arXiv:1309.0617v1 [math.AG] 3 Sep 2013

LOCAL COHOMOLOGY WITH SUPPORT IN GENERIC DETERMINANTAL IDEALS
CLAUDIU RAICU AND JERZY WEYMAN
To the memory of Andrei Zelevinsky
Abstract. For positive integers m ≥ n ≥ p, we compute the GLm × GLn –equivariant description of the local
cohomology modules of the polynomial ring S = Sym(Cm ⊗ Cn ) with support in the ideal of p × p minors of
the generic m × n matrix. Our techniques allow us to explicitly compute all the modules Ext•S (S/Ix , S), for x a
partition and Ix the ideal generated by the irreducible sub-representation of S indexed by x. In particular we
determine the regularity of the ideals Ix , and we deduce that the only ones admitting a linear free resolution
are the powers of the ideal of maximal minors of the generic matrix, as well as the products between such
powers and the maximal ideal of S.

1. Introduction
Given positive integers m ≥ n and a field K of characteristic zero, we consider the space Km×n of m × n
matrices, and the ring S of polynomial functions on this space. For each p = 1, · · · , n we define the ideal
Ip ⊂ S generated by the polynomial functions in S that compute the p × p minors of the matrices in Km×n .
The goal of this paper is to describe for each p the local cohomology modules HI•p (S) of S with support in the
ideal Ip . The case p = n was previously analyzed by the authors in joint work with Emily Witt [RWW13].
There is a natural action of the group GLm × GLn on Km×n and hence on S, and this action preserves each
of the ideals Ip . This makes HI•p (S) into GLm × GLn –representations, and our results describe the characters
of these representations explicitly. Our methods also allow us to determine explicitly the characters of all the
modules Ext•S (S/I, S), where I is an ideal of S generated by an irreducible GLm × GLn –subrepresentation
of S, and in particular determine the regularity of such ideals. It is an interesting problem to determine the
minimal free resolutions of such ideals I, which is unfortunately only answered in a small number of cases.
We hope that our results will help shed some light on this problem in the future.
We will adopt a basis-independent notation throughout the paper, writing F (resp. G), for K–vector
spaces of dimension m (resp. n), and thinking of F ∗ ⊗ G∗ as the space Km×n of m × n matrices, and of
S = Sym(F ⊗ G) as the ring of polynomial functions on this space. S is graded by degree, with the space
of linear forms F ⊗ G sitting in degree 1. The Cauchy formula [Wey03, Cor. 2.3.3]
M
Sx F ⊗ Sx G
(1.1)
S=
x=(x1 ≥···≥xn ≥0)

describes the decomposition of S into a sum of irreducible GL(F ) × GL(G)–representations, indexed by
partitions x with at most n parts (Sx denotes the Schur functor associated to x). This decomposition
respects the grading, the term corresponding to x being situated in degree |x| = x1 + · · · + xn . We denote
by Ix the ideal generated by Sx F ⊗ Sx G. If we write (1p ) for the partition x with x1 = · · · = xp = 1, xi = 0
Date: March 14, 2018.
2010 Mathematics Subject Classification. Primary 13D45, 14M12.
Key words and phrases. Local cohomology, determinantal ideals, regularity.
1

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CLAUDIU RAICU AND JERZY WEYMAN

for i > 0, then I(1p ) is just another notation for the ideal Ip of p × p minors. Our first result gives an explicit
formula for the regularity of the ideals Ix :
Theorem on Regularity of Equivariant Ideals (Theorem 5.1). For a partition x with at most n parts,
letting xn+1 = −1 we have the following formula for the regularity of the ideal Ix :
reg(Ix ) = max (n · xp + (p − 2) · (n − p)).
p=1,··· ,n
xp >xp+1

In particular, the only ideals Ix which have a linear resolution are those for which x1 = · · · = xn (i.e. powers
Inx1 of the ideal In of maximal minors) or x1 − 1 = x2 = · · · = xn (i.e. Inx1 −1 · I1 ).
The theorem above is a consequence of the explicit description of the modules Ext•S (S/Ix , S) that we
obtain in Theorem 4.3. This description is somewhat involved, so we avoid stating it for the moment. A
key point is that the modules Ext•S (S/Ix , S) grow as we append new columns to the end of the partition x.
More precisely, we can identify a partition x with its pictorial realization as a Young diagram consisting of
left-justified rows of boxes, with xi boxes in the i-th row: for example, x = (5, 5, 5, 3) corresponds to
,
and adding two columns of size 2 and three columns of size 1 to the end of x yields y = (10, 7, 5, 3).
Theorem on the Growth of Ext Modules (Theorem 4.2). Let d ≥ 0 and consider partitions x, y, where
x consists of the first d columns of y, i.e. xi = min(yi , d) for all i = 1, · · · , n. The natural quotient map
S/Iy ։ S/Ix induces injective maps
ExtiS (S/Ix , S) ֒−→ ExtiS (S/Iy , S),
for all i = 0, 1, · · · , m · n.
We warn the reader that the naive generalization of the statement above fails: if y is a partition containing
x (i.e. yi ≥ xi for all i), then it is not always the case that the induced maps ExtiS (S/Ix , S) → ExtiS (S/Iy , S)
are injective. In fact, a general partition x has the property that most modules ExtiS (S/Ix , S) are non-zero,
but it is always contained in some partition y with y1 = · · · = yn ; for such a y, all but n of the modules
ExtiS (S/Iy , S) will vanish.
We next give the explicit description of Ext•S (S/Ix , S), which requires some piece of notation. We write
R for the representation ring of the group GL(F ) × GL(G). Given a Z–graded S–module M , admitting an
action of GL(F ) × GL(G) compatible with the natural one on S, we define its character χM (z) to be the
element in the Laurent power series ring R((z)) given by
X
χM (z) =
[Mi ] · z i ,
i∈Z

where [Mi ] denotes the class in R of the GL(F ) × GL(G)–representation Mi . We will often work with
doubly-graded modules Mij , where the second grading (in j) is a cohomological one, and M•j 6= 0 only for
finitely many values of j: for us they will be either Ext modules, or local cohomology modules. We define
the character of such M to be the element χM (z, w) ∈ R((z))[w±1 ] given by
X
χM (z, w) =
[Mij ] · z i · wj .
i,j∈Z

LOCAL COHOMOLOGY WITH SUPPORT IN GENERIC DETERMINANTAL IDEALS

3

We will refer to an r–tuple λ = (λ1 , · · · , λr ) ∈ Zr (for r = m or n) as a weight. We say that λ is dominant
if λ1 ≥ λ2 ≥ · · · ≥ λr , and denote by Zrdom the set of dominant weights. Note that a partition is just a
dominant weight with non-negative entries. We will usually use the notation x, y, z etc. to refer to partitions
indexing the subrepresentations of S, and λ, µ etc. to denote the weights describing the characters of other
equivariant modules (Ext modules or local cohomology modules).
For λ ∈ Zndom and 0 ≤ s ≤ n, we define (note that in [RWW13] this was called λ(n − s))
λ(s) = (λ1 , · · · , λs , s − n, · · · , s − n, λs+1 + (m − n), · · · , λn + (m − n)) ∈ Zm .
{z
}
|

(1.2)

m−n

Theorem on the Characters of Ext Modules (Theorem 4.3). With the above notation, the character
of the doubly–graded module Ext•S (S/Ix , S) is given by
Pn−p
X
2
χExt•S (S/Ix ,S) (z, w) =
[Sλ(s) F ⊗ Sλ G] · z |λ| · wm·n+1−p −s·(m−n)−2·( j=1 tj ) ,
1≤p≤n
0≤s≤t1 ≤···≤tn−p ≤p−1
λ∈W ′ (x,p;t,s)

where W ′ (x, p; t, s) is the set of dominant weights λ ∈ Zn satisfying


λn ≥ p − xp − m,
λtj +j ≤ tj − xn+1−j − m, for j = 1, · · · , n − p,


λs ≥ s − n and λs+1 ≤ s − m.

Our proof of this theorem starts with the observation in [dCEP80] that even though the algebraic set
defined by Ix is somewhat simple (it is the set of matrices of rank smaller than the number of non-zero parts
of x), its scheme theoretic structure is more complicated: it is generally non-reduced, and has embedded
components supported on Ip for each size p of some column of x. Our approach is then to filter S/Ix with
subquotients Jz,p (defined in Section 2.2) whose scheme theoretic support is the (reduced) space of matrices
of rank at most p, hence they are less singular and easier to resolve. In fact, each Jz,p is the push-forward of
a locally free sheaf on some product of flag varieties, which allows us to compute Ext•S (Jz,p , S) via duality
theory. Solving the extension problem to deduce the formulas for Ext•S (S/Ix , S) turns out to be then trivial,
due to the restrictions imposed by the equivariant structure of the modules.
We end this introduction with our main theorem on local cohomology modules, whose statement needs
some more notation. For 0 ≤ s ≤ n, we define (with the convention λ0 = ∞, λn+1 = −∞)
X
hs (z) =
[Sλ(s) F ⊗ Sλ G] · z |λ| ,
(1.3)
λ∈Zn
dom
λs ≥s−n
λs+1 ≤s−m

so that hn (z) is just the character of S. The other hs (z)’s are characters of local cohomology modules (in the
case when m > n). More precisely, for p = 1, · · · , n we write Hp (z, w) for the character of the doubly-graded
module HI•p (S). In [RWW13] we proved that for m > n
Hn (z, w) =

n−1
X

hs (z) · w(n−s)·(m−n)+1 ,

s=0

and it is easy to see that the same formula holds for m = n (in this case, the only non-zero local cohomology
module is HI1n (S) = Sdet /S, where det denotes the determinant of the generic n × n matrix, and Sdet is the
localization of S at det).

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CLAUDIU RAICU AND JERZY WEYMAN

We write p(a, b; c) for the number of partitions of c contained in an a × b rectangle, and define the Gauss
polynomial a+b
b (w) to be the generating function for the sequence p(a, b; c)c≥0 :


X
X
a+b
(w) =
p(a, b; c) · wc =
wt1 +···+ta .
(1.4)
a
c≥0

b≥t1 ≥t2 ≥···≥ta ≥0

Gauss polynomials have previously appeared in [AW07] in connection to the closely related problem of
understanding the minimal free resolutions of the ideals I(pd ) .
Theorem on Local Cohomology with Support in Generic Determinantal Ideals (Theorem 6.1).
With the above notation, we have for each p = 1, · · · , n,


p−1
X
n−s−1
(n−p+1)2 +(n−s)·(m−n)
hs (z) · w
·
(w2 ).
Hp (z, w) =
p−s−1
s=0

The theorem implies that the maximal cohomological index for which HI•p (S) is non-zero (the cohomological
dimension of the ideal Ip ) is obtained for s = 0 and is equal to
(n − p + 1)2 + n · (m − n) + (p − 1) · (n − p) = m · n − p2 + 1.

This was first observed in [BS90]. Our result says that many of the local cohomology modules HIjp (S) are
non-zero, for j between the depth (m−p+1)·(n−p+1) of the ideal Ip and its cohomological dimension. This
contrasts with the positive characteristic situation where the only non-vanishing local cohomology module
appears in degree j = (m − p + 1) · (n − p + 1) (see [HE71, Cor. 4] or [BV88, Cor. 5.18] where it is shown that
Ip is perfect, and [PS73, Prop. 4.1] where a local cohomology vanishing result for perfect ideals in positive
characteristic is proved). For determinantal ideals over arbitrary rings one can’t expect such explicit results
as Theorem 6.1; for the latest advances in this general context, the reader should consult [LSW13] and the
references therein.
Our paper is organized as follows. In Section 2 we give some representation-theoretic preliminaries: in
Section 2.1 we fix some notation for Schur functors, weights and partitions; in Section 2.2 we recall from
[dCEP80] some properties of the ideals Ix , and introduce certain associated subquotients Jx,p that will play
an important role in the sequel; in Section 2.3 we recall the definition of flag varieties and formulate some
consequences of Bott’s theorem in a form that will be useful to us; we also recall in Section 2.4 a method
described in [RWW13] for computing extension groups for certain modules that arise as push-forwards of
vector bundles with vanishing higher cohomology. In Section 3 we compute explicitly the characters of
the modules Ext•S (Jx,p , S), and in Section 4 we use this calculation to deduce the main result about the
characters of the modules Ext•S (S/Ix , S) for all partitions x. In Section 5 we derive the formulas for the
regularity of the ideals Ix , while in Section 6 we describe the characters of the local cohomology modules
with support in determinantal varieties.
2. Preliminaries
2.1. Representation Theory [FH91], [Wey03, Ch. 2]. Throughout the paper, K will denote a field of
characteristic 0. If W is a K–vector space of dimension dim(W ) = N , a choice of basis determines an
isomorphism between GL(W ) and GLN (K). We will refer to N –tuples λ = (λ1 , · · · , λN ) ∈ ZN as weights
of the corresponding maximal torus of diagonal matrices. We say that λ is a dominant weight if λ1 ≥
λ2 ≥ · · · ≥ λN . Irreducible representations of GL(W ) are in one-to-one correspondence with dominant
weights λ. We denote by Sλ W the irreducible representation associated to λ, often referred to as a Schur
functor. We write (aN ) for the weight with all parts equal to a, and define the determinant of W by

LOCAL COHOMOLOGY WITH SUPPORT IN GENERIC DETERMINANTAL IDEALS

5

V
det(W ) = S(1N ) W = N W . We have Sλ W ⊗ det(W ) = Sλ+(1N ) W , and Sλ W ∗ = S(−λN ,··· ,−λ1 ) W . We write
|λ| for the total size λ1 + · · · + λN of λ.
When x is a dominant weight with xN ≥ 0, we say that x is a partition of r = |x|. Note that when we’re
dealing with partitions we often omit the trailing zeros, so x = (5, 2, 1) is the same as x = (5, 2, 1, 0, 0, 0). If
y is another partition, we write x ⊂ y to indicate that xi ≤ yi for all i.
2.2. The ideals Ix and the subquotients Jx,p . Recall the Cauchy formula (1.1) and the definition of the
ideals Ix ⊂ S = Sym(F ⊗ G) as the ideals generated by subrepresentations Sx F ⊗ Sx G of S. It is shown in
[dCEP80] that
M
Ix =
Sy F ⊗ Sy G,
(2.1)
x⊂y

and in particular Iy ⊂ Ix if and only if x ⊂ y. More generally, for arbitrary partitions x, y, we let z =
max(x, y) be defined by zi = max(xi , yi ) for all i, and get
Ix ∩ Iy = Iz .

(2.2)

Even more generally, for any set T of partitions we let
X
IT =
Iy ,

(2.3)

y∈T

and have
Ix ∩ IT =

X

Imax(x,y) .

(2.4)

y∈T

For p ∈ {0, 1, · · · , n} and x a partition, we write

Succ(x, p) = {y : x ⊂ y, and yi > xi for some i > p}.

(2.5)

By the discussion above, Iy ⊂ Ix for all y ∈ Succ(x, p). We define
Jx,p = Ix /ISucc(x,p)
It follows from (2.1) that
Jx,p =

M

Sy F ⊗ Sy G.

(2.6)
(2.7)

x⊂y
yi =xi for all i>p

If p = n then Jx,p = Ix , while if p = 0 then Jx,p = Sx F ⊗ Sx G is just a vector space (it is annihilated by the
maximal ideal of S). We have
Lemma 2.1. Fix an index p ∈ {0, 1, · · · , n − 1}, and consider a partition x with x1 = · · · = xp+1 . Let
Z = {z : z1 = · · · = zp+1 = x1 }.
We have



ISucc(x,p) = 

X

z∈Z, x(z



Iz  + Imax(x,(x1 +1)p+1 ) .

(2.8)

(2.9)

Proof. “⊃”: Consider z ∈ Z, x ( z. We have zi > xi for some i, and since xi = zi for i ≤ p + 1, we
conclude that zi > xi for some i > p + 1, thus z ∈ Succ(x, p). Writing y = max(x, (x1 + 1)p+1 ) we have that
yp+1 > xp+1 and y ⊃ x, so y ∈ Succ(x, p), proving that the RHS of (2.9) is contained in the LHS.
“⊂”: Consider a partition y ∈ Succ(x, p). If yp+1 > xp+1 = x1 then y ⊃ max(x, (x1 + 1)p+1 ), so Iy is
contained in the RHS of (2.9). Otherwise yp+1 = xp+1 , so by possibly shrinking some of the first p rows of

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CLAUDIU RAICU AND JERZY WEYMAN

y (which would enlarge Iy ), we may assume that y ∈ Z. Clearly y ) x, since yi > xi for some i > p + 1, so

it follows again that Iy is contained in the RHS of (2.9).
The following result will be used in Section 4:
Lemma 2.2. Fix an index p ∈ {0, 1, · · · , n − 1}, and consider a partition x with x1 = · · · = xp+1 . For a
non-negative integer d ≥ 0, let y be the partition defined by yi = xi + d + 1 for i = 1, · · · , p + 1, and yi = xi
for i > p + 1 (y = max(x, (x1 + d + 1)p+1 )). The quotient Ix /Iy admits a filtration with successive quotients
Jz,p , where z runs over all partitions with
(
x1 ≤ z1 = · · · = zp+1 ≤ x1 + d,
zi ≥ xi , for i > p + 1.
Proof. By induction, it suffices to prove the result when d = 0. We consider Z as in (2.8) and define
I(Z) = {IT : T ⊂ Z}.
For I ∈ I(Z), we write
Z(I) = {z ∈ Z : Iz ⊂ I}.
Note that if

z0

∈ Z(I) then

(2.10)
if z ∈ Z and z 0 ⊂ z then z ∈ Z(I).
We let I0 = I((x1 +1)p+1 ) and prove by induction on |Z(I)| that for I ∈ I(Z), the quotient (I + I0 )/I0 has
a filtration with successive quotients Jz,p , where z varies over the set of elements of Z(I). Once we do this,
we can take I = Ix and observe that Ix ∩ I0 = Iy (by (2.2)), which yields
(I + I0 )/I0 ≃ I/(I ∩ I0 ) = Ix /Iy ,
concluding the proof of the lemma.
For the induction, assume first that |Z(I)| = 1, so that I = Iz with z1 = · · · = zn = x1 . We have
(Iz + I0 )/I0 = Jz,p so the base case for the induction follows.
Suppose now that |Z(I)| > 1 and consider a maximal element z 0 in Z(I), i.e. a partition z 0 with the
property that Iz0 6⊂ Iz for any z ∈ Z(I) \ {z 0 }. Define
I ′ = IZ(I)\{z 0 } ,
and note that |Z(I ′ )| = |Z(I)| − 1, I = I ′ + Iz 0 , and
(I + I0 )/(I ′ + I0 ) ≃ Jz 0 ,p ,

(2.11)

which is proved as follows. The equality I = I ′ + Iz 0 implies that the natural map
Iz 0 → (I + I0 )/(I ′ + I0 )
is surjective. Its kernel is


(2.4)
Iz 0 ∩ (I ′ + I0 ) = 

X

z∈Z(I)\{z 0 }

(2.10)



= 

X

z∈Z, z 0 (z



Imax(z 0 ,z)  + Imax(z 0 ,(x1 +1)p+1 )


(2.9)
Iz  + Imax(z 0 ,(x1 +1)p+1 ) = ISucc(z 0 ,p),

from which (2.11) follows. Since by induction (I ′ + I0 )/I0 has a filtration with successive quotients Jz,p for

z ∈ Z(I ′ ), we get the corresponding statement for (I + I0 )/I0 , finishing the induction step.

LOCAL COHOMOLOGY WITH SUPPORT IN GENERIC DETERMINANTAL IDEALS

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2.3. Partial flag varieties and Bott’s Theorem [Wey03, Ch. 4]. Consider a K–vector space V with
dim(V ) = d, and positive integers q ≤ n ≤ d. We denote by F lag([q, n]; V ) the variety of partial flags
V• :

V ։ Vn ։ Vn−1 · · · ։ Vq ։ 0,

where Vp is a p–dimensional quotient of V for each p = q, q + 1, · · · , n. For p ∈ [q, n] we write Qp (V ) for
the tautological rank p quotient bundle on F lag([q, n]; V ) whose fiber over a point V• ∈ F lag([q, n]; V ) is
Vp . For each p there is a natural surjection of vector bundles
V ⊗ OF lag([q,n];V ) ։ Qp (V ).

(2.12)

Note that for q = n, F lag([q, n]; V ) = G(n, V ) is the Grassmannian of n–dimensional quotients of V .
We consider the natural projection maps
π (q) : F lag([q, n]; V ) → F lag([q + 1, n]; V ),

(2.13)

defined by forgetting Vq from the flag V• . For q ≤ n−1, this map identifies F lag([q, n]; V ) with the projective
bundle PF lag([q+1,n];V ) (Qp+1 (V )), which comes with a tautological surjection
Qp+1 (V ) ։ Qp (V ).
(q)

For q = n we make the convention F lag([q + 1, n]; V ) = Spec(K). With the usual notation R• π∗ for derived
push-forward, we have the following consequence of Bott’s theorem:
Theorem 2.3. (a) Suppose that q ≤ n − 1 and consider a dominant weight µ ∈ Zq . For q < p ≤ n
(
Sµ Qp (V ) if j = 0,
(q)
Rj π∗ (Sµ Qp (V )) =
0
otherwise.
If µq−t + t = −1 for some t = 0, · · · , q − 1, then
(q)

Rj π∗ (Sµ Qq (V )) = 0 for all j.
Otherwise (with the convention µ0 = ∞, µq+1 = −∞), there exists a unique index 0 ≤ t ≤ q such that
µq−t+1 + t + 1 ≤ 0 ≤ µq−t + t.
Letting
µ̃ = (µ1 , · · · , µq−t , −t, µq−t+1 + 1, · · · , µq + 1),
we have
(q)
Rj π∗ (Sµ Qq (V

(
Sµ̃ Qq+1 (V )
)) =
0

if j = t,
otherwise.

(b) Consider a dominant weight µ ∈ Zn . If n − d ≤ µn−s + s ≤ −1 for some s = 0, · · · , n − 1 then
(n)

Rj π∗ (Sµ Qn (V )) = 0 for all j.
Otherwise, (with the convention µ0 = ∞, µn+1 = −∞), there exists a unique index 0 ≤ s ≤ n such that
µn−s ≥ −s and µn−s+1 ≤ −s − d + n.
Letting
µ̃ = (µ1 , · · · , µn−s , −s, · · · , −s, µn−s+1 + (d − n), · · · , µn + (d − n)) ∈ Zd ,
|
{z
}
d−n

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CLAUDIU RAICU AND JERZY WEYMAN

(compare to (1.2)) we have
(
Sµ̃ V
)) =
0

(n)
Rj π∗ (Sµ Qn (V

if j = s · (d − n),
otherwise.

2.4. Computing Ext modules via duality. In this section we recall [RWW13, Thm. 3.1] as a tool to
compute Ext•S (M, S) when M comes as the push-forward of certain vector bundles with vanishing higher
cohomology. More precisely, we have
Theorem 2.4. Let X be a projective variety, and let W be a finite dimensional K–vector space. Suppose
W ⊗ OX ։ η,
is a surjective map, where η is locally free, and let k = dim(W ) − rank(η). Consider a locally free sheaf V
on X and define
M(V) = V ⊗ Sym(η),

M∗ (V) = V ⊗ det(W ) ⊗ det(η ∗ ) ⊗ Sym(η ∗ ).

Giving V internal degree v, and η and W degree 1, we think of M(V) and M∗ (V) as graded sheaves, with
M(V)i+v = V ⊗ Symi (η), M∗ (V)i+v = V ⊗ det(W ) ⊗ det(η ∗ ) ⊗ Sym−i+k (η ∗ ).
Suppose that H j (X, M(V)) = 0 for j > 0, and let
M (V) = H 0 (X, M(V)).
We have for each j ≥ 0 a graded isomorphism
ExtjS (M (V), S) = H k−j (X, M∗ (V))∗ ,

(2.14)

where (−)∗ stands for the graded dual.
3. Ext modules for the subquotients Jx,p
The goal of this section is to compute explicitly the character of Ext•S (Jx,p , S) for all p and all partitions
x with x1 = · · · = xp , where Jx,p is defined as in (2.6). We will achieve this by realizing Jx,p as the global
sections of a vector bundle with vanishing higher cohomology on a certain product of flag varieties, and then
using the duality Theorem 2.4 and Bott’s Theorem 2.3.
Consider as before vector spaces F, G, with dim(F ) = m, dim(G) = n, m ≥ n. For q = 1, · · · , n, we
consider the projective varieties
X (q) = F lag([q, n]; F ) × F lag([q, n]; G), X = X (∞) = Spec K,
and the locally free sheaves (see Section 2.3)
η (p) = Qp (F ) ⊗ Qp (G), p = 1, · · · , n, η = η (∞) = F ⊗ G.
Note that η (p) can be thought of as a sheaf on X (q) whenever p ≥ q. We consider for q ≤ n − 1 (resp. q = n)
the natural maps π (q) : X (q) → X (q+1) (resp. π (n) : X (n) → X) induced from (2.13). We define
S (q) = Sym η (q)
as relative versions of the polynomial ring S = S (∞) = Sym(F ⊗ G). We will always work implicitly with
quasi-coherent sheaves on the affine bundles
Y (q) = AX (q) (η (q) ) = SpecX (q) (S (q) ),

LOCAL COHOMOLOGY WITH SUPPORT IN GENERIC DETERMINANTAL IDEALS

9

which we identify with S (q) –modules on X (q) as in [Har77, Ex. II.5.17]. The Cauchy formula (1.1) becomes
in the relative setting
M
Sx Qq (F ) ⊗ Sx Qq (G),
(3.1)
S (q) =
x=(x1 ≥···≥xq ≥0)

(q)
Ix

(q)

and we can define the ideals
⊂ S (q) and subquotients Jx,p for 0 ≤ p ≤ q, analogously to (2.1) and (2.6).
(q)
(q)
For 1 ≤ p ≤ q, we write Ip for I(1p ) , the ideal of p × p minors in S (q) . We define the line bundle
det(q) = det(Qq (F )) ⊗ det(Qq (G)),

(3.2)

(q)

and note that the ideal Iq is generated by det(q) . It follows easily from (3.1) and Theorem 2.3 that
(
(
(q+1)
S (p) if j = 0,
S (q+1) /Iq+1
if j = 0,
(p)
(q)
j (q)
j (q)
and for p > q, R π∗ (S ) =
R π∗ (S ) =
(3.3)
0
otherwise.
0
otherwise,
Lemma 3.1. (a) For a partition x = (x1 ≥ · · · ≥ xq ), there exist natural identifications
(q)

det(q) ⊗ Ix(q) = Ix+(1q ) , and

(3.4)

(q)

(q)
= Jx+(1q ),p , for 0 ≤ p ≤ q.
det(q) ⊗ Jx,p

(b) For a partition x = (x1 ≥ · · · ≥ xq ) we have
(
(q+1)
(q+1)
(q+1)
(Ix
+ Iq+1 )/Iq+1
j (q) (q)
R π ∗ Ix =
0

if j = 0,
otherwise.

(c) For a partition x = (x1 ≥ · · · ≥ xq ) and 0 ≤ p ≤ q we have
(
(q+1)
if j = 0,
Jx,p
j (q) (q)
R π∗ Jx,p =
0
otherwise.

(3.5)

(3.6)

(3.7)

Proof. (a) The multiplication map det(q) ⊗ S (q) → S (q) is injective: if we think of S (q) as locally the ring of
polynomial functions on q × q matrices, then det(q) is the determinant of the generic q × q matrix. It follows
(q)
(q)
that det(q) ⊗ Ix = det(q) · Ix is in fact an ideal in S (q) . (3.4) then follows from the fact that multiplying
by the determinant corresponds to adding a column of maximal size to the Young diagram (a special case
of Pieri’s rule). In fact, the same argument shows that for any set of partitions Z


X (q)
X
det(q) ⊗ 
Iz(q)  =
Iz+(1q ) .
z∈Z

z∈Z

(q)
Jx,p

as the analogue of (2.6), (3.5) follows by taking Z = Succ(q) (x, p) (the analogue
Given the definition of
of (2.5)) in the formula above, and using (3.4) and the exactness of tensoring with det(q) .
Part (b) follows from (3.3), while (c) follows from the fact that if x = (x1 , · · · , xq ) and 0 ≤ p ≤ q then
[

Succ(q+1) (x, p) = Succ(q) (x, p) {z : z ⊃ x, zp+1 ≥ 1}.
For each partition x = (x1 = · · · = xp ≥ xp+1 · · · ≥ xn ≥ xn+1 = 0), we define the locally free sheaf Mx,p
on X (p) by
!
n
O
(3.8)
(det(q) )⊗(xq −xq+1 ) ⊗ S (p) .
Mx,p =
q=p

10

CLAUDIU RAICU AND JERZY WEYMAN

Lemma 3.2. With the notation above, we have
j

H (X

(p)

(
Jx,p
, Mx,p ) =
0

if j = 0,
otherwise.

(p)

Proof. Note that S (p) = J0,p , so using (3.4) we get


n
O
(p)
Mx,p = 
(det(q) )⊗(xq −xq+1 )  ⊗ J((xp −xp+1 )p ),p .
q=p+1

It follows that



(3.7)
(p)
(p)
Rπ∗ Mx,p = π∗ Mx,p = 
(3.4)

(p+1)

Applying Rπ∗

(p+2)

, Rπ∗

(n)

, · · · , Rπ∗



= 

n
O

(p+1)



(p+1)

(det(q) )⊗(xq −xq+1 )  ⊗ J((xp −xp+1 )p ),p

q=p+1
n
O



(det(q) )⊗(xq −xq+1 )  ⊗ J((xp −xp+2 )p ,xp+1 −xp+2 ),p .

q=p+2

iteratively, and using (3.7) and (3.4) as above, we obtain

Rπ∗ Mx,p = π∗ Mx,p = J(xpp ,xp+1 ,··· ,xn ),p

(x1 =···=xp )

=

Jx,p ,

where π = π (n) ◦ · · · ◦ π (p) is the structure map X (p) → Spec K, concluding the proof of the lemma.



We are now ready to prove the main result of this section:
Theorem 3.3. The character of the doubly–graded module Ext•S (Jx,p , S) is given by
Pn−p
X
2
χExt•S (Jx,p ,S)(z, w) =
[Sλ(s) F ⊗ Sλ G] · z |λ| · wm·n−p −s·(m−n)−2·( j=1 tj ) ,

(3.9)

0≤s≤t1 ≤···≤tn−p ≤p
λ∈W (x,p;t,s)

where W (x, p; t, s) is the set of dominant weights λ ∈ Zn with the properties

 λn ≥ p − xp − m,
λt +j = tj − xn+1−j − m, for j = 1, · · · , n − p,
 j
λs ≥ s − n and λs+1 ≤ s − m.

(3.10a)
(3.10b)
(3.10c)

Remark 3.4. If we take p = n and x1 = · · · = xn = d in the above theorem, we recover [RWW13, Thm. 4.3].
The character of Jx,n = Ix = Ind is
X
[Sλ(s) F ⊗ Sλ G] · z |λ| · w(n−s)·(m−n) .
χExt•S (Jx,n ,S) (z, w) =
0≤s≤n
λn ≥n−d−m
λs ≥s−n
λs+1 ≤s−m

When p = 0, since Jx,0 = Sx F ⊗ Sx G is just a vector space, the only non-vanishing Ext-module is
∗
Extmn
S (Jx,0 , S) = Sx F ⊗ Sx G ⊗ det(F ⊗ G) .

LOCAL COHOMOLOGY WITH SUPPORT IN GENERIC DETERMINANTAL IDEALS

11

Proof of Theorem 3.3. We apply Theorem 2.4 with
X=X

(p)

, η=η

(p)

, W = F ⊗ G, V =

n
O

(det(q) )⊗(xq −xq+1 ) ,

q=p

so that M(V) = Mx,p (see (3.8)). Lemma 3.2 insures that the hypotheses of the duality theorem hold, and
M (V) = Jx,p . We have rank(η (p) ) = p2 , dim(W ) = m · n, so k = m · n − p2 . We give V internal degree
v = |x| and get
!∗
n
O
2
2
j
(det(q) )⊗(xq −xq+1 ) ⊗ det(F ⊗ G) ⊗ det(η ∗ ) ⊗ Symr+m·n−p (η ∗ ) .
ExtS (Jx,p , S)r−|x| = H m·n−p −j X (p) ,
q=p

(3.11)

Formula (3.9) now follows from a direct application of Theorem 2.3, which we sketch below.
Using Cauchy’s formula and the fact that det(η ∗ ) = det(Qp (F ))−p ⊗ det(Qp (G))−p we get
M
2
det(η ∗ ) ⊗ Symr+m·n−p (η ∗ ) =
Sµ Qp (F ) ⊗ Sµ Qp (G).
µ∈Zpdom
|µ|=r+m·n
µ1 ≤−p

For each µ in the formula above, we have to first compute
Rπ∗

!
n
O
(q) ⊗(xq −xq+1 )
(det )
⊗ Sµ Qp (F ) ⊗ Sµ Qp (G) ,

(3.12)

q=p

π (n)

π (p)

X (p)

where π =
◦ ··· ◦
:
→ Spec K is the structure map, then tensor with det(F ⊗ G) and dualize,
in order to get the corresponding contribution to (3.11). If (3.12) is non-zero, then there exists uniquely
determined dominant weights µ(q) , δ(q) ∈ Zq for q = p, · · · , n, and non-negative integers tn−q , q = p, · · · , n−1,
and s, such that µ(p) = µ, and if we write
M(q) = Sµ(q) Qq (F ) ⊗ Sµ(q) Qq (G),

N (q) = Sδ(q) Qq (F ) ⊗ Sδ(q) Qq (G), for q = p, · · · , n,

then
N (q) = M(q) ⊗ (det(q) )⊗(xq −xq+1 ) , for q = p, · · · , n,
2 · tn−q is the unique integer j for which

(q)
Rj π∗ (N (q) )

(3.13)

6= 0, and

(q)

R2·tn−q π∗ (N (q) ) = M(q+1) for q = p, · · · , n − 1,
(n)
Rj π∗ (N (n) )

and finally, s · (m − n) is the unique integer j for which
δ(q) is easy to determine, namely we get from (3.13) that

(3.14)

6= 0.

δ(q) = µ(q) + ((xq − xq+1 )q ).

(3.15)

Assuming we know δ(q) , (3.14) determines tn−q and µ(q) according to Theorem 2.3(a): tn−q is the unique
integer t with the property
(q)
(q)
(3.16)
δq−t+1 + t + 1 ≤ 0 ≤ δq−t + t
and
(q)
(q)
(q)
(3.17)
µ(q+1) = (δ1 , · · · , δq−tn−q , −tn−q , δq−tn−q +1 + 1, · · · , δq(q) + 1).
It follows from (3.17) and (3.15) that
(q+1)

δq+1−tn−q = −tn−q + xq+1 − xq+2 ≥ −tn−q ,

12

CLAUDIU RAICU AND JERZY WEYMAN

so t = tn−q satisfies the RHS inequality in (3.16) with q replaced by (q + 1), which forces tn−(q+1) ≤ tn−q .
It follows easily that
(i)
δq+1−tn−q = −tn−q + xq+1 − xi+1 , for i = q + 1, · · · , n.
(3.18)
We have so far seen how to calculate µ(q) , δ(q) for q = p, · · · , n, and tn−q for q = p, · · · , n − 1, so we’re left
with determining s. By Theorem 2.3(b), s is uniquely determined by the inequalities
(n)

(n)

δn−s ≥ −s and δn−s+1 ≤ −s − m + n,
and moreover
where δ =

δ(n)

(n)

Rs·(m−n) π∗ (N (n) ) = Sδ̃ F ⊗ Sδ G,

(3.19)
(3.20)

and
δ̃ = (δ1 , · · · , δn−s , (−s)m−n , δn−s+1 + (m − n), · · · , δn + (m − n)).

(n)

Since δn−t1 = −t1 + xn ≥ −t1 , it follows as before that s ≤ t1 . Tensoring (3.20) with det(F ⊗ G) =
det(F )⊗n ⊗ det(G)⊗m and dualizing, we obtain by putting everything together that there exist integers
0 ≤ s ≤ t1 ≤ · · · ≤ tn−p ≤ p such that
!∗
n
P
O
s·(m−n)+2· n−p
t
(q)
⊗(x
−x
)
j
q
q+1
j=1
(det )
⊗ det(F ⊗ G) ⊗ Sµ Qp (F ) ⊗ Sµ Qp (G)
= Sλ(s) F ⊗ Sλ G,
R
π∗
q=p

where λ(s) is defined as in (1.2) and

λi = −m − δn−i+1 for all i = 1, · · · , n.

(3.21)

We next check that λ ∈ W (x, p; t, s). Since µ1 ≤ −p, it follows from (3.15) and (3.17) that δ1 ≤ −p + xp , so
λn = −δ1 − m ≥ p − xp − m, i.e. (3.10a) holds. Letting i = n in (3.18) we get δq+1−tn−q = −tn−q + xq+1 , so
λn−q+tn−q = tn−q − xq+1 − m, i.e. (3.10b) holds. Finally, (3.10c) follows from (3.19).
We conclude from the discussion above that (3.9) holds, after possibly replacing W (x, p; t, s) by a smaller
set. To see that all weights λ ∈ W (x, p; t, s) in fact appear, one has to reverse the steps above in order to
show that each λ can be reached from a certain weight µ. We give the formula for µ, and leave the details to
the interested reader. We first define δ ∈ Zndom by reversing (3.21), δi = −m − λn+1−i . Letting tn−p+1 = p
and t0 = 0, we let for each i = 0, · · · , n − p, and j = 1, · · · , ti+1 − ti ,
µp−ti+1 +j = δn−i−ti+1 +j + xn−i + (n − p − i).



Corollary 3.5. Fix an index p ∈ {0, 1, · · · , n}. Suppose that M is an S–module with a compatible GL(F ) ×
GL(G) action, admitting a finite filtration with successive quotients isomorphic to Jxj ,p , for j = 1, · · · , r,
where each xj is a partition satisfying xj1 = xj2 = · · · = xjp . We have a decomposition as GL(F ) × GL(G)–
representations
r
M
ExtiS (Jxj ,p , S),
ExtiS (M, S) =
(3.22)
j=1

for each i = 0, 1, · · · , m · n. Equivalently, if

0 −→ A −→ B −→ C −→ 0

(3.23)

is a GL(F ) × GL(G)–equivariant short exact sequence of S–modules admitting filtrations as above, then for
each i = 0, 1, · · · , m · n, the induced sequences
0 −→ ExtiS (C, S) −→ ExtiS (B, S) −→ ExtiS (A, S) −→ 0
are exact.

(3.24)

LOCAL COHOMOLOGY WITH SUPPORT IN GENERIC DETERMINANTAL IDEALS

13

Proof. Suppose that the conclusion of the Corollary fails, and consider modules A, B, C sitting in an exact
sequence (3.23) such that (3.22) holds for A and C, but fails for B. In particular, not all sequences (3.24)
are exact, so there exists an index i and a non-trivial connecting homomorphism
δ

ExtiS (A, S) −→ Exti+1
S (C, S).
It follows that some irreducible representation of GL(F )×GL(G) appears in both ExtiS (A, S) and Exti+1
S (C, S).
This is clearly impossible when m = n, because from (3.22) and (3.9) it follows that the cohomological degrees
j for which ExtjS (A, S) (resp. ExtjS (C, S)) are non-zero satisfy
j ≡ m · n − p2 (mod 2).
When m > n, a similar argument applies: if [Sλ(s) F ⊗ Sλ G] = [Sµ(t) F ⊗ Sµ G] (with λ, µ, λ(s) and µ(t)
dominant) then it follows from (1.2) that λ = µ and s = t; moreover, we get from (3.22) and (3.9) that the
cohomological degrees j for which Sλ(s) F ⊗ Sλ G appears in ExtjS (A, S) (resp. ExtjS (C, S)) satisfy
j ≡ m · n − p2 − s · (m − n) (mod 2).



4. Ext modules for S/Ix
In this section we will use the explicit calculation of Ext•S (Jx,p , S) from the previous section in order
to deduce a formula for the characters of Ext•S (S/Ix , S) for all ideals Ix . We begin with an important
consequence of the results in the preceding section:
Corollary 4.1. Fix an index p ∈ {0, 1, · · · , n − 1}, and positive integers b > c > 0. If we let x = (cp+1 ),
y = (bp+1 ), then the natural quotient map S/Iy ։ S/Ix induces injective maps
ExtiS (S/Ix , S) ֒−→ ExtiS (S/Iy , S),
for all i = 0, 1, · · · , m · n. More generally, if z is any partition with z1 = · · · = zp+1 and x = z + (cp+1 ),
y = z + (bp+1 ), then the quotient map Iz /Iy ։ Iz /Ix induces injective maps
ExtiS (Iz /Ix , S) ֒−→ ExtiS (Iz /Iy , S),
for all i = 0, 1, · · · , m · n.
Proof. Note that the former statement follows from the latter by taking z = 0 and noting that S = I0 .
By Lemma 2.2, the modules A = Ix /Iy , B = Iz /Iy , and C = Iz /Ix have finite filtrations with quotients
isomorphic to Jt,p for various partitions t. Corollary 3.5 applies to yield the desired conclusion.

Theorem 4.2. Let d ≥ 0 and consider partitions x, y, where x consists of the first d columns of y, i.e.
xi = min(yi , d) for all i = 1, · · · , n. The natural quotient map S/Iy ։ S/Ix induces injective maps
ExtiS (S/Ix , S) ֒−→ ExtiS (S/Iy , S),

(4.1)

for all i = 0, 1, · · · , m · n.
Proof. Arguing inductively, it suffices to prove the result when all the columns of y outside x have the same
size (say p + 1, for p ∈ {0, 1, · · · , n − 1}), i.e. y = x + (ap+1 ) for some positive integer a. Note that this
forces x1 = x2 = · · · = xp+1 = d. We prove by descending induction on p that the induced map (4.1) is
injective. When p = n − 1, x = (dn ) and y = ((d + a)n ), so the conclusion follows from Corollary 4.1 (or
from the results in [RWW13]).
Suppose now that p < n − 1 and y = x + (ap+1 ), x1 = · · · = xp+1 = d. Let z be the partition consisting
of the columns of x of size strictly larger than p + 1, i.e. zi = min(xi , xp+2 ) for all i = 1, · · · , n. Consider

14

CLAUDIU RAICU AND JERZY WEYMAN

the partitions x̃ (resp. ỹ), defined by letting x̃i = xi (resp. ỹi = yi ) for i 6= p + 2, and x̃p+2 = xp+1 (resp.
ỹp+2 = yp+1 ). Alternatively, x̃ = z + ((d − xp+2 )p+2 ), ỹ = z + ((d + a − xp+2 )p+2 ). By induction, we obtain
for all i = 0, 1, · · · , m · n, inclusions
ExtiS (S/Iz , S) ֒−→ ExtiS (S/Ix̃ , S), and ExtiS (S/Iz , S) ֒−→ ExtiS (S/Iỹ , S).

(4.2)

The natural commutative diagrams
S/Ix̃


S/Ix

/ S/Iz

S/Iỹ

/ S/Iz

S/Iy



/ S/Iz

/ S/Iz

induce commutative diagrams
ExtiS (S/Iz , S)

/ Exti (S/Ix̃ , S)
S O

ExtiS (S/Iz , S)

/ Exti (S/Iỹ , S)
S
O

ExtiS (S/Iz , S)

/ Exti (S/Ix , S)
S

ExtiS (S/Iz , S)

/ Exti (S/Iy , S)
S

Since the top maps are injective by (4.2), the bottom ones must be injective as well. We get for all
i = 0, 1, · · · , m · n, commutative diagrams
0

/ Exti (S/Iz , S)
S

/ Exti (S/Ix , S)
S

/ Exti (Ix /Iz , S)
S

αi

0

/ Exti (S/Iz , S)
S



/ Exti (S/Iy , S)
S

/0

βi



/ Exti (Iy /Iz , S)
S

/0

where the rows are exact. The maps βi are injective by Corollary 4.1, forcing the maps αi to be injective as
well. We conclude that the inclusion (4.1) holds, finishing the proof of the theorem.

Theorem 4.3. The character of the doubly–graded module Ext•S (S/Ix , S) is given by
Pn−p
X
2
χExt•S (S/Ix ,S)(z, w) =
[Sλ(s) F ⊗ Sλ G] · z |λ| · wm·n+1−p −s·(m−n)−2·( j=1 tj ) ,

(4.3)

1≤p≤n
0≤s≤t1 ≤···≤tn−p ≤p−1
λ∈W ′ (x,p;t,s)

where W ′ (x, p; t, s) is the set of dominant weights λ ∈ Zn satisfying

 λn ≥ p − xp − m,
λt +j ≤ tj − xn+1−j − m, for j = 1, · · · , n − p,
 j
λs ≥ s − n and λs+1 ≤ s − m.

(4.4a)
(4.4b)
(4.4c)

Remark 4.4. The condition tn−p ≤ p − 1 in (4.3) combined with the inequalities
tn−p − xp+1 − m ≥ λtn−p +n−p ≥ λn ≥ p − xp − m
obtained from (4.4b) by letting j = n−p, shows that the only values of p for which there may be a non-trivial
contribution to (4.3) are the ones for which xp > xp+1 , or p = n. It follows that for x1 = · · · = xn , the only
interesting value of p is p = n, in which case Ix = Jx,n and (4.3) follows from (3.9) and the standard long
exact sequence relating Ext•S (S/Ix , S) to Ext•S (Ix , S).

LOCAL COHOMOLOGY WITH SUPPORT IN GENERIC DETERMINANTAL IDEALS

15

Proof of Theorem 4.3. We do induction on the number of columns of x. When x = 0, S/Ix = 0, so
Ext•S (S/Ix , S) = 0. It follows that (4.3) is verified in this case, since W ′ (0, p; t, s) is empty whenever
0 ≤ s ≤ p − 1: to see this, note that
s−m

(4.4c)

≥

λs+1 ≥ λn

(4.4a )

≥

p−m

implies s ≥ p, which is incompatible with the condition s ≤ p − 1.
Suppose now that y is obtained from x by appending a column of size (q + 1), for some q = 0, · · · , n − 1.
This implies that x1 = · · · = xq+1 , and yi = xi + 1 for 1 ≤ i ≤ q + 1. It follows from Theorem 4.2 that
χExt•S (S/Iy ,S) (z, w) = χExt•S (S/Ix ,S) (z, w) + χExt•S (Ix /Iy ,S) (z, w),

(4.5)

and from Lemma 2.2 that
X

χExt•S (Ix /Iy ,S) (z, w) =

χExt•S (Jz,q ,S) (z, w).

(4.6)

z=(z1 ≥···≥zn ≥0)
z1 =···=zq+1 =x1
zi ≥xi , i>q+1

By Remark 4.4, since x1 = · · · = xq+1 and y1 = · · · = yq+1 , the only relevant terms in (4.3) (for both x and
y) are those for which p ≥ q + 1. For such p, xn+1−j = yn+1−j whenever 1 ≤ j ≤ n − p, so condition (4.4b)
is the same for x as it is for y. (4.4c) is clearly the same for both x and y, and the same is true for (4.4a)
when p ≥ q + 2. We conclude that W ′ (x, p; t, s) = W ′ (y, p; t, s) for p 6= q + 1, from which it follows using
(4.5) and the induction hypothesis that in order to prove (4.3) for y, it suffices to show that
Pn−p
X
2
χExt•S (Ix /Iy ,S) (z, w) =
[Sλ(s) F ⊗ Sλ G] · z |λ| · wm·n+1−p −s·(m−n)−2·( j=1 tj )
p=q+1
0≤s≤t1 ≤···≤tn−p ≤p−1
λ∈W ′ (y,p;t,s)\W ′ (x,p;t,s)

X

=

[Sλ(s) F ⊗ Sλ G] · z |λ| · wm·n−q

2 −2·q−s·(m−n)−2·

Pn−q−1

(

j=1

tj )

0≤s≤t1 ≤···≤tn−q−1 ≤q
λ∈W ′ (y,q+1;t,s)\W ′ (x,q+1;t,s)

(4.7)
Note that since (4.4b) and (4.4c) are the same for x and y when p = q + 1, it follows that


λn = q + 1 − yq+1 − m = q − x1 − m,
′
′
λ ∈ W (y, q + 1; t, s) \ W (x, q + 1; t, s) ⇔ λtj +j ≤ tj − xn+1−j − m, for j = 1, · · · , n − q − 1,


λs ≥ s − n and λs+1 ≤ s − m.

(4.8)

Consider now a partition z appearing in (4.6). We claim that W (z, q; t, s) is empty unless tn−q = q.
Furthermore, if λ ∈ W (z, q; t, s), then λn = q − x1 − m. To see this, note that
λn ≤ λtn−q +n−q

(3.10b )

=

tn−q − zq+1 − m = tn−q − zq − m ≤ q − zq − m

(3.10a )

≤

λn ,

which forces equalities throughout, and in particular
tn−q = q and λn = tn−q − zq+1 − m = q − x1 − m.
We get from (3.9) that
χExt•S (Jz,q ,S) (z, w) =

X

0≤s≤t1 ≤···≤tn−q−1 ≤tn−q =q
λ∈W (x,q;t,s)

[Sλ(s) F ⊗ Sλ G] · z |λ| · wm·n−q

2 −2q−s·(m−n)−2·

Pn−q−1

(

j=1

tj )

. (4.9)

16

CLAUDIU RAICU AND JERZY WEYMAN

Combining (4.6), (4.7) and (4.9), it remains to show that
W ′ (y, q + 1; t, s) \ W ′ (x, q + 1; t, s) =

[

W (x, q; t, s).

z=(z1 ≥···≥zn ≥0)
z1 =···=zq+1 =x1
zi ≥xi , i>q+1

This follows immediately from (4.8) and the fact that the condition
λtj +j ≤ tj − xn+1−j − m, for j = 1, · · · , n − q − 1
in (4.8) is equivalent to the existence of a partition z satisfying z1 = · · · = zq+1 = x1 , and zn+1−j ≥ xn+1−j
for j = 1, · · · , n − q − 1 (or equivalently zi ≥ xi for i > q + 1), such that
λtj +j = tj − zn+1−j − m, for j = 1, · · · , n − q − 1.



5. Regularity of the ideals Ix
The explicit description of the character of Ext•S (Ix , S) obtained in Theorem 4.3 allows us to obtain the
following result on the regularity of every ideal Ix , whose proof will be the focus of the current section.
Theorem 5.1. For a partition x with at most n parts, letting xn+1 = −1 we have the following formula for
the regularity of the ideal Ix :
reg(Ix ) = max (n · xp + (p − 2) · (n − p)).

(5.1)

p=1,··· ,n
xp >xp+1

In particular, the only ideals Ix which have a linear resolution are those for which x1 = · · · = xn (i.e. powers
Inx1 of the ideal In of maximal minors) or x1 − 1 = x2 = · · · = xn (i.e. Inx1 −1 · I1 ).
By [Eis95, Prop. 20.16], one can compute the regularity of a finitely generated S–module M by the formula
reg(M ) = max{−r − j : ExtjS (M, S)r 6= 0}.
Since reg(Ix ) = reg(S/Ix ) + 1, we get by combining (5.2) and (4.3) that

reg(Ix ) =

max

1≤p≤n
0≤s≤t1 ≤···≤tn−p ≤p−1
λ∈W ′ (x,p;t,s)

(5.2)


n−p
X

−|λ| − mn + p2 + s · (m − n) + 2 · 

j=1



tj   .

(5.3)

It is then important to decide when W ′ (x, p; t, s) is non-empty, which we do in the following lemma.
Lemma 5.2. Fix p ∈ {1, · · · , n} and 0 ≤ s ≤ t1 ≤ · · · ≤ tn−p ≤ p − 1. The set W ′ (x, p; t, s) is non-empty if
and only if
(
xp − xn+1−j ≥ p − tj , for j = 1, · · · , (n − p),
(5.4)
s ≥ p − xp .
Moreover, the weight λ ∈ W ′ (x, p; t, s) of minimal size (i.e. for which the quantity −|λ| is maximal) is given
by
(
λ1 = · · · = λs = (s − n),
(5.5)
λs+1 = · · · = λn = (p − xp − m).

LOCAL COHOMOLOGY WITH SUPPORT IN GENERIC DETERMINANTAL IDEALS

17

Proof. If W ′ (x, p; t, s) is non-empty, then for any λ ∈ W ′ (x, p; t, s) we have
tj − xn+1−j − m

(4.4b )

≥

λtj +j ≥ λn

(4.4a )

≥

p − xp − m,

for j = 1, · · · , (n − p), and
s−m

(4.4c)

≥

λs+1 ≥ λn

(4.4a )

≥

p − xp − m,

so (5.4) holds.
Conversely, assume that (5.4) holds, and define λ via (5.5). It is immediate to check that λ satisfies (4.4a–
4.4c), so λ ∈ W ′ (x, p; t, s) and the set is non-empty. The fact that this λ has minimal size follows from
the fact that any other λ ∈ W ′ (x, p; t, s) is dominant and thus satisfies λ1 ≥ · · · ≥ λs ≥ (s − n), and
λs+1 ≥ · · · ≥ λn ≥ (p − xp − m), so
|λ| ≥ s · (s − n) + (n − s) · (p − xp − m) = (n − s) · (p − xp − s − m).
Lemma 5.2 allows us to rewrite (5.3) in the form


reg(Ix ) =

=

=

max

1≤p≤n
0≤s≤t1 ≤···≤tn−p ≤p−1
xp −xn+1−j ≥p−tj
s≥p−xp

max

1≤p≤n
0≤s≤t1 ≤···≤tn−p ≤p−1
xp −xn+1−j ≥p−tj
s≥p−xp

max

1≤p≤n
0≤s≤p−1
xp −xp+1 ≥1
s≥p−xp




n−p
X

−(n − s) · (p − xp − s − m) − mn + p2 + s · (m − n) + 2 · 




s · (p − xp − s) + n · (xp − p) + p2 + 2 · 

n−p
X
j=1



tj  

j=1



tj  
(5.6)

(s · (p − xp − s) + n · xp + (p − 2) · (n − p).)

Since s ≥ p − xp , we have s · (p − xp − s) ≤ 0, with equality if s = 0 or s = p − xp . For 1 ≤ p ≤ n − 1,
the condition xp − xp+1 ≥ 1 forces xp ≥ 1, so p − xp ≤ p − 1. It follows that we can take s = max(0, p − xp )
in order to maximize the expression above. Likewise, if p = n and xn ≥ 1, we take s = max(0, n − xn ). It
follows that when xn ≥ 1, (5.6) reduces to (5.1). However, if xn = 0 then for p = n the conditions s ≤ p − 1
and s ≥ p − xp are incompatible, so (5.6) reduces to
reg(Ix ) =

max

(n · (xp − p) + p2 + 2 · (p − 1) · (n − p)).

p=1,··· ,n−1
xp >xp+1

To see that this is the same as (5.1) it suffices to observe that reg(Ix ) ≥ nxn = 0 (which is the term
corresponding to p = n).
To finish the proof of the theorem, we need to verify the last assertion. Note that Ix is generated in degree
x1 + · · · + xn , so it has a linear resolution if and only if
reg(Ix ) = x1 + · · · + xn .

(5.7)

When x1 = · · · = xn , (5.1) reduces to the term with p = n, whose value is nxn = x1 + · · · + xn . For
x1 − 1 = x2 = · · · = xn , the only surviving terms in (5.1) are those with p = 1 and p = n, so we get
reg(Ix ) = max(n · (x1 − 1) + 1, nxn ) = n · (x1 − 1) + 1 = x1 + · · · + xn .

18

CLAUDIU RAICU AND JERZY WEYMAN

Conversely, assume that (5.7) holds, and that the xi ’s aren’t all equal. Take p minimal with the property
that xp > xp+1 , so that p < n, x1 = · · · = xp and xi ≤ xp − 1 for i > p. We have
reg(Ix ) ≥ n · (xp − p) + p2 + 2 · (p − 1) · (n − p) = pxp + (n − p) · (xp − 1) + (n − p) · (p − 1) ≥ x1 + · · · + xn ,
with equality when xi = xp −1 for i > p and (n−p)·(p−1) = 0. This forces p = 1 and x1 −1 = x2 = · · · = xn ,
concluding the proof of the theorem.
6. Local cohomology with support in determinantal ideals
In this section we prove our main theorem on local cohomology with support in determinantal ideals.
Recall that S = Sym(Cm ⊗ Cn ) denotes the polynomial ring of functions on the space of m × n matrices,
Ip ⊂ S is the ideal of p × p minors of the generic m × n matrix, and Hp (z, w) is the character of the
doubly graded module HI•p (S). Recall also the definition (1.3) of hs (z), and the notation (1.4) for Gauss
polynomials.
Theorem 6.1. We have for each p = 1, · · · , n,
Hp (z, w) =

p−1
X

hs (z) · w

(n−p+1)2 +(n−s)·(m−n)

s=0




n−s−1
·
(w2 ).
p−s−1

To prove the theorem, note that since the system of ideals {I(dp ) : d ≥ 0} is cofinal with the one consisting
of powers of the ideal of p × p minors, we obtain from [Eis05, Ex. A1D.1] that for each i = 0, 1, · · · , m · n,
HIip (S) = lim ExtiS (S/I(dp ) , S),
−→
d

where the successive maps in the directed system are induced from the natural quotient maps S/I((d+1)p ) ։
S/I(dp ) . By Theorem 4.2, all these maps are injective, so the description of the character of HIip (S) can be
deduced from Theorem 4.3. Note that the partitions x to which we apply Theorem 4.3 have the property
that x1 = · · · = xp = d, and xi = 0 for i > p. Since we are interested in the limit as d → ∞, we might as
well assume that x1 = · · · = xp = ∞, in which case (4.4a) becomes vacuous. In what follows, λ will always
be assumed to be a dominant weight.
If s ≤ tj then s + 1 ≤ tj + j for every j = 1, · · · , n − p, so we get
λtj +j

(λ∈Zn
dom )

≤

λs+1

(4.4c)

≤

(s≤tj )

s − m ≤ tj − m,

i.e (4.4c) implies (4.4b) (note that xn+1−j = 0 for j ≤ n − p). We conclude that
Pn−p
X
2
Hp (z, w) =
[Sλ(s) F ⊗ Sλ G] · z |λ| · wm·n+1−p −s·(m−n)−2·( j=1 tj )
0≤s≤t1 ≤···≤tn−p ≤p−1
λs ≥s−n
λs+1 ≤s−m

(1.3)

=

X

hs (z) · wm·n+1−p

2 −s·(m−n)−2·

Pn−p

(

j=1

tj )

0≤s≤t1 ≤···≤tn−p ≤p−1
p−1
(t′j :=p−1−tj ) X

=

2 +(n−s)·(m−n)

hs (z) · w(n−p+1)

s=0

p−1
(1.4) X

=

s=0

X

·

p−1−s≥t′1 ≥···≥t′n−p ≥0
2 +(n−s)·(m−n)

hs (z) · w(n−p+1)

·




n−s−1
(w2 ).
p−s−1

w2·(

Pn−p
j=1

t′j )

LOCAL COHOMOLOGY WITH SUPPORT IN GENERIC DETERMINANTAL IDEALS

19

Acknowledgments
This work was initiated while we were visiting the Mathematical Sciences Research Institute, for whose
hospitality we are grateful. Special thanks go to Emily Witt who participated in the initial stages of this
project. We would also like to thank David Eisenbud, Steven Sam, Anurag Singh and Uli Walther for helpful
conversations. Experiments with the computer algebra software Macaulay2 [GS] have provided numerous
valuable insights. The first author acknowledges the support of the National Science Foundation Grant
No. 1303042. The second author acknowledges the support of the Alexander von Humboldt Foundation,
and of the National Science Foundation Grant No. 0901185.
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Department of Mathematics, Princeton University, Princeton, NJ 08544
Institute of Mathematics “Simion Stoilow” of the Romanian Academy
E-mail address: craicu@math.princeton.edu
Department of Mathematics, University of Connecticut, Storrs, CT 06269
E-mail address: jerzy.weyman@uconn.edu

