AN INTRODUCTION TO D-MODULES WITH APPLICATIONS TO LOCAL COHOMOLOGY

WITH APPLICATIONS TO LOCAL COHOMOLOGY

Craig Huneke, in a Research Notes in Mathematics of 1992, raised the following question:

R commutative noetherian ring, I ⊆ R an ideal. If M is a finitely

generated (f.g.) R-module, is Ass(H _I ⁱ (M)) a finite set for all i ?

The point of the question above is that H _I ⁱ (M) may be not finitely generated. In some cases however the answer is clearly positive:

(i) M f.g. ⇒ H _I ⁰ (M) f.g.; in particular, Ass(H _I ⁰ (M)) is finite.

(ii) If I = m is maximal, then Ass(H _m ⁱ (M)) ⊆ {m} for all i .

A bit of history of the problem...

(i) Huneke-Sharp, Trans. Amer. Math. Soc. 1993: If R is regular and char(R) > 0, then then for any ideal | Ass(H _I ⁱ (R))| < ∞.

(ii) Lyubeznik, Invent. Math. 1993: If R is a regular local ring containing Q, then for any ideal | Ass(H _I ⁱ (R))| < ∞.

(iii) Lyubeznik, Comm. Algebra 2000: If R is an unramified regular local ring of mixed characteristic, then for any ideal

| Ass(H _I ⁱ (R))| < ∞.

(iv) Bhatt-Blickle-Lyubeznik-Singh-Zhang, 2012: If

R = Z[x 1 , . . . , x n ], then for any ideal | Ass(H _I ⁱ (R))| < ∞.

(iv) Brodmann-Lashgari, Proc. Amer. Math. Soc. 2000: If g is grade(I , M), then Ass(H _I ¹ (M)) and Ass(H _I ^g (M)) are finite.

(v) Singh, Math. Res. Lett. 2000: If R = Z[u, v , w , x , y , z ] (ux + vy + wz) and I = (x , y , z) ⊆ R, then H _I ³ (R) has p-torsion elements for infinitely many primes. Consequently Ass(H _I ³ (R)) is infinite.

(vi) Katzman, J. Algebra 2002: For any field K , if R is the ring K [s, t, u, v , x , y ]

(su ² x ² + (s + t)uvxy + tv ² y ² ) , then | Ass(H _{(x ,y )} ² (R))| = ∞.

(vii) Takagi-Takahashi, Math. Res. Lett. 2007: If R has positive

characteristic, is Gorenstein and has F -finite representation

type, then for any ideal | Ass(H _I ⁱ (R))| < ∞.

In the proof of Huneke-Sharp (R regular of positive characteristic), it is crucial that the iterated Frobenius maps F ^e : R → R , sending r to r ^p

, are flat and that {F ^e (I )R} _e∈N is cofinal with {I ^k } _k∈N . A family of ring homomorphisms with these two properties does not exist for R = Q[[x 1 , . . . , x n ]]. In these lectures we will explore the machinery used by Lyubeznik to prove his first theorem:

If R is a regular local ring containing Q and I ⊆ R is an ideal, then Ass(H _I ⁱ (R)) is finite for every i .

To this aim, we need to introduce the theory of D-modules.

In these seminars I’ll follow some lectures given by Anurag Singh at

the Local Cohomology Workshop held in Mumbai (India) in 2011.

Let A be a (possibly noncommutative) ring. A left A-module M is noetherian if one of the following equivalent conditions holds:

(i) All submodules are finitely generated.

(ii) ACC holds on submodules.

(iii) ACC holds on f.g. submodules.

EXAMPLE: The free K -algebra A = K hx , y i on x and y is not noetherian: In fact the left ideal P

i ∈N Axy ⁱ is not f.g..

Let K be a field in the center of A.

A filtration on A is a family F = {F _i } _{i ≥−1} such that:

(i) F _i are finite-dimensional K -vector spaces;

(ii) F −1 = {0} and 1 ∈ F 0 ;

(iii) F −1 ⊆ F ₀ ⊆ F ₁ ⊆ . . . ⊆ A and S

i ≥−1 F _i = A.

(iv) F _i · F _j ⊆ F _{i +j} .

From now on we will fix the filtration F and assume that gr ^F A = L

i ≥0 F _i /F _{i −1} is a commutative noetherian ring.

Such rings are called almost commutative.

Let M be a left A-module.

A filtration on M is a family G = {G _i } _{i ≥m} , m ∈ Z, such that:

(i) G _i are finite F 0 -modules;

(ii) G _m = {0};

(iii) G _m ⊆ G _m+1 ⊆ G _m+2 ⊆ . . . ⊆ M and S

i ≥m G _i = M.

(iv) F _i · G _j ⊆ G _{i +j} .

We will denote by gr ^G M the gr ^F A-module L

i ≥m+1 G _i /G _{i −1} . G is called a good filtration if gr ^G M is finitely generated over gr ^F A.

REMARK: If M is finitely generated as a left A-module by m 1 , . . . , m k , then G s = P k

i =1 F _s m i is a good filtration on M.

LEMMA: Let M be a left A-module, and G and H filtrations on it.

(i) G is good if and only if there is a j ₀ ∈ Z such that:

F _i · G _j = G _{i +j} ∀ i ≥ 0, j ≥ j ₀ .

(ii) If G is good, then M is finitely generated over A.

(iii) If G is good, then there exists r ∈ Z such that:

G _t ⊆ H _t+r ∀ t ∈ Z.

If also H is good, then there exists s ∈ Z such that:

H _t−s ⊆ G _t ⊆ H _t+s ∀ t ∈ Z.

Proof: (i). Put f F _j = L

i ≤j F _i /F _{i −1} and e G _j = L

i ≤j G _i /G _{i −1} . If G is good, then there exists j 0 ∈ Z such that

f F _i · e G _j = g G _{i +j} ∀ i ≥ 0, j ≥ j ₀

This means that, for every i ≥ 0, j ≥ j ₀ , F _i · G _j + G _{i +j −1} = G _{i +j} . By inducting on i we can assume G _{i +j −1} = F _{i −1} · G _j (1 ∈ F ₀ ).

This yields F i · G _j = G i +j since F i −1 ⊆ F _i . The converse is trivial.

(ii) follows immediately from (i).

(iii) Let j 0 ∈ Z be as in (i). Since G and H are filtrations, there exists r ∈ Z such that G j

⊆ H _r . So

G _t ⊆ G _t+j

= F t · G _j

⊆ F _t · H _r ⊆ H _t+r . 

Assume that F ₀ = K and gr ^F A is generated in degree 1 over K. If G is a good filtration on a left A-module M, then dim _K G _i /G i −1 is a polynomial in i for i  0. This implies that dim _K G _i is also a polynomial in i for i  0. Let us call such a polynomial P G . The polynomial P G is not an invariant of M, depending on the good filtration G. However, given two good filtrations G and H on M, the degrees and the leading coefficients of P G and of P H are the same! This follows immediately from point (iii) of the lemma.

DEFINITION: Let M be a finitely generated left A-module. By considering a good filtration G on it, we define:

I dim(M) = degree of P G .

I e(M) = dim(M)!·(leading coefficient of P G ).

LEMMA: Let M be a left A-module with a good filtration G.

Given a short exact sequence of left A-modules 0 → M ⁰ → M → M ⁰⁰ → 0, set G ⁰ = G ∩ M ⁰ and G ⁰⁰ = Im(G). Then

0 → gr ^G

M ⁰ → gr ^G M → gr ^G

M ⁰⁰ → 0 is an exact sequence of gr ^F A-modules, G ⁰ and G ⁰⁰ are good filtrations and:

I dim(M) = max{dim(M ⁰ ), dim(M ⁰⁰ )}.

I If dim(M ⁰ ) = dim(M ⁰⁰ ), then e(M) = e(M ⁰ ) + e(M ⁰⁰ ).

In particular, if M is a f.g. left A-module, then it is noetherian.

Let K be a field of characteristic 0 and D be the Weil algebra, namely the free K -algebra K hx ₁ , . . . , x _n , ∂ ₁ , . . . , ∂ _n i mod out by



 

 

[x _i , x _j ] = x _i x _j − x _j x _i = 0 [∂ i , ∂ j ] = ∂ i ∂ j − ∂ _j ∂ i = 0 [∂ _i , x _j ] = ∂ _i x _j − x _i ∂ _j = δ _ij

Given two vectors α = (α 1 , . . . , α n ) and β = (β 1 , . . . , β n ) in N ⁿ , we write x ^α ∂ ^β for x ₁ ^α

· · · x _n ^α

∂ ₁ ^β

· · · ∂ ^β _n

. It is easy to see that:

{x ^α ∂ ^β : α, β ∈ N ⁿ }

is a K -basis of D, known as the Poincar´ e-Birkhoff-Witt basis.

The Bernstein filtration F on D is defined as F −1 = 0 and F _s = hx ^α ∂ ^β : |α| + |β| ≤ si ∀ s ∈ N.

Since the only nonzero bracket decreases the “degrees”, gr ^F D is commutative. More precisely:

gr ^F D = K [x ₁ , . . . , x _n , ∂ ₁ , . . . , ∂ _n ].

In particular, dim(D) = 2n and e(D) = 1.

THEOREM (Bernstein) Let M 6= 0 be a finitely generated left D-module. Then n ≤ dim(M) ≤ 2n.

Proof: Let G be a good filtration on M. We want to show that F _s −→ Hom _K (G _s , G _2s )

mapping f to m 7→ fm is injective for all s ∈ N. If s = 0 it is OK.

Suppose t is the least positive integer such that exists 0 6= f ∈ F t

with f · G _t = 0. For sure there exists i such that either x _i or ∂ _i occurs in f . If ∂ _i occurs in f , then [x _i , f ] is a nonzero element of F _t−1 . Then there is m ∈ G t−1 ⊆ G _t such that [x i , f ] · m 6= 0.

However [x _i , f ] · m = x _i fm − fx _i m ∈ x _i (f · G _t ) − f · G _t = 0. If x _i occurs in f ... So the maps above are injective. In particular:

pol. of degree 2n ∼

dim _K (F _s ) ≤ (dim _K (G _s )) · (dim _K (G _2s ))



A finitely generated left D-module M is called holonomic if either M = 0 or dim(M) = n.

EXAMPLES: (i) S = K [z 1 , . . . , z n ] is a left D-module by putting:

x _i · f = z _i f , ∂ _i · f = ∂f

∂z i

∀ f ∈ S.

Obviously G _s = hf ∈ S : deg(f ) ≤ si defines a good filtration on S . So S is holonomic.

(ii) Let m = (z 1 , . . . , z m ) ⊆ S . H _m ⁿ (S ) is a left D-module by:

∂ _i ^k · 1

z ₁ · · · z _n = (−1) ^k k!

z ₁ · · · z _{i −1} z _i ^k+1 z _{i +1} · · · x _n .

Clearly G _s =

 1

z ₁ · · · z _n · u : u monomial of S , deg(u) ≤ s

defines

a good filtration on H _m ⁿ (S ). In particular H _m ⁿ (S ) is holonomic.

Let S = K [z ₁ , . . . , z _n ] be the polynomial ring in n variables over K . Clearly there is a K -algebra homomorphism (indeed an inclusion):

S − → D ^ι z _i 7→ x _i

From this, we can (and will) view any left D-module M as an S -module via restriction by ι. In particular, we are allowed to define the set of associated primes over S of any D-module M:

Ass _S (M) = {p ∈ Spec(S ) : p = 0 : _S m for some 0 6= m ∈ M}

The first goal of today is to show the following:

THEOREM: If M is holonomic, then | Ass _S (M)| ≤ e(M) < ∞.

f · ∂ _i = ∂ _i · f − ∂f

∂z i

∀ i ∈ {1, . . . , n}.

Doing an induction on s, we get

f ^s · ∂ _i = ∂ _i · f ^s − sf ^s−1 ∂f

∂z _i . This shows the following:

REMARK: Let M be a left D-module and I an ideal of S . By considering M as an S -module, we can form the S -module

H _I ⁰ (M) = {m ∈ M : I ^s m = 0 for some s ∈ N}.

Then H _I ⁰ (M) is a D-submodule of M. 

Given a shot exact sequence of left D-modules 0 → M ⁰ → M → M ⁰⁰ → 0 we have that:

(i) M is holonomic if and only if M ⁰ and M ⁰⁰ are holonomic.

(ii) If (i) holds, then e(M) = e(M ⁰ ) + e(M ⁰⁰ ).

PROPOSITION: If M is holonomic, then length _D (M) ≤ e(M).

Proof: Take a chain of D-modules:

0 = M 0 ( M 1 ( . . . ( M ` = M.

From the exact sequences 0 → M _{i −1} → M _i → M _i /M _{i −1} → 0, we infer that M i and M i /M i −1 are nonzero holonomic D-modules for all i = 1, . . . , ` and that

0 < e(M ₁ ) < e(M ₂ ) < . . . < e(M _` ) = e(M).



We are now ready to show the following for left D-modules M:

THEOREM: If M is holonomic, then | Ass S (M)| ≤ e(M) < ∞.

Proof: We want to induct on ` = length <sub>D</sub> (M) (≤ e(M)). If ` = 1, take p ∈ Ass _S (M). Then H _p ⁰ (M) is a nonzero D-submodule of M.

Because ` = 1, then we have H _p ⁰ (M) ∼ = M. If q ∈ Ass S (M) = Ass _S (H _p ⁰ (M)) one has p ⊆ q, so by symmetry we deduce q = p.

If ` > 1, take a (nonzero) simple D-submodule N ⊆ M. Of course length <sub>S</sub> (M/N) < `, and by the short exact sequence of D-modules

0 → N → M → M/N → 0

we get that N and M/N are holonomic D-modules. However that above is also an exact sequence of S -modules, therefore we have:

Ass _S (M) ⊆ Ass _S (N) ∪ Ass _S (M/N)



Let I = (f 1 , . . . , f _k ) be an ideal of S = K [z 1 , . . . , z n ], and M an S -module. The i th local cohomology module H _I ⁱ (M) is the (k − i )th homology of the ˇ Chec complex:

0 → M → M

i

M _f

→ M

i <j

M _f

_f

→ . . . → M _f

_···f

→ 0,

where the maps are the natural ones multiplied by a suitable sign.

If each module of the above complex were a holonomic D-module, then also the cohomology would be holonomic, since holonomicity is closed under short exact sequences. So, in this case, H _I ⁱ (M) would have a finite number of associated primes.

Our goal now is to show that S f is a holonomic D-module ∀ f ∈ S.

Recall that, if G is a good filtration on a left D-module, then for any other filtration H on M, there is r ∈ Z such that, for all t ∈ Z, G _t ⊆ H _t+r . In particular, ∀  > 0 ∃ t 0 such that:

(∗) dim _K (H t ) ≥ e(M) − 

dim(M)! t ^dim(M) ∀t ≥ t ₀

LEMMA: Let M be a left D-module with a filtration G. If there is c ∈ R >0 such that dim _K (G _t ) ≤ ct ⁿ for t  0, then M is

holonomic. In particular, it is finitely generated!

Proof: Let M ₀ be a finitely generated D-submodule of M and set

G _t ⁰ = G t ∩ M ₀ ∀ t ∈ Z, which defines a filtration on M 0 . Then

dim(M ₀ ) ≤ n and e(M ₀ ) ≤ n!c by (*). Therefore M has ACC for

finitely generated submodules. So it is finitely generated, and

holonomic again by (*). 

THEOREM: Let M be a holonomic D-module and f ∈ S . Then M f is holonomic.

Proof: Let G be a good filtration on M, and δ the maximum degree of a monomial in the support of f . Define H = {H t } _t on M _f as:

H

=  m

f

: m ∈ G

 .

That ∪ _t H _t = M _f and H _t ⊆ H _t+1 is easy. If m/f ^t ∈ H _t , then obviously x _i · m/f ^t ∈ H _t+1 . Furthermore:

∂

· m

f

= f

· (∂

· m) − tf

∂f /∂z

· m

f

= f · (∂

· m) − t∂f /∂z

· m

f

∈ H

.

So H is a filtration of M _f such that:

dim

(H

) ≤ dim

(G

) ∼ e(M)(1 + δ)

n! t

By the previous lemma M _f is holonomic. 

We saw last week that S has a structure of holonomic D-module.

So by the above theorem S _f is holonomic for all f ∈ S . If I is an ideal of S generated by f 1 , . . . , f k then, by meaning of the ˇ Cech complex, the local cohomology module H _I ⁱ (S ) is a subquotient of:

M

1≤`

<...<`

≤k

S _f

···f

.

In particular H _I ⁱ (S ) is a holonomic D-module. As a consequence:

THEOREM: For any ideal I ⊆ S the set Ass _S (H _I ⁱ (S )) is finite.

More generally, Ass _S (H _I ⁱ (M)) is finite ∀ holonomic D-modules M.

THEOREM: Let m = (z

, . . . , z

) ⊆ S = K [z

, . . . , z

].

(i) Then H

(S ) ∼ = D/Dm as left D-modules. In particular, D/Dm is isomorphic to the injective hull E

(K ) of K = S /m.

(ii) If M is an m-torsion D-module, then M ∼ = L

λ∈Λ

D/Dm as left D-modules. In particular M is an injective S-module.

Proof: (i) The map of D-modules D → H

(S ) sending 1 to [1/z

· · · z

] is surjective, and one can check that its kernel is Dm.

(ii) Consider soc(M) ⊆ M and a K -basis {m

}

of soc(M). This gives rise to the following diagram of S -modules:

L

λ∈Λ

D/Dm − −−−−

→ M

x





x



 L

λ∈Λ

K − −−−−

→ soc(M)

The inclusion on the left is essential, so f is injective. Therefore M ∼ = L

λ∈Λ

D/Dm L C . Since soc(C ) = 0, we infer that C = 0. 

The above result implies that, if H _I ⁱ (S ) is m-torsion, then ∃ s ∈ N:

H _I ⁱ (S ) ∼ = E S (K ) ^s .

There are several interesting situations in which H _I ⁱ (S ) is m-torsion, for example if i > ht(I ) and I defines a smooth projective scheme.

So s is an interesting number. Quite surprisingly, it is an invariant of S /I , we will soon discuss this aspect in more generality. First, let me say that, with not much more effort, we could prove:

THEOREM: injdim(H _I ⁱ (S )) ≤ dim(Supp(H _I ⁱ (S )) (≤ i ).

Recall that the Bass numbers of an S -module M are defined as:

µ _i (p, M) = dim _κ(p) (Ext ⁱ _S

(κ(p), M _p )),

where p is a prime ideal of S . Another way to think at them is the following: Every S -module M admits a minimal injective resolution

0 → M → E ⁰ → E ¹ → E ² . . . Then E ⁱ ∼ = L

p∈Spec(S) E _S (S /p) ^µ

^(p,M) .

THEOREM: A holonomic D-module has finite Bass numbers. In

particular, µ i (p, H _I ^j (S )) is a finite number for all triples p, i , j .

All the results stated for S hold true for any regular local ring R containing K . The point is that b R ∼ = K [[x 1 , . . . , x _n ]]. The algebra of differentials D(K [[x ]], K ) of b R is left-Noetherian and well described, so one can play a similar game to the previous one replacing D by D(K [[x ]], K ). Finally one can descend everything to R, essentially because K [[x ]] is a faithfully flat R-algebra.

Besides all the previous beautiful results, Lyubeznik supplied us new invariants to play with:

DEFINITION-THEOREM: Let A be a local ring containing K . By Cohen-structure theorem we have a surjection K [[x 1 , . . . , x n ]] − → b ^π A.

Denoting by I = Ker(π) and m the maximal ideal of K [[x ]], the

finite numbers µ p (m, H _I ⁿ⁻ⁱ (K [[x ]]) depend only on A, p and i .

These invariants of A are usually denoted by λ _p,i (A) and called the

Lyubeznik’s numbers of A.

(i) The conjecture of Lyubeznik is still unsolved: Ass _R (H _I ⁱ (R)) are finite sets for any regular ring R.

(ii) A question of Lyubeznik: If A is a standard graded K -algebra with maximal ideal m, are λ p,i (A m ) invariants of Proj(A)? (Zhang, Adv. Math. 2011: Yes in positive characteristic).

(iii) One can show that λ _p,i = 0 if i > d = dim(A), p > i , or p ≥ i − 1 and i < depth(A). Is λ i −2,i (A) = 0 for all i < depth(A)?

(iv) Compute the entire table λ _p,i (A m ) for determinantal rings A.