to this case at the end of this section.

In degree $\leq 0$, Proposition 3.5 tells us that differential modules with
finite length homology cannot be generated in fewer degrees than minimal free
complexes with finite length homology. In particular, their Betti vectors have
at least as many nonzero entries as the degree 0 flattenings of Betti tables
of pure resolutions. On the other hand, we know that any flattening of a Betti
table of a finite length module over $S$ exists as the Betti vector of a
degree 0 differential module with finite length homology. This leads us to
suspect a Boij-Söderberg type conjecture for differential modules.

######  Conjecture 3.7.

Every differential module over $S$ with homology of finite length has a Betti
vector that can be expressed as a positive rational combination of Betti
vectors of folds of pure resolutions with finite length homology whose degree
sequences form a chain.

Conjecture 3.7 posits that the rational cone of Betti vectors of differential
modules with finite length homology is the “flattening” of the cone of Betti
tables of finite length modules over $S$. That is, _up to scalar multiple_ ,
every Betti vector of a differential module with finite length homology can be
obtained by taking the flattening the Betti table of a finite length module
over $S$, as we saw in Example 3.3. We return to this example now in light of
Conjecture 3.7 and compare the pure decomposition of the Betti vector to that
of the Betti table of the homology.

######  Example 3.8.

Let $R=\mathbbm{k}[x,y]$ and $(D,{\partial})$ be as in Example 3.3. The Betti
table of the homology has pure decomposition

$\begin{matrix}&0&1&2\\\ \text{0:}&1&2&1\\\
\text{1:}&\text{-}&\text{-}&\text{-}\\\ \text{2:}&1&2&1\\\
\end{matrix}\quad=\quad\begin{matrix}&0&1&2\\\ \text{0:}&1&2&1\\\
\text{1:}&\text{-}&\text{-}&\text{-}\\\
\text{2:}&\text{-}&\text{-}&\text{-}\\\
\end{matrix}\quad+\quad\begin{matrix}&0&1&2\\\
\text{0:}&\text{-}&\text{-}&\text{-}\\\
\text{1:}&\text{-}&\text{-}&\text{-}\\\ \text{2:}&1&2&1\\\ \end{matrix}$

while the pure decomposition of $\beta^{\text{DM}}(D)$ is

$\beta^{\text{DM}}(D)=\frac{1}{2}(2^{\circ},3,0,1,0)+\frac{1}{2}(0^{\circ},1,0,3,2)$

On the other hand, consider the differential module
$(D^{\prime},{\partial}^{\prime})$ where $D^{\prime}=D$ is the same underlying
module, but where ${\partial}^{\prime}$ is now given by the matrix

$\begin{bmatrix}0&0&x&y&0&0&0&0\\\ 0&0&0&0&x&y&0&0\\\ 0&0&0&0&0&0&-y&0\\\
0&0&0&0&0&0&x&0\\\ 0&0&0&0&0&0&0&-y\\\ 0&0&0&0&0&0&0&x\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\end{bmatrix}$

(this matrix is _almost_ the same, except that the 1 has been replaced with a
0). The homology of $(D^{\prime},{\partial}^{\prime})$ is the same as the
homology of $(D,{\partial})$, but the Betti vector is

$\beta^{\text{DM}}(D^{\prime})=(1^{\circ},2,2,2,1)=(1^{\circ},2,1,0,0)+(0^{\circ},0,1,2,1).$

In particular, the pure decomposition of the Betti vector of a differential
module does not necessarily come from the pure decomposition of the Betti
table of its homology.

###  3.2. Differential modules with positive degree

We now return to the case where our differential modules have positive degree.
As we will see, this case demonstrates behavior that differs starkly from that
of $\operatorname{DM}(S,a)$ for $a\leq 0$, so much so that we may handle it
with methods that are much more direct and elementary. We begin with an
example of Remark 3.6.

######  Example 3.9.

Let $R=\mathbbm{k}[x,y]$ and consider the degree 1 differential module $R^{4}$
with differential

${\partial}=\begin{bmatrix}0&x&y&0\\\ 0&0&0&-y\\\ 0&0&0&x\\\
0&0&0&0\end{bmatrix}\colon R^{4}\to R^{4}(1).$

This differential module is a minimal free flag, so its Betti numbers are the
number of generators in each degree. That is, its Betti vector is
$(4,0,0,\ldots)$.

One consequence of Example 3.9 is that the rational cone of Betti vectors of
degree 2 differential modules over $\mathbbm{k}[x,y]$ with finite length
homology includes the entire positive orthant. It turns out that this is not
an isolated phenomenon.

######  Proposition 3.10.

For any $a>0$, the rational cone of Betti vectors of degree $a$ differential
modules over $S$ with finite length homology is equal to the positive orthant
in $\mathbbm{B}$.

###### Proof.

It suffices to show that the standard basis vectors are in the cone, i.e. that
every standard basis vector is a rational multiple of the Betti vector of some
$D\in\operatorname{DM}(S,a)$ with finite length homology. To show this, we
show that we can get multiples of basis vectors as the degree $a$ folds of
certain minimal free complexes.

Let $\mathbf{K}$ be the koszul complex on
$(x_{1}^{a},x_{2}^{a},\ldots,x_{n}^{a})$, and define $K$ to be
$\operatorname{Fold}_{a}(\mathbf{K})\in\operatorname{DM}(S,a)$. Then $K$ has
underlying module $S^{2^{n}}$ and has minimal differential, so its Betti
vector is $(2^{n},0,0,\ldots)$. By twisting appropriately, we obtain a
differential module whose Betti vector has $2^{n}$ in the $i^{th}$ entry and
$0$ elsewhere. ∎

######  Remark 3.11.

Any resolution may be folded into a degree $a$ differential module for any
$a$, and the resulting Betti vector varies for different values of $a$. We can
picture the vectors coming from pure resolutions as moving around while $a$
varies. For positive $a$, some of these vectors “flare out” as far as possible
resulting in a cone that fills the entire positive orthant, losing some of its
combinatorial structure.

As an immediate corollary of Proposition 3.10, we obtain the following:

######  Theorem 3.12.

Conjecture 3.7 is true for $a>0$.

###### Proof.

We will prove this by constructing the extremal rays of the cone explicitly.
By Proposition 3.10, extremal rays of the Betti cone are spanned by the
standard basis vectors $\mathbf{e}_{i}$. The ray generated by $e_{i}$ contains
the Betti vector of
$\operatorname{Fold}_{a}(\mathbf{K}(x_{1}^{a},\ldots,x_{n}^{a})[-i])$. Since
the Koszul complex on $(x_{1}^{a},\ldots,x_{n}^{a})$ is pure of type
$(0,a,2a,\ldots,na)$ with finite length homology, the Betti cone is spanned by
Betti vectors of differential modules whose homology have pure resolutions.
Furthermore, the shifted degree sequences of these resolutions form a chain. ∎

##  4\. Categorified Duality

In this section, we restrict to the case where $a=0$. We give a generalization
of the categorical pairing described in [EE17]. Our version is a pairing
between the derived category of differential $S$-modules and the derived
category of coherent sheaves on $\mathbbm{P}^{n-1}$ and takes values in the
derived category of differential $A$-modules where $A=\mathbbm{k}[t]$.

######  Definition 4.1.

By formally inverting all quasi-ismorphisms in $\operatorname{DM}(S,a)$, we
form the _derived category of differential $S$-modules_, denoted
$\operatorname{D_{DM}}(S,a)$. The subcategory of $\operatorname{D_{DM}}(S,a)$
whose objects have finitely generated homology is the _bounded derived
category of differential $S$-modules_, denoted
$\operatorname{D^{b}_{DM}}(S,a)$.

######  Notation.

We denote by $\operatorname{DM}(\mathbbm{P}^{n-1})$ the category of sheaves of
differential ${\mathcal{O}}_{\mathbbm{P}^{n-1}}$-modules, and denote the
corresponding derived and bounded derived categories in the same way as above.
We denote the bounded derived category of sheaves of (not necessarily
differential) ${\mathcal{O}}_{\mathbbm{P}^{n-1}}$-modules by
$\operatorname{D^{b}}(\mathbbm{P}^{n-1})$.

A key ingredient of the categorical pairing is the derived pushforard of a
differential module, which we now describe here in some detail, following the
same idea as [BE21b, pp. 11-12].

######  Definition 4.2 (Derived pushforward of a differential module).

Let $({\mathcal{D}},\delta)$ be a representative of a class in
$\operatorname{D^{b}_{DM}}(\mathbbm{P}^{n-1}_{A})$. Take
$\pi\colon\mathbbm{P}^{n-1}_{A}=\mathbbm{P}^{n-1}\otimes\mathbbm{A}^{1}\to\mathbbm{A}^{1}$
to be projection.

We exploit the equivalence in [BE21b, Remark 2.3] between the category of
differential modules over a ring $R$ and the category of 1-periodic
$R$-complexes. Taking the expansion of $({\mathcal{D}},\delta)$ we get a
1-periodic complex $\textbf{Ex}({\mathcal{D}})=$

$\cdots\leftarrow{\mathcal{D}}\xleftarrow{\delta}{\mathcal{D}}\xleftarrow{\delta}{\mathcal{D}}\leftarrow\cdots$

Now we compute the derived pushforward $R\pi_{*}$ of this complex. In each
position we take the Čech resolution of ${\mathcal{D}}$ with respect to the
usual affine cover of $\mathbbm{P}^{n-1}$ which gives a bicomplex

${\cdots}$${{\mathcal{C}}^{n-1}}$${{\mathcal{C}}^{n-1}}$${{\mathcal{C}}^{n-1}}$${\cdots}$${\cdots}$${\vdots}$${\vdots}$${\vdots}$${\cdots}$${\cdots}$${{\mathcal{C}}^{1}}$${{\mathcal{C}}^{1}}$${{\mathcal{C}}^{1}}$${\cdots}$${\cdots}$${{\mathcal{C}}^{0}}$${{\mathcal{C}}^{0}}$${{\mathcal{C}}^{0}}$${\cdots}$$\scriptstyle{\pm\delta}$$\scriptstyle{\pm\delta}$$\scriptstyle{\pm\delta}$$\scriptstyle{\pm\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{-\delta}$$\scriptstyle{-\delta}$$\scriptstyle{-\delta}$$\scriptstyle{-\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$

where the vertical arrows are the maps in the Čech resolution and the
horizontal arrows are induced by the differential $\delta$ (and thus still
square to 0). We apply $\pi_{*}$ to this complex and totalize. Since the Čech
complex is 0 after $n$ steps, each diagonal of the bicomplex above contains
one copy each of
${\mathcal{C}}^{0},{\mathcal{C}}^{1},\ldots,{\mathcal{C}}^{n-1}$, so this
yields a 1-periodic complex

$\cdots\leftarrow\bigoplus_{i=0}^{n-1}\pi_{*}({\mathcal{C}}^{i})\leftarrow\bigoplus_{i=0}^{n-1}\pi_{*}({\mathcal{C}}^{i})\leftarrow\bigoplus_{i=0}^{n-1}\pi_{*}({\mathcal{C}}^{i})\leftarrow\cdots$

We consider this as a sheaf of differential
${\mathcal{O}}_{\mathbbm{A}^{1}}$-modules, and the class represented by this
object is what we define to be $R\pi_{*}({\mathcal{D}})$.

######  Remark 4.3.

The degree 0 condition is necessary in this definition of $R\pi_{*}$. In
particular, we need the total complex after taking the Čech resolution to be
one-periodic, but this fails to be the case if the twists of ${\mathcal{D}}$
are not the same at each spot, since twisting a sheaf can genuinely change its
cohomology.

With this definition in hand, we now define our categorical pairing. Let
$\varphi\colon S\to S\otimes_{\mathbbm{k}}A$ be the homomorphism defined by
$x_{i}\mapsto tx_{i}$. This defineds an $S$-module structure on
$S\otimes_{\mathbbm{k}}A=S[t]$ which in turn yields an action of
$\mathbbm{P}^{n-1}$ on $\mathbbm{P}^{n-1}_{A}$. Using this action, we define
for $F\in\operatorname{DM}(S)$, a bigraded sheaf of differential
${\mathcal{O}}_{\mathbbm{P}^{n-1}_{A}}$-modules $\Sigma
F=\tilde{F}\otimes^{DM}_{\mathbbm{P}^{n-1}}{\mathcal{O}}_{\mathbbm{P}^{n-1}_{A}}$,
where the tensor product is taken using the structure given by $\varphi$. If
we write the differential ${\partial}$ on $F$ as a matrix, the differental
$\Sigma{\partial}$ on $\Sigma F$ is given by the matrix obtained from
${\partial}$ by replacing every entry $f$ by $t^{\deg(f)}f$. The purpose of
this step is that it will allow us to push forward to $A$ while still
retaining all of the graded data of our differential module over $S$.

######  Definition 4.4.

For $F\in\operatorname{D^{b}_{DM}}(S,0)$ and
${\mathcal{E}}\in\operatorname{D^{b}}(\mathbbm{P}^{n-1})$, we define the
functor

$\Phi\colon\operatorname{D^{b}_{DM}}(S,0)\times\operatorname{D^{b}}(\mathbbm{P}^{n-1})\to\operatorname{D^{b}_{DM}}(A,0)$

by

$\Phi(F,{\mathcal{E}})=R\pi_{*}\left(\Sigma
F\otimes^{DM}_{\mathbbm{P}^{n-1}_{A}}({\mathcal{E}}\boxtimes{\mathcal{O}}_{\mathbbm{A}^{1}})\right).$

where $\pi$ is the projection
$\mathbbm{P}^{n-1}\times\mathbbm{A}^{1}\to\mathbbm{A}^{1}$.

######  Definition 4.5.

Let $\gamma({\mathcal{E}})$ denote the vector whose $j{{}^{\text{th}}}$ entry
is the sum of the ranks of the hypercohomology modules of ${\mathcal{E}}(j)$.
We define the _absolute Hilbert function_ of ${\mathcal{E}}$ to be the
function $j\mapsto\gamma_{j}({\mathcal{E}})$. This should be thought of as a
cousin of the Hilbert polynomial of ${\mathcal{E}}$, but where the Euler
characteristic is replaced by the sum $\sum_{i\geq 0}h^{i}({\mathcal{E}}(j))$.

######  Theorem 4.6.

$\beta^{\text{DM}}(\Phi(F,{\mathcal{E}}))$ depends only on
$\beta^{\text{DM}}(F)$ and $\gamma({\mathcal{E}})$, in particular the Betti
vector of $\Phi(F,{\mathcal{E}})$ is given by the formula

$\beta^{\text{DM}}_{j}(\Phi(F,{\mathcal{E}}))=\beta^{\text{DM}}_{j}(F)\gamma_{-j}({\mathcal{E}})$

###### Proof.

Replacing $F$ with its minimal free resolution, we may assume that up to
quasi-isomorphism that its underlying module is
$F=\bigoplus_{j\in\mathbbm{Z}}S(-j)^{\beta^{\text{DM}}(F)_{j}}$. Let
${\mathcal{F}}=\Sigma
F\otimes^{DM}_{\mathbbm{P}^{n-1}_{A}}({\mathcal{E}}\boxtimes{\mathcal{O}}_{\mathbbm{A}^{1}})$.
We can rewrite the underlying module of ${\mathcal{F}}$ as

$\displaystyle{\mathcal{F}}$ $\displaystyle=\Sigma
F\otimes_{\mathbbm{P}^{n-1}_{A}}({\mathcal{E}}\boxtimes{\mathcal{O}}_{\mathbbm{A}^{1}})$
$\displaystyle=\left(\bigoplus_{j\in\mathbbm{Z}}{\mathcal{O}}_{\mathbbm{P}^{n-1}_{A}}(-j,-j)^{\beta^{\text{DM}}(F)_{j}}\right)\otimes_{\mathbbm{P}^{n-1}_{A}}({\mathcal{E}}\boxtimes
A)$
$\displaystyle=\bigoplus_{j\in\mathbbm{Z}}\left({\mathcal{O}}_{\mathbbm{P}^{n-1}}(-j)\boxtimes
A(-j)\right)^{\beta^{\text{DM}}(F)_{j}}\otimes_{\mathbbm{P}^{n-1}_{A}}({\mathcal{E}}\boxtimes
A)$
$\displaystyle=\bigoplus_{j\in\mathbbm{Z}}\left({\mathcal{E}}(-j)^{\beta^{\text{DM}}(F)_{j}}\boxtimes
A(-j)\right)$

By Definition 4.2, to compute $R\pi_{*}({\mathcal{F}})$ we expand
${\mathcal{F}}$ to a one-periodic complex, resolve it with a Čech resolution
at each step, then compute $\pi_{*}$. Letting ${\mathcal{C}}^{i}(-)$ denote
the $i{{}^{\text{th}}}$ sheaf in the Čech resolution using the standard affine
cover of $\mathbbm{P}^{n-1}$, we get the following bicomplex

${\cdots}$${\bigoplus_{j\in\mathbbm{Z}}{\mathcal{C}}^{n-1}({\mathcal{E}}(-j)^{\beta^{\text{DM}}(F)_{j}})\boxtimes
A(-j)}$${\bigoplus_{j\in\mathbbm{Z}}{\mathcal{C}}^{n-1}({\mathcal{E}}(-j)^{\beta^{\text{DM}}(F)_{j}})\boxtimes
A(-j)}$${\cdots}$${\vdots}$${\vdots}$${\cdots}$${\bigoplus_{j\in\mathbbm{Z}}{\mathcal{C}}^{1}({\mathcal{E}}(-j)^{\beta^{\text{DM}}(F)_{j}})\boxtimes
A(-j)}$${\bigoplus_{j\in\mathbbm{Z}}{\mathcal{C}}^{1}({\mathcal{E}}(-j)^{\beta^{\text{DM}}(F)_{j}})\boxtimes
A(-j)}$${\cdots}$${\cdots}$${\bigoplus_{j\in\mathbbm{Z}}{\mathcal{C}}^{0}({\mathcal{E}}(-j)^{\beta^{\text{DM}}(F)_{j}})\boxtimes
A(-j)}$${\bigoplus_{j\in\mathbbm{Z}}{\mathcal{C}}^{0}({\mathcal{E}}(-j)^{\beta^{\text{DM}}(F)_{j}})\boxtimes
A(-j)}$${\cdots}$$\scriptstyle{\pm\delta}$$\scriptstyle{\pm\delta}$$\scriptstyle{\pm\delta}$$\scriptstyle{-\delta}$$\scriptstyle{-\delta}$$\scriptstyle{-\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$

Now we apply $\pi_{*}$, but since every sheaf in the bicomplex is a box tensor
of a sheaf on $\mathbbm{P}^{n-1}$ with a twist of the structure sheaf on
$\mathbbm{A}^{1}$ and $\pi$ is a proper map, applying $\pi_{*}$ to
${\mathcal{C}}^{i}({\mathcal{E}}(-j)^{\beta^{\text{DM}}(F)_{j}})\boxtimes
A(-j)$ amounts to taking global sections of
${\mathcal{C}}^{i}({\mathcal{E}}(-j)^{\beta^{\text{DM}}(F)_{j}})$ tensored
with the appropriate twists of $A$. This turns each column into a direct sum
of complexes
$\Gamma\left(\mathbbm{P}^{n-1},{\mathcal{C}}^{\bullet}({\mathcal{E}}(-j))\right)\otimes_{\mathbbm{k}}A(-j)$.
Since
$\Gamma\left(\mathbbm{P}^{n-1},{\mathcal{C}}^{\bullet}({\mathcal{E}}(-j))\right)$
is a complex of $\mathbbm{k}$-vector spaces, the vertical maps in the
bicomplex are split. By [EFS03, Lemma 3.5], the totalization of the bicomplex
is homotopic to

$\cdots\leftarrow\bigoplus_{j\in\mathbbm{Z}}H^{tot}({\mathcal{E}}(-j)^{\beta^{\text{DM}}(F)_{j}})\otimes_{\mathbbm{k}}A(-j)\leftarrow\bigoplus_{j\in\mathbbm{Z}}H^{tot}({\mathcal{E}}(-j)^{\beta^{\text{DM}}(F)_{j}})\otimes_{\mathbbm{k}}A(-j)\leftarrow\cdots$

where $H^{tot}$ denotes the sum of the hypercohomology modules. The
differential is a sum of maps $H^{i}\to H^{(j<i)}$, where each map comes from
taking repeated compositions of the horizontal map $\delta$ with the section
that splits the vertical map. In particular, this complex is minimal and
1-periodic. The minimal free differential $A$-module that this corresponds to
is a representative of the class
$R\pi_{*}({\mathcal{F}})\in\operatorname{D^{b}_{DM}}(A,0)$, and we can compute
its Betti vector by counting the number of generators of the underlying module
in each degree. We have

$R\pi_{*}({\mathcal{F}})\simeq\bigoplus_{j\in\mathbbm{Z}}H^{tot}({\mathcal{E}}(-j)^{\beta^{\text{DM}}(F)_{j}})\otimes_{\mathbbm{k}}A(-j)=\bigoplus_{j\in\mathbbm{Z}}A(-j)^{\beta^{\text{DM}}(F)_{j}\gamma({\mathcal{E}})_{-j}}$

So the Betti vector of $\Phi(F,{\mathcal{E}})$ has $j{{}^{\text{th}}}$ entry

$\beta^{\text{DM}}_{j}(\Phi(F,{\mathcal{E}}))=\beta^{\text{DM}}_{j}(F)\gamma_{-j}({\mathcal{E}}).$

∎

######  Corollary 4.7.

Let $F\in\operatorname{D^{b}_{DM}}(S)$ and
${\mathcal{E}}\in\operatorname{D^{b}}(\mathbbm{P}^{n-1})$. Then

  1. (i)

$\beta^{\text{DM}}_{j}(F)=\beta^{\text{DM}}_{j}(\Phi(F,{\mathcal{O}}_{\mathbbm{P}^{n-1}}(j)).$

  2. (ii)

$\gamma_{j}({\mathcal{E}})=\beta^{\text{DM}}_{-j}(\Phi(S(j),{\mathcal{E}})).$

######  Lemma 4.8.

As sheaves of differential
${\mathcal{O}}_{\mathbbm{P}^{n-1}_{\mathbbm{k}(t)}}$-modules, there is an
isomorphism

$\left(\Sigma
F\otimes_{\mathbbm{P}^{n-1}_{A}}({\mathcal{E}}\boxtimes{\mathcal{O}}_{\mathbbm{A}^{1}})\right)\otimes_{A}\mathbbm{k}(t)\simeq\left(\tilde{F}\otimes_{\mathbbm{P}^{n-1}_{\mathbbm{k}}}{\mathcal{E}}\right)\otimes_{\mathbbm{k}}\mathbbm{k}(t).$

###### Proof.

We have a commutative diagram of rings

${S}$${S\otimes_{\mathbbm{k}}A}$${S\otimes_{\mathbbm{k}}\mathbbm{k}(t)}$${S\otimes_{\mathbbm{k}}\mathbbm{k}(t)}$$\scriptstyle{\varphi}$$\scriptstyle{1\otimes_{\mathbbm{k}}\mathbbm{k}(t)}$$\scriptstyle{1\otimes_{A}\mathbbm{k}(t)}$$\scriptstyle{\tilde{\varphi}}$

$\displaystyle\varphi\colon x_{i}\mapsto x_{i}\otimes t$
$\displaystyle\tilde{\varphi}\colon x_{i}\mapsto x_{i}\otimes t$

where the map $\tilde{\varphi}$ is invertible. This induces a diagram

${\operatorname{DM}(\mathbbm{P}^{n-1}_{\mathbbm{k}})}$${\operatorname{DM}(\mathbbm{P}^{n-1}_{A})}$${\operatorname{DM}(\mathbbm{P}^{n-1}_{\mathbbm{k}(t)})}$${\operatorname{DM}(\mathbbm{P}^{n-1}_{\mathbbm{k}(t)})}$$\scriptstyle{\varphi}$$\scriptstyle{-\otimes_{\mathbbm{k}}\mathbbm{k}(t)}$$\scriptstyle{-\otimes_{A}\mathbbm{k}(t)}$$\scriptstyle{\tilde{\varphi}}$

Unraveling definitions, we have
$\varphi\left(\tilde{F}\otimes_{\mathbbm{P}^{n-1}}{\mathcal{E}}\right)=\Sigma
F\otimes_{\mathbbm{P}^{n-1}_{A}}\left({\mathcal{E}}\boxtimes{\mathcal{O}}_{\mathbbm{A}^{1}}\right)$
as differential modules—that is, $\varphi$ is a map of modules that is
compatible with the differential (if we consider the differentials to be given
by matrices, the map $\varphi$ just replaces each entry $f$ with $t^{\deg
f}f$). By commutativity of the diagram, the claim follows since
$\tilde{\varphi}$ is invertible. ∎

######  Theorem 4.9.

If $\tilde{F}\otimes{\mathcal{E}}$ is exact then $\Phi(F,{\mathcal{E}})$ has
finite length homology.

###### Proof.

First we note that the conditions of exactness and finite length are not
altered by the grading on $A$, so in what follows we will always forget the
$A$-grading. Because flat pullback commutes with proper pushforward [Sta22],
we have a commutative diagram

${\operatorname{D^{b}_{DM}}(\mathbbm{P}^{n-1}_{A})}$${\operatorname{D^{b}_{DM}}(A)}$${\operatorname{D^{b}_{DM}}(\mathbbm{P}^{n-1}_{\mathbbm{k}(t)})}$${\operatorname{D^{b}_{DM}}(\mathbbm{k}(t))}$$\scriptstyle{-\otimes_{A}\mathbbm{k}(t)}$$\scriptstyle{R\pi_{*}}$$\scriptstyle{-\otimes_{\mathbbm{k}}\mathbbm{k}(t)}$$\scriptstyle{R\pi_{*}}$

The differential module $\Phi(F,{\mathcal{E}})$ is the image under $R\pi_{*}$
of the sheaf ${\mathcal{F}}=\Sigma
F\otimes_{\mathbbm{P}^{n-1}_{A}}({\mathcal{E}}\boxtimes{\mathcal{O}}_{\mathbbm{A}^{1}})$.
On the other hand, taking the tensor product
${\mathcal{F}}\otimes_{A}\mathbbm{k}(t)$ gives $\left(\Sigma
F\otimes_{\mathbbm{P}^{n-1}_{A}}({\mathcal{E}}\boxtimes{\mathcal{O}}_{\mathbbm{A}^{1}})\right)\otimes_{A}\mathbbm{k}(t)$.
By Lemma 4.8, this is isomorphic to
$(\tilde{F}\otimes{\mathcal{E}})\otimes_{\mathbbm{k}}\mathbbm{k}(t)$. But
since $\tilde{F}\otimes{\mathcal{E}}$ is exact,
$(\tilde{F}\otimes{\mathcal{E}})\otimes_{\mathbbm{k}}\mathbbm{k}(t)$ is 0 in
the derived category. Thus
$\Phi(F,{\mathcal{E}})\otimes_{\mathbbm{k}}\mathbbm{k}(t)$ must be 0 in
$\operatorname{D^{b}_{DM}}(\mathbbm{k}(t))$. Since inverting $t$ yields a
differential module with no homology, it therefore follows that
$\Phi(F,{\mathcal{E}})$ has finite length homology. ∎

The implication of Theorems 4.6 and 4.9 is that the categorical pairing $\Phi$
descends to a pairing on cones

$BS_{\text{DM}}(S,0)\times C_{\text{vb}}(\mathbbm{P}^{n-1})\to
BS_{\text{DM}}(A,0)$

where $C_{\text{vb}}(\mathbbm{P}^{n-1})$ is the cone of cohomology vectors of
vector bundles on $\mathbbm{P}^{n-1}$. In Section 5, we will describe the
exterior facets of $BS_{\text{DM}}(A,0)$ as linear functionals on
$\mathbbm{B}$, and in Section 6 we will use the base case of
$BS_{\text{DM}}(A,0)$ alongside some computational data to outline a strategy
for how this pairing on cones could be used to prove Conjecture 3.7 in full
for degree 0 differential modules.

We finish this section with a concrete example to demonstrate how $\Phi$
works.

######  Example 4.10.

Let $S=\mathbbm{k}[x,y]$ and let $(F,{\partial})$ be the fold of the Koszul
complex on the variables, so

$\displaystyle F=S\oplus S(-1)^{2}\oplus
S(-2),\quad{\partial}=\begin{bmatrix}0&x&y&0\\\ 0&0&0&-y\\\ 0&0&0&x\\\
0&0&0&0\end{bmatrix}$

and let ${\mathcal{E}}={\mathcal{O}}_{\mathbbm{P}^{1}}$. Then $\Sigma F$ has
underlying sheaf
${\mathcal{O}}_{\mathbbm{P}^{1}_{A}}\oplus{\mathcal{O}}_{\mathbbm{P}^{1}_{A}}(-1,-1)^{2}\oplus{\mathcal{O}}_{\mathbbm{P}^{1}_{A}}(-2,-2)$
with differential

$\Sigma{\partial}=\begin{bmatrix}0&tx&ty&0\\\ 0&0&0&-ty\\\ 0&0&0&tx\\\
0&0&0&0\end{bmatrix}.$

Tensoring with ${\mathcal{E}}\boxtimes{\mathcal{O}}_{\mathbbm{A}^{1}}$ does
not change the underlying sheaf or differential since
${\mathcal{E}}={\mathcal{O}}_{\mathbbm{P}^{1}}$.

Now computing $R\pi_{*}$, we get a differential $A$-module whose underying
module is

$\bigoplus_{j\in\mathbbm{Z}}A(-j)^{\beta^{\text{DM}}(F)_{j}\gamma({\mathcal{E}})_{-j}}=A\oplus
A(-2)$

since ${\mathcal{O}}_{\mathbbm{P}^{1}}(-1)$ has no cohomology. To compute the
differential on $\Phi(F,{\mathcal{E}})$, we use [EFS03, Lemma 3.5], which
tells us that our differential should look like

$\begin{bmatrix}d_{0,0}&d_{1,0}\\\ 0&d_{1,1}\end{bmatrix}$

where $d_{0,0}:A\to A$ and $d_{1,1}:A(-2)\to A(-2)$ are induced by the pieces
of $\Sigma{\partial}$ that map
${\mathcal{O}}_{\mathbbm{P}^{1}_{A}}\to{\mathcal{O}}_{\mathbbm{P}^{1}_{A}}$
and
${\mathcal{O}}_{\mathbbm{P}^{1}_{A}}(-2)\to{\mathcal{O}}_{\mathbbm{P}^{1}_{A}}(-2)$
and are therefore 0 since $\Sigma F$ was flag. By 4.9 the homology is finite
length, so the map $d_{1,0}:A(-2)\to A$ must be of the form $at^{2}$ for some
unit $a\in A$, so our differential is

$\begin{bmatrix}0&at^{2}\\\ 0&0\end{bmatrix}$

##  5\. Differential Modules Over $\mathbbm{k}[t]$

In this section we investigate in detail the case of degree 0 differential
modules over the polynomial ring in one variable. In particular, we prove
Conjecture 3.7 for differential modules over $\mathbbm{k}[t]$ with finite
length homology. When combined with the results of Section 4, this gives a
base case for the theory in $n>1$ variables.

######  Theorem 5.1.

Every degree 0 differential modules over $A=\mathbbm{k}[t]$ with finite length
homology has a Betti vector that can be expressed as a positive rational
combination of Betti vectors of folds of pure resolutions whose degree
sequences form a chain.

To prove Theorem 5.1, we will define three cones in $\mathbbm{B}$. The first
is the cone $B$ of Betti vectors of differential modules with finite length
homology. The second is the cone $C$ spanned by Betti vectors of folds of pure
resolutions. We will show these two cones are equal via a third cone $T$
defined by the nonnegativity of the following linear functionals:

$\left\\{\tau_{j}\colon\beta\mapsto-\beta_{j}+\sum_{i\neq
j}\beta_{i}\quad\text{and}\quad\sigma_{j}\colon\beta\mapsto\beta_{j}\quad\text{for}\quad
j\in\mathbbm{Z}\right\\}$ (1)

Let $\mathbbm{B}_{(p,q)}$ be the subspace of vectors $\mathbf{b}$ such that
$b_{n}=0$ for $n>p$ or $n<p$, i.e. all nonzero entries occur between the
indices $p$ and $q$. We set $B_{(p,q)}=\mathbbm{B}_{(p,q)}\cap
B,C_{(p,q)}=\mathbbm{B}_{(p,q)}\cap C,$ and $T_{(p,q)}=\mathbbm{B}_{(p,q)}\cap
T$. We first show that for any $p,q$, $T_{(p,q)}\subseteq C_{(p,q)}$ by giving
an explicit algorithm for decomposing vectors in $T$ as positive rational sums
of Betti vectors of folds of pure resolutions.

Algorithm 1 Simplicial Decomposition Algorithm

vector $\mathbf{b}=(b_{p},b_{p+1},\ldots,b_{q})\in T$

decomposition of $\mathbf{b}$ as rational combination of pure Betti vectors
$c\mathbf{e}_{a,b}$

$L:=$ empty list

PHASE I:

while $\tau_{i}(\mathbf{b})>0$ for all $p\leq i\leq q$ do

Let $k,\ell$ be the first two indices such that $b_{k},b_{\ell}\neq 0$

Set $j$ to be the an index where $\tau_{j}(\mathbf{b})$ is minimal

$c:=\min\\{b_{k},b_{\ell},\frac{1}{2}\tau_{j}\cdot b\\}$

$\mathbf{b}:=\mathbf{b}-c\mathbf{e}_{\\{k,\ell\\}}$

add $(c,\\{k,l\\})$ to $L$

end while

PHASE II:

if $\tau_{j}(\mathbf{b})=0$ then

for $i=p,\ldots,q$ do

if $b_{i}\neq 0$ then

add $(b_{i},\\{i,j\\})$ to $L$

end if

$\mathbf{b}:=\mathbf{b}-b_{i}\mathbf{e}_{\\{i,j\\}}$

end for

end ifreturn $L$

We demonstrate the above algorithm with an example:

######  Example 5.2.

Take $\mathbf{b}=(3,4,2,5)$ and assume our nonzero window is between indicies
$0$ and $3$. We can quickly see that $\tau_{i}(\mathbf{b})>0$ for
$i=0,\ldots,3$, so we proceed with ‘Phase I’, which is essentially a greedy
algorithm. We take $(3,4,2,5)-(1,1,0,0)=(2,3,2,5)$, which is our new
$\mathbf{b}$. Again we check that $\tau_{i}(\mathbf{b})>0$ for each $i$. Since
the first two entries are still nonzero, we take our new $\mathbf{b}$ to be
$(2,3,2,5)-(1,1,0,0)=(1,2,2,5)$. Now $\tau_{3}(\mathbf{b})=0$, so we move on
to ‘Phase II’, which tells us to decompose $(1,2,2,5)$ as
$(1,0,0,1)+2(0,1,0,1)+2(0,0,1,1)$. This gives us a final decomposition of
$\mathbf{b}$ as

$\mathbf{b}=2(1,1,0,0)+(1,0,0,1)+2(0,1,0,1)+2(0,0,1,1).$

The correctness of the decomposition algorithm is proved in the following
lemmas.

######  Lemma 5.3.

If $\tau_{j}\mathbf{b}=0$ for some $j$ and $b_{n}=0$ for $k<n<l$ and $n<k$,
then Phase II of the algorithm gives a decomposition of $\mathbf{b}$ as a
rational combination of pure Betti vectors.

###### Proof.

By definition, $\tau_{j}\mathbf{b}=0$ if and only if $b_{j}=\sum_{i\neq
j}b_{i}$, so $\mathbf{b}=\sum_{i\neq j}b_{i}\mathbf{e}_{\\{i,j\\}}$. ∎

The next two lemmas combine to say that after each iteration of the while loop
in Phase I of the algorithm, the vector $\mathbf{b}$ remains in the cone $T$.

######  Lemma 5.4.

Let $\mathbf{b}\in T$ and $c=\min\\{b_{k},b_{\ell},\frac{1}{2}\tau_{j}\cdot
b\\}$ where $j$ is an index such that $\tau_{j}\mathbf{b}$ is minimal. Let
$\mathbf{b}^{\prime}=\mathbf{b}-c\mathbf{e}_{\\{k,l\\}}$. Then
$\tau_{i}\mathbf{b}^{\prime}\geq 0$ for all $i$.

###### Proof.

By definition,

$\displaystyle\tau_{i}\mathbf{b}^{\prime}$
$\displaystyle=\tau_{i}\mathbf{b}-c\tau_{i}\mathbf{e}_{\\{k,l\\}}$
$\displaystyle=\max\begin{cases}\tau_{i}\mathbf{b}-b_{k}\tau_{i}\mathbf{e}_{\\{k,l\\}}\\\
\tau_{i}\mathbf{b}-b_{l}\tau_{i}\mathbf{e}_{\\{k,l\\}}\\\
\tau_{i}\mathbf{b}-(\frac{1}{2}\tau_{j}\mathbf{b})\tau_{i}\mathbf{e}_{\\{k,l\\}}\end{cases}$

Note that $\tau_{i}\mathbf{e}_{\\{k,l\\}}=0$ if $i=k,l$ and
$\tau_{i}\mathbf{e}_{\\{k,l\\}}=2$ otherwise, so we have

$\displaystyle\tau_{i}\mathbf{b}^{\prime}=\begin{cases}\tau_{i}\mathbf{b}&\text{
if }i=k,l\\\
\max\\{\tau_{i}\mathbf{b}-2b_{k},\tau_{i}\mathbf{b}-2b_{l},\tau_{i}\mathbf{b}-\tau_{j}\mathbf{b}\\}&\text{
if }i\neq k,l\end{cases}$

By assumption, $\tau_{i}\mathbf{b}\geq 0$, so in the case where $i=k,l$ we’re
done. In the case where $i\neq k,l$,
$\tau_{i}\mathbf{b}-\tau_{j}\mathbf{b}\geq 0$ since $j$ is an index where
$\tau_{j}\mathbf{b}$ is minimal. Since
$\tau_{i}\mathbf{b}-\tau_{j}\mathbf{b}\geq 0$, we must have that

$\max\\{\tau_{i}\mathbf{b}-2b_{k},\tau_{i}\mathbf{b}-2b_{l},\tau_{i}\mathbf{b}-\tau_{j}\mathbf{b}\\}\geq
0.$

∎

######  Lemma 5.5.

Let $\mathbf{b}\in T$ and $c=\min\\{b_{k},b_{\ell},\frac{1}{2}\tau_{j}\cdot
b\\}$ where $j$ is an index such that $\tau_{j}\mathbf{b}$ is minimal. Let
$\mathbf{b}^{\prime}=\mathbf{b}-c\mathbf{e}_{\\{k,l\\}}$. Then
$\sigma_{i}\mathbf{b}^{\prime}\geq 0$ for all $i$.

###### Proof.

$\displaystyle\sigma_{i}\mathbf{b}^{\prime}=\sigma_{i}\mathbf{b}-c\sigma_{i}\mathbf{e}_{\\{k,l\\}}=\begin{cases}b_{i}-c&\text{
if }i=k,l\\\ b_{i}&\text{ if }i\neq k,l\end{cases}$

By assumption, $b_{i}\geq 0$, so in the case where $i\neq k,l$
$\sigma_{i}\mathbf{b}^{\prime}\geq 0$. If $i=k,l$,

$\displaystyle\sigma_{i}\mathbf{b}^{\prime}=b_{i}-c=\max\\{b_{i}-b_{k},b_{i}-b_{l},b_{i}-\frac{1}{2}\tau_{j}\mathbf{b}\\}.$

If $i=k$, then $b_{i}-b_{k}=0$ and if $i=l$ then $b_{i}-b_{l}=0$, so in eather
case
$\sigma_{i}\mathbf{b}^{\prime}=\max\\{b_{i}-b_{k},b_{i}-b_{l},b_{i}-\frac{1}{2}\tau_{j}\mathbf{b}\\}\geq
0$. ∎

We have seen already that Phase II of the algorithm terminates with a
decomposition by pure vectors, so to show that the algorithm as a whole
terminates we just need the following:

######  Lemma 5.6.

Phase I of the algorithm terminates, that is, after a finite number of
iterations, $\mathbf{b}$ will satisfy $\tau_{j}\mathbf{b}=0$ for some $j$.

###### Proof.

First notice that if we multiply $\mathbf{b}$ by a constant, the only impact
on the algorithm is that at each step the choice of $c$ will be multiplied by
said constant, no other aspect of the decomposition will be altered.
Multiplying by a constant, we may assume that $b_{i}\in\mathbbm{Z}$ for all
$i$ and, furthermore, that $\sum b_{i}$ is even. This implies that
$\frac{1}{2}\tau_{i}\mathbf{b}\in\mathbbm{Z}$ for all $i$,so all entries of
$\mathbf{b}$ are integers throughout every step of the algorithm.

We proceed by induction on the number of nonzero entries of $\mathbf{b}$. If
all entries are 0, the result is trivial. Exactly one nonzero entry is
impossible by the assumption that $\tau_{i}\mathbf{b}\geq 0$ for all $i$. If
$b_{k}$ and $b_{\ell}$ are the only two nonzero entries of $\mathbf{b}$, then
$\tau_{k}\mathbf{b}=\tau_{\ell}\mathbf{b}=0$. Suppose that $\mathbf{b}$ has
more than 2 nonzero entries and take
$c=\min\\{b_{k},b_{l},\frac{1}{2}\tau_{j}\mathbf{b}\\}$ where $j$ is an index
such that $\tau_{j}\mathbf{b}$ is minimal as in the algorithm. If $c=b_{k}$ or
$b_{l}$, then replacing $\mathbf{b}$ by $\mathbf{b}-c\mathbf{e}_{\\{k,l\\}}$
results in either $b_{l}=0$ or $b_{k}=0$, so the number of nonzero entries
decreases and we may apply the induction hypothesis to get the result.

Now consider the case where $c=\frac{1}{2}\tau_{j}\mathbf{b}$ and let
$\mathbf{b}^{\prime}=\mathbf{b}-c\mathbf{e}_{\\{k,l\\}}$. Then we have

$\tau_{i}\mathbf{b}^{\prime}=\tau_{i}\mathbf{b}-\left(\frac{1}{2}\tau_{j}\mathbf{b}\right)\tau_{i}\mathbf{e}_{\\{k,l\\}}=\begin{cases}\tau_{i}\mathbf{b}&\text{
if }i=k,l\\\ \tau_{i}\mathbf{b}-\tau_{j}\mathbf{b}&\text{ if }i\neq
k,l\end{cases}.$

If $j\neq k,l$ then
$\tau_{j}\mathbf{b}^{\prime}=\tau_{j}\mathbf{b}-\tau_{j}\mathbf{b}=0$ and
we’re done.

If $j=k$ or $l$, then
$\tau_{i}\mathbf{b}^{\prime}=\tau_{i}\mathbf{b}-\tau_{j}\mathbf{b}$, in
particular $\tau_{i}\mathbf{b}$ strictly decreases for all $i\neq k,l$ every
iteration. Since $\tau_{i}\mathbf{b}\in\mathbbm{Z}$ for all $i$, after enough
iterations there will be some $i\neq k,l$ for which
$\tau_{i}\mathbf{b}^{\prime}<\tau_{j}\mathbf{b}$. Since in each iteration, $j$
is picked so that $\tau_{j}$ is minimal, this means that in the subsequent
iteration of the algorithm we will be in the case where $j\neq k,l$, so the
result follows. ∎

The above lemmas combine to show that for any $p<q$, we have containment
$T_{(p,q)}\subseteq C_{(p,q)}$, i.e. that the cone spanned by the pure Betti
vectors is contained in the cone defined by the facet equations in (1). The
following proposition says that additionally any vector in $T$ can be
decomposed via pure Betti vectors whose degree sequences _form a chain_.

######  Proposition 5.7.

The degree sequences in $L$ form a chain.

###### Proof.

First we consider the two phases of the algorithm separately. In Phase I, the
degree sequence added to the list at each iteration is $(k,\ell)$ where
$k,\ell$ are the first two indices of nonzero entries in $\mathbf{b}$. In
order for the values of $k,\ell$ to change at the next iteration, one of
$b_{k}$ or $b_{\ell}$ must become zero, meaning that the first two indices of
nonzero entries in $\mathbf{b}$ give a degree sequence strictly greater than
the previous $(k,\ell)$. In Phase II, $j$ is fixed and $i$ increases, so the
degree sequences $(i,j)$ for $i<j$ and $(j,i)$ for $i>j$ form a chain.

What remains to prove is that the first degree sequence added to $L$ in Phase
II is greater than the last degree sequence added to $L$ in Phase I. Let
$(k,\ell)$ be the final degree sequence added in Phase I. This means that for
all $i<k$ and all $k<i<\ell$, $b_{i}=0$ since $k$ and $\ell$ are the first two
nonzero indices. If $\tau_{j}\cdot\mathbf{b}=0$ then either $j>\ell$ or
$\mathbf{b}=0$. In the latter case, no more degree sequences are added, so
we’re done. In the former case, the first degree sequence added in Phase II is
either $(i,j)$ for $i=k$ or $\ell$, in which case we have $(i,j)>(k,\ell)$ or
$i>\ell$ in which case we also have $(i,j)>(k,\ell)$. ∎

We can now prove Theorem 5.1 by establishing the equality of the three cones
$B,C$, and $T$.

###### Proof of Theorem 5.1.

The containment $C\subseteq B$ follows from the existence of differential
modules with pure Betti vectors.

By the simplicial decomposition algorithm and subsequent lemmas, we have
$T_{(a,b)}\subseteq C_{(a,b)}$. Since any vector in $\mathbbm{B}$ has finitely
many nonzero entries, this gives the containment $T\subseteq C$.

To show the containment $B\subseteq T$, we will prove that the functionals
$\tau_{j}$ and $\sigma_{j}$ are nonnegative on the Betti vector
$\beta^{\text{DM}}(D)$ for any graded, degree 0 differential module $D$ with
finite length homology over $\mathbbm{k}[t]$. Let $D\in\operatorname{DM}(A,0)$
have finite length homology $H$. Since $H$ has finite length over a PID, it is
torsion. Thus $H$ is of the form $\bigoplus A(-p)/(t^{q})$, so it suffices to
check nonnegativity for $H=A(-p)/(t^{q})$. In this case, $H$ has minimal free
resolution $A(-p-q)\xrightarrow{t^{q}}A(-p)\to 0$. By [BE21a, Theorem 3.2],
$D$ has a free flag resolution $F$ of the form $A(-p-q)\oplus A(-p)$ with
differential $\begin{bmatrix}0&t^{q}\\\ 0&0\end{bmatrix}$, which is minimal.
Thus $\beta^{\text{DM}}(D)_{i}=1$ for $i=p,p+q$ and 0 otherwise, so
$\tau_{j}\beta^{\text{DM}}(D)=0$ or $2$ and $\sigma_{j}\beta^{\text{DM}}(D)=0$
or $1$.

This establishes the containments $B\subseteq T\subseteq C$ and $C\subseteq
B$. Thus the cone $B$ of Betti vectors of differential modules with finite
length homology over $A$ is equal to the cone $C$ spanned by the pure Betti
vectors.

Furthermore, by Proposition 5.7, every Betti vector of a differential module
with finte length homology has a decomposition by pure Betti vectors whose
degree sequences form a chain. ∎

######  Remark 5.8.

For Betti diagrams of finite length modules, every Betti table has a _unique_
decomposition by pure Betti tables whose degree sequences form a chain. For
differential modules however, the Betti vector my have a nonunique
decomposition even after insisting that the pure vectors come from a chain of
degree sequences. As an easy example, consider the Betti vector $(1,1,1,1)$.
It can be decomposed either as $(1,1,0,0)+(0,0,1,1),(1,0,1,0)+(0,1,0,1),\text{
or }(1,0,0,1)+(0,1,1,0)$. Of these three decompositions, the first two both
correspond to degree sequences that form a chain. Our algorithm would yield
the first decomposition, corresponding to the chain of degree sequences
$(0,1)<(2,3)$. The second decomposition corresponds to the chain
$(0,2)<(1,3)$.

##  6\. Predictions for Differential Modules Over $S$

We conjecture that Theorem 5.1 gives a potential base case for the general
theory in $n$ variables. Because the $\tau$ and $\sigma$ functionals are non-
negative on differential modules in $\operatorname{D^{b}_{DM}}(A,0)$ with
finite length homology, Theorem 4.9 implies that they are non-negative on
$\beta^{\text{DM}}(\Phi(F,{\mathcal{E}}))$ for any coherent sheaf
${\mathcal{E}}$ and any differential module
$F\in\operatorname{D^{b}_{DM}}(S,0)$ with finite length homology. In the
original theory, similar machinery provided a method for proving nonnegativity
of facet equations since the facet equations of the cone spanned by the pure
Betti tables came from cohomology tables of vector bundles on
$\mathbbm{P}^{n}$. Due to the apparent similarity between the original theory
and the generalization to differential modules in the degree 0 case, we expect
that something similar might hapen here as well. This illuminates a strategy
for proving Conjecture 3.7 in general. The idea is that once we know the
exterior facets of the cone spanned by Betti vectors of differential modules
with pure homology, this will enable us to prove non-negativity of the facet
equations on the Betti vectors of differential modules in
$\operatorname{DM}(S,0)$, provided the facet equations come from cohomology
tables of sheaves on $\mathbbm{P}^{n-1}$ as in the original theory. Proving a
facet description has remained elusive thus far, but we can use Macaulay2 to
compute facet descriptions of the cone in finite dimensional windows.

For instance, let’s examine the case where $S=\mathbbm{k}[x,y,z]$ and look at
the cone spanned by Betti vectors of differential modules with homology that
is pure with degree sequence between $(0,1,2,3)$ and $(4,5,6,7)$. A Fourier
Motzkin elimination computation gives the columns of the following matrix as a
subset of the linear functionals in the facet description of the cone:

$\setcounter{MaxMatrixCols}{14}\begin{bmatrix}&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\\\
\cdots&15&15&-10&5&6&15&2&3&5&1&8&3&\cdots\\\
\cdots&10&10&6&2&-3&-8&0&1&0&0&3&0&\cdots\\\
\cdots&-6&6&3&0&1&3&1&0&3&0&0&1&\cdots\\\
\cdots&3&-3&1&1&0&0&1&0&-4&1&1&0&\cdots\\\
\cdots&1&1&0&1&0&1&0&1&3&-3&0&3&\cdots\\\
\cdots&0&0&0&0&1&0&2&-3&0&6&3&8&\cdots\\\
\cdots&0&0&1&2&3&3&5&6&5&10&8&-15&\cdots\\\
\cdots&1&1&3&-5&6&8&-9&10&12&15&-15&24&\cdots\\\
&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\end{bmatrix}$
(2)

One might recognize the entries of each column as the ranks of the sheaf
cohomology modules of _supernatural_ vector bundles on $\mathbbm{P}^{2}$ (see
[ES09]) appearing with a single negative entry in each column. For instance,
twists of the line bundle ${\mathcal{O}}_{\mathbbm{P}^{2}}$ have cohomology
modules whose ranks are given by the entries in the first three columns of
(2), with some shift. If we look at the sign pattern, we notice that there is
a single negative entry in each column, which is exactly what we saw in the
facet equations for $BS_{\text{DM}}(A,0)$, where the facets were given by
taking inner product with vectors of the form
$\tau_{j}=(\ldots,1,1,-1,1,\ldots)$, with the negative entry occurring in the
$j{{}^{\text{th}}}$ entry. What is more, we see this same phenomenon repeated
for all examples computed.

This leads us to a conjecture that, similarly to [EE17] the facets
$BS_{\text{DM}}(S,0)$ come from combining the cohomology tables of certain
vector bundles on $\mathbbm{P}^{n-1}$ with the facets for
$BS_{\text{DM}}(A,0)$.

######  Conjecture 6.1.

The exterior facets of the cone of Betti vectors of degree 0 differential
modules with finite length homology over $S$ are given by the vanishing of
linear functionals that arise as the composition

$\operatorname{D^{b}_{DM}}(S)\xrightarrow{\Phi(-,{\mathcal{E}})}\operatorname{D^{b}_{DM}}(A)\xrightarrow{f}\mathbbm{Z}$

where ${\mathcal{E}}$ is a supernatural vector bundle on $\mathbbm{P}^{n-1}$
and $f$ is one of the linear functionals $\tau_{j}$ or $\sigma_{i}$ defined in
(1).

To see this in action, consider the first column of (2), and suppose we index
so that the $-6$ is the entry in position 0. The vector

$(\ldots,15,10,6,3,1,0,0,1,\ldots)$

is equal to $\gamma({\mathcal{O}}_{\mathbbm{P}^{n-1}}(-5))$. Dot product with
this vector defines a linear functional that sends $\beta^{\text{DM}}(F)$ to
$\beta^{\text{DM}}(\Phi(F,{\mathcal{O}}_{\mathbbm{P}^{n-1}}(-5)))$. Composing
this with $\tau_{0}$ as defined in (1) gives the linear function that is dot
product with the first column of (2).

## References

  * [ABI07] Luchezar L. Avramov, Ragnar-Olaf Buchweitz, and Srikanth Iyengar, _Class and rank of differential modules_ , Invent. Math. 169 (2007), no. 1, 1–35. MR 2308849
  * [BBEG12] Christine Berkesch, Jesse Burke, Daniel Erman, and Courtney Gibbons, _The cone of betti diagrams over a hypersurface ring of low embedding dimension_ , Journal of Pure and Applied Algebra 216 (2012), no. 10, 2256–2268.
  * [BD10] Adam Boocher and Justin W DeVries, _On the rank of multigraded differential modules_ , arXiv preprint arXiv:1011.2167 (2010).
  * [BE21a] Michael K. Brown and Daniel Erman, _Minimal free resolutions of differential modules_ , 2021.
  * [BE21b] Michael K. Brown and Daniel Erman, _Tate resolutions on toric varieties_ , 2021\.
  * [BEKS11] Christine Berkesch, Daniel Erman, Manoj Kummini, and Steven V. Sam, _Shapes of free resolutions over a local ring_ , Mathematische Annalen 354 (2011), no. 3, 939–954.
  * [BS08] Mats Boij and Jonas Söderberg, _Graded betti numbers of cohen-macaulay modules and the multiplicity conjecture_ , Journal of the London Mathematical Society 78 (2008), no. 1, 85–106.
  * [BS12] Mats Boij and Jonas Söderberg, _Betti numbers of graded modules and the multiplicity conjecture in the non-cohen–macaulay case_ , Algebra & Number Theory 6 (2012), no. 3, 437–454.
  * [BS15] Mats Boij and Gregory G. Smith, _Cones of hilbert functions_ , International Mathematics Research Notices 2015 (2015), no. 20, 10314–10338.
  * [BV22] Maya Banks and Keller VandeBogert, _Differential modules with complete intersection homology_ , 2022.
  * [CE16] Henry Cartan and Samuel Eilenberg, _Homological algebra (pms-19), volume 19_ , Princeton university press, 2016.
  * [EE17] David Eisenbud and Daniel Erman, _Categorified duality in boij–söderberg theory and invariants of free complexes_ , Journal of the European Mathematical Society 19 (2017), no. 9, 2657–2695.
  * [EFS03] David Eisenbud, Gunnar Fløystad, and Frank-Olaf Schreyer, _Sheaf cohomology and free resolutions over exterior algebras_ , Transactions of the American Mathematical Society 355 (2003), no. 11, 4397–4426.
  * [Erm09] Daniel Erman, _The semigroup of betti diagrams_ , Algebra & Number Theory 3 (2009), no. 3, 341–365.
  * [ES] David Eisenbud and Frank-Olaf Schreyer, _Betti numbers of graded modules and cohomology of vector bundles_ , J. Amer. Math. Soc. 22, no. 3, 859–888.
  * [ES09] David Eisenbud and Frank-Olaf Schreyer, _Cohomology of coherent sheaves and series of supernatural bundles_ , 2009.
  * [ES16] Daniel Erman and Steven V Sam, _Supernatural analogues of beilinson monads_ , Compositio Mathematica 152 (2016), no. 12, 2545–2562.
  * [FJK11] Gunnar Fløystad, Trygve Johnsen, and Andreas Leopold Knutsen (eds.), _Combinatorial aspects of commutative algebra and algebraic geometry_ , Springer Berlin Heidelberg, 2011.
  * [FL18] Nicolas Ford and Jake Levinson, _Foundations of boij–söderberg theory for grassmannians_ , Compositio Mathematica 154 (2018), no. 10, 2205–2238.
  * [Flø10] Gunnar Fløystad, _The linear space of betti diagrams of multigraded artinian modules_ , Mathematical Research Letters 17 (2010), no. 5, 943–958.
  * [FLS18] Nicolas Ford, Jake Levinson, and Steven Sam, _Towards boij–söderberg theory for grassmannians : the case of square matrices_ , Algebra & Number Theory 12 (2018), no. 2, 285–303.
  * [GS] Daniel R. Grayson and Michael E. Stillman, _Macaulay2, a software system for research in algebraic geometry_ , Available at urlhttp://www.math.uiuc.edu/Macaulay2/.
  * [Hoc74] Melvin Hochster, _The equicharacteristic case of some homological conjectures on local rings_ , Bulletin of the American Mathematical Society 80 (1974), no. 4, 683 – 686.
  * [IMW22] Srikanth B. Iyengar, Linquan Ma, and Mark E. Walker, _Lim ulrich sequences and boij-söderberg cones_ , 2022.
  * [IW18] Srikanth B Iyengar and Mark E Walker, _Examples of finite free complexes of small rank and small homology_ , Acta Mathematica 221 (2018), no. 1, 143–158.
  * [PS73] Christian Peskine and Lucien Szpiro, _Dimension projective finie et cohomologie locale_ , Publications Mathématiques de l’IHÉS 42 (1973), 47–119 (fr).
  * [Rob89] Paul Roberts, _Intersection theorems_ , Commutative Algebra (New York, NY) (M. Hochster, C. Huneke, and J. D. Sally, eds.), Springer New York, 1989, pp. 417–436.
  * [SS21] Alessandro De Stefani and Ilya Smirnov, _Decomposition of graded local cohomology tables_ , Mathematische Zeitschrift 297 (2021), no. 1, 1–24.
  * [Sta22] The Stacks project authors, _The stacks project_ , https://stacks.math.columbia.edu, 2022.
  * [ŞÜ19] Berrin Şentürk and Özgün Ünlü, _Carlsson’s rank conjecture and a conjecture on square-zero upper triangular matrices_ , Journal of Pure and Applied Algebra 223 (2019), no. 6, 2562–2584.