scholarly journals An intrinsic definition of the Rees algebra of a module

2017 ◽  
Vol 61 (1) ◽  
pp. 13-30
Author(s):  
Gustav Sædén Ståhl
Keyword(s):  
Noetherian Ring ◽  
Rees Algebra ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Definition Of ◽  
Free Modules ◽  
Divided Powers ◽  
Intrinsic Definition

This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring. Their definition is in terms of maps to free modules. We give an intrinsic definition using divided powers.

Some properties of a family of quotients of the Rees algebra

2019 ◽  
Vol 18 (06) ◽  
pp. 1950113 ◽  
Author(s):  
Elham Tavasoli
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Proper Ideal ◽  
Rees Algebra ◽  
Finitely Generated ◽  
Flat Ideal

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a nonzero proper ideal of [Formula: see text]. In this paper, we study the properties of a family of rings [Formula: see text], with [Formula: see text], as quotients of the Rees algebra [Formula: see text], when [Formula: see text] is a semidualizing ideal of Noetherian ring [Formula: see text], and in the case that [Formula: see text] is a flat ideal of [Formula: see text]. In particular, for a Noetherian ring [Formula: see text], it is shown that if [Formula: see text] is a finitely generated [Formula: see text]-module, then [Formula: see text] is totally [Formula: see text]-reflexive as an [Formula: see text]-module if and only if [Formula: see text] is totally reflexive as an [Formula: see text]-module, provided that [Formula: see text] is a semidualizing ideal and [Formula: see text] is reducible in [Formula: see text]. In addition, it is proved that if [Formula: see text] is a nonzero flat ideal of [Formula: see text] and [Formula: see text] is reducible in [Formula: see text], then [Formula: see text], for any [Formula: see text]-module [Formula: see text].

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 Lower and upper bounds for Cohen-Macaulay dimension

2005 ◽  
Vol 71 (2) ◽  
pp. 337-346
Author(s):  
J. Asadollahi ◽  
Sh. Salarain
Keyword(s):  
Noetherian Ring ◽  
Upper Bounds ◽  
Finitely Generated ◽  
Lower And Upper Bounds ◽  
Finitely Generated Module ◽  
Basic Properties ◽  
Homological Dimensions

Lower and upper bounds for CM-dimension, called CM*-dimension and CM*-dimension, will be defined for any finitely generated module M over a local Noetherian ring R. Both CM* and CM*-dimension reflect the Cohen–Macaulay property of rings. Our results will show that these dimensions have the expected basic properties parallel to those of the homological dimensions. In particular, they satisfy an analog of the Auslander-Buchsbaum formula.

scholarly journals REFLEXIVITY AND CONNECTEDNESS

2014 ◽  
Vol 57 (1) ◽  
pp. 231-240 ◽  
Author(s):  
SEAN SATHER-WAGSTAFF
Keyword(s):  
Noetherian Ring ◽  
Prime Spectrum ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Commutative Noetherian Ring

AbstractGiven a finitely generated module over a commutative noetherian ring that satisfies certain reflexivity conditions, we show how failure of the semidualizing property for the module manifests in a disconnection of the prime spectrum of the ring.

scholarly journals A note on the countable union of prime submodules

2001 ◽  
Vol 27 (10) ◽  
pp. 641-643 ◽  
Author(s):  
M. R. Pournaki ◽  
M. Tousi
Keyword(s):  
Noetherian Ring ◽  
Countable Family ◽  
Prime Ideals ◽  
Countable Union ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Prime Submodules

LetMbe a finitely-generated module over a Noetherian ringR. Suppose𝔞is an ideal ofRand letN=𝔞Mand𝔟=Ann(M/N). If𝔟⫅J(R),Mis complete with respect to the𝔟-adic topology,{Pi}i≥1is a countable family of prime submodules ofM, andx∈M, thenx+N⫅∪i≥1Piimplies thatx+N⫅Pjfor somei≥1. This extends a theorem of Sharp and Vámos concerning prime ideals to prime submodules.

scholarly journals s-Sequences and Monomial Modules

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2659
Author(s):  
Gioia Failla ◽  
Paola Lea Staglianó
Keyword(s):  
Noetherian Ring ◽  
Symmetric Algebra ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Commutative Noetherian Ring ◽  
Syzygy Module ◽  
Homological Invariants

In this paper we study a monomial module M generated by an s-sequence and the main algebraic and homological invariants of the symmetric algebra of M. We show that the first syzygy module of a finitely generated module M, over any commutative Noetherian ring with unit, has a specific initial module with respect to an admissible order, provided M is generated by an s-sequence. Significant examples complement the results.

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 THE STRUCTURE OF BALANCED BIG COHEN–MACAULAY MODULES OVER COHEN–MACAULAY RINGS

2016 ◽  
Vol 59 (3) ◽  
pp. 549-561 ◽  
Author(s):  
HENRIK HOLM
Keyword(s):  
Local Ring ◽  
Structure Theorem ◽  
Direct Limit ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Flat Modules ◽  
Direct Limits ◽  
Free Modules

AbstractOver a Cohen–Macaulay (CM) local ring, we characterize those modules that can be obtained as a direct limit of finitely generated maximal CM modules. We point out two consequences of this characterization: (1) Every balanced big CM module, in the sense of Hochster, can be written as a direct limit of small CM modules. In analogy with Govorov and Lazard's characterization of flat modules as direct limits of finitely generated free modules, one can view this as a “structure theorem” for balanced big CM modules. (2) Every finitely generated module has a pre-envelope with respect to the class of finitely generated maximal CM modules. This result is, in some sense, dual to the existence of maximal CM approximations, which has been proved by Auslander and Buchweitz.

On a Class of Automaton Algebras

2016 ◽  
Vol 60 (1) ◽  
pp. 31-38 ◽  
Author(s):  
Ferran Cedó ◽  
Jan Okniński
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Module ◽  
Commutative Subalgebra

AbstractWe show that every finitely generated algebra that is a finitely generated module over a finitely generated commutative subalgebra is an automaton algebra in the sense of Ufnarovskii.

scholarly journals Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras

2016 ◽  
Vol 23 (04) ◽  
pp. 701-720 ◽  
Author(s):  
Xiangui Zhao ◽  
Yang Zhang
Keyword(s):  
Computational Method ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Polynomial Algebras ◽  
Ore Extensions ◽  
Basis Method ◽  
Finitely Generated Modules ◽  
Kirillov Dimension

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

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