A note on the Annihilator of local cohomology modules in characteristic zero
aa r X i v : . [ m a t h . A C ] J a n . A NOTE ON THE ANNIHILATOR OF LOCAL COHOMOLOGYMODULES IN CHARACTERISTIC ZERO
MAJID EGHBALI
Abstract.
We give an alternative proof to the annihilator of local cohomologyin characteristic zero which was proved by Lyubeznik. Introduction
Let R be a commutative Noetherian ring and I ⊂ R be an ideal. The i thlocal cohomology module of R with support in I is denoted by H iI ( R ). In thepresent note, our main result (Theorem 2.3) provides a different point of view ofAnn R H iI ( R ) = 0 credited to Lyubeznik [3, Corollary 3.6], where R is a regular localring containing a field of characteristic 0. Our way to prove the results is to use theso-called D -modules. This method has played a decisive role in many subsequentstudies in the rings of characteristic zero.Let R be a commutative algebra over a field k of characteristic zero. We de-note by End k ( R ) the k -linear endomorphism ring. The ring of k -linear differentialoperators D R | k ⊆ End k ( R ) generated by the k -linear derivations R → R and themultiplications by elements of R . By a D R | k -module we always mean a left D R | k -module. The injective ring homomorphism R → D R | k that sends r to the map R → R which is the multiplication by r , gives D R | k a structure of R -algebra. Every D R | k -module M is automatically an R -module via this map. The natural actionof D R | k on R makes R a D R | k -module. If R = k [[ x , . . . , x n ]] is a formal powerseries ring of n variables x , . . . , x n over k , then D R | k is left and right Noether-ian. Moreover, D R | k is a simple ring. Noteworthy, the local cohomology module H iI ( R ) , i ∈ Z is a finitely generated D R | k -module. For a more advanced expositionbased on differential operators and undefined concepts the interested reader mightconsult [1]. For brevity we often write D R for D R | k when there is no ambiguityabout the field k . Mathematics Subject Classification.
Key words and phrases.
Annihilator of Local cohomology, Characteristic zero, D-modules.The author is supported in part by a grant from IPM. results Lemma 2.1.
Let R be k [ | x , . . . , x n | ] , a formal power series ring of n variables x , . . . , x n over a field k of characteristic zero. Suppose that H iI ( R ) = 0 , then Ann D R H iI ( R ) = 0 .Proof. Consider the homomorphism(2.1) D R f −→ Hom k ( H iI ( R ) , H iI ( R ))of D R -modules defined by f ( P )( m ) = P m, P ∈ D R and m ∈ H iI ( R ) for all i ∈ Z .The homomorphism f is injective, as D R is a simple ring, i.e. Ann D R H iI ( R ) =ker f = 0. (cid:3) Lemma 2.2.
Let R be as in Lemma 2.1. Let M be both an R module and a D R -module. Then Ann D R M = 0 implies Ann R M = 0 .Proof. Let r ∈ Ann R M be an arbitrary element. As the endomorphism ϕ r : R → R with ϕ r ( s ) = rs for all s ∈ R is an element of D R so from rsM = 0 (for all s ∈ R )we have ϕ r ( s ) M = 0. That is ϕ r ( s ) ∈ Ann D R M = 0, i.e. rs = ϕ r ( s ) = 0, for all s ∈ R . Hence, we have r = 0, as desired. (cid:3) Theorem 2.3.
Let ( R, m ) be a regular local ring containing a field of characteristiczero. Suppose that H iI ( R ) = 0 . Then Ann R H iI ( R ) = 0 .Proof. Suppose k = R/ m , where m is the maximal ideal of R and char ( k ) = 0. Byvirtue of [2, Chapter IX, Appendice 2.] there exists a faithfully flat homomorphismfrom ( R, m ) to a regular local ring ( S, n ) such that S/ n is the algebraic closureof k . As S is faithfully flat over R then the homomorphism is injective so S contains a field. Moreover, it is known that Ann R H iI ( R ) = (Ann S H iIS ( S )) ∩ R and H iI ( R ) ⊗ R S ∼ = H iIS ( S ) = 0, because of faithfully flatness of S . Then, we mayassume that k is algebraically closed. As ˆ R is also faithfully flat R -module, we mayassume that R is complete, so that R = k [[ x , . . . , x n ]] by the Cohen StructureTheorem, where n = dim R . Since, k [[ x , . . . , x n ]] has a D R -module structure so,we are done by Lemma 2.2 and Lemma 2.1. (cid:3) Acknowledgements.
The author is grateful to Gennady Lyubeznik for pointingout an error in a previous version of the paper. Many thanks to Josep `Alvarez-Montaner, who taught me the theory of D-modules, generously.
References [1] J. E. Bj¨ork ,
Rings of differential operators.
Amsterdam North-Holland (1979).[2] N. Bourbaki , ´El´ements de math´ematique.
Alg`ebre commutative. Chapters 8 et 9, SpringerVerlag, (2006).
NNIHILATOR OF LOCAL COHOMOLOGY MODULES 3 [3] G. Lyubeznik,
Finiteness properties of local cohomology modules (an application of D-modules to Commutative Algebra).
Invent. math. , 41–55 (1993).
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran-Iran.
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