A polynomial identity via differential operators

arXiv:1703.04167v1 [] 12 Mar 2017

Anurag K. Singh

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

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 HI32 (Z[X]) where X is a 2 × 3
matrix of indeterminates, and I2 the ideal generated by its size 2 minors. In that
paper, we constructed a polynomial identity that shows that the local cohomology
module HI32 (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 HIkt (Z[X]),
where X is a matrix of indeterminates, It the ideal generated by its size t minors,
and k an integer with k > heightIt , [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 Z can have p-torsion for at most finitely
many p; we record a special case of [BBLSZ, Theorem 3.1]:
Theorem 1. Let R be a polynomial ring over the ring of integers, and let f1 , . . . , fm
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 ( f1 , . . . , fm ; R) is also a nonzerodivisor on the local cohomology module H(nf ,..., fm ) (R).
1

A. K. Singh
Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake City, UT 84112, USA
e-mail: singh@math.utah.edu

1

2

Anurag K. Singh

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 HIkt (Z[X]): a matrix of indeterminates X specializes to a given matrix of that size over an arbitrary commutative
noetherian ring (this is where 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 D-module,
we point the reader towards [Ly1] and [BBLSZ].

Koszul and Čech cohomology
For an element f in a commutative ring R, the Koszul complex K • ( f ; R) has a
natural map to the Čech complex C• ( f ; R) as follows:
f

K • ( f ; R) := 0 −−→ R −−→


y

R −−→ 0

1
yf

C• ( f ; R) := 0 −−→ R −−→ R f −−→ 0.
For a sequence of elements f = f1 , . . . , fm in R, one similarly obtains
K • ( f ; R) :=

N

iK

•( f

i;

R) −−→

N

iC

•( f

i;

R)

=: C• ( f ; R),

and hence, for each n > 0, an induced map on cohomology modules
H n ( f ; R) −−→ H(nf ) (R).

(1)

A polynomial identity via differential operators

3

Now suppose R is a polynomial ring over a field F of characteristic p > 0. The
Frobenius endomorphism ϕ of R induces an additive map
H(nf ) (R) −−→ H(nf p ) (R) = H(nf ) (R),
where f p = f1p , . . . , fmp . Set R{ϕ } to be the extension ring of R obtained by adjoining
the Frobenius operator, i.e., adjoining a generator ϕ subject to the relations ϕ r = r p ϕ
for each r ∈ R; see [Ly2, Section 4]. By an R{ϕ }-module we will mean a left R{ϕ }module. The map displayed above gives H(nf ) (R) an R{ϕ }-module structure. It is not
hard to see that the image of H n ( f ; R) in H(nf ) (R) generates the latter as an R{ϕ }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 ( f ; R) in H(nf ) (R) generates the
latter as a D(R, 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
H 1 ( f ; R) = R/ f R −→ R f /R = H(1f ) (R)
[1] 7−→ [1/ f ] .
By [ABL], the element 1/ f generates R f as a D(R, F)-module. It is of course evie
dent that 1/ f generates R f as an R{ϕ }-module since the elements ϕ e (1/ f ) = 1/ f p
with e > 0 serve as R-module generators for R f . See [BDV] for an algorithm to
e
explicitly construct a differential operator δ with δ (1/ f ) = 1/ f p , 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
1 ∂k
k! ∂ xk
is the A-linear endomorphism of R with
 
1 ∂k m
m m−k
(x ) =
x
k
k! ∂ x
k

for m > 0,


where we use the convention that the binomial coefficient mk vanishes if m < k.
Note that


r+s
1
∂ r+s
1 ∂r 1 ∂s
·
=
.
r! ∂ xr s! ∂ xs
(r + s)! ∂ xr+s
r
For the purposes of this paper, if R is a polynomial ring over A in the indeterminates x1 , . . . , xd , we define the ring of A-linear differential operators on R, de-

4

Anurag K. Singh

noted D(R, A), to be the free R-module with basis
1 ∂ k1
1 ∂ kd
· ··· ·
k
k1 ! ∂ x 1
kd ! ∂ xkd
1

for ki > 0,

d

with the ring structure coming from composition. This is consistent with more general definitions; see [Gr, 16.11]. By a D(R, A)-module, we will mean a left D(R, A)module; the ring R has a natural D(R, A)-module structure, as do localizations
of R. For a sequence of elements f in R, the Čech complex C• ( f ; R) is a complex of D(R, A)-modules, and hence so are its cohomology modules H(nf ) (R). Note
that for m > 1, one has


 
1 ∂k
1
1
k m+k−1
= (−1)
.
m
k
m+k
k! ∂ x x
k
x
We also recall the Leibniz rule, which states that
1 ∂i
1 ∂j
1 ∂k
(
f
g)
=
(
f
)
(g).
∑
i
k! ∂ xk
j! ∂ x j
i+ j=k i! ∂ x

3 The identity
Let R be the ring of polynomials with integer coefficients in the indeterminates


uvw
.
xy z
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 HI3 (R) is a rational vector
space. We label the minors as ∆1 = vz − wy, ∆2 = wx − uz, and ∆3 = uy − vx. Fix a
prime integer p, and consider the exact sequence
p

0 −−→ R −−→ R −−→ R −−→ 0,
where R = R/pR. This induces an exact sequence of local cohomology modules
π

p

−−→ HI2 (R) −−→ HI2 (R) −−→ HI3 (R) −−→ HI3 (R) −−→ HI3 (R) −−→ 0.
The ring R/IR is Cohen-Macaulay of dimension 4, so [PS, Proposition III.4.1] implies that HI3 (R) = 0. As p is arbitrary, it follows that HI3 (R) is a divisible abelian
group. To prove that it is a rational vector space, one needs to show that multiplication by p on HI3 (R) is injective, equivalently that π is surjective. We first prove this
using the identity (2) below, and then proceed with the proof of the identity.
For each k > 0, one has

A polynomial identity via differential operators

∑

i, j>0



k
i+ j

5



k + i k + j (−wx)i (vx) j uk+1
k
k
∆2k+1+i ∆3k+1+ j




k
k + i k + j (−uy)i (wy) j vk+1
+ ∑
k
k
∆3k+1+i ∆1k+1+ j
i, j>0 i + j




k
k + i k + j (−vz)i (uz) j wk+1
= 0. (2)
+ ∑
k
k
∆1k+1+i ∆2k+1+ j
i, j>0 i + j




Since the binomial coefficient i+k j vanishes if i or j exceeds k, this equation may
be rewritten as an identity in the polynomial ring Z[u, v, w, x, y, z] after multiplying
by (∆1 ∆2 ∆3 )2k+1 .
Computing HI2 (R) as the cohomology of the Čech complex C• (∆1 , ∆2 , ∆3 ; R),
equation (2) gives a 2-cocycle in
C2 (∆1 , ∆2 , ∆3 ; R) = R∆1 ∆2 ⊕ R∆1∆3 ⊕ R∆2∆3 ;
we denote the cohomology class of this cocycle in HI2 (R) by ηk . When k = pe − 1,
one has




k
k+i k+ j
≡ 0 mod p for (i, j) 6= (0, 0),
i+ j
k
k
so (2) reduces modulo p to
up
e

e

∆2p ∆3p

e

+

vp
e

e

∆3p ∆1p

e

+

wp
e

e

∆1p ∆2p

e

≡0

mod p,

and the cohomology class η pe −1 has image
!#
"
e
e
e
−v p
up
wp
π (η pe −1 ) =
e
e,
e
e,
e
e
∆1p ∆2p ∆1p ∆3p ∆2p ∆3p

in HI2 (R).

Since R is a regular ring of positive characteristic, HI2 (R) is generated as an R{ϕ }module by the image of
H 2 (∆1 , ∆2 , ∆3 ; R) −−→ HI2 (R).
The Koszul cohomology module H 2 (∆1 , ∆2 , ∆3 ; R) is readily seen to be generated,
as an R-module, by elements corresponding to the relations
u∆1 + v∆2 + w∆3 = 0

and

x∆1 + y∆2 + z∆3 = 0.

These two generators of H 2 (∆1 , ∆2 , ∆3 ; R) map, respectively, to




−v
u
−y
x
w
z
α :=
,
,
and β :=
,
,
∆1 ∆2 ∆1 ∆3 ∆2 ∆3
∆1 ∆2 ∆1 ∆3 ∆2 ∆3

6

Anurag K. Singh

in HI2 (R). Thus, HI2 (R) is generated over R by ϕ e (α ) and ϕ e (β ) for e > 0. But

ϕ e (α ) = π (η pe −1 )
is in the image of π , and hence so is ϕ e (β ) by symmetry. Thus, π is surjective.

The proof of the identity
We start by observing that C2 (∆1 , ∆2 , ∆3 ; R) is a D(R, Z)-module. The element


w
−v
u
,
,
∆1 ∆2 ∆1 ∆3 ∆2 ∆3
is a 2-cocycle in C2 (∆1 , ∆2 , ∆3 ; R) since
w
v
u
+
+
= 0.
∆1 ∆2 ∆1 ∆3 ∆2 ∆3

(3)

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

1 ∂k 1 ∂k 1 ∂k
·
·
k! ∂ uk k! ∂ yk k! ∂ zk

applied termwise to (3); we first explain the choice of this operator: set k = pe − 1,
and consider D = D mod p as an element of
D(R, Z)/pD(R, Z) = D(R/pR, Z/pZ).
It is an elementary verification that
D(u∆2p

e −1

D(v∆3p

∆3p

e −1

e −1

∆1p

) ≡ up

e −1

) ≡ vp

e
e
D(w∆1p −1 ∆2p −1 )

≡ w

e
e

mod p.

pe

pe

Since k < pe , the differential operator D is R -linear; dividing the above equations
e
e
e
e
e
e
by ∆2p ∆3p , ∆3p ∆1p , and ∆1p ∆2p respectively, we obtain
!


e
e
e
−v
u
wp
w
−v p
up
mod p,
D
,
,
≡
e
e,
e
e,
e
e
∆1 ∆2 ∆1 ∆3 ∆2 ∆3
∆1p ∆2p ∆1p ∆3p ∆2p ∆3p
which maps to the desired cohomology class ϕ e (α ) in HI2 (R). Of course, the operator D is not unique in this regard.
Using elementary properties of differential operators recorded in §2, we have

A polynomial identity via differential operators

D



v
∆3 ∆1



7

1 ∂k 1 ∂k 1 ∂k
v
·
·
k! ∂ uk k! ∂ yk k! ∂ zk (uy − vx)(vz − wy)


1 ∂k 1 ∂k
v(−v)k
=
·
k! ∂ uk k! ∂ yk (uy − vx)(vz − wy)k+1


v(−v)k (−y)k
1 ∂k
=
k! ∂ yk (uy − vx)k+1 (vz − wy)k+1


k
yk
k+1 1 ∂
=v
k! ∂ yk (uy − vx)k+1(vz − wy)k+1




∂ k−i− j k
1 ∂i
1
1
1 ∂j
1
= vk+1 ∑
y
i
k+1
j! ∂ y j (vz − wy)k+1 (k − i − j)! ∂ yk−i− j
i, j i! ∂ y (uy − vx)






k+i
(−u)i
k+ j
k
wj
= vk+1 ∑
yi+ j
k+1+i
k+1+ j i + j
i
j
(uy
−
vx)
(vz
−
wy)
i, j




k+i k+ j
k
(−uy)i (wy) j
.
= vk+1 ∑
i
j
i + j ∆ k+1+i ∆ k+1+ j
i, j
1
3


=



A similar calculation shows that






w
k+i k+ j
k
(−vz)i (uz) j
D
.
= wk+1 ∑
∆1 ∆2
i
j
i + j ∆ k+1+i ∆ k+1+ j
i, j
1

2


u
; we reduce this to the previous calculation as
It remains to evaluate D
∆2 ∆3
∂ ∂
∂ ∂
follows. First note that the differential operators
·
and
·
commute; it
∂u ∂y
∂v ∂x
u
. Consequently the operators
is readily checked that they agree on
∆2 ∆3


1 ∂k 1 ∂k 1 ∂k
·
·
k! ∂ uk k! ∂ yk k! ∂ zk
agree on

and

1 ∂k 1 ∂k 1 ∂k
·
·
k! ∂ vk k! ∂ zk k! ∂ xk

u
as well. But then
∆2 ∆3




u
u
1 ∂k 1 ∂k 1 ∂k
·
·
D
=
∆2 ∆3
k! ∂ vk k! ∂ zk k! ∂ xk (wx − uz)(uy − vx)

which, using the previous calculation and symmetry, equals




k+i k+ j
k
(−wx)i (vx) j
uk+1 ∑
.
i
j
i + j ∆ k+1+i ∆ k+1+ j
i, j
2
3

8

Anurag K. Singh

Identities in general
Suppose f = f1 , . . . , fm are elements of a polynomial ring R over Z, and g1 , . . . , gm
are elements of R such that
g1 f1 + · · · + gm fm = 0.
Then, for each prime integer p and e > 0, the Frobenius map on R = R/pR gives
e

e

e

e

g1p f1p + · · · + gmp fmp ≡ 0 mod p.

(4)

Now suppose p is a nonzerodivisor on the Koszul cohomology module H m ( f ; R).
Then Theorem 1 implies that (4) lifts to an equation
G1 f1N + · · · + Gm fmN = 0

(5)

in R in the sense that the cohomology class corresponding to (5) in H(m−1
(R) maps
f)
to the cohomology class corresponding to (4) in H(m−1
f ) (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.

References
ABL.

J. Àlvarez Montaner, M. Blickle, and G. Lyubeznik, Generators of D-modules in characteristic p > 0, Math. Res. Lett. 12 (2005), 459–473.
BBLSZ. B. Bhatt, M. Blickle, G. Lyubeznik, A. K. Singh, and W. Zhang, Local cohomology
modules of a smooth Z-algebra have finitely many associated primes, Invent. Math. 197
(2014), 509–519.
BDV.
A. F. Boix, A. De Stefani, and D. Vanzo, An algorithm for constructing certain differential operators in positive characteristic, Matematiche (Catania) 70 (2015), 239–271.
BH.
W. Bruns and J. Herzog, Cohen-Macaulay rings, revised edition, Cambridge Stud. Adv.
Math. 39, Cambridge Univ. Press, Cambridge, 1998.
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W. Bruns and R. Schwänzl, The number of equations defining a determinantal variety,
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G. Lyubeznik, Finiteness properties of local cohomology modules (an application of Dmodules to commutative algebra), Invent. Math. 113 (1993), 41–55.
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G. Lyubeznik, F-modules: applications to local cohomology and D-modules in characteristic p > 0, J. Reine Angew. Math. 491 (1997), 65–130.
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G. Lyubeznik, A. K. Singh, and U. Walther, Local cohomology modules supported at
determinantal ideals, J. Eur. Math. Soc. 18 (2016), 2545–2578.
PS.
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A polynomial identity via differential operators
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9

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