Adic approximation of complexes, and multiplicities

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.

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An Euler characteristic for modules of finite G-dimension

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-15059 ◽

2008 ◽

Vol 102 (2) ◽

pp. 206 ◽

Cited By ~ 2

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.

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On modules of finite projective dimension

Nagoya Mathematical Journal ◽

10.1017/s0027763000027100 ◽

2015 ◽

Vol 219 ◽

pp. 87-111 ◽

Cited By ~ 1

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.

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On modules of finite projective dimension

Nagoya Mathematical Journal ◽

10.1215/00277630-3140702 ◽

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.

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Finite projective dimension over pullbacks of local rings

Communications in Algebra ◽

10.1080/00927879108824319 ◽

1991 ◽

Vol 19 (12) ◽

pp. 3271-3294 ◽

Cited By ~ 2

Author(s):

Hing Shing Man

Keyword(s):

Projective Dimension ◽

Local Rings ◽

Finite Projective Dimension

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ABSOLUTE, RELATIVE, AND TATE COHOMOLOGY OF MODULES OF FINITE GORENSTEIN DIMENSION

Proceedings of the London Mathematical Society ◽

10.1112/s0024611502013527 ◽

2002 ◽

Vol 85 (2) ◽

pp. 393-440 ◽

Cited By ~ 156

Author(s):

LUCHEZAR L. AVRAMOV ◽

ALEX MARTSINKOVSKY

Keyword(s):

Exact Sequence ◽

Projective Dimension ◽

Subject Classification ◽

Local Rings ◽

Tate Cohomology ◽

Gorenstein Dimension ◽

Relative Cohomology ◽

Numerical Invariants ◽

The One ◽

Finitely Generated Modules

We study finitely generated modules $M$ over a ring $R$, noetherian on both sides. If $M$ has finite Gorenstein dimension $\mbox{G-dim}_RM$ in the sense of Auslander and Bridger, then it determines two other cohomology theories besides the one given by the absolute cohomology functors ${\rm Ext}^n_R(M,\ )$. Relative cohomology functors ${\rm Ext}^n_{\mathcal G}(M,\ )$ are defined for all non-negative integers $n$; they treat the modules of Gorenstein dimension $0$ as projectives and vanish for $n > \mbox{G-dim}_RM$. Tate cohomology functors $\widehat{\rm Ext}^n_R(M,\ )$ are defined for all integers $n$; all groups $\widehat{\rm Ext}^n_R(M,N)$ vanish if $M$ or $N$ has finite projective dimension. Comparison morphisms $\varepsilon_{\mathcal G}^n \colon {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ )$ and $\varepsilon_R^n \colon {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ )$ link these functors. We give a self-contained treatment of modules of finite G-dimension, establish basic properties of relative and Tate cohomology, and embed the comparison morphisms into a canonical long exact sequence $0 \to {\rm Ext}^1_{\mathcal G}(M,\ ) \to \cdots \to {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ ) \to {\rm Ext}^{n+1}_{\mathcal G}(M,\ ) \to \cdots$. We show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings. 2000 Mathematical Subject Classification: 16E05, 13H10, 18G25.

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Cohomological dimensions with respect to sum and intersection of ideals

Publications de l Institut Mathematique ◽

10.2298/pim1716115v ◽

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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EMBEDDING MODULES OF FINITE HOMOLOGICAL DIMENSION

Glasgow Mathematical Journal ◽

10.1017/s0017089512000353 ◽

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.

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