Duality and Tameness
arXiv:0704.0102v1  [math.AC]  1 Apr 2007
DUALITY AND TAMENESS
MARC CHARDIN, STEVEN DALE CUTKOSKY, JÜRGEN HERZOG AND HEMA SRINIVASAN
Abstract. We prove a duality theorem for certain graded algebras and show by various
examples different kinds of failure of tameness of local cohomology.
Introduction
The purpose of this paper is to construct examples of strange behavior of local coho-
mology. In these constructions we follow a strategy that was already used in [CH] and
which relates, via a spectral sequence introduced in [HR], the local cohomology for the
two distinguished bigraded prime ideals in a standard bigraded algebra.
In the first part we consider algebras with rather general gradings and deduce a similar
spectral sequence in this more general situation. A typical example of such an algebra
is the Rees algebra of a graded ideal. The proof for the spectral sequence given here is
simpler than that of the corresponding spectral sequence in [HR].
In the second part of this paper we construct examples of standard graded rings A,
which are algebras over a field K, such that the function
(1) j 7→ dimK(H iA+(A)−j)
is an interesting function for j ≫ 0. In our examples, this dimension will be finite for all
Suppose that A0 is a Noetherian local ring, A =
j≥0Aj is a standard graded ring and
set A+ :=
j>0Aj. Let M be a finitely generated graded A-module and F := M̃ be the
sheafification of M on Y = Proj(A). We then have graded A-module isomorphisms
H i+1A+ (M)
H i(Y,F(n))
for i ≥ 1, and a similar expression for i = 0 and 1.
By Serre vanishing, H iA+(M)j = 0 for all i and j ≫ 0. However, the asymptotic
behaviour of H iA+(M)−j for j ≫ 0 is much more mysterious.
In the case when A0 = K is a field, the function (1) is in fact a polynomial for large
enough j. The proof is a consequence of graded local duality ([BS, 13.4.6] or [BH, 3.6.19])
or follows from Serre duality on a projective variety.
For more general A0, HA+(M)−j are finitely generated A0 modules, but need not have
finite length.
The following problem was proposed by Brodmann and Hellus [BrHe].
The second author was partially supported by NSF.
http://arxiv.org/abs/0704.0102v1
Tameness problem: Are the local cohomology modules H iA+(M) tame? That is, is it
true that either
{H iA+(M)j 6= 0, ∀j ≪ 0} or {H
A+(M)j = 0, ∀j ≪ 0}?
The problem has a positive solution for A0 of small dimension (some of the references
are Brodmann [Br], Brodmann and Hellus [BrHe], Lim [L], Rotthaus and Sega [RS]).
Theorem 0.1 ([BrHe]). If dim(A0) ≤ 2, then M is tame.
However, it has recently been shown by two of the authors that tameness can fail if
dim(A0) = 3.
Theorem 0.2 ([CH]). There are examples with dim(A0) = 3 where M is not tame.
The statement of this example is reproduced in Theorem 3.1 of this paper. The function
(1) is periodic for large j. Specifically, the function (1) is 2 for large even j and is 0 for
large odd j.
In Theorem 3.3 we construct an example of failure of tameness of local cohomology
which is not periodic, and is not even a quasi polynomial (in −j) for large j. Specifically,
we have for j > 0,
dimK(H
(A)−j) =
1 if j ≡ 0 (mod) (p+ 1),
1 if j = pt for some odd t ≥ 0,
0 otherwise,
where the characteristic of K is p. We have pt ≡ −1 (mod) (p + 1) for all odd t ≥ 0.
We also give an example (Theorem 3.5) of failure of tameness where (1) is a quasi
polynomial with linear growth in even degree and is 0 in odd degree.
In Theorem 3.6, we give a tame example, but we have
dimK(H
(A)−j)
so (1) is far from being a quasi polynomial in −j for large j.
While the example of [CH] is for M = ωA, where ωA is the canonical module of A, the
examples of the paper are all for M = A. This allows us to easily reinterpret our examples
as Rees algebras in Section 4, and thus we have examples of Rees algebras over local rings
for which the above failure of tameness holds.
In the final section, Section 5, we give an analysis of the explicit and implicit role of
bigraded duality in the construction of the examples, and some comments on how it effects
the geometry of the constructions.
1. Duality for polynomial rings in two sets of variables
Let K be any commutative ring (with unit). In later applications K will be mostly
a field. Furthermore let S = K[x1, . . . , xm, y1, . . . , yn], P = (x1, . . . , xm) and Q =
(y1, . . . , yn).
The homology of the Čech complex CP ( ) (resp. CQ( )) will be denoted by HP ( ) (resp.
HQ( )). Notice that for any commutative ring K, this homology is the local cohomology
supported in P (resp. Q), as P and Q are generated by a regular sequences.
Assume that S is Γ-graded for some abelian group Γ, and that deg(a) = 0 for a ∈ K.
If xsyp ∈ R, deg(xsyp) = l(s) + l′(p) with l(s) :=
i si deg(xi) and l
′(p) :=
j pj deg(yj).
Definition 1.1. Let I ⊂ S be a Γ-graded ideal. The Γ-grading of S is I-sharp if H iI(S)γ
is a finitely generated K-module, for every i and γ ∈ Γ.
Lemma 1.2. The following conditions are equivalent:
(i) the Γ-grading of S is P -sharp.
(ii) the Γ-grading of S is Q-sharp.
(iii) for all γ ∈ Γ, |{(α, β) : α ≥ 0, β ≥ 0, l(α) = γ + l′(β)}| < ∞.
Note that if K is Noetherian, M is a finitely generated Γ-graded S-module, and the
Γ-grading of S is I-sharp, then H iI(M)γ is a finite K-module, for every i and γ ∈ Γ. This
follows from the converging Γ-graded spectral sequence Hp−q(H
I (F)) ⇒ H
I (M), where F
is a Γ-graded free S-resolution of M with Fi finite for every i.
We will assume from now on that the Γ-grading of S is P -sharp (equivalently Q-
sharp). Set σ = deg(x1 · · · xmy1 · · · yn), and if N is a Γ-graded module, then let N∨ =
HomS(N,S(−σ)) and N∗ = ∗HomK(N,K) where the Γ-grading of N∗ is given by (N∗)γ =
HomK(N−γ ,K). More generally, we always denote the graded K-dual of a graded mod-
ule N (over what graded K-algebra soever) by N∗. Finally we denote by ϕαβ the map
S(−a) → S(−b) induced by multiplication by xαyβ where a = deg xα and b = − deg yβ.
Lemma 1.3. HmP (ϕαβ)γ
∼= HnQ(ϕ∨αβ)∗.
Proof. The free K-moduleHmP (S)γ is generated by the elements x
−s−1yp with s, p ≥ 0 and
−l(s)− l(1) + l′(p) = γ, and HnQ(S)γ′ is generated by the elements xty−q−1 with t, q ≥ 0
and l(t)− l′(q)− l′(1) = γ′.
Let dγ : H
P (S)γ → (HnQ(S∨)∗)γ = HnQ(S)−γ−σ be the K-linear map defined by
−s−1yp)(xty−q−1) =
1, if s = t and p = q,
0, else.
Then dγ is an isomorphism (because the Γ-grading of R is Q-sharp) and there is a com-
mutative diagram
HmP (S)γ−a
(ϕαβ)γ−−−−−−−→ HmP (S)γ−b
y dγ−b
(HnQ(S)−γ+a−σ)
−−−−−−−−−→ (HnQ(S)−γ+b−σ)∗.
The assertion follows. �
As an immediate consequence we obtain
Corollary 1.4. (a) Let f ∈ S be an homogeneous element of degree a−b, and ϕ : S(−a) →
S(−b) the graded degree zero map induced by multiplication with f . Then
HmP (ϕ) ≃ HnQ(ϕ∨)∗.
(b) Let F be a Γ-graded complex of finitely generated free S-modules. Then
(i) H iP (F) = 0 for i 6= m and H
Q(F) = 0 for j 6= n,
(ii) HmP (F) ≃ HnQ((F)∨)∗.
As the main result of this section we have
Theorem 1.5. Assume that K is Noetherian, the Γ-grading of S is P -sharp (equivalently
Q-sharp) and M is a finitely generated Γ-graded S-module. Set ωS/K := S(−σ). Let F be
a minimal Γ-graded S-resolution of M . Then,
(a) For all i, there is a functorial isomorphism
H iP (M) ≃ Hm−i(HmP (F)).
(b) There is a convergent Γ-graded spectral sequence,
H iQ(Ext
S(M,ωS/K)) ⇒ H
i+j−n(HmP (F)
In particular, if K is a field, there is a convergent Γ-graded spectral sequence,
H iQ(Ext
S(M,ωS)) ⇒ H
dimS−(i+j)
P (M)
Proof. Claim (a) is an immediate consequence of Corollary 1.4 via the Γ-graded spec-
tral sequence Hp−i(H
P (F)) ⇒ H iP (M). For (b), the two spectral sequences arising from
the double complex CQF∨ have as second terms respectively ′Eij2 = H iQ(Ext
S(M,ωS/K)),
2 = 0 for i 6= n and ′′E
2 = H
j(HnQ(F
∨)) ≃ Hj(HmP (F)∗). If further K is a field,
Hj(HmP (F)
∗) ≃ (Hj(HmP (F)))∗ ≃ H
P (F)
Corollary 1.6. Under the hypotheses of the theorem, if K is a field, then for any γ ∈ Γ,
there are convergent spectral sequences of finite dimensional K-vector spaces
H iQ(Ext
S(M,ωR))γ ⇒ H
dimS−(i+j)
P (M)−γ ,
H iP (Ext
S(M,ωR))γ ⇒ H
dimS−(i+j)
Q (M)−γ .
We now consider the special case that Γ = Z2, S := K[x1, . . . , xm, y1, . . . , yn] with
deg(xi) = (1, 0) and deg(yj) = (dj , 1) with dj ≥ 0. Set T := K[x1, . . . , xm] and let M be
a Γ-graded S-module. We view M as a Z-graded module by defining Mk =
j M(j,k).
Observe that each Mk itself is a graded T -module with (Mk)j = M(j,k) for all j. We also
note that H iP (M)k
∼= H iP0(Mk), as can been seen from the definition of local cohomology
using the Čech complex. Here P0 = (x1, . . . , xm) is the graded maximal ideal of T .
Corollary 1.7. With the notation introduced, let s := dimS = m + n and d := dimM .
(a) H0P (Ext
S (M,ωS))
∼= HdQ(M)∗ for any k,
(b) there is an exact sequence
0→H1P (Exts−dS (M,ωS))→H
Q (M)
∗→H0P (Exts−d+1S (M,ωS)).
(c) Let i ≥ 2. If ExtjS(M,ωS) is annihilated by a power of P for all s−d < j < s−d+i,
then there is an exact sequence
Exts−d+i−1S (M,ωS)→H
P (Ext
S (M,ωS))→H
Q (M)
∗→H0P (Exts−d+iS (M,ωS)).
In particular, if Ext
S(M,ωS) has finite length for all s − d < j ≤ s − d+ i0, for some
integer i0, then
H iP0(Ext
S (M,ωS)k)
∼= (Hd−iQ (M)−k)
∗ for all i ≤ i0 and k ≫ 0.
Consequently, if M is a generalized Cohen-Macaulay module (i.e. Exts−iS (M,ωS) has
finite length for all i 6= d), and if we set N = Exts−dS (M,ωS), then
H iP0(Nk)
∼= (Hd−iQ (M)−k)
∗ for all i and all k ≫ 0.
Proof. (a), (b) and (c) are direct consequences of Corollary 1.6. For the application, notice
that if γ = (ℓ, k) ∈ Γ with k ≫ 0 one has ExtjS(M,ωS)γ = 0 for all s− d < j ≤ s− d+ i0.
Therefore, for such γ, the desired conclusion follows. �
A typical example to which this situation applies is the Rees algebra of a graded ideal
I in the standard graded polynomial ring T = K[x1, . . . , xm]. Say, I is generated be the
homogeneous polynomials f1, . . . , fn with deg fj = dj for j = 1, . . . , n. Then the Rees
algebra R(I) ⊂ T [t] is generated the elements fjt. If we set deg fjt = (dj , 1) for all j
and deg xi = (1, 0) for all i, then R(I) becomes a Γ-graded S-module via the K-algebra
homomorphism S → R(I) with xi 7→ xi and yj 7→ fjt.
According to this definition we have R(I)k = Ik for all k.
Since dimR(I) = m+1, the module ωR(I) = Extn−1S (R(I), ωS) is the canonical module
of R(I) (in the sense of [HK, 5. Vortrag]). Recall that if a ring R is a finite S-module
of dimension m + 1, the natural finite map R→Hom(ωR, ωR) ∼= Extn−1S (ωR, ωS) is an
isomorphism if and only if R is S2. Thus in combination with Corollary 1.7 we obtain
Corollary 1.8. Let R := R(I). Suppose that Rp is Cohen-Macaulay for all p 6= (m, R+)
where m = (x1, . . . , xm) and R+ =
k>0 I
ktk. Then
H im(I
k) ∼= (Hm+1−iR+ (ωR)−k)
∗ for all i and all k ≫ 0.
Proof. Since ωR localizes, the conditions imply that (ωR)p is Cohen-Macaulay for all p 6=
(m, R+). Hence the natural into map R→R′ := Extn−1S (ωR, ωS) has a cokernel of finite
length. In particular, R′k = Rk = I
k for k ≫ 0. Thus Corollary 1.7 applied to M = ωR
gives the desired conclusion. �
Remark 1.9. Let R := R(I). If the cokernel of R→Hom(ωR, ωR) is annihilated by a
power of R+ (in other words, the blow-up is S2, as a projective scheme over Spec(T )),
then R′k = I
k for k ≫ 0 and therefore one has an exact sequence
0→H0m(T/Ik)→(HmR+(ωR)−k)
∗→H0m(ExtnS(ωR, ωS)k)→H1m(T/Ik)→(Hm−1R+ (ωR)−k)
for such a k.
2. A method of constructing examples
Suppose that R =
i,j≥0Rij is a standard bigraded algebra over a ring K = R00.
Define Ri =
j≥0Rij and Rj =
i≥0Rij . Define ideals P =
i and Q =
j>0Rj
in R. Suppose that M =
ij∈ZMij is a finitely generated, bigraded R-module. Define
M i =
j∈ZMij and Mj =
i∈ZMij. M
i is a graded R0-module and Mj is a graded
R0-module. Let Q0 = R01R
0, so that Q = Q0R. Let P0 = R10R0 so that P = R10R. We
have K module isomorphisms
H lQ(M)m,n
∼= H lQ0(M
for m,n ∈ Z. Let M̃m be the sheafification of the graded R0-module Mm on Proj(R0).
We have K module isomorphisms
H lQ0(M
m)n ∼= H l−1(Proj(R0), M̃m(n))
for l ≥ 2 and exact sequences
0 → H0Q0(M
m)n → (Rm)n = Rm,n → H0(Proj(R0), M̃m(n)) → H1Q0(M
m)n → 0.
We have similar formulas for the calculation of H lP (M).
Now assume that X is a projective scheme over K and F1 and F2 are very ample line
bundles on X. Let
Rm,n = Γ(X,F⊗m1 ⊗F
We require that R =
m,n≥0Rm,n be a standard bigraded K-algebra. We have
X ∼= Proj(R0) ∼= Proj(R0).
The sheafification of the graded R0-module Rm on X is R̃m = F⊗m1 , and the sheafification
of the graded R0-module Rn on X is R̃n ∼= F⊗n2 (Exercise II.5.9 [Ha]).
For l ≥ 2 we have bigraded isomorphisms
H lQ(R)
H lQ0(R
m)n ∼=
m≥0,n∈Z
H l−1(X,F⊗m1 ⊗F
Viewing R as a graded R0 algebra, we thus have graded isomorphisms
(2) H lQ(R)n
H l−1(X,F⊗m1 ⊗F
for l ≥ 2 and n ∈ Z. Let d = dim(R) = dim(X) + 2.
We now further assume that K is an algebraically closed field and X is a nonsingular
K variety. Let
V = P(F1 ⊕F2),
a projective space bundle over X with projection π : V → X. Since F1 ⊕ F2 is an ample
bundle on X, OV (1) is ample on V . Since
Γ(V,OV (t))
Γ(V,OV (t)) ∼= Γ(X,St(F1 ⊕F2)) ∼=
i+j=t
and R is generated in degree 1 with respect to this grading, OV (1) is very ample on V and
R is the homogeneous coordinate ring of the nonsingular projective variety V , so that R
is generalized Cohen Macaulay (all local cohomology modules H iR+(R) of R with respect
to the maximal bigraded ideal R+ of R have finite length for i < d). We further have that
V is projectively normal by this embedding (Exercise II.5.14 [Ha]) so that R is normal.
3. Strange behavior of local cohomology
In [CH], we constructed the following example of failure of tameness of local cohomology.
In the example, R0 has dimension 3, which is the lowest possible for failure of tameness
[Br].
Theorem 3.1. Suppose that K is an algebraically closed field. Then there exists a normal
standard graded K-algebra R0 with dim(R0) = 3, and a normal standard graded R0-algebra
R with dim(R) = 4 such that for j ≫ 0,
dimK(H
Q(ωR)−j) =
2 if j is even,
0 if j is odd,
where ωR is the canonical module of R, Q =
n>0Rn.
We first show that the above theorem is also true for the local cohomology of R.
Theorem 3.2. Suppose that K is an algebraically closed field. Then there exists a normal
standard graded K-algebra R0 with dim(R0) = 3, and a normal standard graded R0-algebra
R with dim(R) = 4 such that for j > 0,
dimK(H
Q(R)−j) =
2 if j is even,
0 if j is odd,
where Q =
n>0Rn.
Proof. We compute this directly for the R of Theorem 3.1 from (2) and the calculations of
[CH]. Translating from the notation of this paper to the notation of [CH], we have X = S
is an Abelian surface, F1 = OS(r2laH) and F2 = OS(r2(D + alH)).
By (2) of this paper, for n ∈ N, we have
dimK(H
Q(R)n) =
m≥0 h
1(X,F⊗m1 ⊗F
m≥0 h
1(S,OS((m+ n)r2alH + nr2D)).
Formula (1) of [CH] tells us that for m,n ∈ Z,
(3) h1(S,OS(mH + nD)) =
2 if m = 0 and n is even,
0 otherwise.
Thus for n < 0, we have
dimK(H
Q(R)n) =
2 if n is even,
0 if n is odd,
giving the conclusions of the theorem. �
The following example shows non periodic failure of tameness.
Theorem 3.3. Suppose that p is a prime number such that p ≡ 2 (mod) 3 and p ≥ 11.
Then there exists a normal standard graded K-algebra R0 over a field K of characteristic
p with dim(R0) = 4, and a normal standard graded R0-algebra R with dim(R) = 5 such
that for j > 0,
dimK(H
Q(R)−j) =
1 if j ≡ 0 (mod) (p+ 1),
1 if j = pt for some odd t ≥ 0,
0 otherwise,
where Q =
n>0Rn. We have p
t ≡ −1(mod)(p + 1) for all odd t ≥ 0.
To establish this, we need the following simple lemma.
Lemma 3.4. Suppose that C is a non singular curve of genus g over an algebraically closed
field K, and M, N are line bundles on C. If deg(M) ≥ 2(2g+1) and deg(N ) ≥ 2(2g+1),
then the natural map
Γ(C,M) ⊗ Γ(C,N ) → Γ(C,M⊗N )
is a surjection.
Proof. If L is a line bundle on C, then H1(C,L) = 0 if deg(L) > 2g − 2 and L is very
ample if deg(L) ≥ 2g + 1 (Chapter IV, Section 3 [Ha]).
Suppose that L is very ample and G is another line bundle on C. If deg(G) > 2g − 2−
deg(L), then G is 2-regular for L (Lecture 14, [M1]). Thus if deg(G) > 2g − 2 + deg(L),
Γ(C,G) ⊗ Γ(C,L) → Γ(C,G ⊗ L)
is a surjection by Castelnuovo’s Proposition, Lecture 14, page 99 [M1].
We now apply the above to prove the lemma. Write M ∼= A⊗q ⊗ B where A is a
line bundle such that deg(A) = 2g + 1, and 2g + 1 ≤ deg(B) < 2(2g + 1). deg(N ) >
2g − 2 + deg(A). Thus there exists a surjection
Γ(C,N ) ⊗ Γ(C,A) → Γ(C,A⊗N ).
We iterate to get surjections
Γ(C,A⊗i ⊗N )⊗ Γ(C,A) → Γ(C,A⊗(i+1) ⊗N )
for i ≤ q, and a surjection
Γ(C,A⊗q ⊗N )⊗ Γ(C,B) → Γ(C,M⊗N ).
We now prove Theorem 3.3. For the construction, we start with an example from
Section 6 of [CS]. There exists an algebraically closed field K of characteristic p, a curve
C of genus 2 over K, a point q ∈ C and a line bundle M on C of degree 0, such that for
n ≥ 0,
H1(C,OC (q)⊗M⊗n) =
1 if n = pt for some t ≥ 0,
0 otherwise.
Further, H1(C,OC (2q)⊗M⊗n) = 0 for all n > 0.
Let a = p+ 1. Let E be an elliptic curve over K, and let T = E × E, with projections
πi : T → E. Let b ∈ E be a point and let A = π∗1(OE(b)) ⊗ π∗2(OE(b)). Let X = T × C,
with projections ϕ1 : X → T , ϕ2 : X → C. Let L = OC(q). Let
F1 = ϕ∗1(A)⊗a ⊗ ϕ∗2(L)⊗a,
F2 = ϕ∗1(A)⊗(1+a) ⊗ ϕ∗2(L⊗(1+a) ⊗M−1).
For m,n ≥ 0, we have
Γ(X,F⊗m1 ⊗F
2 ) = Γ(T,A⊗(ma+n(1+a)))⊗ Γ(C,L⊗(ma+n(1+a)) ⊗M−⊗n)
= Γ(T,A⊗a)⊗m ⊗ Γ(T,A⊗(1+a))⊗n ⊗ Γ(C,La)⊗m ⊗ Γ(C,L⊗(1+a) ⊗M−1)⊗n
= Γ(X,F1)⊗m ⊗ Γ(X,F2)⊗n
by the Künneth formula (IV of Lecture 11 [M1]) and Lemma 3.4.
Let Rm,n = Γ(X,F⊗m1 ⊗F
2 ). R =
m,n≥0Rm,n is a standard bigraded K-algebra by
(4). Thus (2) holds.
By the Riemann Roch Theorem, we compute,
(5) h0(C,L⊗r ⊗M−⊗s) = h1(C,L⊗r ⊗M−⊗s) + r − 1,
and for s < 0,
(6) h1(C,L⊗r ⊗M−⊗s) =


1− r r < 0,
1 r = 0, s < 0,
1 r = 1, s = −pt, for some t ∈ N,
0 r = 1, s 6= −pt for some t ∈ N,
0 r = 2, s < 0,
0 r ≥ 3.
We further have
(7) h1(T,A⊗r) =
0 r 6= 0,
2 r = 0,
(8) h0(T,A⊗r) =
0 r < 0,
1 r = 0,
r2 r > 0.
By (2), for n ∈ Z, we have
dimK(H
Q(R)n) =
h1(X,F⊗m1 ⊗F
By the Künneth formula,
H1(X,F⊗m1 ⊗F
∼= H0(T,A⊗(ma+n(1+a)))⊗H1(C,L⊗(ma+n(1+a)) ⊗M−⊗n)
⊕H1(T,A⊗(ma+n(1+a)))⊗H0(C,L⊗(ma+n(1+a)) ⊗M−⊗n).
Thus by (5) - (8), we have for j > 0,
dimK(H
Q(R)−j) =
1 j ≡ 0 (mod) a,
1 j = pt for some odd t ∈ N,
0 otherwise.
and we have the conclusions of Theorem 3.3.
Theorem 3.5 gives an example of failure of tameness of local cohomology with larger
growth.
Theorem 3.5. Suppose that K is an algebraically closed field. Then there exists a normal
standard graded K-algebra R0 over K with dim(R0) = 4, and a normal standard graded
R0-algebra R with dim(R) = 5 such that for j > 0,
dimK(H
Q(R)−j) =
6j if j is even,
0 if j is odd,
where Q =
n>0Rn.
Proof. Let E be an elliptic curve over K, and let q ∈ E be a point. Let L = OE(3q). By
Proposition IV.4.6 [Ha], L is very ample on E, and
(9) ⊕n≥0Γ(E,L⊗n)
is generated in degree 1 as a K-algebra. For n ∈ N,
(10) h0(C,L⊗n) =
0 n < 0,
1 n = 0,
3n n > 0.
(11) h1(C,L⊗n) =
−3n n < 0,
1 n = 0,
0 n > 0.
Let X = E3, with the three canonical projections πi : X → E. Define
F1 = π∗1(L⊗2)⊗ π∗2(L⊗2)⊗ π∗3(L⊗2)
F2 = π∗1(L)⊗ π∗2(L)⊗ π∗3(L⊗2).
Rm,n = Γ(X,F⊗m1 ⊗F
m,n≥0
Rm,n.
By (9) and the Künneth formula, R is standard bigraded. By (2), the fact that ωX ∼= OX
and Serre duality,
dimK(H
Q(R)−j) =
h2(X,F⊗m1 ⊗F
2 ) =
h1(X,F⊗m1 ⊗F
for j ∈ Z.
Now by (10), (11) and the Künneth formula, we have that for n > 0,
h1(X,F⊗m1 ⊗F
2 ) =
0 if 2m+ n 6= 0,
2h0(X,L⊗n) if 2m+ n = 0.
Thus the conclusions of Theorem 3.5 hold. �
The following theorem gives an example of tame, but still rather strange local cohomol-
ogy. Let [x] be the greatest integer in a real number x.
Theorem 3.6. Suppose that K is an algebraically closed field. Then there exists a normal
standard graded K-algebra R0 with dim(R0) = 3, and a normal standard graded R0-algebra
R with dim(R) = 4 such that for j > 0,
dimK(H
Q(R)−j) = 162
dimK(H
Q(R)−j)
where Q =
n>0Rn.
Proof. We use the method of Example 1.6 [Cu]. Let E be an elliptic curve over an
algebraically closed field K, and let p ∈ E be a point. Let X = E × E with projections
πi : X → E. Let C1 = π∗1(p), C2 = π∗2(p) and
∆ = {(q, q) | q ∈ E}
be the diagonal of X. We compute (as in [Cu]) that
(12) (C21 ) = (C
2 ) = (∆
2) = 0
(13) (∆ · C1) = (∆ · C2) = (C1 · C2) = 1.
If N is an ample line bundle on X, then
(14) H i(X,N ) = 0 for i > 0
by the vanishing theorem of Section 16 [M2].
Suppose that L is a very ample line bundle on X, and M is a numerically effective (nef)
line bundle. Then M is 3 regular for L, so that
Γ(X,M⊗L⊗n)⊗ Γ(X,L) → Γ(X,M⊗L⊗(n+1))
is a surjection if n ≥ 3. C1 + 2C2 is an ample divisor by the Moishezon Nakai criterion
(Theorem V.1.10 [Ha]), so that 3(C1+2C2) is very ample by Lefschetz’s theorem (Theorem,
Section 17 [M2]). Let
F1 = OX(9(C1 + 2C2)).
Then OX is 3 regular for OX(3(C1 + 2C2)), so we have surjections
Γ(X,F⊗n1 )⊗ Γ(X,F1) → Γ(X,F
⊗(n+1)
for all n ≥ 1.
∆+C2 is ample by the Moishezon Nakai criterion. Let D = 3(∆+C2). D is very ample
by Lefschetz’s theorem, and thus OX(D)⊗F1 is very ample. Let
F2 = OX(3D)⊗F⊗31 .
OX is 3 regular for OX(D)⊗F1, so we have surjections
Γ(X,F2⊗n)⊗ Γ(X,F2) → Γ(X,F⊗(n+1)2 )
for all n ≥ 1.
for n > 0 and m ≥ 0, we have
F⊗m1 ⊗F
∼= OX(3nD)⊗F⊗(m+3n)1 .
Since D is nef, it is 3 regular for F1, and we have a surjection for all m ≥ 0, n > 0,
Γ(X,F⊗m1 ⊗F
2 )⊗ Γ(X,F1) → Γ(X,F
⊗(m+1)
Rm,n = Γ(X,F⊗m1 ⊗F
We have shown that ⊕m,n≥0Rm,n is a standard bigraded K-algebra. Thus (2) holds.
For m,n ∈ Z, let G = F⊗m1 ⊗ F
2 . As in Example 1.6 [Cu], and by (14) and Serre
Duality (ωX ∼= OX since X is an Abelian variety), we deduce that
1. (G2) > 0 and (G · F1) > 0 imply G is ample and h1(X,G) = h2(X,G) = 0.
2. (G2) < 0 implies h0(X,G) = h2(X,G) = 0.
3. (G2) > 0 and (G · F1) < 0 imply G−1 is ample and h0(X,G) = h1(X,G) = 0.
Let τ2 = −4−
and τ1 = −4 +
Using (12) and (13), we compute
(F21 ) = 2 · 162, (F2)2 = 31 · 162, (F1 · F2) = 8 · 162.
We have
(G2) = 324(m2 + 8mn+ 31
= 324(m− τ1n)(m− τ2n).
(G · F1) = 324(m + 4n).
Since τ2 < −4 < τ1 < 0, for n < 0 and m ∈ Z, we have
1. m > τ2n if and only if G2 > 0 and G · F1 > 0
2. τ1n < m < τ2n if and only if (G2) < 0
3. m < τ1n if and only if (G2) > 0 and (G · F1) < 0.
By the Riemann Roch Theorem for an Abelian surface (Section 16 [M2]),
χ(G) = 1
(G2).
Thus for m ∈ Z and n < 0,
h1(X,G) =
(G2) = −162(m2 + 8mn+ 31
n2) if τ1n < m < τ2n,
0 otherwise.
For n ∈ Z, let σ(n) = dimK(H2Q(Rn)). By (2),
σ(n) =
h1(X,F⊗m1 ⊗F
For n < 0, we have
σ(n) = −162(
τ1n<m<τ2n
(m2 + 8mn+
n2)).
Setting r = m+ 4n, we have
σ(n) = −162(
n<r<−
(r2 − 1
= −324
r=1 (r
2 − 1
n2) + 81n2
= −324
+ 81n2
= 162
We thus have the conclusions of the theorem. �
4. Strange examples of Rees Algebras
Let notation and assumptions be as in Section 2. Since F1 is ample, there exists l > 0
such that Γ(X,F⊗l1 ⊗ F
2 ) 6= 0. Thus we have an embedding F2 ⊗ F
1 ⊂ OX . Let
A = F2 ⊗F−l1 , which we have embedded as an ideal sheaf of X. For j ≥ 0 and i ≥ jl, let
Tij = Γ(X,F⊗i1 ⊗A⊗j) = Ri−jl,j.
For j ≥ 0, let Tj =
i≥jl Tij and T =
j≥0 Tj . Let B =
j>0 Tj. R
∼= T as graded rings
over R0 ∼= T0, although they have different bigraded structures. Thus for all i, j we have
(15) H iB(T )j
∼= H iQ(R)j .
T1 is a homogeneous ideal of T0, and T is the Rees algebra of T1. Thus all of the exam-
ples of Section 3 can be interpreted as Rees algebras over normal rings T0 with isolated
singularities.
We thus obtain the following theorems from Theorems 3.2 - 3.6. Theorems 4.1, 4.2 and
4.3 give examples of Rees algebras with non tame local cohomology.
Theorem 4.1. Suppose that K is an algebraically closed field. Then there exists a normal,
standard graded K algebra T0 with dim(T0) = 3 and a graded ideal A ⊂ T0 such that the
Rees algebra T = T0[At] of A is normal, and for j > 0,
dimK(H
B(T )−j) =
2 if j is even,
0 if j is odd.
where B is the graded ideal AtT of T .
Theorem 4.2. Suppose that p is a prime number such that p ≡ 2(mod)3 and p ≥ 11.
Then there exists a normal standard graded K-algebra T0 over a field K of characteristic
p with dim(T0) = 4, and a graded ideal A ⊂ T0 such that the Rees algebra T = T0[At] of
A is normal, and for j > 0,
dimK(H
Q(T )−j) =
1 if j ≡ 0(mod)(p + 1),
1 if j = pt for some odd t ≥ 0,
0 otherwise,
where B is the graded ideal AtT of T . We have pt ≡ −1(mod)(p + 1) for all odd t ≥ 0.
Theorem 4.3. Suppose that K is an algebraically closed field. Then there exists a normal,
standard graded K-algebra T0 with dim(T0) = 4 and a graded ideal A ⊂ T0 such that the
Rees algebra T = R0[At] of A is normal, and for j > 0,
dimK(H
B(T )−j) =
6j if j is even,
0 if j is odd,
where B is the graded ideal AtT of T .
Theorem 4.4. Suppose that K is an algebraically closed field. Then there exists a normal
standard graded K algebra T0 with dim(T0) = 3, and a graded ideal A ⊂ T0 such that the
Rees algebra T = T0[At] of A is normal, and for j > 0,
dimK(H
B(T )−j) = 162
dimK(H
B(T )−j)
where B is the graded ideal AtT of T .
By localizing at the graded maximal ideal of T0, we obtain examples of Rees algebras
of local rings with strange local cohomology.
In all of these examples, T0 is generalized Cohen Macaulay, but is not Cohen Macaulay.
This follows since in all of these examples,
H2P0(R0)0
∼= H1(X,OX ) 6= 0.
5. Local duality in the examples
The example of [CH], giving failure of tameness of local cohomology, is stated in The-
orem 3.1 of this paper. The proof of [CH] uses the bigraded local duality theorem of
[HR], which now follows from the much more general bigraded local duality theorem, The-
orem 1.5 and Corollary 1.7 of this paper, to conclude that in our situation, where R is
generalized Cohen Macaulay,
(16) (Hd−iQ (ωR)−j)
∗ ∼= H iP (R)j
for j ≫ 0.
In [CH], the formula
H iP (R)j
∼= H iP0(Rj)
i−1(X, R̃j(m))
i−1(X,F⊗m1 ⊗F
for i ≥ 2 and j ≥ 0 is then used with formula (1) of [CH] ((3) of this paper) to prove
Theorem 3.1.
In Section 2 we derive (2) from which we directly compute the local cohomology in the
examples of this paper. We make essential use of Serre duality on X in computing the
examples.
In this section, we show how (16) can be obtained directly from the geometry of X and
V , and how this formula can be directly interpreted as Serre duality on X.
Let notation be as in Section 2, so that K is an algebraically closed field, F1 and F2
are very ample line bundles on the nonsingular variety X.
Let ωR be the dualizing module of R, and let ωX be the canonical bundle of X (which
is a dualizing sheaf on X). For a K module W , let W ′ = HomK(W,K).
Lemma 5.1. We have that
(ωR)ij =
Γ(X,F⊗i1 ⊗F
2 ⊗ ωX) if i ≥ 1 and j ≥ 1
0 otherwise.
Set (ωR)
j∈Z(ωR)i,j , a graded R
0 module. The sheafification of (ωR)
i on X is
(18) (̃ωR)i =
F⊗i1 ⊗ ωX if i ≥ 1
0 if i ≤ 0.
Set (ωR)j =
i∈Z(ωR)i,j , a graded R0 module. The sheafification of (ωR)j on X is
(19) (̃ωR)j =
F⊗j2 ⊗ ωX if j ≥ 1
0 if j ≤ 0.
Proof. Give R the grading where the elements of degree e in R are [R]e =
i+j=eRij.
We have realized R (with this grading) as the coordinate ring of the projective embed-
ding of V = P(F1 ⊕F2) by the very ample divisor OV (1), with projection π : V → X.
Let ωV be the canonical line bundle on V . We first calculate ωV . Let f be a fiber of
the map π : V → X. By adjunction, we have that (f · ωV ) = −2. Since
Pic(V ) ∼= ZOV (1) ⊕ π∗(Pic(X)),
we see that there exists a line bundle G on X such that
ωV ∼= OV (−2)⊗ π∗(G).
The natural split exact sequence
(20) 0 → F2 → F1 ⊕F2 → F1 → 0
determines a section X0 of X, such that π∗ of the exact sequence
0 → OV (1)⊗OV (−X0) → OV (1) → OV (1) ⊗OX0 → 0
is (20) (Proposition II.7.12 [Ha]). Thus
OV (1) ⊗OV (−X0) ∼= π∗(F2)
OV (1)⊗OX0 ∼= F1.
By adjunction, we have that the canonical line bundle of X0 is
∼= ωV ⊗OV (X0)⊗OX0 .
Putting the above together, we see that
G ∼= F1 ⊗F2 ⊗ ωX .
ωV ∼= OV (−2)⊗ π∗(F1 ⊗F2 ⊗ ωX).
We realize R as a bigraded quotient of a bigraded polynomial ring
S = K[x1, . . . , xm, y1, . . . , yn],
with deg(xi) = (1, 0) for all i and deg(yj) = (0, 1) for all j. Viewing S as a graded K-
algebra with the grading determined by d(xi) = d(yj) = 1 for all i, j, we have a projective
embedding V ⊂ P = Proj(S). Since V is nonsingular, we see from Section III.7 of [Ha]
ωV ∼= ExtrP(OV ,Op(−e))
where e = m+ n is the dimension of S, and r = e− dim(R). ωR is defined as
ωR = *Ext
S(R,S(−e)) ∼=
ExtrP(OV ,OP(m− e)).
For m ≫ 0,
Γ(P, ExtrP(OV ,Op(m− e))) ∼= ExtrP(OV ,OP(m− e))
(by Proposition III.6.9 [Ha]). Thus ωR and
Γ∗(ωV ) =
Γ(V, ωV (m))
are isomorphic in high degree. Since both modules have depth≥ 2 at the maximal bigraded
ideal of R, we see that
ωR ∼= Γ∗(ωV ).
m∈Z Γ(V, ωV (m))
m∈Z Γ(V,OV (m− 2)⊗ π∗(F1 ⊗F2 ⊗ ωX)).
Since a fiber f of π satisfies (f ·OV (m− 2)⊗π∗(F1⊗F2)) < 0 if m < 2, we see that (with
this grading) [ωR]m = 0 if m < 2 and For m ≥ 2, we have
[ωR]m = Γ(X,S
m−2(F1 ⊕F2)⊗F1 ⊗F2)
i+j=m−2 Γ(X,F
⊗(i+1)
⊗(j+1)
2 ⊗ ωX).
The conclusions of the lemma now follow. �
Suppose that 2 ≤ i ≤ d − 2. Since F1 and F2 are ample, and d − (i + 1) > 0, there
exists a natural number n0 such that
(21) Hd−(i+1)(X,F⊗m1 ⊗Fn2 ⊗ ωX) = 0
for n ≥ n0 and all m ≥ 0.
By (18), we have graded isomorphisms
(22) H iQ(ωR)n
H i−1(X,F⊗m1 ⊗F
2 ⊗ ωX)
for n ∈ Z.
By Serre duality,
(23) H iQ(ωR)n
(Hd−i−1(X,F−⊗m1 ⊗F
By (21), there exists n0 such that
(24) H iQ(ωR)−n
(Hd−i−1(X,F−⊗m1 ⊗F
for n ≥ n0.
Now apply the functor L∗ = HomK(L,K) on graded R0-modules, with the grading
(L∗)i = HomK(L−i,K)
to (24), and compare with (17), to obtain
(25) Hd−iP (R)n
∼= (H iQ(ωR)−n)∗
for n ≥ n0, from which (16) immediately follows.
We can now verify that Theorem 3.1 is in fact true for all j > 0, using (22) and (3).
We finally comment that an alternate proof of Theorem 3.2 for j ≫ 0 is obtained
from Theorem 3.1, Formulas (2) and (22), the fact that X is an Abelian variety so that
ωX ∼= OX , and the observation that
h1(X,F−⊗n2 ) = h
1(X,F⊗n2 ) = 0
for n > 0.
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E-mail address: chardin@math.jussieu.fr
Dale Cutkosky, Mathematics Department, 202 Mathematical Sciences Bldg, University
of Missouri, Columbia, MO 65211 USA
E-mail address: dale@math.missouri.edu
Jürgen Herzog, Fachbereich Mathematik und Informatik, Universität Duisburg-Essen,
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Hema Srinivasan, Mathematics Department, 202 Mathematical Sciences Bldg, University
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E-mail address: srinivasanh@math.missouri.edu <srinivasan@math.missouri.edu>
