scholarly journals Edge rings satisfying Serre’s condition $(R_{1})$

2014 ◽  
Vol 142 (7) ◽  
pp. 2537-2541 ◽  
Author(s):  
Takayuki Hibi ◽  
Lukas Katthän
Keyword(s):  
Serre’S Condition

scholarly journals On the h-vector of a simplicial complex with Serre’s condition

2012 ◽  
Vol 216 (1) ◽  
pp. 91-94 ◽  
Author(s):  
A. Goodarzi ◽  
M.R. Pournaki ◽  
S.A. Seyed Fakhari ◽  
S. Yassemi
Keyword(s):  
Simplicial Complex ◽  
Serre’S Condition

Serre’s Condition R k for Sums of Geometric Links

2015 ◽  
Vol 40 (3) ◽  
pp. 393-401
Author(s):  
Mark Johnson ◽  
Bernd Ulrich
Keyword(s):  
Serre’S Condition

A CONSTRUCTION OF QUASI-GORENSTEIN RINGS

2012 ◽  
Vol 11 (01) ◽  
pp. 1250013 ◽  
Author(s):  
AMIR BAGHERI ◽  
MARYAM SALIMI ◽  
ELHAM TAVASOLI ◽  
SIAMAK YASSEMI
Keyword(s):  
Local Ring ◽  
Noetherian Ring ◽  
Commutative Noetherian Ring ◽  
Gorenstein Rings ◽  
Canonical Ideal ◽  
Amalgamated Duplication ◽  
Serre’S Condition

Let R be a commutative Noetherian ring and let I be an ideal of R. In this paper, we study the amalgamated duplication ring R ⋈ I which is introduced by D'Anna and Fontana. It is shown that if R satisfies Serre's condition (Sn) and I𝔭 is a maximal Cohen–Macaulay R𝔭-module for every 𝔭 ∈ Spec (R), then R ⋈ I satisfies Serre's condition (Sn). Moreover if R ⋈ I satisfies Serre's condition (Sn), then so does R. This gives a generalization of the same result for Cohen–Macaulay rings in [D'Anna, A construction of Gorenstein rings, J. Algebra306 (2006) 507–519]. In addition it is shown that if R is a local ring and Ann R(I) = 0, then R ⋈ I is quasi-Gorenstein if and only if [Formula: see text] satisfies Serre's condition (S2) and I is a canonical ideal of R. This result improves the result of D'Anna which is corrected by Shapiro and states that if R is a Cohen–Macaulay local ring, then R ⋈ I is Gorenstein if and only if the canonical ideal of R exists and is isomorphic to I, provided Ann R(I) = 0.

scholarly journals A generalized Serre’s condition

2019 ◽  
Vol 47 (7) ◽  
pp. 2689-2701
Author(s):  
Brent Holmes
Keyword(s):  
Serre’S Condition

scholarly journals Rank Selection and Depth Conditions for Balanced Simplicial Complexes

10.37236/9299 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Brent Holmes ◽  
Justin Lyle
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Barycentric Subdivision ◽  
Selection Theorems ◽  
Serre’S Condition ◽  
Balanced Simplicial Complex

We prove some new rank selection theorems for balanced simplicial complexes. Specifically, we prove that if a balanced simplicial complex satisfies Serre's condition $(S_{\ell})$ then so do all of its rank selected subcomplexes.  We also provide a formula for the depth of a balanced simplicial complex in terms of reduced homologies of its rank selected subcomplexes. By passing to a barycentric subdivision, our results give information about Serre's condition and the depth of any simplicial complex. Our results extend rank selection theorems for depth proved by Stanley, Munkres, and Hibi. 

scholarly journals F-THRESHOLDS OF GRADED RINGS

2016 ◽  
Vol 229 ◽  
pp. 141-168 ◽  
Author(s):  
ALESSANDRO DE STEFANI ◽  
LUIS NÚÑEZ-BETANCOURT
Keyword(s):  
Characteristic Zero ◽  
Projective Dimension ◽  
Graded Rings ◽  
Projective Varieties ◽  
Regular Sequences ◽  
Serre’S Condition ◽  
Regular Rings

The $a$-invariant, the $F$-pure threshold, and the diagonal $F$-threshold are three important invariants of a graded $K$-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly $F$-regular rings. In this article, we prove that these relations hold only assuming that the algebra is $F$-pure. In addition, we present an interpretation of the $a$-invariant for $F$-pure Gorenstein graded $K$-algebras in terms of regular sequences that preserve $F$-purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo–Mumford regularity, and Serre’s condition $S_{k}$. We also present analogous results and questions in characteristic zero.

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