scholarly journals A FREENESS CRITERION WITHOUT PATCHING FOR MODULES OVER LOCAL RINGS

2021 ◽  
pp. 1-13
Author(s):  
Sylvain Brochard ◽  
Srikanth B. Iyengar ◽  
Chandrashekhar B. Khare
Keyword(s):  
Complete Intersection ◽  
Local Rings ◽  
Finitely Generated ◽  
Flat Dimension ◽  
Special Case ◽  
Local Homomorphism

Abstract It is proved that if $\varphi \colon A\to B$ is a local homomorphism of commutative noetherian local rings, a nonzero finitely generated B-module N whose flat dimension over A is at most $\operatorname {edim} A - \operatorname {edim} B$ is free over B and $\varphi $ is a special type of complete intersection. This result is motivated by a ‘patching method’ developed by Taylor and Wiles and a conjecture of de Smit, proved by the first author, dealing with the special case when N is flat over A.

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 Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups

2015 ◽  
Vol 58 (4) ◽  
pp. 787-798 ◽  
Author(s):  
Yu Kitabeppu ◽  
Sajjad Lakzian
Keyword(s):  
Diameter Growth ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Finite Generation ◽  
Alexandrov Spaces ◽  
Geodesic Spaces ◽  
Special Case ◽  
Linear Diameter

AbstractIn this paper, we generalize the finite generation result of Sormani to non-branching RCD(0, N) geodesic spaces (and in particular, Alexandrov spaces) with full supportmeasures. This is a special case of the Milnor’s Conjecture for complete non-compact RCD(0, N) spaces. One of the key tools we use is the Abresch–Gromoll type excess estimates for non-smooth spaces obtained by Gigli–Mosconi.

scholarly journals Proof of de Smit’s conjecture: a freeness criterion

2017 ◽  
Vol 153 (11) ◽  
pp. 2310-2317
Author(s):  
Sylvain Brochard
Keyword(s):  
Complete Intersection ◽  
Local Rings ◽  
Embedding Dimension ◽  
Fermat's Last Theorem ◽  
Multiplicity One ◽  
Fermat’S Last Theorem ◽  
Invent Math

Let $A\rightarrow B$ be a morphism of Artin local rings with the same embedding dimension. We prove that any $A$-flat $B$-module is $B$-flat. This freeness criterion was conjectured by de Smit in 1997 and improves Diamond’s criterion [The Taylor–Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379–391, Theorem 2.1]. We also prove that if there is a nonzero $A$-flat $B$-module, then $A\rightarrow B$ is flat and is a relative complete intersection. Then we explain how this result allows one to simplify Wiles’s proof of Fermat’s last theorem: we do not need the so-called ‘Taylor–Wiles systems’ any more.

scholarly journals Linearity defect of the residue field of short local rings

2016 ◽  
Vol 16 (09) ◽  
pp. 1750163
Author(s):  
Rasoul Ahangari Maleki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Residue Field ◽  
Positive Answer ◽  
Local Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Linear Resolution

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text] and residue field [Formula: see text]. The linearity defect of a finitely generated [Formula: see text]-module [Formula: see text], which is denoted [Formula: see text], is a numerical measure of how far [Formula: see text] is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether [Formula: see text] implies [Formula: see text], in the case when [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.

Hilbert rings and G(oldman)-rings issued from amalgamated algebras

2018 ◽  
Vol 17 (02) ◽  
pp. 1850023 ◽  
Author(s):  
L. Izelgue ◽  
O. Ouzzaouit
Keyword(s):  
New York ◽  
Commutative Algebra ◽  
Quotient Ring ◽  
Sufficient Conditions ◽  
Ring Homomorphism ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Finitely Generated ◽  
Necessary And Sufficient ◽  
The Way

Let [Formula: see text] and [Formula: see text] be two rings, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. The ring [Formula: see text] is called the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]. It was proposed by D’anna and Fontana [Amalgamated algebras along an ideal, Commutative Algebra and Applications (W. de Gruyter Publisher, Berlin, 2009), pp. 155–172], as an extension for the Nagata’s idealization, which was originally introduced in [Nagata, Local Rings (Interscience, New York, 1962)]. In this paper, we establish necessary and sufficient conditions under which [Formula: see text], and some related constructions, is either a Hilbert ring, a [Formula: see text]-domain or a [Formula: see text]-ring in the sense of Adams [Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17(1) (1974)]. By the way, we investigate the transfer of the [Formula: see text]-property among pairs of domains sharing an ideal. Our results provide original illustrating examples.

scholarly journals Acyclic complexes of finitely generated free modules over local rings

2009 ◽  
Vol 105 (1) ◽  
pp. 85 ◽  
Author(s):  
Meri T. Hughes ◽  
David A. Jorgensen ◽  
Liana M. Sega
Keyword(s):  
Local Ring ◽  
Local Rings ◽  
Finitely Generated ◽  
Reflexive Module ◽  
Commutative Local Ring ◽  
Standard Construction ◽  
Acyclic Complexes ◽  
Free Modules

We consider the question of how minimal acyclic complexes of finitely generated free modules arise over a commutative local ring. A standard construction gives that every totally reflexive module yields such a complex. We show that for certain rings this construction is essentially the only method of obtaining such complexes. We also give examples of rings which admit minimal acyclic complexes of finitely generated free modules which cannot be obtained by means of this construction.

scholarly journals Linearity defect and regularity over a Koszul algebra

2009 ◽  
Vol 104 (2) ◽  
pp. 205 ◽  
Author(s):  
Kohji Yanagawa
Keyword(s):  
Complete Intersection ◽  
Polynomial Ring ◽  
Commutative Algebra ◽  
Exterior Algebra ◽  
Finitely Generated ◽  
Koszul Algebra ◽  
Algebra Over A Field

Let $A = \bigoplus_{i\in \mathsf{N}}A_i$ be a Koszul algebra over a field $K = A_0$, and $*\operatorname{mod} A$ the category of finitely generated graded left $A$-modules. The linearity defect $\mathrm{ld}_A(M)$ of $M \in *\operatorname{mod} A$ is an invariant defined by Herzog and Iyengar. An exterior algebra $E$ is a Koszul algebra which is the Koszul dual of a polynomial ring. Eisenbud et al. showed that $\mathrm{ld}_E(M) < \infty$ for all $M \in *\operatorname{mod} E$. Improving this, we show that the Koszul dual $A^!$ of a Koszul commutative algebra $A$ satisfies the following. Let $M \in *\operatorname{mod} A^!$. If $\{\dim_K M_i \mid i \in {\mathsf Z}\}$ is bounded, then $\mathrm{ld}_{A^!}(M) < \infty$. If $A$ is complete intersection, then $\mathrm{reg}_{A^!}(M) < \infty$ and $\mathrm{ld}_{A^!}(M) < \infty$ for all $M \in *\operatorname{mod} A^!$. If $E=\bigwedge \langle y_1, \ldots, y_n\rangle$ is an exterior algebra, then $\mathrm{ld}_E(M)\leq c^{n!} 2^{(n-1)!}$ for $M \in *\operatorname{mod} E$ with $c := \max \{\dim_K M_i \mid i \in{\mathsf Z}\}$.

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