Commutative ring

Philosophy

Pure mathematics

Discrete mathematics

Commutative property

Local ring

Prime (order theory)

Noetherian ring

Computer science

Integer (computer science)

Organic chemistry

Ideal (ethics)

Mathematics

Direct limit

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Local cohomology

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Algebra homomorphism

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Finitely-generated abelian group

Ring (chemistry)

Journal of Pure and Applied Algebra

Peskine and Szpiro [7] pioneered the use of the Frobenius homomorphism in commutative algebra over commutative Noetherian rings of prime characteristic; in particular, they used it to obtain results about local cohomology modules over an n-dimensional regular local ring A of characteristic p > 0 with respect to an arbitrary ideal 6 of A. For example, their finiteness theorem [7, III, ThCor&me 4.91 gives conditions under which, for a suitable integer s, the local cohomology modules H’,(M) are Artinian for all j 2 n s and all finitely generated A-modules M. Also, they showed in [7, III, Proposition 4.1 and Remarque] that, if 6 is proper and depth(Alb) = d, then H’,( ) = 0 for all j > R d. The purpose of this short note is to show that the methods of Peskine and Szpiro can be used to prove some related results in a short and direct manner: we shall prove that if j is a nonnegative integer for which Exta(A /6, A) = 0, then the local cohomology module HI(A) = 0, and also that, if ExtL(Ai0, A) has finite length, then H{,(A) is Artinian. Part of the interest of these results lies in the fact that H{,(A) is isomorphic to a direct limit of Extension modules,

The Frobenius homomorphism, and local cohomology in regular local rings of positive characteristic