Hom and Ext, Revisited (original) (raw)
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A generalization of the finiteness problem in local cohomology modules
Proceedings Mathematical Sciences, 2009
Let a be an ideal of a commutative Noetherian ring R with non-zero identity and let N be a weakly Laskerian R-module and M be a finitely generated R-module. Let t be a non-negative integer. It is shown that if H i a (N) is a weakly Laskerian R-module for all i < t, then Hom R (R/a, H t a (M, N)) is weakly Laskerian R-module. Also, we prove that Ext i R (R/a, H t a (N)) is weakly Laskerian R-module for all i = 0, 1. In particular, if Supp R (H i a (N)) is a finite set for all i < t, then Ext i R (R/a, H t a (N)) is weakly Laskerian R-module for all i = 0, 1.
Proc Amer Math Soc, 1974
Let M be a finitely generated module over a (not necessarily commutative) local Artin algebra (R, 3JÎ) with 9Jl2=0. It is known that when R is Gorenstein (i.e. of finite injective dimension) M=2AffiSA/3JÎ. For R not Gorenstein we describe all M with ExtHW, R)=0 and show that Ext'W, R)=0 for some i>l if and only if M is free. It follows that for R not Gorenstein all reflexives are free. We also calculate the lengths of all the Ext*'(A^, R). As an application we show that if {R, 9Ji) is a commutative Cohen-Macaulay local ring of dimension d which is not Gorenstein, if R/Wl2 is Artin and (x¡, ■ ■ ■, xd) is a system of parameters with 9JÏ2 contained in the ideal (*i, ■ • • , xd) and if M is a finitely generated .R-moduIe with ExV(M, R)=0 for 1 ¿i<2d+2, then M is free. We call (R, 93Î) a local Artin algebra if A/93Î is a division ring, where 9JÍ is the lacobson radical of R, if the center of R is an Artin ring, and if R if a finitely generated module over its center. We say R is Gorenstein if it is of finite injective dimension as an i?-module. Throughout this paper all modules will be finitely generated. Let (R, 931) be a local Artin algebra. It is well known that every finitely generated left Ä-module M has a projective cover (i.e. an epimorphism <p:P^>-M minimal in the sense that Ker tp£93ÎP) which, is unique up to isomorphism [3], and that M has no projective (free) direct summands if and only if M* = Homie(M, R) is isomorphic to Homñ(M, 931) [2]. Further, for each finitely generated left Ä-module M there is a minimal presentation-i.e. an exact sequence Fl-^-F0-^-M->0 with <p0:F0-*-M and (px : i*\^>-Ker q>0 projective covers, F0 and Fx free and finitely generated. We use this (unique) minimal presentation to define the parameters Sm and rM. Definition. Let M be a finitely generated left module over a local Artin algebra (R, 93Î). Let Sm-^(^o/93i^o) tne number of generators of M, rM = /(i71/93ÎF,i) the number of relations of M, where /() means the length as a left i?-module.
On vanishing of certain Ext modules
Journal of the Mathematical Society of Japan, 2008
Let R be a Noetherian local ring with the maximal ideal m and dim R = 1. In this paper, we shall prove that the module Ext 1 R (R/Q, R) does not vanish for every parameter ideal Q in R, if the embedding dimension v(R) of R is at most 4 and the ideal m 2 kills the 0 th local cohomology module H 0 m (R). The assertion is no longer true unless v(R) ≤ 4. Counterexamples are given. We shall also discuss the relation between our counterexamples and a problem on modules of finite G-dimension.
On the finiteness properties of local cohomology modules
2006
Let a denote an ideal in a regular local (Noetherian) ring R and let N be a finitely generated R-module with support in V (a). The purpose of this paper is to show that all homomorphic images of the R-modules Ext j R (N, H i a (R)) have only finitely many associated primes, for all i, j ≥ 0, whenever dim R ≤ 4 or dim R/a ≤ 3 and R contains a field. In addition, we show that if dim R = 5 and R contains a field, then the R-modules Ext j R (N, H i a (R)) have only finitely many associated primes, for all i, j ≥ 0.
On the cofiniteness properties of certain general local cohomology modules
Acta Scientiarum Mathematicarum
Let R be a commutative Noetherian ring, Φ a system of ideals of R, and M a finitely generated R-module. Suppose that a∈Φ and t is a non-negative integer. It is shown that if Ext R i (R/a,H Φ j (M)) is finitely generated for all i and all j<t, then Ext R i (R/a,H Φ t (M)) is finitely generated for i=0,1. In particular, if R is a local ring of dimension at most 2, then Ext R i (R/a,H Φ j (M)) is finitely generated for all i, j.
A Criterion for Regularity of Local Rings
It is proved that a noetherian commutative local ring A containing a field is regular if there is a complex M of free A-modules with the following properties: M i = 0 for i / ∈ [0, dim A]; the homology of M has finite length; H 0 (M) contains the residue field of A as a direct summand. This result is an essential component in the proofs of the McKay correspondence in dimension 3 and of the statement that threefold flops induce equivalences of derived categories.
Finiteness of local cohomology modules over rings of small dimension
Houston journal of mathematics, 2011
Let R be a commutative Noetherian local ring of dimension d, a an ideal of R, and M , N two finitely generated R-modules. We prove that if d ≤ 2, then Ext p R (M, H q a (N)) is a-cofinite for all p, q ≥ 0. Also, if d ≤ 3 then the set of associated primes of any quotient of Ext p R (R/a, H q a (M, N)) and Ext t R (R/a, Ext p R (M, H q a (N))) are finite for all p, q, t ≥ 0.
Big projective modules are free
Illinois Journal of Mathematics - ILL J MATH, 1963
BY HYMA BASS 1. Introduction Finitely generated proiective modules rise significantly in certain geometric nd rithmetic questions. We shll show here that nonfinitely generated proiective modules, in contrast, invite little interest; for we show that n obviously necessary "connectedness" condition for such module to be free is lso sufficient. More precisely, cll n R-module P uniformly-big, where b is n infinite crdinl, if (i) P cn be generated by b elements, nd (ii) PlOP requires generators for ll two-sided ideals R). A free module with bsis of elements is mnifestly uniformly-big. Our min result (Corollary 3.2) sserts, conversely, that, with suitable chain conditions on R, a uniformly big projective R-module is free. Finally, we sk, for wht R re 11 nonfinitely generated proiective modules uniformly big? For commutative rings, the nswer is quite stisfctory; with mild ssumptions one requires only that spec (R) be connected (i.e., that there exist no nontrivil idempotents). For R Zr, with r finite group, Swan (unpublished) hs established this conclusion when r is solvable, ad it is undoubtedly true in general. Our method relies on two bsic tools. One, nturlly enough, is Kplnsky's remarkable theorem [2, Theorem 1] which sserts that every pro-]ective module is direct sum of countbly generated modules. The second is n elegant little swindle, observed several years go by Eilenberg, nd which might well hve sprung from the brow of Brry M:zur. It is this result, recorded below, which permits us to wive the delicate rithmetic questions which plague the finitely generated cse. EILENBERG'S LEMMA. If P @ Q F with F a nonfinitely generated free module, then P @ F _N F.
Finitely Generated Flat Modules and a Characterization of Semiperfect Rings
Communications in Algebra, 2003
For a ring S, let K 0 (FGFl(S)) and K 0 (FGPr(S)) denote the Grothendieck groups of the category of all finitely generated flat S-modules and the category of all finitely generated projective S-modules respectively. We prove that a semilocal ring R is semiperfect if and only if the group homomorphism K 0 (FGFl(R)) → K 0 (FGFl(R/J(R))) is an epimorphism and K 0 (FGFl(R)) = K 0 (FGPr(R)).