A polynomial identity via differential operators

TL;DR

This paper presents a novel proof of a polynomial identity related to matrix minors, which has applications in understanding integer torsion in local cohomology modules.

Contribution

It introduces a new proof technique for a polynomial identity involving minors, connecting algebraic identities with local cohomology.

Findings

New proof of a polynomial identity involving minors

Insights into integer torsion in local cohomology modules

Potential applications in algebraic geometry and commutative algebra

Abstract

We give a new proof of a polynomial identity involving the minors of a matrix, that originated in the study of integer torsion in a local cohomology module.

Equations

$$\setcounter{MaxMatrixCols}{11}\begin{CD}K^{\bullet}(f;\,R)\ \colonequals\quad 0@>{}>{}>R@>{f}>{}>R@>{}>{}>0\phantom{.}\\ @V{}V{}V\Big{\|}@V{}V{\frac{1}{f}}V\\ C^{\bullet}(f;\,R)\ \colonequals\quad 0@>{}>{}>R@>{}>{}>R_{f}@>{}>{}>0.\end{CD}$$

$$\begin{array}{ccc}{K}^{\bullet }(\mathbf{f};R):=\underset{i}{⨂}{K}^{\bullet }(f_i;R)& \to & \underset{i}{⨂}{C}^{\bullet }(f_i;R)=:C^{\bullet }(\mathbf{f};R),\end{array}$$

$$\begin{array}{ccc}{H}^n(\mathbf{f};R)& \to & H_{(\mathbf{f})}^n(R).\end{array}$$

$$\begin{array}{ccc}{H}_{(\mathbf{f})}^n(R)& \to & H_{({\mathbf{f}}^p)}^n(R)=H_{(\mathbf{f})}^n(R),\end{array}$$

$$H^1(f;R)=R/fR$$

$$\left[1\right]$$

$$\frac{1}{k!}\frac{{\partial }^k}{\partial x^k}$$

$$\frac{1}{k!}\frac{{\partial }^k}{\partial x^k}(x^m)=\left(\genfrac{}{}{0ex}{}{m}{k}\right)x^{m-k}\text{ for }m⩾0,$$

$$\frac{1}{r!}\frac{{\partial }^r}{\partial x^r}\cdot \frac{1}{s!}\frac{{\partial }^s}{\partial x^s}=\left(\genfrac{}{}{0ex}{}{r+s}{r}\right)\frac{1}{(r+s)!}\frac{{\partial }^{r+s}}{\partial x^{r+s}}.$$

$$\frac{1}{k_1!}\frac{{\partial }^{k_1}}{\partial x_1^{k_1}}\cdot \cdots \cdot \frac{1}{k_d!}\frac{{\partial }^{k_d}}{\partial x_d^{k_d}}\text{ for }{k}_i⩾0,$$

$$\frac{1}{k!}\frac{{\partial }^k}{\partial x^k}\left(\frac{1}{x^m}\right)=(-1{)}^k(\genfrac{}{}{0ex}{}{m+k-1}{k})\frac{1}{x^{m+k}}.$$

$$\frac{1}{k!}\frac{{\partial }^k}{\partial x^k}(fg)=\sum _{i+j=k}\frac{1}{i!}\frac{{\partial }^i}{\partial x^i}(f)\frac{1}{j!}\frac{{\partial }^j}{\partial x^j}(g).$$

$$\left(\begin{array}{ccc}u& v& w\\ x& y& z\end{array}\right).$$

$$\begin{array}{ccccccccc}0& \to & R& \stackrel{p}{\to }& R& \to & \overline{R}& \to & 0,\end{array}$$

$$\text{setcounter}MaxMatrixCols13\begin{array}{ccccccccccccc}& \to & H_I^2(R)& \stackrel{\pi }{\to }& H_I^2(\overline{R})& \to & H_I^3(R)& \stackrel{p}{\to }& H_I^3(R)& \to & H_I^3(\overline{R})& \to & 0.\end{array}$$

$$\begin{gathered} \sum _{i,j⩾0}\left(\genfrac{}{}{0ex}{}{k}{i+j}\right)\left(\genfrac{}{}{0ex}{}{k+i}{k}\right)\left(\genfrac{}{}{0ex}{}{k+j}{k}\right)\frac{(-wx{)}^i(vx{)}^j{u}^{k+1}}{{\Delta }_2^{k+1+i}{\Delta }_3^{k+1+j}} \\ +\sum _{i,j⩾0}\left(\genfrac{}{}{0ex}{}{k}{i+j}\right)\left(\genfrac{}{}{0ex}{}{k+i}{k}\right)\left(\genfrac{}{}{0ex}{}{k+j}{k}\right)\frac{(-uy{)}^i(wy{)}^j{v}^{k+1}}{{\Delta }_3^{k+1+i}{\Delta }_1^{k+1+j}} \\ +\sum _{i,j⩾0}\left(\genfrac{}{}{0ex}{}{k}{i+j}\right)\left(\genfrac{}{}{0ex}{}{k+i}{k}\right)\left(\genfrac{}{}{0ex}{}{k+j}{k}\right)\frac{(-vz{)}^i(uz{)}^j{w}^{k+1}}{{\Delta }_1^{k+1+i}{\Delta }_2^{k+1+j}}=0. \end{gathered}$$

$$C^2({\Delta }_1,{\Delta }_2,{\Delta }_3;R)=R_{{\Delta }_1{\Delta }_2}\oplus R_{{\Delta }_1{\Delta }_3}\oplus R_{{\Delta }_2{\Delta }_3};$$

$$\left(\genfrac{}{}{0ex}{}{k}{i+j}\right)\left(\genfrac{}{}{0ex}{}{k+i}{k}\right)\left(\genfrac{}{}{0ex}{}{k+j}{k}\right)\equiv 0\mathrm{mod}p\text{ for }(i,j)\ne (0,0),$$

$$\frac{u^{p^e}}{{\Delta }_2^{p^e}{\Delta }_3^{p^e}}+\frac{v^{p^e}}{{\Delta }_3^{p^e}{\Delta }_1^{p^e}}+\frac{w^{p^e}}{{\Delta }_1^{p^e}{\Delta }_2^{p^e}}\equiv 0\mathrm{mod}p,$$

$$\pi ({\eta }_{p^e-1})=\left[\left(\frac{w^{p^e}}{{\Delta }_1^{p^e}{\Delta }_2^{p^e}},\frac{-v^{p^e}}{{\Delta }_1^{p^e}{\Delta }_3^{p^e}},\frac{u^{p^e}}{{\Delta }_2^{p^e}{\Delta }_3^{p^e}}\right)\right]\text{ in }{H}_I^2(\overline{R}).$$

$$\begin{array}{ccc}{H}^2({\Delta }_1,{\Delta }_2,{\Delta }_3;\overline{R})& \to & H_I^2(\overline{R}).\end{array}$$

$$u{\Delta }_1+v{\Delta }_2+w{\Delta }_3=0\text{ and }x{\Delta }_1+y{\Delta }_2+z{\Delta }_3=0.$$

$$\alpha :=\left[\left(\frac{w}{{\Delta }_1{\Delta }_2},\frac{-v}{{\Delta }_1{\Delta }_3},\frac{u}{{\Delta }_2{\Delta }_3}\right)\right]\text{ and }\beta :=\left[\left(\frac{z}{{\Delta }_1{\Delta }_2},\frac{-y}{{\Delta }_1{\Delta }_3},\frac{x}{{\Delta }_2{\Delta }_3}\right)\right]$$

$${\phi }^e(\alpha )=\pi ({\eta }_{p^e-1})$$

$$\left(\frac{w}{{\Delta }_1{\Delta }_2},\frac{-v}{{\Delta }_1{\Delta }_3},\frac{u}{{\Delta }_2{\Delta }_3}\right)$$

$$\frac{w}{{\Delta }_1{\Delta }_2}+\frac{v}{{\Delta }_1{\Delta }_3}+\frac{u}{{\Delta }_2{\Delta }_3}=0.$$

$$D=\frac{1}{k!}\frac{{\partial }^k}{\partial u^k}\cdot \frac{1}{k!}\frac{{\partial }^k}{\partial y^k}\cdot \frac{1}{k!}\frac{{\partial }^k}{\partial z^k}$$

$$\mathcal{D}(R,\mathbb{Z})/p\mathcal{D}(R,\mathbb{Z})=\mathcal{D}(R/pR,\mathbb{Z}/p\mathbb{Z}).$$

$$\overline{D}(u{\Delta }_2^{p^e-1}{\Delta }_3^{p^e-1})\equiv u^{p^e}$$

$$\overline{D}(v{\Delta }_3^{p^e-1}{\Delta }_1^{p^e-1})\equiv v^{p^e}$$

$$\overline{D}(w{\Delta }_1^{p^e-1}{\Delta }_2^{p^e-1})\equiv w^{p^e}$$

$$\overline{D}\left(\frac{w}{{\Delta }_1{\Delta }_2},\frac{-v}{{\Delta }_1{\Delta }_3},\frac{u}{{\Delta }_2{\Delta }_3}\right)\equiv \left(\frac{w^{p^e}}{{\Delta }_1^{p^e}{\Delta }_2^{p^e}},\frac{-v^{p^e}}{{\Delta }_1^{p^e}{\Delta }_3^{p^e}},\frac{u^{p^e}}{{\Delta }_2^{p^e}{\Delta }_3^{p^e}}\right)\mathrm{mod}p,$$

Full text

Dedicated to Professor Winfried Bruns, on the occasion of his 70th birthday

1 Introduction

Our study of integer torsion in local cohomology modules began in the paper Si , where we constructed a local cohomology module that has $p$-torsion for each prime integer $p$, and also studied the determinantal example $H^{3}_{I_{2}}(\mathbb{Z}[X])$ where $X$ is a $2\times 3$ matrix of indeterminates, and $I_{2}$ the ideal generated by its size $2$ minors. In that paper, we constructed a polynomial identity that shows that the local cohomology module $H^{3}_{I_{2}}(\mathbb{Z}[X])$ has no integer torsion; it then follows that this module is a rational vector space. Subsequently, in joint work with Lyubeznik and Walther, we showed that the same holds for all local cohomology modules of the form $H^{k}_{I_{t}}(\mathbb{Z}[X])$, where $X$ is a matrix of indeterminates, $I_{t}$ the ideal generated by its size $t$ minors, and $k$ an integer with $k>\operatorname{height}I_{t}$, (LSW, , Theorem 1.2). In a related direction, in joint work with Bhatt, Blickle, Lyubeznik, and Zhang, we proved that the local cohomology of a polynomial ring over $\mathbb{Z}$ can have $p$-torsion for at most finitely many $p$; we record a special case of (BBLSZ, , Theorem 3.1):

Theorem 1.1

Let $R$ be a polynomial ring over the ring of integers, and let $f_{1},\dots,f_{m}$ be elements of $R$. Let $n$ be a nonnegative integer. Then each prime integer that is a nonzerodivisor on the Koszul cohomology module $H^{n}(f_{1},\dots,f_{m};\,R)$ is also a nonzerodivisor on the local cohomology module $H^{n}_{(f_{1},\dots,f_{m})}(R)$.

These more general results notwithstanding, a satisfactory proof or conceptual understanding of the polynomial identity from Si had previously eluded us; extensive calculations with Macaulay2 had led us to a conjectured identity, which we were then able to prove using the hypergeometric series algorithms of Petkovšek, Wilf, and Zeilberger PWZ , as implemented in Maple. The purpose of this note is to demonstrate how techniques using differential operators underlying the papers BBLSZ and LSW provide the “right” proof of the identity, and, indeed, provide a rich source of similar identities.

We remark that there is considerable motivation for studying local cohomology of rings of polynomials with integer coefficients such as $H^{k}_{I_{t}}(\mathbb{Z}[X])$: a matrix of indeterminates $X$ specializes to a given matrix of that size over an arbitrary commutative noetherian ring (this is where $\mathbb{Z}$ is crucial), which turns out to be useful in proving vanishing theorems for local cohomology supported at ideals of minors of arbitrary matrices. See (LSW, , Theorem 1.1) for these vanishing results, that build upon the work of Bruns and Schwänzl BS .

2 Preliminary remarks

We summarize some notation and facts. As a reference for Koszul cohomology and local cohomology, we mention BH ; for more on local cohomology as a $\mathcal{D}$-module, we point the reader towards Ly (1) and BBLSZ .

Koszul and Čech cohomology

For an element $f$ in a commutative ring $R$, the Koszul complex $K^{\bullet}(f;\,R)$ has a natural map to the Čech complex $C^{\bullet}(f;\,R)$ as follows:

[TABLE]

For a sequence of elements ${\bm{f}}=f_{1},\dots,f_{m}$ in $R$, one similarly obtains

[TABLE]

and hence, for each $n\geqslant 0$, an induced map on cohomology modules

[TABLE]

Now suppose $R$ is a polynomial ring over a field $\mathbb{F}$ of characteristic $p>0$. The Frobenius endomorphism $\varphi$ of $R$ induces an additive map

[TABLE]

where ${\bm{f}}^{p}=f_{1}^{p},\dots,f_{m}^{p}$. Set $R\{\varphi\}$ to be the extension ring of $R$ obtained by adjoining the Frobenius operator, i.e., adjoining a generator $\varphi$ subject to the relations $\varphi r=r^{p}\varphi$ for each $r\in R$; see (Ly, 2, Section 4). By an $R\{\varphi\}$-module we will mean a left $R\{\varphi\}$-module. The map displayed above gives $H^{n}_{({\bm{f}})}(R)$ an $R\{\varphi\}$-module structure. It is not hard to see that the image of $H^{n}({\bm{f}};\,R)$ in $H^{n}_{({\bm{f}})}(R)$ generates the latter as an $R\{\varphi\}$-module; what is much more surprising is a result of Àlvarez, Blickle, and Lyubeznik, (ABL, , Corollary 4.4), by which the image of $H^{n}({\bm{f}};\,R)$ in $H^{n}_{({\bm{f}})}(R)$ generates the latter as a $\mathcal{D}(R,\mathbb{F})$-module; see below for the definition. The result is already notable in the case $m=1=n$, where the map (1) takes the form

[TABLE]

By ABL , the element $1/f$ generates $R_{f}$ as a $\mathcal{D}(R,\mathbb{F})$-module. It is of course evident that $1/f$ generates $R_{f}$ as an $R\{\varphi\}$-module since the elements $\varphi^{e}(1/f)=1/f^{p^{e}}$ with $e\geqslant 0$ serve as $R$-module generators for $R_{f}$. See BDV for an algorithm to explicitly construct a differential operator $\delta$ with $\delta(1/f)=1/f^{p^{e}}$, along with a Macaulay2 implementation.

Differential operators

Let $A$ be a commutative ring, and $x$ an indeterminate; set $R=A[x]$. The divided power partial differential operator

[TABLE]

is the $A$-linear endomorphism of $R$ with

[TABLE]

where we use the convention that the binomial coefficient $\binom{m}{k}$ vanishes if $m<k$. Note that

[TABLE]

For the purposes of this paper, if $R$ is a polynomial ring over $A$ in the indeterminates $x_{1},\dots,x_{d}$, we define the ring of $A$-linear differential operators on $R$, denoted $\mathcal{D}(R,A)$, to be the free $R$-module with basis

[TABLE]

with the ring structure coming from composition. This is consistent with more general definitions; see (Gr, , 16.11). By a $\mathcal{D}(R,A)$-module, we will mean a left $\mathcal{D}(R,A)$-module; the ring $R$ has a natural $\mathcal{D}(R,A)$-module structure, as do localizations of $R$. For a sequence of elements ${\bm{f}}$ in $R$, the Čech complex $C^{\bullet}({\bm{f}};\,R)$ is a complex of $\mathcal{D}(R,A)$-modules, and hence so are its cohomology modules $H^{n}_{({\bm{f}})}(R)$. Note that for $m\geqslant 1$, one has

[TABLE]

We also recall the Leibniz rule, which states that

[TABLE]

3 The identity

Let $R$ be the ring of polynomials with integer coefficients in the indeterminates

[TABLE]

The ideal $I$ generated by the size $2$ minors of the above matrix has height $2$; our interest is in proving that the local cohomology module $H^{3}_{I}(R)$ is a rational vector space. We label the minors as $\Delta_{1}=vz-wy$, $\Delta_{2}=wx-uz$, and $\Delta_{3}=uy-vx$. Fix a prime integer $p$, and consider the exact sequence

[TABLE]

where $\overline{R}=R/pR$. This induces an exact sequence of local cohomology modules

[TABLE]

The ring $\overline{R}/I\overline{R}$ is Cohen-Macaulay of dimension $4$, so (PS, , Proposition III.4.1) implies that $H^{3}_{I}(\overline{R})=0$. As $p$ is arbitrary, it follows that $H^{3}_{I}(R)$ is a divisible abelian group. To prove that it is a rational vector space, one needs to show that multiplication by $p$ on $H^{3}_{I}(R)$ is injective, equivalently that $\pi$ is surjective. We first prove this using the identity (2) below, and then proceed with the proof of the identity.

For each $k\geqslant 0$, one has

[TABLE]

Since the binomial coefficient $\binom{k}{i+j}$ vanishes if $i$ or $j$ exceeds $k$, this equation may be rewritten as an identity in the polynomial ring $\mathbb{Z}[u,v,w,x,y,z]$ after multiplying by $(\Delta_{1}\Delta_{2}\Delta_{3})^{2k+1}$.

Computing $H^{2}_{I}(R)$ as the cohomology of the Čech complex $C^{\bullet}(\Delta_{1},\Delta_{2},\Delta_{3};\,R)$, equation (2) gives a $2$-cocycle in

[TABLE]

we denote the cohomology class of this cocycle in $H^{2}_{I}(R)$ by $\eta_{k}$. When $k=p^{e}-1$, one has

[TABLE]

so (2) reduces modulo $p$ to

[TABLE]

and the cohomology class $\eta_{p^{e}-1}$ has image

[TABLE]

Since $\overline{R}$ is a regular ring of positive characteristic, $H^{2}_{I}(\overline{R})$ is generated as an $\overline{R}\{\varphi\}$-module by the image of

[TABLE]

The Koszul cohomology module $H^{2}(\Delta_{1},\Delta_{2},\Delta_{3};\,\overline{R})$ is readily seen to be generated, as an $\overline{R}$-module, by elements corresponding to the relations

[TABLE]

These two generators of $H^{2}(\Delta_{1},\Delta_{2},\Delta_{3};\,\overline{R})$ map, respectively, to

[TABLE]

in $H^{2}_{I}(\overline{R})$. Thus, $H^{2}_{I}(\overline{R})$ is generated over $\overline{R}$ by $\varphi^{e}(\alpha)$ and $\varphi^{e}(\beta)$ for $e\geqslant 0$. But

[TABLE]

is in the image of $\pi$, and hence so is $\varphi^{e}(\beta)$ by symmetry. Thus, $\pi$ is surjective.

The proof of the identity

We start by observing that $C^{2}(\Delta_{1},\Delta_{2},\Delta_{3};\,R)$ is a $\mathcal{D}(R,\mathbb{Z})$-module. The element

[TABLE]

is a $2$-cocycle in $C^{2}(\Delta_{1},\Delta_{2},\Delta_{3};\,R)$ since

[TABLE]

We claim that the identity (2) is simply the differential operator

[TABLE]

applied termwise to (3); we first explain the choice of this operator: set $k=p^{e}-1$, and consider $\overline{D}=D\mod p$ as an element of

[TABLE]

It is an elementary verification that

[TABLE]

Since $k<p^{e}$, the differential operator $\overline{D}$ is $\overline{R}^{p^{e}}$-linear; dividing the above equations by $\Delta_{2}^{p^{e}}\Delta_{3}^{p^{e}}$, $\Delta_{3}^{p^{e}}\Delta_{1}^{p^{e}}$, and $\Delta_{1}^{p^{e}}\Delta_{2}^{p^{e}}$ respectively, we obtain

[TABLE]

which maps to the desired cohomology class $\varphi^{e}(\alpha)$ in $H^{2}_{I}(\overline{R})$. Of course, the operator $D$ is not unique in this regard.

Using elementary properties of differential operators recorded in §2, we have

[TABLE]

A similar calculation shows that

[TABLE]

It remains to evaluate $\displaystyle{D\left(\frac{u}{\Delta_{2}\Delta_{3}}\right)}$; we reduce this to the previous calculation as follows. First note that the differential operators $\displaystyle{\frac{\partial}{\partial u}\cdot\frac{\partial}{\partial y}}$ and $\displaystyle{\frac{\partial}{\partial v}\cdot\frac{\partial}{\partial x}}$ commute; it is readily checked that they agree on $\displaystyle{\frac{u}{\Delta_{2}\Delta_{3}}}$. Consequently the operators

[TABLE]

agree on $\displaystyle{\frac{u}{\Delta_{2}\Delta_{3}}}$ as well. But then

[TABLE]

which, using the previous calculation and symmetry, equals

[TABLE]

Identities in general

Suppose ${\bm{f}}=f_{1},\dots,f_{m}$ are elements of a polynomial ring $R$ over $\mathbb{Z}$, and $g_{1},\dots,g_{m}$ are elements of $R$ such that

[TABLE]

Then, for each prime integer $p$ and $e\geqslant 0$, the Frobenius map on $\overline{R}=R/pR$ gives

[TABLE]

Now suppose $p$ is a nonzerodivisor on the Koszul cohomology module $H^{m}({\bm{f}};\,R)$. Then Theorem 1.1 implies that (4) lifts to an equation

[TABLE]

in $R$ in the sense that the cohomology class corresponding to (5) in $H^{m-1}_{({\bm{f}})}(R)$ maps to the cohomology class corresponding to (4) in $H^{m-1}_{({\bm{f}})}(\overline{R})$.

Acknowledgements.

NSF support under grant DMS 1500613 is gratefully acknowledged. This paper owes an obvious intellectual debt to our collaborations with Bhargav Bhatt, Manuel Blickle, Gennady Lyubeznik, Uli Walther, and Wenliang Zhang; we take this opportunity to thank our coauthors.