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ON THE HARTSHORNE–SPEISER–LYUBEZNIK THEOREM ABOUT ARTINIAN MODULES WITH A FROBENIUS ACTION
, 2006
"... Abstract. Let R be a commutative Noetherian local ring of prime characteristic. The purpose of this paper is to provide a short proof of G. Lyubeznik’s extension of a result of R. Hartshorne and R. Speiser about a module over the skew polynomial ring R[x, f] (associated to R and the Frobenius homomo ..."
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Abstract. Let R be a commutative Noetherian local ring of prime characteristic. The purpose of this paper is to provide a short proof of G. Lyubeznik’s extension of a result of R. Hartshorne and R. Speiser about a module over the skew polynomial ring R[x, f] (associated to R and the Frobenius homomorphism f, in the indeterminate x) that is both xtorsion and Artinian over R. In the theory of tight closure of ideals in a ddimensional commutative (Noetherian) local ring (R, m) of prime characteristic p, study of properties of the ‘top ’ local cohomology module Hd m(R) related to the Frobenius homomorphism f: R − → R has been a very effective tool: see, for example, K. E. Smith [10, 11]. Some of the properties of Hd m(R) related to f can be neatly described in terms of a natural
Test exponents for modules with finite phantom projective dimension
, 2011
"... Let (R,m) be an equidimensional excellent local ring of prime characteristic p> 0. We give an alternate proof of the existence of a uniform test exponent for any given c ∈ R ◦ and all ideals generated by (full or partial) systems of parameters. This follows from a more general result about the ..."
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Cited by 3 (3 self)
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Let (R,m) be an equidimensional excellent local ring of prime characteristic p> 0. We give an alternate proof of the existence of a uniform test exponent for any given c ∈ R ◦ and all ideals generated by (full or partial) systems of parameters. This follows from a more general result about the existence of a test exponent for any given Artinian Rmodule. If we further assume R is CohenMacaulay, then there exists a test exponent for any given c ∈ R ◦ and all finite length modules with finite (phantom) projective dimension.
[Preliminary Version] AN EMBEDDING THEOREM FOR MODULES OF FINITE (G)PROJECTIVE DIMENSION
"... Abstract. Let M be any finitely generated module of finite projective dimension (respectively, finite Gdimension) over a commutative Noetherian ring R. Then M embeds into a finite direct sum Z of cyclic Rmodules each of which is the quotient of R by an ideal generated by an Rregular sequence. Thi ..."
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Abstract. Let M be any finitely generated module of finite projective dimension (respectively, finite Gdimension) over a commutative Noetherian ring R. Then M embeds into a finite direct sum Z of cyclic Rmodules each of which is the quotient of R by an ideal generated by an Rregular sequence. This can be done so that both Z/M and hence Z have projective dimension (respectively, Gdimension) no more than the projective dimension (respectively, Gdimension) of M. Consequently, we also get a similar embedding theorem for finitely generated modules of finite injective dimension over any CohenMacaulay ring that has a global canonical module. Throughout this paper R is a commutative Noetherian ring with 1. It is wellknown that any quotient module of R modulo an ideal generated by an Rregular sequence has finite projective dimension. The main theorem of the paper is to embed any finitely generated Rmodule with
UNIFORM TEST EXPONENTS FOR RINGS OF FINITE FREPRESENTATION TYPE
"... Let R be a commutative Noetherian ring of prime characteristic p. Assume R (or, more generally, a finitely generated Rmodule N with SuppR(N) = Spec(R)) has finite Frepresentation type (abbreviated FFRT) by finitely generated Rmodules. Then, for every c ∈ R◦, there is a uniform test exponent Q = ..."
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Let R be a commutative Noetherian ring of prime characteristic p. Assume R (or, more generally, a finitely generated Rmodule N with SuppR(N) = Spec(R)) has finite Frepresentation type (abbreviated FFRT) by finitely generated Rmodules. Then, for every c ∈ R◦, there is a uniform test exponent Q = pE for c and for all Rmodules. As a consequence, we show the existence of uniform test exponents over binomial rings (in particular, affine semigroup rings). The existence of uniform test exponents (for all modules) implies that the tight closure coincides with the finitistic tight closure, and tight closure commutes with localization for all Rmodules.