2.1 Spaces of interest

Throughout, let $V,W$ be fixed $\mathbb{C}$ -vector spaces of dimensions $k$ and $n$ , with $k\leqslant n$ . We set

so $X=\operatorname{Spec}(R_{k,n})$ , the affine variety of $k\times n$ matrices $(x_{ij})$ , and $R_{k,n}$ is the polynomial ring whose variables are the entries of the matrix. We also consider the subvarieties of full-rank and rank-deficient matrices,

$$\begin{eqnarray}U=\text{Emb}(V,W)=\{T:\ker (T)=0\},\quad X_{k-1}=X-U,\end{eqnarray}$$

which are open and closed, respectively. The locus $X_{k-1}$ is integral and has codimension $n-k+1$ . Its prime ideal $P_{k}$ is generated by the $\binom{n}{k}$ maximal minors $\unicode[STIX]{x1D6E5}_{J}$ of the $k\times n$ matrix, one for each $k$ -tuple $J\subset [n]$ . Each of the spaces $X$ , $R_{k,n}$ , $U$ , and $X_{k-1}$ has an action of $\text{GL}(V)$ and $\text{GL}(W)$ ; we will primarily care about the $\text{GL}(V)$ -action.

2.2 $\text{GL}$ -representation theory

A good introduction to these notions is [Reference FultonFul96]. The irreducible algebraic representations of $\text{GL}(V)$ are indexed by weakly decreasing integer sequences $\unicode[STIX]{x1D706}=(\unicode[STIX]{x1D706}_{1}\geqslant \cdots \geqslant \unicode[STIX]{x1D706}_{k})$ , where $k=\dim (V)$ . We write $\mathbb{S}_{\unicode[STIX]{x1D706}}(V)$ for the corresponding representation, and $d_{\unicode[STIX]{x1D706}}(k)$ for its dimension. We call $\mathbb{S}_{\unicode[STIX]{x1D706}}$ a Schur functor. If $\unicode[STIX]{x1D706}$ has all nonnegative parts, we write $\unicode[STIX]{x1D706}\geqslant 0$ and say $\unicode[STIX]{x1D706}$ is a partition. In this case, $\mathbb{S}_{\unicode[STIX]{x1D706}}(V)$ is functorial for linear transformations $V\rightarrow W$ . If $\unicode[STIX]{x1D706}$ has negative parts, $\mathbb{S}_{\unicode[STIX]{x1D706}}$ is only functorial for isomorphisms $V\xrightarrow[{}]{{\sim}}W$ .

We often represent partitions by their Young diagrams:

We write  for the rectangular partition $(n-k)^{k}$ , with $k$ rows and $n-k$ columns. We partially order partitions and integer sequences by containment:

$$\begin{eqnarray}\unicode[STIX]{x1D706}\subseteq \unicode[STIX]{x1D707}\quad \text{if }\unicode[STIX]{x1D706}_{i}\leqslant \unicode[STIX]{x1D707}_{i}\text{ for all }i.\end{eqnarray}$$

We write $\mathbb{Y}$ for the poset of partitions with at most $k$ parts, with this ordering, called Young’s lattice. We write $\mathbb{Y}_{\pm }$ for the set of length- $k$ weakly decreasing integer sequences; we call it the extended Young’s lattice. Schur functors include symmetric and exterior powers:

We will write $\det (V)$ for the one-dimensional representation $\bigwedge ^{\dim (V)}(V)=\mathbb{S}_{1^{k}}(V)$ . We may always twist a representation by powers of the determinant:

$$\begin{eqnarray}\det (V)^{\otimes a}\otimes \mathbb{S}_{\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{k}}(V)=\mathbb{S}_{\unicode[STIX]{x1D706}_{1}+a,\ldots ,\unicode[STIX]{x1D706}_{k}+a}(V)\end{eqnarray}$$

for any integer $a\in \mathbb{Z}$ . This operation is invertible and can sometimes be used to reduce to considering the case when $\unicode[STIX]{x1D706}$ is a partition.

2.3 Equivariant rings and modules

If $R$ is a $\mathbb{C}$ -algebra with an action of $\text{GL}(V)$ and $S$ is any $\text{GL}(V)$ -representation, then $S\otimes _{\mathbb{C}}R$ is an equivariant free $R$ -module; it has the universal property

$$\begin{eqnarray}\operatorname{Hom}_{\text{GL}(V),R}(S\otimes _{\mathbb{C}}R,M)\cong \operatorname{Hom}_{\text{GL}(V)}(S,M)\end{eqnarray}$$

for all equivariant $R$ -modules $M$ . The basic examples will be the modules $\mathbb{S}_{\unicode[STIX]{x1D706}}(V)\,\otimes \,R$ .

Let $R=R_{k,n}$ be the polynomial ring defined above. Its structure as a $\text{GL}(V)\times \text{GL}(W)$ representation is known as the Cauchy identity:

$$\begin{eqnarray}R_{k,n}=\operatorname{Sym}^{\bullet }(\operatorname{Hom}(V,W)^{\ast })\cong \bigoplus _{\unicode[STIX]{x1D706}\geqslant 0}\mathbb{S}_{\unicode[STIX]{x1D706}}(V)\otimes \mathbb{S}_{\unicode[STIX]{x1D706}}(W^{\ast }).\end{eqnarray}$$

Note that the prime ideal $P_{k}$ and the maximal ideal $\mathfrak{m}=(x_{ij})$ of the zero matrix are $\text{GL}(V)$ - and $\text{GL}(W)$ -equivariant.

Let $M$ be a finitely generated $\text{GL}(V)$ -equivariant $R$ -module. The module $\operatorname{Tor}_{R}^{i}(R/\mathfrak{m},M)$ naturally has the structure of a finite-dimensional $\text{GL}(V)$ -representation. We define the equivariant Betti number $\unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}(M)$ as the multiplicity of the Schur functor $\mathbb{S}_{\unicode[STIX]{x1D706}}(V)$ in this Tor module, that is,

$$\begin{eqnarray}\operatorname{Tor}_{R}^{i}(R/\mathfrak{m},M)~\cong ~\bigoplus _{\unicode[STIX]{x1D706}}\mathbb{S}_{\unicode[STIX]{x1D706}}(V)^{\oplus \unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}(M)}\quad (\text{as }\text{GL}(V)\text{-representations}).\end{eqnarray}$$

By semisimplicity of $\text{GL}(V)$ -representations, any minimal free resolution of $M$ can be made equivariant, so we may instead define $\unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}$ as the multiplicity of the equivariant free module $\mathbb{S}_{\unicode[STIX]{x1D706}}(V)\,\otimes \,R$ in the $i$ th step of an equivariant minimal free resolution of $M$ :

$$\begin{eqnarray}M\leftarrow F_{0}\leftarrow F_{1}\leftarrow \cdots \leftarrow F_{d}\leftarrow 0\quad \text{where }F_{i}=\bigoplus _{\unicode[STIX]{x1D706}}\mathbb{S}_{\unicode[STIX]{x1D706}}(V)^{\unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}(M)}\otimes R.\end{eqnarray}$$

All other notation on Betti tables is as defined in § 1.2.

3.1 K-theory rings

For background on equivariant K-theory, we refer to the original paper by Thomason [Reference ThomasonTho87]; a more recent discussion is [Reference MerkurjevMer05].

Excision in equivariant K-theory [Reference ThomasonTho87, Theorem 2.7] gives the right-exact sequence of abelian groups

$$\begin{eqnarray}K^{\text{GL}(V)}(X_{k-1})\xrightarrow[{}]{i_{\ast }}K^{\text{GL}(V)}(\operatorname{Hom}(V,W))\xrightarrow[{}]{j^{\ast }}K^{\text{GL}(V)}(U)\rightarrow 0.\end{eqnarray}$$

The pullback $j^{\ast }$ , induced by the open inclusion $j:U{\hookrightarrow}X$ , is a map of rings. The pushforward $i_{\ast }$ , induced by the closed embedding $i:X_{k-1}{\hookrightarrow}X$ , is only a map of abelian groups. Its image is the ideal $I$ generated by the classes of modules supported along the rank-deficient locus $X_{k-1}$ .

We do not attempt to describe the first term. For the second term, we have ([Reference ThomasonTho87, Theorem 4.1] or [Reference MerkurjevMer05, Example 2 and Corollary 12])

$$\begin{eqnarray}K^{\text{GL}(V)}(\operatorname{Hom}(V,W))\cong \mathbb{Z}[t_{1}^{\pm },\ldots ,t_{k}^{\pm }]^{S_{k}},\end{eqnarray}$$

the ring of symmetric Laurent polynomials in $k$ variables, essentially the representation ring of $\text{GL}(V)$ . Here, the class of the equivariant $R$ -module $\mathbb{S}_{\unicode[STIX]{x1D706}}(V)\otimes _{\mathbb{C}}R$ is identified with the Schur polynomial $s_{\unicode[STIX]{x1D706}}(t_{1},\ldots ,t_{k})$ . We note that the exterior powers $\bigwedge ^{d}(V)\otimes _{\mathbb{C}}R$ correspond to the elementary symmetric polynomials $e_{d}(t)$ , while symmetric powers $\text{Sym}^{d}(V)\otimes _{\mathbb{C}}R$ correspond to homogeneous symmetric polynomials $h_{d}(t)$ .

If $M$ is a finitely generated $R$ -module, its equivariant minimal free resolution expresses the K-class $[M]$ as a finite alternating sum of Schur polynomials. In other words, the equivariant Betti table determines the K-class

$$\begin{eqnarray}[M]=\mathop{\sum }_{i,\unicode[STIX]{x1D706}}(-1)^{i}\unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}(M)s_{\unicode[STIX]{x1D706}}(t).\end{eqnarray}$$

An equivalent approach is to write

$$\begin{eqnarray}M\cong \bigoplus _{\unicode[STIX]{x1D706}}\mathbb{S}_{\unicode[STIX]{x1D706}}(V)^{c_{\unicode[STIX]{x1D706}}(M)}\quad \text{as a }\text{GL}(V)\text{-representation},\end{eqnarray}$$

and define the equivariant Hilbert series of $M$ ,

$$\begin{eqnarray}\displaystyle H_{M}(t) & = & \displaystyle \mathop{\sum }_{\unicode[STIX]{x1D706}}c_{\unicode[STIX]{x1D706}}(M)s_{\unicode[STIX]{x1D706}}(t)\nonumber\\ \displaystyle & = & \displaystyle \frac{f(t)}{\mathop{\prod }_{i=1}^{k}(1-t_{i})^{n}}\nonumber\end{eqnarray}$$

for some symmetric function $f(t)$ . Then $f(t)$ is the K-theory class of $M$ . (If we forget the $\text{GL}(V)$ -action and remember only the grading of $M$ , we recover the usual Hilbert series.)

To see that these definitions agree, note that the second definition is additive in short exact sequences, hence is well-defined on K-classes. Replacing $M$ by an equivariant minimal free resolution, it suffices to consider indecomposable free modules $M=\mathbb{S}_{\unicode[STIX]{x1D706}}(V)\otimes _{\mathbb{C}}R$ . This tensor product multiplies the entire series by $s_{\unicode[STIX]{x1D706}}(t)$ and does the same to $f(t)$ , so it suffices to consider the case $M=R$ . For this case, we observe

$$\begin{eqnarray}R=\text{Sym}(V\otimes W)\cong \text{Sym}(\overbrace{V\oplus \cdots \oplus V}^{n\;\text{times}})=\text{Sym}(V)^{\otimes n},\end{eqnarray}$$

so the equivariant Hilbert series of $R$ is the $n$ th power of the series for $\text{Sym}(V)$ , which is

$$\begin{eqnarray}\mathop{\sum }_{d\geqslant 0}h_{d}(t)=\mathop{\prod }_{i=1}^{k}\frac{1}{1-t_{i}},\end{eqnarray}$$

where $h_{d}(t)$ is the $d$ th homogeneous symmetric polynomial.

Finally, we have for the third term (cf. [Reference ThomasonTho87, Proposition 6.2])

$$\begin{eqnarray}K^{\text{GL}(V)}(U)\cong K(U/\text{GL}(V))=K(\text{Gr}(k,W)),\end{eqnarray}$$

because the action of $\text{GL}(V)$ is free on $U$ . Note that, under this correspondence, $j^{\ast }$ sends the module $M$ to the class of the induced sheaf $\widetilde{M}$ (see Remark 2.1). Notably, $j_{\ast }[\mathbb{S}_{\unicode[STIX]{x1D706}}(V)\,\otimes \,R]=[\mathbb{S}_{\unicode[STIX]{x1D706}}({\mathcal{S}})]$ . The ring structure of $K(\text{Gr}(k,W))$ is well-known from K-theoretic Schubert calculus (e.g. [Reference Kostant and KumarKK90] or [Reference BuchBuc02]). We will only need to know the following: it is a free abelian group with an additive basis consisting of $\binom{n}{k}$ generators, indexed by partitions $\unicode[STIX]{x1D707}$ fitting inside a $k\times (n-k)$ rectangle, that is, . These correspond to the classes $[{\mathcal{O}}_{\unicode[STIX]{x1D707}}]$ of structuresheaves of Schubert varieties. It is easy to check that $K_{+}^{\text{GL}(V)}(\operatorname{Hom}(V,W))\rightarrow K(\text{Gr}(k,W))$ is also surjective (because, for example, matrix Schubert varieties are generated in positive degree).

3.2 Modules on the rank-deficient locus and the equivariant Herzog-Kühl equations

From the surjection $K_{+}^{\text{GL}(V)}(\operatorname{Hom}(V,W))\rightarrow K(\text{Gr}(k,W))$ , we see that the ideal

$$\begin{eqnarray}I^{\prime }:=I\cap K_{+}^{\text{GL}(V)}(\operatorname{Hom}(V,W)),\end{eqnarray}$$

as a linear subspace, has co-rank $\binom{n}{k}$ . We wish to find exactly this many linear equations cutting out the ideal, indexed appropriately by partitions. That is, given a K-class written in the Schur basis,

$$\begin{eqnarray}f=\mathop{\sum }_{\unicode[STIX]{x1D706}\geqslant 0}a_{\unicode[STIX]{x1D706}}s_{\unicode[STIX]{x1D706}}\in K_{+}^{\text{GL}(V)}(\operatorname{Hom}(V,W)),\end{eqnarray}$$

we wish to have coefficients $b_{\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}}$ for each , such that

We will then apply these equations in the case where $f$ is the class of a module $M$ , and

$$\begin{eqnarray}a_{\unicode[STIX]{x1D706}}=\mathop{\sum }_{i}(-1)^{i}\unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}(M)\end{eqnarray}$$

comes from the equivariant Betti table of $M$ . Our approach is to prove the following. We write $1-t$ as shorthand for the tuple $(1-t_{1},\ldots ,1-t_{k})$ and we show the following theorem.

We will prove Theorem 3.2 in the next section. Here is how it leads to the desired equations. Let $b_{\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}}$ be the change-of-basis coefficients defined by sending $t_{i}\mapsto 1-t_{i}$ . So, by definition,

$$\begin{eqnarray}s_{\unicode[STIX]{x1D706}}(1-t)=\mathop{\sum }_{\unicode[STIX]{x1D707}}b_{\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}}s_{\unicode[STIX]{x1D707}}(t).\end{eqnarray}$$

Note that we have, equivalently,

$$\begin{eqnarray}s_{\unicode[STIX]{x1D706}}(t)=\mathop{\sum }_{\unicode[STIX]{x1D707}}b_{\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}}s_{\unicode[STIX]{x1D707}}(1-t).\end{eqnarray}$$

Thus

$$\begin{eqnarray}f=\mathop{\sum }_{\unicode[STIX]{x1D706}}a_{\unicode[STIX]{x1D706}}s_{\unicode[STIX]{x1D706}}(t)=\mathop{\sum }_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}}a_{\unicode[STIX]{x1D706}}b_{\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}}s_{\unicode[STIX]{x1D707}}(1-t).\end{eqnarray}$$

The polynomials $s_{\unicode[STIX]{x1D707}}(1-t)$ for all $\unicode[STIX]{x1D707}\geqslant 0$ form an additive basis for the $K_{+}^{\text{GL}(V)}(\operatorname{Hom}(V,W))$ . Thus, $f\in I^{\prime }$ if and only if the coefficient of $s_{\unicode[STIX]{x1D707}}(1-t)$ is $0$ for all . That is,

The following description of $b_{\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}}$ is due to Stanley. Recall that, if $\unicode[STIX]{x1D707}\subseteq \unicode[STIX]{x1D706}$ are partitions, the skew shape $\unicode[STIX]{x1D706}/\unicode[STIX]{x1D707}$ is the Young diagram of $\unicode[STIX]{x1D706}$ with the squares of $\unicode[STIX]{x1D707}$ deleted. A standard Young tableau is a filling of a (possibly skew) shape by the numbers $1,2,\ldots ,t$ (with $t$ boxes in all), such that the rows increase from left to right, and the columns increase from top to bottom. We write $f^{\unicode[STIX]{x1D70E}}$ for the number of standard Young tableaux of shape $\unicode[STIX]{x1D70E}$ .

Proposition 3.3 [Reference StanleySta99].

If $\unicode[STIX]{x1D707}\not \subseteq \unicode[STIX]{x1D706}$ then $b_{\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}}=0$ . If $\unicode[STIX]{x1D707}\subseteq \unicode[STIX]{x1D706}$ , then

$$\begin{eqnarray}b_{\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}}=(-1)^{|\unicode[STIX]{x1D707}|}\frac{f^{\unicode[STIX]{x1D706}/\unicode[STIX]{x1D707}}f^{\unicode[STIX]{x1D707}}}{f^{\unicode[STIX]{x1D706}}}\binom{|\unicode[STIX]{x1D706}|}{|\unicode[STIX]{x1D707}|}\frac{d_{\unicode[STIX]{x1D706}}(k)}{d_{\unicode[STIX]{x1D707}}(k)}.\end{eqnarray}$$

An equivalent formulation is

$$\begin{eqnarray}b_{\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}}=(-1)^{|\unicode[STIX]{x1D707}|}\frac{f^{\unicode[STIX]{x1D706}/\unicode[STIX]{x1D707}}}{|\unicode[STIX]{x1D706}/\unicode[STIX]{x1D707}|!}\mathop{\prod }_{(i,j)\in \unicode[STIX]{x1D706}/\unicode[STIX]{x1D707}}(k+j-i).\end{eqnarray}$$

Corollary 3.4 (Equivariant Herzog–Kühl equations).

Let $M$ be an equivariant $R$ -module with equivariant Betti table $\unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}$ . Assume $M$ is generated in positive degree.

The set-theoretic support of $M$ is contained in the rank-deficient locus if and only if,

(3.1)

Note that $\unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}$ is the multiplicity of the $\unicode[STIX]{x1D706}$ -isotypic component of the resolution of $M$ (in cohomological degree $i$ ), whereas $\unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}d_{\unicode[STIX]{x1D706}}(k)=\widetilde{\unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}}$ is the rank of this isotypic component.

Example 3.6 (Projective spaces).

When $k=1$ , the equivariant Herzog–Kühl equations specialize to the usual equations: they are indexed by the row partitions  for $j=1,\ldots ,n$ . We have $f^{\unicode[STIX]{x1D706}/\unicode[STIX]{x1D707}}=f^{\unicode[STIX]{x1D707}}=f^{\unicode[STIX]{x1D706}}=d_{\unicode[STIX]{x1D706}}(1)=1$ (since the partitions are single rows), so the equation for $\unicode[STIX]{x1D707}=(j)$ just says

$$\begin{eqnarray}\mathop{\sum }_{i,d\geqslant j}(-1)^{i}\unicode[STIX]{x1D6FD}_{i,d}\binom{d}{j}=\mathop{\sum }_{i,d\geqslant j}(-1)^{i}\unicode[STIX]{x1D6FD}_{i,d}\biggl(\frac{1}{j!}d^{j}+\cdots \,\biggr)=0.\end{eqnarray}$$

These are upper-triangular to the familiar Herzog–Kühl equations for graded modules,

$$\begin{eqnarray}\mathop{\sum }_{i,d}(-1)^{i}\unicode[STIX]{x1D6FD}_{i,d}~d^{j}=0.\end{eqnarray}$$

The coefficient in equation (3.1) has the following interpretation. Consider a uniformly random filling $T$ of the shape $\unicode[STIX]{x1D706}$ by the numbers $1,\ldots ,|\unicode[STIX]{x1D706}|$ . Say that $T$ splits along $\unicode[STIX]{x1D707}\sqcup \unicode[STIX]{x1D706}/\unicode[STIX]{x1D707}$ if the numbers $1,\ldots ,|\unicode[STIX]{x1D707}|$ lie in the subshape $\unicode[STIX]{x1D707}$ . Then:

(3.2) $$\begin{eqnarray}\frac{f^{\unicode[STIX]{x1D706}/\unicode[STIX]{x1D707}}f^{\unicode[STIX]{x1D707}}}{f^{\unicode[STIX]{x1D706}}}\binom{|\unicode[STIX]{x1D706}|}{|\unicode[STIX]{x1D707}|}=\frac{\text{Prob}(T\text{ splits along }\unicode[STIX]{x1D707}\sqcup \unicode[STIX]{x1D706}/\unicode[STIX]{x1D707}~|~T\text{ is standard})}{\text{Prob}(T\text{ splits along }\unicode[STIX]{x1D707}\sqcup \unicode[STIX]{x1D706}/\unicode[STIX]{x1D707})}.\end{eqnarray}$$

3.3 Proof of Theorem 3.2

First, we recall the following fact about K-theory of Grassmannians.

Proposition 3.8 [Reference Fink and SpeyerFS12, p. 21].

The following identity holds of formal power series in the variable $u$ over $K(\text{Gr}(k,W))$ :

$$\begin{eqnarray}\biggl(\mathop{\sum }_{p}\biggl[\bigwedge ^{p}{\mathcal{S}}\biggr]u^{p}\biggr)\cdot \biggl(\mathop{\sum }_{q}\biggl[\bigwedge ^{q}{\mathcal{Q}}\biggr]u^{q}\biggr)=(1+u)^{n}.\end{eqnarray}$$

It is a consequence of the tautological exact sequence of vector bundles on $\text{Gr}(k,W)$ ,

$$\begin{eqnarray}0\rightarrow {\mathcal{S}}\rightarrow W\rightarrow {\mathcal{Q}}\rightarrow 0.\end{eqnarray}$$

We rearrange the above identity as

$$\begin{eqnarray}\biggl(\mathop{\sum }_{q}\biggl[\bigwedge ^{q}{\mathcal{Q}}\biggr]u^{q}\biggr)=(1+u)^{n}\cdot \frac{1}{(\mathop{\sum }_{p}[\bigwedge ^{p}{\mathcal{S}}]u^{p})}.\end{eqnarray}$$

Recall that $\bigwedge ^{p}{\mathcal{S}}$ is the sheaf on $\text{Gr}(k,W)$ induced by the equivariant free module $\bigwedge ^{p}(V)\,\otimes \,R$ . Thus, viewing $K(\text{Gr}(k,W))$ as a quotient of the ring of symmetric functions, $\bigwedge ^{p}{\mathcal{S}}$ comes from the $p$ th elementary symmetric polynomial $e_{p}(t)$ . So, we consider the coefficients $f_{\ell }$ of $u^{\ell }$ in the analogous expression over $\mathbb{Z}[t_{1}^{\pm },\ldots ,t_{k}^{\pm }]^{S_{k}}$ :

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\ell \geqslant 0}f_{\ell }u^{\ell } & = & \displaystyle (1+u)^{n}\cdot \frac{1}{(\mathop{\sum }_{p}e_{p}(t)u^{p})}\nonumber\\ \displaystyle & = & \displaystyle (1+u)^{n}\cdot \mathop{\prod }_{i=1}^{k}\frac{1}{1+ut_{i}}\nonumber\\ \displaystyle & = & \displaystyle (1+u)^{n}\mathop{\sum }_{p}(-1)^{p}h_{p}(t)u^{p},\nonumber\end{eqnarray}$$

where $h_{p}$ is the $p$ th homogeneous symmetric polynomial. The key observation is that, in $K(\text{Gr}(k,W))$ , this expression becomes a polynomial in $u$ of degree $n-k$ : all higher terms vanish because ${\mathcal{Q}}$ has rank $n-k$ . In particular, $f_{\ell }\in I^{\prime }$ for $\ell >n-k$ .

We compute the coefficient $f_{\ell }$ . We have

$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\ell }f_{\ell }u^{\ell } & = & \displaystyle (1+u)^{n}\mathop{\sum }_{p}(-1)^{p}h_{p}u^{p}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{q=0}^{n}\mathop{\sum }_{p=0}^{\infty }u^{p+q}(-1)^{p}h_{p}\binom{n}{q}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{\ell =0}^{\infty }u^{\ell }\mathop{\sum }_{p=\ell -n}^{\ell }(-1)^{p}h_{p}\binom{n}{\ell -p},\nonumber\end{eqnarray}$$

so our desired coefficients are

$$\begin{eqnarray}f_{\ell }=\mathop{\sum }_{p=\ell -n}^{\ell }(-1)^{p}h_{p}\binom{n}{\ell -p},\end{eqnarray}$$

where in the last two lines we use the convention $h_{p}=0$ for $p<0$ . We next show the following lemma.

Proof. Equivalently, we change basis $t\mapsto 1-t$ , calling the (new) ideal $J$ , and we show

$$\begin{eqnarray}J\supseteq (h_{n-k+1},\ldots ,h_{n}).\end{eqnarray}$$

We consider the elements $f_{n-k+i}(1-t)\in J$ for $i=1,\ldots ,k$ :

$$\begin{eqnarray}f_{n-k+i}(1-t)=\mathop{\sum }_{p=-k+i}^{n-k+i}(-1)^{p}h_{p}(1-t)\binom{n}{n-k+i-p}.\end{eqnarray}$$

Since $i\leqslant k$ , we have

$$\begin{eqnarray}=\mathop{\sum }_{p=0}^{n-k+i}(-1)^{p}h_{p}(1-t)\binom{n}{n-k+i-p}.\end{eqnarray}$$

We apply the second formula from Proposition 3.3. Note that all terms are single-row partitions, $\unicode[STIX]{x1D706}=(p)$ and $\unicode[STIX]{x1D707}=(s)$ , with $s\leqslant p$ , so $f^{\unicode[STIX]{x1D706}/\unicode[STIX]{x1D707}}=1$ and the change of basis is

$$\begin{eqnarray}b_{\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}}=(-1)^{s}\frac{f^{\unicode[STIX]{x1D706}/\unicode[STIX]{x1D707}}}{|\unicode[STIX]{x1D706}/\unicode[STIX]{x1D707}|!}\cdot (k+s)\cdots (k+p-1)=(-1)^{s}\binom{k+p-1}{k+s-1}.\end{eqnarray}$$

Hence,

$$\begin{eqnarray}\displaystyle f_{n-k+i}(1-t) & = & \displaystyle \mathop{\sum }_{p=0}^{n-k+i}\mathop{\sum }_{s=0}^{p}(-1)^{p+s}\binom{n}{n-k+i-p}\binom{k+p-1}{k+s-1}h_{s}\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{s=0}^{n-k+i}(-1)^{s}h_{s}\mathop{\sum }_{p=s}^{n-k+i}(-1)^{p}\binom{n}{n-k+i-p}\binom{k+p-1}{k+s-1}.\nonumber\end{eqnarray}$$

We reindex, sending $p\mapsto n-k+i-p$ , and reverse the order of the inner sum:

$$\begin{eqnarray}=(-1)^{n-k+i}\mathop{\sum }_{s=0}^{n-k+i}(-1)^{s}h_{s}\mathop{\sum }_{p=0}^{n-k+i-s}(-1)^{p}\binom{n}{p}\binom{n+i-p-1}{k+s-1}.\end{eqnarray}$$

The terms $h_{s}$ for $s\leqslant n-k$ . First, we show that all the lower terms $h_{s}$ , with $s\leqslant n-k$ , vanish. For these terms, we view the large binomial coefficient as a polynomial function of $p$ . It has degree $k+s-1$ , with zeros at $p=(n-k+i-s)+1,\ldots ,n+i-1$ , so we may freely include these terms in the inner sum. It is convenient to extend the inner sum only as far as $p=n$ , obtaining

$$\begin{eqnarray}\mathop{\sum }_{p=0}^{n}(-1)^{p}\binom{n}{p}\binom{n-i-p-1}{k+s-1}.\end{eqnarray}$$

Recall from the theory of finite differences that

$$\begin{eqnarray}\mathop{\sum }_{p=0}^{d}(-1)^{p}\binom{d}{p}g(p)=0\end{eqnarray}$$

whenever $g$ is a polynomial of degree less than $d$ . Since the above sum has degree $k+s-1\leqslant n-1$ , it vanishes. Thus, dropping the lower terms, we are left with

(3.3) $$\begin{eqnarray}f_{n-k+i}(1-t)=(-1)^{i}\mathop{\sum }_{s=1}^{i}(-1)^{s}h_{n-k+s}\mathop{\sum }_{p=0}^{i-s}(-1)^{p}\binom{n}{p}\binom{n+i-p-1}{n+s-1}.\end{eqnarray}$$

Showing $h_{n-k+i}\in J$ for $i=1,\ldots ,k$ . From equation (3.3), we see directly that the coefficient of $h_{n-k+i}$ in $f_{n-k+i}(1-t)$ is 1. This is the leading coefficient, so the claim follows by induction on $i$ .◻

Corollary 3.10. We have

Proof. The equality of ideals

$$\begin{eqnarray}(h_{n-k+1},\ldots ,h_{n})=(h_{i}:i>n-k)\end{eqnarray}$$

follows from Newton’s identities and induction. The equality

follows from the Pieri rule (for $\subseteq$ ) and the Jacobi–Trudi formula (for $\supseteq$ ). See [Reference FultonFul96] for these identities. This shows that $J$ contains this linear span. But then quotienting by $J$ leaves at most $\binom{n}{k}$ classes. This is already the rank of $K(\text{Gr}(k,W))$ , so we must have equality.◻

Changing bases $t\mapsto 1-t$ a final time completes the proof of Theorem 3.2.

4.1 Prior work on $\widetilde{\mathit{BS}}_{k,k}$  [Reference Ford, Levinson and SamFLS18]

The rank-deficient locus $\{\det (T)=0\}\subset \operatorname{Hom}(V,W)$ is codimension 1. Thus, modules satisfying Condition 1.2 have free resolutions of length 1,

$$\begin{eqnarray}M\leftarrow F^{0}\leftarrow F^{1}\leftarrow 0.\end{eqnarray}$$

There is only one equivariant Herzog–Kühl equation, labeled by the empty partition $\unicode[STIX]{x1D707}=\varnothing$ :

(4.1) $$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D706}}\widetilde{\unicode[STIX]{x1D6FD}_{0,\unicode[STIX]{x1D706}}}=\mathop{\sum }_{\unicode[STIX]{x1D706}}\widetilde{\unicode[STIX]{x1D6FD}_{1,\unicode[STIX]{x1D706}}},\quad \text{that is},\operatorname{rank}(F^{0})=\operatorname{rank}(F^{1}).\end{eqnarray}$$

Algebraically, this simply says that $M$ is a torsion module.

The extremal rays and supporting hyperplanes of $\widetilde{\mathit{BS}}_{k,k}$ are as follows.

Definition 4.2 (Pure tables).

Fix $\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}\in \mathbb{Y}_{\pm }$ with $\unicode[STIX]{x1D706}\subsetneq \unicode[STIX]{x1D707}$ . The pure table $\widetilde{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D706}\subsetneq \unicode[STIX]{x1D707})$ is defined by setting

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6FD}_{0,\unicode[STIX]{x1D706}}}=\widetilde{\unicode[STIX]{x1D6FD}_{1,\unicode[STIX]{x1D707}}}=1\end{eqnarray}$$

and all other entries 0.

It is nontrivial to show that each pure table $\widetilde{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D706}\subsetneq \unicode[STIX]{x1D707})$ is realizable up to scalar multiple [Reference Ford, Levinson and SamFLS18, Theorem 4.1]. Any such table generates an extremal ray of $\widetilde{\mathit{BS}}_{k,k}$ . The construction relies on an adapted version of Weyman’s geometric technique. We do not reproduce the proof, but see Example 6.1 for a similar construction in the case $k=2,n=3$ .

It is, by contrast, easy to establish the following inequalities on $\widetilde{\mathit{BS}}_{k,k}$ . Recall that a subset $S$ of a poset $(P,\preccurlyeq )$ is downwards closed if, whenever $x\in S$ and $y\preccurlyeq x$ , it follows that $y\in S$ .

Definition 4.3 (Antichain inequalities).

Let $S\subseteq \mathbb{Y}_{\pm }$ be a downwards closed set. Let

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}=\{\unicode[STIX]{x1D706}:\unicode[STIX]{x1D706}\subsetneq \unicode[STIX]{x1D707}\text{ for some }\unicode[STIX]{x1D707}\in S\}.\end{eqnarray}$$

For any rank Betti table $(\widetilde{\unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}})$ , the antichain inequality (for $S$ ) is then

(4.2) $$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6E4}}\widetilde{\unicode[STIX]{x1D6FD}_{0,\unicode[STIX]{x1D706}}}\geqslant \mathop{\sum }_{\unicode[STIX]{x1D706}\in S}\widetilde{\unicode[STIX]{x1D6FD}_{1,\unicode[STIX]{x1D706}}}.\end{eqnarray}$$

(The terminology of antichains is due to [Reference Ford, Levinson and SamFLS18], where the inequality (4.2) is stated in terms of the maximal elements of $S$ , which form an antichain in $\mathbb{Y}_{\pm }$ .)

These conditions follow directly from minimality of the underlying maps of modules: the summands corresponding to $S$ in $F_{1}$ must map into those corresponding to $\unicode[STIX]{x1D6E4}$ in $F_{0}$ .

Finally, we recall the graph-theoretic model of $\widetilde{\unicode[STIX]{x1D6FD}}$ introduced in § 1.3.2. This construction was implicit in [Reference Ford, Levinson and SamFLS18, Lemma 3.6]. It is especially simple in this case.

The Boij–Söderberg cone $\widetilde{\mathit{BS}}_{k,k}$ is characterized as follows.

A perfect matching on $G(\widetilde{\unicode[STIX]{x1D6FD}})$ expresses $\widetilde{\unicode[STIX]{x1D6FD}}$ as a positive integer sum of pure tables: an edge $\unicode[STIX]{x1D706}\leftarrow \unicode[STIX]{x1D707}$ corresponds to a summand

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6FD}}=\cdots +\widetilde{\unicode[STIX]{x1D6FD}}(\unicode[STIX]{x1D706}\subsetneq \unicode[STIX]{x1D707})+\cdots \,.\end{eqnarray}$$

It is easy to see that the cone spanned by the pure tables is contained in the cone defined by the antichain inequalities. The fact that these cones agree follows from Hall’s matching theorem for bipartite graphs.

In the antichain inequality (4.2), $S$ corresponds to a set of vertex labels on the right-hand side of the Betti graph $G(\widetilde{\unicode[STIX]{x1D6FD}})$ . The set $\unicode[STIX]{x1D6E4}$ consists of the labels of vertices adjacent to $S$ . The numbers of such vertices are then the right- and left-hand sides of the inequality. (The structure of $G(\widetilde{\unicode[STIX]{x1D6FD}})$ implies easily that it suffices to consider inequalities from downwards-closed sets $S$ .)

4.2 The derived cone

We now generalize Theorem 4.5 to describe the derived cone $\widetilde{\mathit{BS}}_{k,k}^{D}$ . We are interested in bounded free equivariant complexes

$$\begin{eqnarray}\cdots \leftarrow F_{i}\leftarrow F_{i+1}\leftarrow F_{i+2}\leftarrow \cdots \,,\end{eqnarray}$$

all of whose homology modules are torsion.

The supporting hyperplanes of $\widetilde{\mathit{BS}}_{k,k}^{D}$ are quite complicated and we do not prove directly that they hold. We instead generalize the descriptions in terms of extremal rays and perfect matchings, which remain fairly simple. We then deduce the inequalities from Hall’s theorem.

The extremal rays of $\widetilde{\mathit{BS}}_{k,k}^{D}$ will be homological shifts of those of $\widetilde{\mathit{BS}}_{k,k}$ .

Definition 4.7 (Homologically shifted pure tables).

Fix $i\in \mathbb{Z}$ and $\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}\in \mathbb{Y}_{\pm }$ with $\unicode[STIX]{x1D706}\subsetneq \unicode[STIX]{x1D707}$ . We define the homologically shifted pure table $\widetilde{\unicode[STIX]{x1D6FD}}[\unicode[STIX]{x1D706}\xleftarrow[{}]{~i~}\unicode[STIX]{x1D707}\,]$ by setting

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}}=\widetilde{\unicode[STIX]{x1D6FD}_{i+1,\unicode[STIX]{x1D707}}}=1\end{eqnarray}$$

and all other entries 0.

The supporting hyperplanes will be defined by the following inequalities. Recall that a convex subset $S$ of a poset $P$ is the intersection of an upwards-closed set with a downwards-closed set.

Definition 4.8 (Convexity inequalities).

For each odd $i$ , let $S_{i}\subseteq \mathbb{Y}_{\pm }$ be any choice of convex set. For each even $i$ , define

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{i}=\{\unicode[STIX]{x1D706}:\unicode[STIX]{x1D707}\subsetneq \unicode[STIX]{x1D706}\text{ for some }\unicode[STIX]{x1D707}\in S_{i-1}\}\cup \{\unicode[STIX]{x1D706}:\unicode[STIX]{x1D706}\subsetneq \unicode[STIX]{x1D707}\text{ for some }\unicode[STIX]{x1D707}\in S_{i+1}\}.\end{eqnarray}$$

For any rank Betti table $(\widetilde{\unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}})$ , the convexity inequality (for the $S_{i}$ ) is then

(4.3) $$\begin{eqnarray}\mathop{\sum }_{i\;\text{even}}\mathop{\sum }_{\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6E4}_{i}}\widetilde{\unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}}\geqslant \mathop{\sum }_{i\;\text{odd}}\mathop{\sum }_{\unicode[STIX]{x1D706}\in S_{i}}\widetilde{\unicode[STIX]{x1D6FD}_{i,\unicode[STIX]{x1D706}}}.\end{eqnarray}$$

(We may, if we wish, switch ‘even’ and ‘odd’ in this definition. We will see that either collection of inequalities yields the same cone.)

We recall the general definition of the Betti graph.

Each segment $S_{i}$ of Definition 4.8 corresponds to a set of vertex labels in $G(\widetilde{\unicode[STIX]{x1D6FD}})$ . The set $\unicode[STIX]{x1D6E4}_{i}$ then contains the labels of vertices adjacent to $S_{i-1}$ and $S_{i+1}$ . The numbers of vertices counted this way give the right- and left-hand sides of inequality (4.3). Thus, the inequality will follow (using Hall’s theorem) from the existence of a perfect matching on $G(\widetilde{\unicode[STIX]{x1D6FD}})$ . Note that if the $S_{i}$ were not convex, we could replace them by their convex hulls without changing the $\unicode[STIX]{x1D6E4}_{i}$ .

We now characterize the derived Boij–Söderberg cone $\widetilde{\mathit{BS}}_{k,k}^{D}$ .

5.1 Perfect matchings in linear and homological algebra

Our approach uses linear maps to produce perfect matchings. The starting point is the following construction.

Definition 5.5. Let $T:V\rightarrow W$ be a map of vector spaces, having specified bases ${\mathcal{V}},{\mathcal{W}}$ . The coefficient graph $G$ of $T$ is the directed bipartite graph with vertex set ${\mathcal{V}}\sqcup {\mathcal{W}}$ and edges

$$\begin{eqnarray}E=\{v\rightarrow w:T(v)\text{ has a nonzero }w\text{-coefficient}\}.\end{eqnarray}$$

Note that the adjacency matrix of $G$ is $T$ with all nonzero coefficients replaced by $1$ s.

We will say the corresponding bijection ${\mathcal{V}}\leftrightarrow {\mathcal{W}}$ is compatible with $T$ , a combinatorial analog of the fact that $T$ is an isomorphism. The proof of existence is simple, but essentially nonconstructive in practice. Here are two ways to do it.

1. (i) (All at once.) Since $\det (T)\neq 0$ , some monomial term of $\det (T)$ is nonzero. This exhibits the perfect matching.

2. (ii) (By induction, using the Laplace expansion.) Expand $\det (T)$ along a row or column; some term $a_{ij}\cdot \text{(complementary minor)}$ is nonzero, and so on.

Method (ii) actually satisfies a slightly stronger condition: the resulting matching is compatible with both $T$ and $T^{-1}$ (since, up to scaling by $\det (T)$ , the complementary minors are the entries of the inverse matrix).

Similarly, if $T$ is merely assumed to be injective or surjective, we may produce a maximal matching in this way (choose some nonvanishing maximal minor).

We generalize Proposition 5.6 to the setting of homological algebra in three ways: to infinite-dimensional vector spaces, to long exact sequences, and to double complexes (motivated by spectral sequences).

We will not need Proposition 5.7 (which uses the axiom of choice) for our proof of Theorem 5.4, so we prove it in the appendix. See Remark 5.14 for additional discussion on our usage of the axiom of choice.

5.1.1 Long exact sequences and the proof of Theorem 4.10

We generalize to the case of long exact sequences. Let

$$\begin{eqnarray}\cdots \leftarrow V_{i}\xleftarrow[{}]{\unicode[STIX]{x1D6FF}}V_{i+1}\leftarrow \cdots\end{eqnarray}$$

be a long exact sequence, with ${\mathcal{V}}_{i}$ a fixed basis for $V_{i}$ . (The vector spaces may be finite- or infinite-dimensional.)

Definition 5.8. The coefficient graph $G$ for $(V_{\bullet },\unicode[STIX]{x1D6FF})$ (with respect to ${\mathcal{V}}_{\bullet }$ ) is the directed graph with vertex set $\bigsqcup _{i}{\mathcal{V}}_{i}$ and an edge $v\rightarrow v^{\prime }$ whenever $\unicode[STIX]{x1D6FF}(v)$ has a nonzero $v^{\prime }$ -coefficient.

Proposition 5.9. The coefficient graph of a long exact sequence has a perfect matching.

Proof. Choose subsets ${\mathcal{F}}_{i}\subset {\mathcal{V}}_{i}$ descending to bases of $\text{im}(\unicode[STIX]{x1D6FF})\subset V_{i-1}$ , using Zorn’s lemma in the infinite case. Let ${\mathcal{G}}_{i}={\mathcal{V}}_{i}-{\mathcal{F}}_{i}$ , and let $F_{i}=\text{span}({\mathcal{F}}_{i})$ and $G_{i}=\text{span}({\mathcal{G}}_{i})$ . The composition $\tilde{\unicode[STIX]{x1D6FF}}:F_{i+1}{\hookrightarrow}V_{i+1}\xrightarrow[{}]{\unicode[STIX]{x1D6FF}}V_{i}{\twoheadrightarrow}G_{i}$ is an isomorphism and has the same coefficients as $\unicode[STIX]{x1D6FF}$ , restricted to ${\mathcal{F}}_{i+1}$ and ${\mathcal{G}}_{i}$ . Thus Proposition 5.6 (or 5.7 in the infinite case) yields a matching of ${\mathcal{F}}_{i+1}$ with  ${\mathcal{G}}_{i}$ .◻

At this point, we complete the proof of Theorem 4.10, characterizing the derived Boij–Söderberg cone $\widetilde{\mathit{BS}}_{k,k}^{D}$ of the square matrices.

Corollary 5.10. If $\widetilde{\unicode[STIX]{x1D6FD}}\in \widetilde{\mathit{BS}}_{k,k}^{D}$ , then the Betti graph $G(\widetilde{\unicode[STIX]{x1D6FD}})$ has a perfect matching.

Proof. Let $\widetilde{\unicode[STIX]{x1D6FD}}$ be the Betti table of a minimal free equivariant complex $(F^{\bullet },\unicode[STIX]{x1D6FF})$ of $R$ -modules, with $R=R_{k,k}$ the coordinate ring of the $k\times k$ matrices, and $F^{\bullet }\,\otimes \,R[1/\!\det ]$ exact.

Choose, for each $F^{i}$ , a $\mathbb{C}$ -basis of each copy of $\mathbb{S}_{\unicode[STIX]{x1D706}}(V)$ occurring in $F^{i}$ . Label the corresponding basis elements $x_{\unicode[STIX]{x1D706}}$ . It follows from minimality that each $\unicode[STIX]{x1D6FF}(x_{\unicode[STIX]{x1D706}})$ is an $R$ -linear combination of basis elements labeled by partitions $\unicode[STIX]{x1D706}^{\prime }\subsetneq \unicode[STIX]{x1D706}$ .

Since the homology modules are torsion, $F_{\bullet }\,\otimes \,\text{Frac}(R)$ is an exact sequence of $\text{Frac}(R)$ -vector spaces, with bases given by the $x_{\unicode[STIX]{x1D706}}$ chosen above. By the previous proposition, its coefficient graph has a perfect matching. This graph has the same vertices as the Betti graph $G(\widetilde{\unicode[STIX]{x1D6FD}})$ , and its edges are a subset of $G(\widetilde{\unicode[STIX]{x1D6FD}})$ ’s edges.◻

Remark 5.11. Rather than tensoring with $\text{Frac}(R)$ , we may instead specialize to any convenient invertible $k\times k$ matrix $T\in \operatorname{Hom}(\mathbb{C}^{k},\mathbb{C}^{k})$ , such as the identity matrix. This approach is useful for computations, since the resulting exact sequence consists of finite-dimensional $\mathbb{C}$ -vector spaces.

5.1.2 Double complexes and the proof of Theorem 5.1

Finally, we generalize to the setting of double complexes and spectral sequences. Let $(E^{\bullet ,\bullet },d_{v},d_{h})$ be a double complex of vector spaces, with differentials pointing up and to the left:

We assume the squares anticommute, so the total differential is

$$\begin{eqnarray}d_{\text{tot}}=d_{h}+d_{v},\quad \text{where }d_{h}d_{v}+d_{v}d_{h}=0.\end{eqnarray}$$

We will always assume the total complex $\text{Tot}(E^{\bullet ,\bullet })$ has a finite number of columns. Note that we do not assume a basis has been specified for each $E^{\bullet ,\bullet }$ .

We recall the complexes $E^{\bullet ,\bullet }$ of interest:

1. (1) each term $E^{p,q}$ has a direct sum decomposition

   $$\begin{eqnarray}E^{p,q}=\bigoplus _{\unicode[STIX]{x1D706}\in P}E^{p,q,\unicode[STIX]{x1D706}},\end{eqnarray}$$
   with labels $\unicode[STIX]{x1D706}$ from a poset $P$ ;

2. (2) the vertical differential $d_{v}$ is graded with respect to this labeling; and

3. (3) the horizontal differential $d_{h}$ is downwards filtered.

Conditions (2) and (3) mean that

$$\begin{eqnarray}d_{v}(E^{p,q,\unicode[STIX]{x1D706}})\subseteq E^{p,q+1,\unicode[STIX]{x1D706}}\quad \text{and}\quad d_{h}(E^{p,q,\unicode[STIX]{x1D706}})\subseteq \bigoplus _{\unicode[STIX]{x1D706}^{\prime }\prec \unicode[STIX]{x1D706}}E^{p-1,q,\unicode[STIX]{x1D706}^{\prime }},\end{eqnarray}$$

so the vertical differential preserves the label and the horizontal differential strictly decreases it.

We are interested in the homology of the vertical map $d_{v}$ . Since $d_{v}$ is $P$ -graded, so is its homology $E_{1}^{p,q,\unicode[STIX]{x1D706}}=H(d_{v})^{p,q,\unicode[STIX]{x1D706}}$ . We recall that the $\text{E}_{1}$ graph $G(E^{\bullet ,\bullet })$ is defined as follows:

1. – the vertex set contains $\dim (E_{1}^{p,q,\unicode[STIX]{x1D706}})$ vertices labeled $(p,q,\unicode[STIX]{x1D706})$ , for each $p,q$ and each $\unicode[STIX]{x1D706}\in P$ ;

2. – the edge set includes all possible edges $(p,q,\unicode[STIX]{x1D706})\rightarrow (p^{\prime },q^{\prime },\unicode[STIX]{x1D706}^{\prime })$ whenever $\unicode[STIX]{x1D706}^{\prime }\prec \unicode[STIX]{x1D706}$ and $(p^{\prime },q^{\prime })=(p-r,q-r+1)$ for some $r>0$ .

The edges of $G$ respect the downwards filtered condition on $P$ -labels, and are shaped like higher-order differentials of the associated spectral sequence, that is, they point downwards and leftwards. We wish to show the following theorem.

Theorem 5.12. If $\text{Tot}(E^{\bullet ,\bullet })$ is exact, the $E_{1}$ graph of $E^{\bullet ,\bullet }$ has a perfect matching.

Remark 5.13. Consider summing the $E_{1}$ page along diagonals. Call the resulting complex $\text{Tot}(E_{1})$ . If it were exact, the matching would exist by Proposition 5.9, and in fact would only use the edges corresponding to $r=1$ . Since $\text{Tot}(E_{1})$ is not exact in general, the proof works by modifying its maps to make it exact.

Explicitly, we will exhibit a quasi-isomorphism from $\text{Tot}(E^{\bullet ,\bullet })$ to a complex with the same terms as $\text{Tot}(E_{1})$ , but different maps – whose nonzero coefficients are only in the spots permitted by the $E_{1}$ graph. Since $\text{Tot}(E^{\bullet ,\bullet })$ is exact, so is the new complex, so we will be done by Proposition 5.9.

Remark 5.14 (The role of the axiom of choice).

In our intended usage (Theorem 5.1), the terms $E_{1}^{p,q,\unicode[STIX]{x1D706}}$ are all finite-dimensional, so we do not need the axiom of choice to produce the perfect matching from Proposition 5.9. The reduction to $E_{1}$ does require choices (on the $E_{0}$ page). For the Čech complex used in the proof of Theorem 5.1, the axiom of countable choice suffices. In general the proof relies on a sufficiently strong choice axiom.

Proof of Theorem 5.12.

First, we split all the vertical maps: for each $p,q,\unicode[STIX]{x1D706}$ , we define subspaces $B,H,B^{\ast }\subseteq E$ (suppressing the indices) as follows. We put $B=\text{im}(d_{v})$ ; we choose $H$ to be linearly disjoint from $B$ and such that $B+H=\ker (d_{v})$ ; then we choose $B^{\ast }$ linearly disjoint from $B+H$ , such that $B+H+B^{\ast }=E$ .

In particular, $d_{v}$ maps the subspace $B^{\ast }$ isomorphically to the subsequent subspace $B$ , and the space $H$ descends isomorphically to $H(d_{v})$ , the $E_{1}$ term. The picture of a single column of the double complex looks like the following:

As for $d_{h}$ , we have $d_{h}(B)\subset B$ and $d_{h}(H)\subset B+H$ , and the poset labels $\unicode[STIX]{x1D706}$ strictly decrease.

Our goal will be to choose bases carefully, so as to match the $H$ basis elements to one another, in successive diagonals, while decreasing the poset labels.

We first choose an arbitrary basis of each $H$ and $B^{\ast }$ space. We descend the basis of $B^{\ast }$ to a basis of the subsequent $B$ using $d_{v}$ . Note that every basis element has a position $(p,q)$ and a label $\unicode[STIX]{x1D706}$ . We will write $x_{\unicode[STIX]{x1D706}}$ if we wish to emphasize that a basis vector $x$ has label $\unicode[STIX]{x1D706}$ .

We now change basis on the entire diagonal $E^{i}:=\bigoplus _{p-q=i}E^{p,q}$ . We leave the $H$ and $B^{\ast }$ bases untouched, but replace all the $B$ basis vectors, as follows. Let $b_{\unicode[STIX]{x1D706}}\in B^{p,q,\unicode[STIX]{x1D706}}$ and let $b_{\unicode[STIX]{x1D706}}^{\ast }=d_{v}^{-1}(b_{\unicode[STIX]{x1D706}})\in (B^{\ast })^{p,q-1,\unicode[STIX]{x1D706}}$ be its ‘twin’. We define

$$\begin{eqnarray}\tilde{b}_{\unicode[STIX]{x1D706}}:=d_{\text{tot}}(b_{\unicode[STIX]{x1D706}}^{\ast })=b_{\unicode[STIX]{x1D706}}+d_{h}(b_{\unicode[STIX]{x1D706}}^{\ast }).\end{eqnarray}$$

We replace $b_{\unicode[STIX]{x1D706}}$ by $\tilde{b}_{\unicode[STIX]{x1D706}}$ , formally labeling the new basis vector by $(p,q,\unicode[STIX]{x1D706})$ . We write $\widetilde{B}^{p,q,\unicode[STIX]{x1D706}}$ for the span of the vectors $\tilde{b}$ , so, in particular, $\widetilde{B}^{p,q,\unicode[STIX]{x1D706}}:=d_{\text{tot}}((B^{\ast })^{p,q-1,\unicode[STIX]{x1D706}})$ .

It is clear that $\widetilde{B},H,B^{\ast }$ collectively gives a new basis for the entire diagonal, unitriangular in the old basis. Notice also that the old basis element $b_{\unicode[STIX]{x1D706}}\in B^{p,q,\unicode[STIX]{x1D706}}$ becomes, in general, a linear combination of $\widetilde{B},H,B^{\ast }$ elements in all positions down and left of $p,q$ , with leading term $\tilde{b}_{\unicode[STIX]{x1D706}}$ :

$$\begin{eqnarray}b_{\unicode[STIX]{x1D706}}=\tilde{b}_{\unicode[STIX]{x1D706}}+\mathop{\sum }_{i>0}x^{p-i,q-i},\quad \text{with }x^{p-i,q-i}\in \bigoplus _{\unicode[STIX]{x1D706}^{\prime }\subsetneq \unicode[STIX]{x1D706}}E^{p-i,q-i,\unicode[STIX]{x1D706}^{\prime }}.\end{eqnarray}$$

The lower terms have strictly smaller labels $\unicode[STIX]{x1D706}^{\prime }\subsetneq \unicode[STIX]{x1D706}$ . (In fact, slightly more is true: if a label $\unicode[STIX]{x1D706}^{\prime }$ occurs in the $i$ th term, the poset $P$ contains a chain of length at least $i$ from $\unicode[STIX]{x1D706}^{\prime }$ to $\unicode[STIX]{x1D706}$ .)

We now inspect the coefficients of $(\text{Tot}(E^{\bullet ,\bullet }),d_{\text{tot}})$ in the new basis. We have

$$\begin{eqnarray}\displaystyle d_{\text{tot}}(b_{\unicode[STIX]{x1D706}}^{\ast }) & = & \displaystyle \tilde{b}_{\unicode[STIX]{x1D706}},\nonumber\\ \displaystyle d_{\text{tot}}(\tilde{b}_{\unicode[STIX]{x1D706}}) & = & \displaystyle 0~(=d_{\text{tot}}^{2}(\tilde{b}_{\unicode[STIX]{x1D706}}^{\ast })),\nonumber\end{eqnarray}$$

so the $B^{\ast }$ elements map one by one onto the $\widetilde{B}$ elements, with the same $\unicode[STIX]{x1D706}$ labels; the latter elements then map to 0.

Next, for a basis element $h_{\unicode[STIX]{x1D706}}\in H$ , the coefficients change but remain ‘filtered’. If $d_{\text{tot}}(h_{\unicode[STIX]{x1D706}})$ included (in the old basis) some nonzero term $t\cdot b_{\unicode[STIX]{x1D707}}$ , then in the new basis we have

$$\begin{eqnarray}d_{\text{tot}}(h_{\unicode[STIX]{x1D706}})=d_{h}(h_{\unicode[STIX]{x1D706}})=\cdots +t\cdot (\tilde{b}_{\unicode[STIX]{x1D707}}-d_{h}(b_{\unicode[STIX]{x1D707}}^{\ast }))+\cdots \,.\end{eqnarray}$$

Since $t$ is nonzero, we have $\unicode[STIX]{x1D707}\subsetneq \unicode[STIX]{x1D706}$ ; and the additional terms coming from $d_{h}(b_{\unicode[STIX]{x1D707}}^{\ast })$ all have labels $\unicode[STIX]{x1D707}^{\prime }\subsetneq \unicode[STIX]{x1D707}$ . Thus all labels occurring in $d_{\text{tot}}(h_{\unicode[STIX]{x1D706}})$ in the new basis are, again, strictly smaller than $\unicode[STIX]{x1D706}$ . We note that $d_{\text{tot}}(h_{\unicode[STIX]{x1D706}})$ is a linear combination of $\widetilde{B},H,B^{\ast }$ elements in all positions below and to the left of $h_{\unicode[STIX]{x1D706}}$ along the subsequent diagonal:

Finally, we observe that the spaces $\widetilde{B}+B^{\ast }$ collectively span a subcomplex of $\text{Tot}(E)$ , so we have a short exact sequence of complexes

$$\begin{eqnarray}0\rightarrow \text{Tot}(\widetilde{B}+B^{\ast })\rightarrow \text{Tot}(E)\rightarrow \text{Tot}(H)\rightarrow 0.\end{eqnarray}$$

By construction, $\text{Tot}(\widetilde{B}+B^{\ast })$ is exact, so $\text{Tot}(E)\rightarrow \text{Tot}(H)$ is a quasi-isomorphism. Note that $\text{Tot}(H)$ and $\text{Tot}(E_{1})$ have ‘the same’ terms, but different maps, as desired. Since $\text{Tot}(E)$ is exact, so is $\text{Tot}(H)$ . The desired matching therefore exists by Proposition 5.9.◻

Remark 5.15. Our initial attempts to establish the Boij–Söderberg pairing (Theorems 5.1 and 5.4) used the higher differentials on the $E_{1},E_{2},\ldots$ pages, rather than the $E_{0}$ page as above – aiming to systematize ‘chasing cohomology of the underlying sheaves’. The following example shows that such an approach fails on general double complexes.

Example 5.16 (A cautionary example).

Consider the following double complex:

Each partition denotes a single basis vector with that label. The vertical map $d_{v}$ preserves labels and the horizontal map $d_{h}$ decreases labels. The unlabeled arrows denote maps taking one indicated basis vector to another, and the map $f$ is given by

Note that the rows are exact, so the total complex is exact as well, and the spectral sequence abuts to zero. The only nonzero higher differentials are on the $E_{1}$ and $E_{3}$ pages. These pages, and (for contrast) the complex $H$ constructed in Theorem 5.4, are as follows.

All the arrows are coefficients of $\pm 1$ . In particular, no combination of the $E_{1}$ and $E_{3}$ differentials gives a valid matching (the $E_{3}$ arrow violates the $P$ -filtered condition). In contrast, $H$ finds the (unique) valid matching .

6.1 Extremal tables and the Herzog–Kühl equations

There are three equivariant Herzog–Kühl equations, corresponding to $\unicode[STIX]{x1D707}=\varnothing$ , , and , which may be simplified to

Note that $d_{\unicode[STIX]{x1D706}}(2)=1+\unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D706}_{2}.$

The key observation is the following. Since there are three Herzog–Kühl equations, if we allow exactly four entries in the Betti table to be nonzero, we expect the equations to pick out one dimension’s worth of valid tables. That is, the resulting table $\unicode[STIX]{x1D6FD}$ will be unique up to scalar multiple. By the same reasoning, such a table cannot be decomposed into a nontrivial positive combination of other valid tables. Thus, if realizable, $\unicode[STIX]{x1D6FD}$ is automatically an extremal ray of $\mathit{BS}_{2,3}$ . In certain cases, the equations will be redundant, and three nonzero entries will suffice. We will call the result a three-term or four-term table accordingly. (Three entries are required or the resolution will be too short.)

The observation above underpins the characterization of pure tables for graded modules, where every choice of increasing degree sequence results in a unique table (up to scaling) with exactly $n+1$ nonzero entries, one in each column. Unlike in the graded case, we see that our extremal tables need not be ‘pure’: each four-term table will have a column with two distinct entries.

6.2 Three-term tables

These tables closely resemble the extremal Betti tables in the square-matrix and graded settings. Each column has one nonzero entry, so the resolution is ‘pure’:

$$\begin{eqnarray}M\leftarrow \mathbb{S}_{\unicode[STIX]{x1D706}}(V)^{\unicode[STIX]{x1D6FD}_{0,\unicode[STIX]{x1D706}}}\otimes R\leftarrow \mathbb{S}_{\unicode[STIX]{x1D707}}(V)^{\unicode[STIX]{x1D6FD}_{1,\unicode[STIX]{x1D707}}}\otimes R\leftarrow \mathbb{S}_{\unicode[STIX]{x1D708}}(V)^{\unicode[STIX]{x1D6FD}_{2,\unicode[STIX]{x1D708}}}\otimes R\leftarrow 0\end{eqnarray}$$

for some triple of partitions $\unicode[STIX]{x1D706}\subsetneq \unicode[STIX]{x1D707}\subsetneq \unicode[STIX]{x1D708}$ .

Example 6.1 (An example with border strips).

We realize the table

We sketch the construction, a version of Weyman’s geometric technique similar to that used in [Reference Ford, Levinson and SamFLS18, Theorem 4.6]. Let $X=\mathbb{P}(V)$ , with tautological line subbundle ${\mathcal{O}}(-1)$ and quotient line bundle ${\mathcal{Q}}$ (we distinguish between ${\mathcal{Q}}$ and the dual of ${\mathcal{O}}(-1)$ since we are tracking the whole $\text{GL}(V)$ -action). By the Borel–Weil–Bott theorem, ${\mathcal{Q}}^{\otimes a}\otimes {\mathcal{O}}(-b)$ has cohomology

$$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}H^{0}=\mathbb{S}_{a,b}(V),H^{1}=0\quad & \text{if }a\geqslant b,\\ H^{0}=H^{1}=0\quad & \text{if }a=b-1,\\ H^{0}=0,H^{1}=\mathbb{S}_{b-1,a+1}(V)\quad & \text{if }a\leqslant b-2.\end{array}\right.\end{eqnarray}$$

Eisenbud et al. [Reference Eisenbud, Fløystad and WeymanEFW11] have constructed, over any polynomial ring $S=\text{Sym}(E)$ with $E$ a vector space of dimension $3$ , a $\text{GL}(E)$ -equivariant free resolution

$$\begin{eqnarray}\mathbb{S}_{(1,0,-1)}(E)\otimes S\leftarrow \mathbb{S}_{(2,0,-1)}(E)\otimes S\leftarrow \mathbb{S}_{(2,2,-1)}(E)\otimes S\leftarrow \mathbb{S}_{(2,2,1)}(E)\otimes S\leftarrow 0,\end{eqnarray}$$

resolving a module of finite length. (In the graded setting, this gives a pure resolution for the degree sequence $(0,1,3,5)$ .) Since the resolution is equivariant, it makes sense for families of vector spaces, so we may replace $E$ by the rank- $3$ vector bundle ${\mathcal{E}}:=W^{\ast }\,\otimes \,{\mathcal{O}}(-1)$ over $X$ . The resulting locally free resolution has terms

$$\begin{eqnarray}\mathbb{S}_{\unicode[STIX]{x1D6FC}}({\mathcal{E}})\otimes \text{Sym}({\mathcal{E}})=\mathbb{S}_{\unicode[STIX]{x1D6FC}}(W^{\ast })\otimes {\mathcal{O}}(-|\unicode[STIX]{x1D6FC}|)\otimes \text{Sym}({\mathcal{E}}).\end{eqnarray}$$

Next, we base-change the resolution along the flat inclusion of ${\mathcal{O}}_{X}$ -algebras

$$\begin{eqnarray}\text{Sym}({\mathcal{E}}){\hookrightarrow}\text{Sym}(V\otimes W^{\ast })\otimes {\mathcal{O}}_{X}=R\otimes {\mathcal{O}}_{X}\end{eqnarray}$$

(locally an inclusion of polynomial rings) induced by the inclusion ${\mathcal{O}}(-1){\hookrightarrow}V\,\otimes \,{\mathcal{O}}_{X}$ . We twist through by ${\mathcal{Q}}^{\otimes 2}$ and take hypercohomology (the relevant terms are ${\mathcal{Q}}^{\otimes 2}\,\otimes \,{\mathcal{O}}(-b)$ for $b=0,1,3,5$ ). Note that the resolved sheaf was supported set-theoretically on

$$\begin{eqnarray}\{(\ell ,T):T(\ell )=0\}\subset \mathbb{P}(V)\times \text{Hom}(V,W),\end{eqnarray}$$

so the resulting $R$ -module is automatically supported on the locus of rank-deficient matrices. The spectral sequence collapses at $E_{3}$ and yields the free resolution

$$\begin{eqnarray}\displaystyle \mathbb{S}_{(2,0)}(V)\otimes \mathbb{S}_{(1,0,-1)}(W^{\ast })\otimes R & \cong & \displaystyle \mathbb{S}_{(2,0)}(V)^{8}\otimes R\nonumber\\ \displaystyle \leftarrow \mathbb{S}_{(2,1)}(V)\otimes \mathbb{S}_{(2,0,-1)}(W^{\ast })\otimes R & \cong & \displaystyle \mathbb{S}_{(2,1)}(V)^{15}\otimes R\nonumber\\ \displaystyle \leftarrow \mathbb{S}_{(4,3)}(V)\otimes \mathbb{S}_{(2,2,1)}(W^{\ast })\otimes R & \cong & \displaystyle \mathbb{S}_{(3,2)}(V)^{3}\otimes R.\nonumber\end{eqnarray}$$

The last arrow is from the $E_{2}$ page. This realizes the desired Betti table.

The construction above produces (based on the input from [Reference Eisenbud, Fløystad and WeymanEFW11]) the three-term tables, up to scalar multiple, for the triples $(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707},\unicode[STIX]{x1D708})$ for which the skew shapes $\unicode[STIX]{x1D707}/\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D708}/\unicode[STIX]{x1D707}$ are border strips:

and where the second border strip is adjacent and to the right of the first. Examples of such triples are:

Here $\unicode[STIX]{x1D706}$ consists of the empty squares, $\unicode[STIX]{x1D707}$ contains the additional squares marked $1$ , and $\unicode[STIX]{x1D708}$ contains the squares marked $1,2$ . We can rule out almost all other possible triples. Indeed, we have the following proposition.

That is, the shapes $\unicode[STIX]{x1D707}/\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D708}/\unicode[STIX]{x1D707}$ are contained in the border strip formed by the squares along the outer edge of $\unicode[STIX]{x1D706}$ :

We do not know if they must be connected or adjacent (see Example 6.3). We prove Proposition 6.2 using the equivariant pairing.

Proof. We pair with the cohomology table $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}({\mathcal{O}}(d))$ for appropriate choice of $d$ . Note that $\unicode[STIX]{x1D706}_{1}\leqslant \unicode[STIX]{x1D707}_{1}\leqslant \unicode[STIX]{x1D708}_{1}$ and that $\unicode[STIX]{x1D708}_{2}\leqslant \unicode[STIX]{x1D708}_{1}$ ; we assume that $\unicode[STIX]{x1D708}_{2}>\unicode[STIX]{x1D706}_{1}+1$ .

Suppose first that $\unicode[STIX]{x1D706}_{1}+1<\unicode[STIX]{x1D708}_{2}\leqslant \unicode[STIX]{x1D707}_{1}+1$ . Then we put $d=\unicode[STIX]{x1D708}_{2}-2$ . By the Borel–Weil–Bott theorem, we have $\unicode[STIX]{x1D6FE}_{0,\unicode[STIX]{x1D706}}\neq 0$ , at most one of $\unicode[STIX]{x1D6FE}_{1,\unicode[STIX]{x1D707}}\neq 0$ or $\unicode[STIX]{x1D6FE}_{2,\unicode[STIX]{x1D707}}\neq 0$ , and all $\unicode[STIX]{x1D708}$ entries and $\unicode[STIX]{x1D6FE}_{0,\unicode[STIX]{x1D707}}$ zero, giving

Since $\unicode[STIX]{x1D706}\subsetneq \unicode[STIX]{x1D707}$ , there is no valid perfect matching regardless of whether $\unicode[STIX]{x1D6FE}_{1,\unicode[STIX]{x1D707}},\unicode[STIX]{x1D6FE}_{2,\unicode[STIX]{x1D707}}$ are zero.

Suppose instead that $\unicode[STIX]{x1D707}_{1}+1<\unicode[STIX]{x1D708}_{2}~({<}\unicode[STIX]{x1D708}_{1}+1)$ . Then we put $d=\unicode[STIX]{x1D707}_{1}-1$ . This time $\unicode[STIX]{x1D6FE}_{2,\unicode[STIX]{x1D708}}\neq 0$ and possibly $\unicode[STIX]{x1D6FE}_{0,\unicode[STIX]{x1D706}}\neq 0$ , but all $\unicode[STIX]{x1D707}$ entries are zero. But then $\langle \unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FE}\rangle$ has only one column:

Since $\unicode[STIX]{x1D6FD}_{2,\unicode[STIX]{x1D708}}\unicode[STIX]{x1D6FE}_{2,\unicode[STIX]{x1D708}}\neq 0$ , there is no perfect matching.◻

The next example shows, however, that not all three-term solutions to the equivariant Herzog–Kühl equations are built from connected border strips.

Example 6.3 (Exceptional three-term tables).

Consider the tables

We do not know whether the tables above (part of an infinite family) are realizable, but we remark that they pair positively with the cohomology tables $\unicode[STIX]{x1D6FE}({\mathcal{O}}(d))$ for all $d\in \mathbb{Z}$ (these are all the tables $\unicode[STIX]{x1D6FE}(\mathbb{S}_{\unicode[STIX]{x1D707}}({\mathcal{Q}}))$ on $\text{Gr}(2,3)\cong \mathbb{P}^{2}$ , since ${\mathcal{Q}}\cong {\mathcal{O}}(1)$ ). The corresponding equivariant Hilbert series are also Schur-positive.

6.3 Four-term tables

In each four-term table, one of the columns has two nonzero entries. We will say the table is diamond-shaped when the middle column has two nonzero entries and Y-shaped otherwise:

We will construct resolutions of these forms starting from modules with three-term resolutions, obtaining diamond-shaped tables as (co)kernels and Y-shaped tables as extensions. For brevity, we will write the Young diagram for $\unicode[STIX]{x1D706}$ to mean the free module $\mathbb{S}_{\unicode[STIX]{x1D706}}(V)\,\otimes \,R$ .

Example 6.4 (Y-shaped tables via extensions).

Consider the following Y-shaped table:

We obtain this table as an extension of the following two three-term resolutions:

Note that the term  appears in both resolutions, and that all the other terms appear in exactly the desired positions for the resolution we wish to construct. We claim that a map  exists such that the following diagram commutes:

Consequently, the module $E$ resolved by the total complex above has (after removing the two  terms) the desired resolution. To show that $t$ exists, it is enough to show $h\circ e=0$ , and so by exactness $\text{im}(e)\subset \text{im}(g)$ . Then $t$ exists because the module  is free, in particular projective. The key fact, which is straightforward to compute, is that the  coefficient is zero in the equivariant Hilbert series of $N$ :

$$\begin{eqnarray}H_{N}(t_{1},t_{2})=\frac{3t_{1}t_{2}-3(t_{1}^{2}t_{2}+t_{1}t_{2}^{2})+(t_{1}^{3}t_{2}+t_{1}^{2}t^{2}+t_{1}t_{2}^{3})}{(1-t_{1})^{3}(1-t_{2})^{3}}.\end{eqnarray}$$

Thus the generators of  must map to zero in $N$ .

The construction above relies only on properties of $M$ and $N$ that are observable from their Betti tables and invariant under scaling. Namely, $M$ and $N$ have three-term resolutions with a term in common, and the appropriate $s_{\unicode[STIX]{x1D706}}$ coefficient in $H_{N}(t)$ is zero. Also, since the constructed module $E$ is Cohen–Macaulay, its dual $\text{Ext}^{2}(E,R)$ has the dual four-term resolution, which is $Y$ -shaped facing the other way.

We construct diamond-shaped tables in a similar way.

Example 6.5 (Diamond shaped tables via (co)kernels).

Consider the table

We obtain (a multiple of) this table by constructing the following double complex:

As in Example 6.4, the rows are three-term resolutions as constructed up to scalar multiple in Example 6.1. (The first row is just three copies of the Eagon–Northcott complex.) Here $\varnothing$ denotes $R$ as a module over itself. The existence of $t_{1}$ follows again from an equivariant Hilbert series computation (in fact the same computation as in the previous example, only twisted by $\det (\mathbb{C}^{2})$ ), showing that the module resolved by the second row has no -isotypic part. Then $t_{2}$ existsautomatically, in fact uniquely since the map  is injective. The total complex is the desired minimal free resolution (after dropping the $\varnothing ^{\oplus 3}$ terms).

As with the three-term tables, we do not know if all four-term tables can be constructed in this way (moreover, it would be natural to construct certain four-term tables from the conjectural three-term tables of Example 6.3). Nonetheless, we can rule out some otherwise plausible four-term tables using the equivariant pairing.

Example 6.6 (A nonrealizable Y-shaped table).

Consider the following table:

Despite satisfying the Herzog–Kühl equations and resembling the unobjectionable examples above, this table is not in $\mathit{BS}_{2,3}$ . This follows from the numerical pairing: when paired with ${\mathcal{O}}(d)$ , the result is in $\widetilde{\mathit{BS}}_{2,2}$ for every $d$ except $d=0$ , for which the output is

Clearly, this table has no perfect matching. We note in this example that the graded Betti table obtained by forgetting the $\text{GL}_{2}$ -action is realizable:

The latter two tables are for the degree sequences $(0,3,5)$ and $(2,3,5)$ . Thus, in this case the equivariant pairing detects that such a table cannot be realized with the additional $\text{GL}_{2}$ -structure given above.

We end with a phenomenon that does not appear in the square-matrix or graded cases.

Example 6.7 (‘Stably realizable’ Betti tables).

Let $\unicode[STIX]{x1D6FD}$ be the nonrealizable table of Example 6.6. Consider tensoring $\unicode[STIX]{x1D6FD}$ with $\mathbb{S}_{1}V$ . (That is, write the Betti table that would result from tensoring such a resolution, if it existed, with $\mathbb{S}_{1}V$ .) The result is as follows:

This table in fact is realizable, a linear combination of the following realizable tables:

Specifically, the large table above is ${\textstyle \frac{3}{2}}A+{\textstyle \frac{1}{2}}B+{\textstyle \frac{3}{2}}C+D.$

This example suggests that it might be easier to study ‘stably realizable’ Betti tables, that is, Betti tables that become realizable after tensoring with some $\text{GL}(V)$ -representation.