scholarly journals On algebra with universal finite module of differentials

1981 ◽  
Vol 83 ◽  
pp. 107-121 ◽  
Author(s):  
Norio Yamauchi
Keyword(s):  
Local Ring ◽  
Positive Characteristic ◽  
Regularity Theorem ◽  
Local Case ◽  
Finite Module ◽  
Characteristic Case ◽  
Differential Modules ◽  
Module Of Differentials

Let k be a field and A a noetherian k-algebra. In this note, we shall study the universal finite module of differentials of A over k, which is denoted by Dk(A). When the characteristic of k is zero, detailed results have been obtained by Scheja and Storch [8]. So we shall treat the positive characteristic case. In § 1, we shall study differential modules of a local ring over subfields. We obtain a criterion of regularity (Theorem (1.14)). In § 2, we shall study the formal fibres and regular locus of A with Dk(A). Our main result is Theorem (2.1) which shows that, if Dk(A) exists, then A is a universally catenary G-ring under a certain assumption. In the local case, this is a generalization of Matsumura’s theorem ([5] Theorem 15), where regularity of A is assumed.

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.

RESIDUAL PROPERTIES OF FREE PRO-P GROUPS

2001 ◽  
Vol 33 (5) ◽  
pp. 578-582 ◽  
Author(s):  
YIFTACH BARNEA
Keyword(s):  
Positive Characteristic ◽  
Congruence Subgroup ◽  
Probabilistic Method ◽  
Algebraic Groups ◽  
Local Fields ◽  
Random Generation ◽  
Zero Characteristic ◽  
Residual Properties ◽  
Characteristic Case ◽  
Lie Methods

Recall that if S is a class of groups, then a group G is residually-S if, for any element 1 ≠ g ∈ G, there is a normal subgroup N of G such that g ∉ N and G/N ∈ S. Let Λ be a commutative Noetherian local pro-p ring, with a maximal ideal M. Recall that the first congruence subgroup of SLd(Λ) is: SL1d(Λ) = ker (SLd(Λ) → SLd(Λ/M)).Let K ⊆ ℕ. We define SΛ(K) = ∪d∈K{open subgroups of SL1d(Λ)}. We show that if K is infinite, then for Λ = [ ]p[[t]] and for Λ = ℤp a finitely generated non-abelian free pro-p group is residually-SΛ(K). We apply a probabilistic method, combined with Lie methods and a result on random generation in simple algebraic groups over local fields. It is surprising that the case of zero characteristic is deduced from the positive characteristic case.

scholarly journals New estimates of Hilbert–Kunz multiplicities for local rings of fixed dimension

2013 ◽  
Vol 212 ◽  
pp. 59-85 ◽  
Author(s):  
Ian M. Aberbach ◽  
Florian Enescu
Keyword(s):  
Lower Bound ◽  
Local Ring ◽  
Positive Characteristic ◽  
Local Rings ◽  
Volume Estimate ◽  
Fixed Dimension ◽  
Samuel Multiplicity

AbstractWe present results on the Watanabe–Yoshida conjecture for the Hilbert–Kunz multiplicity of a local ring of positive characteristic. By improving on a “volume estimate” giving a lower bound for Hilbert–Kunz multiplicity, we obtain the conjecture when the ring has either Hilbert–Samuel multiplicity less than or equal to 5 or dimension less than or equal to 6. For nonregular rings with fixed dimension, a new lower bound for the Hilbert–Kunz multiplicity is obtained.

scholarly journals Dual of the Auslander-Bridger formula and GF-perfectness

2007 ◽  
Vol 101 (1) ◽  
pp. 5 ◽  
Author(s):  
Parviz Sahandi ◽  
Tirdad Sharif
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Krull Dimension ◽  
Local Cohomology Module ◽  
Injective Dimension ◽  
Gorenstein Dimension ◽  
Finite Module ◽  
Gorenstein Injective Dimension ◽  
Gorenstein Injective

Ext-finite modules were introduced and studied by Enochs and Jenda. We prove under some conditions that the depth of a local ring is equal to the sum of the Gorenstein injective dimension and Tor-depth of an Ext-finite module of finite Gorenstein injective dimension. Let $(R,\mathfrak m)$ be a local ring. We say that an $R$-module $M$ with $\dim_R M=n$ is a Grothendieck module if the $n$-th local cohomology module of $M$ with respect to $\mathfrak m$, $\mathrm{H}_{\mathfrak m}^n (M)$, is non-zero. We prove the Bass formula for this kind of modules of finite Gorenstein injective dimension and of maximal Krull dimension. These results are dual versions of the Auslander-Bridger formula for the Gorenstein dimension. We also introduce GF-perfect modules as an extension of quasi-perfect modules introduced by Foxby.

scholarly journals The remaining cases of the Kramer–Tunnell conjecture

2016 ◽  
Vol 152 (11) ◽  
pp. 2255-2268
Author(s):  
Kęstutis Česnavičius ◽  
Naoki Imai
Keyword(s):  
Elliptic Curve ◽  
Local Field ◽  
Positive Characteristic ◽  
Quadratic Extension ◽  
Base Change ◽  
Local Fields ◽  
Root Number ◽  
Characteristic Case ◽  
Local Root

For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$, motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the base change of $E$ to the quadratic extension in terms of a certain norm index. The formula is known in all cases except some where $K$ is of characteristic $2$, and we complete its proof by reducing the positive characteristic case to characteristic $0$. For this reduction, we exploit the principle that local fields of characteristic $p$ can be approximated by finite extensions of $\mathbb{Q}_{p}$: we find an elliptic curve $E^{\prime }$ defined over a $p$-adic field such that all the terms in the Kramer–Tunnell formula for $E^{\prime }$ are equal to those for $E$.

scholarly journals Some Homological Criteria for Regular, Complete Intersection and Gorenstein Rings

2015 ◽  
Vol 59 (2) ◽  
pp. 473-481 ◽  
Author(s):  
Javier Majadas
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Positive Characteristic ◽  
Residue Field ◽  
Canonical Homomorphism ◽  
Gorenstein Rings ◽  
Frobenius Endomorphism

AbstractRegularity, complete intersection and Gorenstein properties of a local ring can be characterized by homological conditions on the canonical homomorphism into its residue field. In positive characteristic, the Frobenius endomorphism (and, more generally, any contracting endomorphism) can also be used for these characterizations. We introduce here a class of local homomorphisms, in some sense larger than all above, for which these characterizations still hold, providing an unified treatment for this class of homomorphisms.

scholarly journals ON A CONJECTURE OF LAN–SHENG–ZUO ON SEMISTABLE HIGGS BUNDLES: RANK 3 CASE

2014 ◽  
Vol 25 (02) ◽  
pp. 1450013
Author(s):  
LINGGUANG LI
Keyword(s):  
Positive Characteristic ◽  
Affirmative Answer ◽  
Higgs Bundle ◽  
Closed Field ◽  
Fundamental Groups ◽  
Product Theorem ◽  
Higgs Bundles ◽  
Characteristic P ◽  
Smooth Projective Curve ◽  
Characteristic Case

Let X be a smooth projective curve of genus g over an algebraically closed field k of characteristic p > 2. We prove that any rank 3 nilpotent semistable Higgs bundle (E, θ) on X is a strongly semistable Higgs bundle. This gives a partially affirmative answer to a conjecture of Lan–Sheng–Zuo [Semistable Higgs bundles and representations of algebraic fundamental groups: positive characteristic case, preprint (2012), arXiv:1210.8280][(Very recently, A. Langer [Semistable modules over Lie algebroids in positive characteristic, preprint (2013), arXiv:1311.2794] and independently Lan–Sheng–Yang–Zuo [Semistable Higgs bundles of small ranks are strongly Higgs semistable, preprint (2013), arXiv:1311.2405] have proven the conjecture for ranks less than or equal to p case.)] In addition, we prove a tensor product theorem for strongly semistable Higgs bundles with p satisfying some bounds (Theorem 4.3). From this we reprove a tensor theorem for semistable Higgs bundles on the condition that the Lan–Sheng–Zuo conjecture holds (Corollary 4.4).

scholarly journals On the Sum of Homological Dimension and Codimension of Modules over a Semi-local Ring

1968 ◽  
Vol 33 ◽  
pp. 165-172
Author(s):  
Samuel S.H. Young
Keyword(s):  
Local Ring ◽  
Global Dimension ◽  
Homological Dimension ◽  
Regular Local Ring ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Arbitrary Ring ◽  
Local Case

Let M be a finitely generated module over a regular local ring R. It is well known that the sum of homological dimension and codimension of M is equal to the global dimension of R. For modules over an arbitrary ring, this is in general not true. The purpose of this paper is to investigate the properties of such sums in the semi-local case.

scholarly journals Strong test ideals associated to Cartier algebras

2019 ◽  
Vol 19 (03) ◽  
pp. 2050044
Author(s):  
Florian Enescu ◽  
Irina Ilioaea
Keyword(s):  
Local Ring ◽  
Positive Characteristic ◽  
Integral Closures

In this note, we use the theory of test ideals and Cartier algebras to examine the interplay between the tight and integral closures in a local ring of positive characteristic. Using work of Schwede, we prove the abundance of strong test ideals, recovering some older fundamental results, and use this approach in concrete computations. In the second part of the paper, the case of Stanley–Reisner rings is fully examined.

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