Complete Intersection Flat Dimension and the Intersection Theorem

2012 ◽  
Vol 19 (spec01) ◽  
pp. 1161-1166
Author(s):  
Parviz Sahandi ◽  
Tirdad Sharif ◽  
Siamak Yassemi
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Intersection Theorem ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Gorenstein Dimension ◽  
Flat Dimension ◽  
Gorenstein Flat ◽  
Similar Extension

Any finitely generated module M over a local ring R is endowed with a complete intersection dimension CI-dim RM and a Gorenstein dimension G-dim RM. The Gorenstein dimension can be extended to all modules over the ring R. This paper presents a similar extension for the complete intersection dimension, and mentions the relation between this dimension and the Gorenstein flat dimension. In addition, we show that in the intersection theorem, the flat dimension can be replaced by the complete intersection flat dimension.

scholarly journals Torsion in tensor powers of modules

2015 ◽  
Vol 219 ◽  
pp. 113-125
Author(s):  
Olgur Celikbas ◽  
Srikanth B. Iyengar ◽  
Greg Piepmeyer ◽  
Roger Wiegand
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Tensor Products ◽  
Central Theme ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Free Modules ◽  
Tensor Powers ◽  
Torsion Free ◽  
Nonzero Torsion

AbstractTensor products usually have nonzero torsion. This is a central theme of Auslander's 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modulesNwith the property thatM ⊗RNhas nonzero torsion unlessMis very special. An important example of such a moduleNis the Frobenius powerpeRover a complete intersection domainRof characteristicp> 0.

scholarly journals Torsion in tensor powers of modules

2015 ◽  
Vol 219 ◽  
pp. 113-125 ◽  
Author(s):  
Olgur Celikbas ◽  
Srikanth B. Iyengar ◽  
Greg Piepmeyer ◽  
Roger Wiegand
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Tensor Products ◽  
Central Theme ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Free Modules ◽  
Tensor Powers ◽  
Torsion Free ◽  
Nonzero Torsion

AbstractTensor products usually have nonzero torsion. This is a central theme of Auslander's 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modules N with the property that M ⊗R N has nonzero torsion unless M is very special. An important example of such a module N is the Frobenius power peR over a complete intersection domain R of characteristic p > 0.

scholarly journals Special homological dimensions and intersection theorem

2005 ◽  
Vol 96 (2) ◽  
pp. 161
Author(s):  
Tirdad Sharif ◽  
Siamak Yassemi
Keyword(s):  
Local Ring ◽  
Projective Dimension ◽  
Intersection Theorem ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Gorenstein Dimension ◽  
Homological Dimensions

Let $(R,m)$ be commutative Noetherian local ring. It is shown that $R$ is Cohen-Macaulay ring if there exists a Cohen-Macaulay finite (i.e. finitely generated) $R$-module with finite upper Gorenstein dimension. In addition, we show that, in the Intersection Theorem, projective dimension can be replaced by quasi-projective dimension.

Artinian modules over commutative rings

1992 ◽  
Vol 111 (1) ◽  
pp. 25-33 ◽  
Author(s):  
R. Y. Sharp
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Artinian Modules ◽  
Noetherian Local Ring ◽  
Matlis Duality ◽  
Artinian Module

In 5, I provided a method whereby the study of an Artinian module A over a commutative ring R (throughout the paper, R will denote a commutative ring with identity) can, for some purposes at least, be reduced to the study of an Artinian module A' over a complete (Noetherian) local ring; in the latter situation, Matlis' duality 1 (alternatively, see 6, ch. 5) is available, and this means that the investigation can often be converted into a dual one about a finitely generated module over a complete (Noetherian) local ring.

𝐿-dimension for modules over a local ring

2021 ◽  
pp. 83-91
Author(s):  
Courtney Gibbons ◽  
David Jorgensen ◽  
Janet Striuli
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Residue Field ◽  
Free Resolution ◽  
Homological Dimension ◽  
Finitely Generated ◽  
Content Type ◽  
Commutative Local Ring ◽  
Finitely Generated Modules

We introduce a new homological dimension for finitely generated modules over a commutative local ring R R , which is based on a complex derived from a free resolution L L of the residue field of R R , and called L L -dimension. We prove several properties of L L -dimension, give some applications, and compare L L -dimension to complete intersection dimension.

Grothendieck groups for categories of complexes

2008 ◽  
Vol 3 (1) ◽  
pp. 165-203 ◽  
Author(s):  
Hans-Bjørn Foxby ◽  
Esben Bistrup Halvorsen
Keyword(s):  
Local Ring ◽  
Finite Length ◽  
Full Subcategory ◽  
Grothendieck Group ◽  
Projective Dimension ◽  
Intersection Theorem ◽  
Projective Modules ◽  
Finitely Generated ◽  
Grothendieck Groups ◽  
Finitely Generated Modules

AbstractThe new intersection theorem states that, over a Noetherian local ring R, for any non-exact complex concentrated in degrees n,…,0 in the category P(length) of bounded complexes of finitely generated projective modules with finite-length homology, we must have n ≥ d = dim R.One of the results in this paper is that the Grothendieck group of P(length) in fact is generated by complexes concentrated in the minimal number of degrees: if Pd(length) denotes the full subcategory of P(length) consisting of complexes concentrated in degrees d,…0, the inclusion Pd(length) → P(length) induces an isomorphism of Grothendieck groups. When R is Cohen–Macaulay, the Grothendieck groups of Pd(length) and P(length) are naturally isomorphic to the Grothendieck group of the category M(length) of finitely generated modules of finite length and finite projective dimension. This and a family of similar results are established in this paper.

Cofiniteness of Local Cohomology Modules

2014 ◽  
Vol 21 (04) ◽  
pp. 605-614 ◽  
Author(s):  
Kamal Bahmanpour ◽  
Reza Naghipour ◽  
Monireh Sedghi
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Partial Answer ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Noetherian Local Ring ◽  
Dimension Formula ◽  
Complete Local Ring ◽  
Local Cohomology Modules ◽  
Cohomology Modules

Let M be a non-zero finitely generated module over a commutative Noetherian local ring (R, 𝔪). In this paper we consider when the local cohomology modules are finitely generated. It is shown that if t ≥ 0 is an integer and [Formula: see text], then [Formula: see text] is not 𝔭-cofinite. Then we obtain a partial answer to a question raised by Huneke. Namely, if R is a complete local ring, then [Formula: see text] is finitely generated if and only if 0 ≤ n ∉ W, where [Formula: see text]. Also, we show that if J ⊆ I are 1-dimensional ideals of R, then [Formula: see text] is J-cominimax, and [Formula: see text] is finitely generated (resp., minimax) if and only if [Formula: see text] is finitely generated for all [Formula: see text] (resp., [Formula: see text]). Moreover, the concept of the J-cofiniteness dimension [Formula: see text] of M relative to I is introduced, and we explore an interrelation between [Formula: see text] and the filter depth of M in I. Finally, we show that if R is complete and dim M/IM ≠ 0, then [Formula: see text].

scholarly journals ON A NEW INVARIANT OF FINITELY GENERATED MODULES OVER LOCAL RINGS

2010 ◽  
Vol 09 (06) ◽  
pp. 959-976 ◽  
Author(s):  
NGUYEN TU CUONG ◽  
DOAN TRUNG CUONG ◽  
HOANG LE TRUONG
Keyword(s):  
Local Ring ◽  
Negative Integer ◽  
Local Rings ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Polynomial Type ◽  
Systems Of Parameters ◽  
Finitely Generated Modules

Let M be a finitely generated module on a local ring R and [Formula: see text] a filtration of submodules of M such that do < d1 < ⋯ < dt = d, where di = dim Mi. This paper is concerned with a non-negative integer [Formula: see text] which is defined as the least degree of all polynomials in n1, …, nd bounding above the function [Formula: see text] We prove that [Formula: see text] is independent of the choice of good systems of parameters [Formula: see text]. When [Formula: see text] is the dimension filtration of M, we can use the polynomial type of Mi/Mi-1 and the dimension of the non-sequentially Cohen–Macaulay locus of M to compute [Formula: see text], and also to study the behavior of it under local flat homomorphisms.

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