scholarly journals Homological properties of the homology algebra of the Koszul complex of a local ring: Examples and questions

2016 ◽  
Vol 465 ◽  
pp. 399-436 ◽  
Author(s):  
Jan-Erik Roos
Keyword(s):  
Local Ring ◽  
Koszul Complex ◽  
Homology Algebra

Graded Complexes Over Power Series Rings

1986 ◽  
Vol 38 (1) ◽  
pp. 158-178 ◽  
Author(s):  
Paul Roberts
Keyword(s):  
Power Series ◽  
Local Ring ◽  
Simple Structure ◽  
Free Resolution ◽  
Koszul Complex ◽  
Local Rings ◽  
Power Series Ring ◽  
Series Ring ◽  
Noetherian Local Ring ◽  
Power Series Rings

A common method in studying a commutative Noetherian local ring A is to find a regular subring R contained in A so that A becomes a finitely generated R-module, and in this way one can obtain some information about the original ring by applying what is known about regular local rings. By the structure theorems of Cohen, if A is complete and contains a field, there will always exist such a subring R, and R will be a power series ring k[[X1, …, Xn]] = k[[X]] over a field k. In this paper we show that if R is chosen properly, the ring A (or, more generally, an A-module M), will have a comparatively simple structure as an R-module. More precisely, A (or M) will have a free resolution which resembles the Koszul complex on the variables (X1, …, Xn) = (X); such a complex will be called an (X)-graded complex and will be given a precise definition below.

Complexes acyclic up to integral closure

1994 ◽  
Vol 116 (3) ◽  
pp. 401-414
Author(s):  
D. Katz
Keyword(s):  
Local Ring ◽  
Homomorphic Image ◽  
Noetherian Ring ◽  
Integral Closure ◽  
Koszul Complex ◽  
Gorenstein Ring ◽  
Tight Closure ◽  
Commutative Noetherian Ring ◽  
Remarkable Result ◽  
Depth Sensitivity

In [R] D. Rees introduced the notions of reduction and integral closure for modules over a commutative Noetherian ring and proved the following remarkable result. Let R be a locally quasi-unmixed Noetherian ring and I an ideal generated by n elements. Suppose that height (I) = h. Then the ith module of cycles in the Koszul complex on a set of n generators for I is contained in the integral closure of the ith module of boundaries for i > n − h. This result should be considered a dimension-theoretic analogue of the famous depth sensitivity property of the Koszul complex demonstrated by Serre and Auslander-Buschsbaum in the 1950s. At roughly the same time, Hoschster and Huneke introduced the notion of tight closure and thereafter gave a number of theorems in the same (though considerably broader) vein for tight closure. In particular, in [HH] they showed that if R is an equidimensional local ring of characteristic p > 0, which is a homomorphic image of a Gorenstein ring, then for all i > 0, the ith module of cycles is contained in the tight closure of the ith module of boundaries for any complex satisfying the so-called standard rank and height conditions (see the definitions below). Since the tight closure is contained in the integral closure for such rings, the result of Hochster and Huneke extends (in characteristic p) considerably the result of Rees. In fact, their result could be considered a dimension-theoretic analogue of the Buchsbaum-Eisenbud exactness theorem ([BE]), which in a certain sense is the ultimate depth sensitivity theorem. Moreover, using the technique of reduction to characteristic p, Hochster and Huneke have shown that their results hold in equicharacteristic zero as well, whenever the tight closure is defined.

scholarly journals Gorenstein quotients by principal ideals of free Koszul homology

2000 ◽  
Vol 42 (1) ◽  
pp. 51-54 ◽  
Author(s):  
JOSÉ J. M. SOTO
Keyword(s):  
Local Ring ◽  
Unit Element ◽  
Subject Classification ◽  
Koszul Complex ◽  
Noetherian Local Ring ◽  
Principal Ideals ◽  
Koszul Homology ◽  
The Ideal

Let A be a noetherian local ring, x a non-unit element of A, B=A/(x). Let E be the Koszul complex associated to an arbitrary set of generators of the ideal (x) of A. Assume that H1(E) is a free B-module. Then A is Gorenstein if and only if B is also.1991 Mathematics Subject Classification 13H10, 13D03.

scholarly journals Homological Invariants of Local Rings

1963 ◽  
Vol 22 ◽  
pp. 219-227 ◽  
Author(s):  
Hiroshi Uehara
Keyword(s):  
Local Ring ◽  
Betti Numbers ◽  
Residue Field ◽  
Unit Element ◽  
Minimal System ◽  
Local Rings ◽  
Homology Algebra ◽  
Homological Invariants ◽  
Minimal System Of Generators ◽  
The Relationship

In this paper R is a commutative noetherian local ring with unit element 1 and M is its maximal ideal. Let K be the residue field R/M and let {t1,t2,…, tn) be a minimal system of generators for M. By a complex R<T1. . ., Tp> we mean an R-algebra* obtained by the adjunction of the variables T1. . ., Tp of degree 1 which kill t1,…, tp. The main purpose of this paper is, among other things, to construct an R-algebra resolution of the field K, so that we can investigate the relationship between the homology algebra H (R < T1,…, Tn>) and the homological invariants of R such as the algebra TorR(K, K) and the Betti numbers Bp = dimk TorR(K, K) of the local ring R. The relationship was initially studied by Serre [5].

Homology algebra of the Koszul complex of a local Gorenstein ring

1971 ◽  
Vol 9 (1) ◽  
pp. 30-32 ◽  
Author(s):  
L. L. Avramov ◽  
E. S. Golod
Keyword(s):  
Koszul Complex ◽  
Gorenstein Ring ◽  
Homology Algebra

Representations of the fundamental group and the torsor of deformations. Global aspects

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros
Keyword(s):  
Fundamental Group ◽  
Koszul Complex ◽  
Finiteness Conditions ◽  
Inverse Systems ◽  
Logarithmic Geometry

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

On Generalized Regular Local Ring

2015 ◽  
Vol 3 (1) ◽  
pp. 145-152
Author(s):  
Zubayda Ibraheem ◽  
Naeema Shereef
Keyword(s):  
Local Ring ◽  
Regular Local Ring

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

On equimultiplicity

1982 ◽  
Vol 91 (2) ◽  
pp. 207-213 ◽  
Author(s):  
M. Herrmann ◽  
U. Orbanz
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Analytic Spread

This note consists of some investigations about the condition ht(A) = l(A) where A is an ideal in a local ring and l(A) is the analytic spread of A (9).In (4) we proved the following: If R is a local ring and P a prime ideal such that R/P is regular then (under some technical assumptions) ht(P) = l(P) is equivalent to the equimultiplicity e(R) = e(RP). Also for a general ideal A (which need not be prime), the condition ht(A) = l(A) can be translated into an equality of certain multiplicities (see Theorem 0).

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