scholarly journals Direct Summands of Syzygy Modules of the Residue Class Field

2008 ◽  
Vol 189 ◽  
pp. 1-25 ◽  
Author(s):  
Ryo Takahashi
Keyword(s):  
Direct Summand ◽  
Local Ring ◽  
Residue Class ◽  
Class Field ◽  
Noetherian Local Ring ◽  
Residue Class Field ◽  
Syzygy Module ◽  
Syzygy Modules

AbstractLet R be a commutative Noetherian local ring. This paper deals with the problem asking whether R is Gorenstein if the nth syzygy module of the residue class field of R has a non-trivial direct summand of finite G-dimension for some n. It is proved that if n is at most two then it is true, and moreover, the structure of the ring R is determined essentially uniquely.

scholarly journals Some criteria for regular and Gorenstein local rings via syzygy modules

2019 ◽  
Vol 18 (05) ◽  
pp. 1950097
Author(s):  
Dipankar Ghosh
Keyword(s):  
Direct Summand ◽  
Local Ring ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Local Rings ◽  
Canonical Module ◽  
Syzygy Module ◽  
Syzygy Modules ◽  
Necessary And Sufficient ◽  
Minimal Multiplicity

Let [Formula: see text] be a Cohen–Macaulay local ring. We prove that the [Formula: see text]th syzygy module of a maximal Cohen–Macaulay [Formula: see text]-module cannot have a semidualizing direct summand for every [Formula: see text]. In particular, it follows that [Formula: see text] is Gorenstein if and only if some syzygy of a canonical module of [Formula: see text] has a nonzero free direct summand. We also give a number of necessary and sufficient conditions for a Cohen–Macaulay local ring of minimal multiplicity to be regular or Gorenstein. These criteria are based on vanishing of certain Exts or Tors involving syzygy modules of the residue field.

scholarly journals The equation y′ = fy in zero residue characteristic

1991 ◽  
Vol 33 (2) ◽  
pp. 149-153
Author(s):  
Alain Escassut ◽  
Marie-Claude Sarmant
Keyword(s):  
Uniform Convergence ◽  
Characteristic Zero ◽  
Residue Class ◽  
Rational Functions ◽  
Closed Field ◽  
Class Field ◽  
Absolute Value ◽  
Residue Class Field ◽  
Analytic Elements ◽  
Residue Characteristic

Let K be an algebraically closed field complete with respect to an ultrametric absolute value |.| and let k be its residue class field. We assume k to have characteristic zero (hence K has characteristic zero too).Let D be a clopen bounded infraconnected set [3] in K, let R(D) be the algebra of the rational functions with no pole in D, let ‖.‖D be the norm of uniform convergence on D defined on R(D), and let H(D) be the algebra of the analytic elements on D i.e. the completion of R(D) for the norm ‖.‖D.

Bounds on Betti Numbers

1982 ◽  
Vol 34 (3) ◽  
pp. 589-592
Author(s):  
Mark Ramras
Keyword(s):  
Natural Number ◽  
Local Ring ◽  
Residue Class ◽  
Asymptotic Properties ◽  
Betti Numbers ◽  
Commutative Local Ring ◽  
Artin Ring ◽  
Residue Class Field ◽  
Image Position ◽  
Free Modules

The Betti numbers βn(k) of the residue class field k = R/m of a commutative local ring (R, m) have been studied for about 20 years, primarily as the coefficients of the Poincaré series of E . Several authors have obtained results about the growth of the sequence {βn(k)}.For example, Gulliksen [3] showed that when R is non-regular, the sequence is non-decreasing. More recently, Avramov [1] studied asymptotic properties of {βn(k)} and found that under certain conditions the growth is exponential, i.e., there is a natural number p such that for all n, βpn(k) ≧ 2n.In this paper, we examine the sequence {βn(M)} for arbitrary finitely generated non-free modules M over any commutative local artin ring R. We establish the following bounds:123where l(X) is the length of X.

A lifting theorem for modular representations

1959 ◽  
Vol 252 (1268) ◽  
pp. 135-142 ◽  
Keyword(s):  
Residue Class ◽  
Sufficient Conditions ◽  
Modular Representations ◽  
Class Field ◽  
Residue Class Field ◽  
Homological Invariants

k is the residue-class field o/p of a ring 0 of p-adic integers. Sufficient conditions are found, I that a given representation p of a group G over k may be lifted to a representation r of G over o, particularly in the case where it is assumed that such a lifting exists for the restriction of p to a given subgroup H of G . The conditions involve certain homological invariants of p .

scholarly journals Ramification Theory for Extensions of Degree p

1971 ◽  
Vol 41 ◽  
pp. 149-168 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Residue Class ◽  
Field Extension ◽  
Quotient Field ◽  
Class Field ◽  
Wild Ramification ◽  
Valuation Rings ◽  
Ramification Theory ◽  
Residue Class Field ◽  
Rank One

The notions of tame and wild ramification lead us to make the following definition.Definition. The quotient field extension of an extension of discrete rank one valuation rings is said to be fiercely ramified if the residue class field extension has a nontrivial inseparable part.

Knowability

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Residue Class ◽  
Class Ii ◽  
Gauss Sums ◽  
Class I ◽  
Class Field ◽  
Residue Class Field

This chapter introduces the Knowability Lemma, which explains when products of Gauss sums associated to elements of a preaccordion are explicitly evaluable as polynomials in q, the order of the residue class field. It considers an episode in the cartoon associated to the short Gelfand-Tsetlin pattern and the three cases that apply according to the Knowability Lemma, two of which are maximality and knowability. Knowability is not important for the proof that Statement C implies Statement B. The chapter discusses the cases where ε‎ is Class II or Class I, leaving the remaining two cases to the reader. It also describes the variant of the argument for the case that ε‎ is of Class I, again leaving the two other cases to the reader.

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