scholarly journals On the Vanishing of Homology with Modules of Finite Length

2013 ◽  
Vol 112 (1) ◽  
pp. 11 ◽  
Author(s):  
Petter Andreas Bergh
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Finite Length ◽  
Homology Groups

We study the vanishing of homology and cohomology of a module of finite complete intersection dimension over a local ring. Given such a module of complexity $c$, we show that if $c$ (co)homology groups with a module of finite length vanish, then all higher (co)homology groups vanish.

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 Complete Intersection in Overrings of a Certain One-Dimensional Gorenstein Graded Local Ring

2000 ◽  
Vol 233 (2) ◽  
pp. 772-790
Author(s):  
Shiro Goto ◽  
Satoshi Haraikawa ◽  
Shin-Ichiro Iai
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
One Dimensional

scholarly journals Local Homology, Cohen–Macaulayness and Cohen–Macaulayfications

2008 ◽  
Vol 15 (01) ◽  
pp. 63-68
Author(s):  
Michael Hellus
Keyword(s):  
Local Ring ◽  
Finite Length ◽  
Necessary Condition ◽  
Direct Generalization ◽  
Local Homology ◽  
Finiteness Result ◽  
Finite Module ◽  
Noetherian Dimension

Let (R,𝔪) be a local ring, X an artinian R-module of noetherian dimension d; let x1,…,xd ∈ 𝔪 be such that 0:X (x1,…,xd)R has finite length. We show by an example that [Formula: see text] is not finite as an R-module in general; it is finite if we assume R is complete. This answers a question posed by Tang. As a first application of the latter finiteness result, we give a necessary condition for a finite module to be Cohen–Macaulay; secondly we propose a notion of Cohen–Macaulayfication and prove its uniqueness; finally we show that this new notion of Cohen–Macaulayfication is a direct generalization of a notion of Cohen–Macaulayfication introduced by Goto.

scholarly journals On Set Theoretically and Cohomologically Complete Intersection Ideals

2014 ◽  
Vol 57 (3) ◽  
pp. 477-484 ◽  
Author(s):  
Majid Eghbali
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Standing Problem

AbstractLet (R;m) be a local ring and a be an ideal of R. The inequalitiesare known. It is an interesting and long-standing problem to determine the cases giving equality. Thanks to the formal grade we give conditions in which the above inequalities become equalities.

scholarly journals Free resolutions of Dynkin format and the licci property of grade $3$ perfect ideals

2019 ◽  
Vol 125 (2) ◽  
pp. 163-178
Author(s):  
Lars Winther Christensen ◽  
Oana Veliche ◽  
Jerzy Weyman
Keyword(s):  
Recent Work ◽  
Local Ring ◽  
Complete Intersection ◽  
Dynkin Diagram ◽  
Free Resolution ◽  
Free Resolutions ◽  
Regular Local Ring ◽  
Minimal Free Resolution ◽  
Grade 3

Recent work on generic free resolutions of length $3$ attaches to every resolution a graph and suggests that resolutions whose associated graph is a Dynkin diagram are distinguished. We conjecture that in a regular local ring, every grade $3$ perfect ideal whose minimal free resolution is distinguished in this way is in the linkage class of a complete intersection.

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.

scholarly journals Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free

2011 ◽  
Vol 148 (1) ◽  
pp. 145-152 ◽  
Author(s):  
Hailong Dao
Keyword(s):  
Abelian Group ◽  
Local Ring ◽  
Complete Intersection ◽  
Homological Algebra ◽  
Local Rings ◽  
Noetherian Local Ring ◽  
Picard Groups ◽  
Crepant Resolutions ◽  
Torsion Free

AbstractLet (R,m) be a Noetherian local ring and UR=Spec(R)−{m} be the punctured spectrum of R. Gabber conjectured that if R is a complete intersection of dimension three, then the abelian group Pic(UR) is torsion-free. In this note we prove Gabber’s statement for the hypersurface case. We also point out certain connections between Gabber’s conjecture, Van den Bergh’s notion of non-commutative crepant resolutions and some well-studied questions in homological algebra over local rings.

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.

Sign in / Sign up

Export Citation Format

Share Document