scholarly journals An Euler characteristic for modules of finite G-dimension

2008 ◽  
Vol 102 (2) ◽  
pp. 206 ◽  
Author(s):  
Sean Sather-Wagstaff ◽  
Diana White
Keyword(s):  
Euler Characteristic ◽  
Betti Numbers ◽  
Projective Dimension ◽  
Finitely Generated ◽  
Finite Projective Dimension ◽  
Extremal Behavior ◽  
Classical Invariant ◽  
Category Of Modules ◽  
Finitely Generated Modules

We extend Auslander and Buchsbaum's Euler characteristic from the category of finitely generated modules of finite projective dimension to the category of modules of finite G-dimension using Avramov and Martsinkovsky's notion of relative Betti numbers. We prove analogues of some properties of the classical invariant and provide examples showing that other properties do not translate to the new context. One unexpected property is in the characterization of the extremal behavior of this invariant: the vanishing of the Euler characteristic of a module $M$ of finite G-dimension implies the finiteness of the projective dimension of $M$. We include two applications of the Euler characteristic as well as several explicit calculations.

scholarly journals Adic approximation of complexes, and multiplicities

1974 ◽  
Vol 54 ◽  
pp. 61-67 ◽  
Author(s):  
David Eisenbud
Keyword(s):  
Euler Characteristic ◽  
General Result ◽  
Noetherian Ring ◽  
Approximation Theorem ◽  
Projective Dimension ◽  
Local Rings ◽  
Free Resolutions ◽  
Intersection Multiplicity ◽  
Finite Projective Dimension ◽  
Finitely Generated Modules

In [2, Section 1.6] Peskine and Szpiro prove a theorem on adic approximations of finite free resolutions over local rings which, together with M. Artin’s Approximation Theorem [1], allows them to “descend” modules of finite projective dimension over the completions of certain local rings to modules of finite projective dimension over finite étale extensions of those rings. In this note we will prove a more general result, which deals with the change in homology under an adic approximation of any complex of finitely generated modules over a noetherian ring, and which allows one to descend not only modules of finite projective dimension, but also the Euler characteristic or intersection multiplicity of two such modules.

scholarly journals EMBEDDING MODULES OF FINITE HOMOLOGICAL DIMENSION

2012 ◽  
Vol 55 (1) ◽  
pp. 85-96
Author(s):  
SEAN SATHER-WAGSTAFF
Keyword(s):  
Projective Dimension ◽  
Homological Dimension ◽  
Finitely Generated ◽  
Injective Dimension ◽  
Locally Finite ◽  
Semidualizing Module ◽  
Finite Projective Dimension ◽  
Finitely Generated Modules ◽  
The Way

AbstractThis paper builds on work of Hochster and Yao that provides nice embeddings for finitely generated modules of finite G-dimension, finite projective dimension or locally finite injective dimension. We extend these results by providing similar embeddings in the relative setting, that is, for certain modules of finite GC-dimension, finite C-projective dimension, locally finite C-injective dimension or locally finite C-injective dimension where C is a semidualizing module. Along the way, we extend some results for modules of finite homological dimension to modules of locally finite homological dimension in the relative setting.

scholarly journals On modules of finite projective dimension

2015 ◽  
Vol 219 ◽  
pp. 87-111 ◽  
Author(s):  
S. P. Dutta
Keyword(s):  
Local Ring ◽  
Projective Dimension ◽  
Order Ideal ◽  
Local Rings ◽  
Minimal Generator ◽  
Finite Projective Dimension ◽  
Special Cases ◽  
Nonzero Divisor ◽  
Finitely Generated Modules ◽  
Minimal Generators

AbstractWe address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ringRof mixed characteristicp> 0, wherepis a nonzero divisor, ifIis an ideal of finite projective dimension overRandp𝜖Iorpis a nonzero divisor onR/I, then every minimal generator ofIis a nonzero divisor. Hence, ifPis a prime ideal of finite projective dimension in a local ringR, then every minimal generator ofPis a nonzero divisor inR.

scholarly journals On modules of finite projective dimension

2015 ◽  
Vol 219 ◽  
pp. 87-111
Author(s):  
S. P. Dutta
Keyword(s):  
Local Ring ◽  
Projective Dimension ◽  
Order Ideal ◽  
Local Rings ◽  
Minimal Generator ◽  
Finite Projective Dimension ◽  
Special Cases ◽  
Nonzero Divisor ◽  
Finitely Generated Modules ◽  
Minimal Generators

AbstractWe address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ring R of mixed characteristic p > 0, where p is a nonzero divisor, if I is an ideal of finite projective dimension over R and p 𝜖 I or p is a nonzero divisor on R/I, then every minimal generator of I is a nonzero divisor. Hence, if P is a prime ideal of finite projective dimension in a local ring R, then every minimal generator of P is a nonzero divisor in R.

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.

scholarly journals Cohomological dimensions with respect to sum and intersection of ideals

2017 ◽  
Vol 102 (116) ◽  
pp. 115-120
Author(s):  
Alireza Vahidi
Keyword(s):  
Noetherian Ring ◽  
Projective Dimension ◽  
Equivalent Condition ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Finite Projective Dimension

Let R be a commutative Noetherian ring with non-zero identity, a and b proper ideals of R,M a finitely generated R-module with finite projective dimension, and X a finitely generated R-module. We study the cohomological dimensions of M and X with respect to a + b and a ? b. We show that the inequality cda+b(M,X) ? cda(M,X) + cdb(X) holds true and we find an equivalent condition for it to be equality.

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