Journal Article10.1080/00927879908826469
Almost finite modules
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TL;DR: In this article, the homological properties of an almost finite module of finite flat dimension over a commutative noetherian local ring R and S have been investigated, and it is shown that an S-module M is almost finite over R if it is finitely generated over S (the R-structure on M is induced by ϕ).
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Abstract: Assume that ϕ(R, m, k) → (S, n, l) is a local homomorphism between commutative noetherian local rings R and S. We say that an S-module M is almost finite over R if it is finitely generated over S (the R-structure on M is induced by ϕ). We investigate the homological behaviour of such modules, as well as various properties of the rings R and S in the presence of an almost finite module of finite flat dimension over R.
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G-dimension over local homomorphisms. Applications to the Frobenius endomorphism
TL;DR: In this article, a theory of G-dimension for modules over local homomorphisms was developed, which encompasses the classical theory of g-dimension of finite modules for finite modules over the local rings, and it was shown that a local ring R of characteristic p is Gorenstein if and only if it pos- sesses a nonzero finite module of finite projective dimension that has finite G-dimensional when considered as an R-module via some power of the Frobenius endomorphism of R.
Finiteness in derived categories of local rings
TL;DR: A number of finiteness results for classical homological invariants, such as flat dimension, injective dimension, and Gorenstein dimension, have been established in this paper.
83
Homology over local homomorphisms
TL;DR: In this paper, the notions of Betti numbers and Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism φ: R → S.
•Posted Content
Homology over local homomorphisms
TL;DR: In this article, the notions of Betti numbers and Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism f: R --> S. Various techniques are developed to study the new invariants and to establish their basic properties.
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•Posted Content
Finiteness in derived categories of local rings
TL;DR: A number of finiteness results for classical homological invariants, such as flat dimension, injective dimension, and Gorenstein dimension, have been established in this article.
39
References
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Commutative Ring Theory
Hideyuki Matsumura,Miles Reid +1 more
- 30 Jun 1989
TL;DR: In this article, the authors introduce the notion of complete local rings and apply it to a wide range of applications, including: I-smoothness, I-flatness revisited, and valuation rings.
4.7K
Cyclic purity versus purity in excellent Noetherian rings
TL;DR: In this paper, the authors give a characterization of Noetherian rings R such that whenever R is ideally closed in an extension algebra S, then R is pure in S. In fact, R has this property if and only if the completion (A, m) of each local ring of R at a maximal ideal has the following equivalent properties: (i) For each integer N > 0 there is an m-primary irreducible ideal IN C mN. (ii) Either dim^ = 0 and A is Gorenstein or else depth A > 1 and there
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