Symbolic powers, Serre conditions and Cohen-Macaulay Rees algebras

1995 ◽  
Vol 86 (1) ◽  
pp. 113-124 ◽  
Author(s):  
Susan Morey ◽  
Sunsook Noh ◽  
Wolmer V. Vasconcelos
Keyword(s):  
Rees Algebras ◽  
Symbolic Powers

scholarly journals Splittings and Symbolic Powers of Square-free Monomial Ideals

2019 ◽  
Author(s):  
Jonathan Montaño ◽  
Luis Núñez-Betancourt
Keyword(s):  
Linear Optimization ◽  
Monomial Ideals ◽  
Sufficient Condition ◽  
Graded Algebra ◽  
Prime Characteristic ◽  
Rees Algebra ◽  
Rees Algebras ◽  
Symbolic Powers ◽  
Frobenius Map

Abstract We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with respect to a morphism that resembles the Frobenius map and that exists in all characteristics. Using these methods, we recover a result by Hoa and Trung that states that the normalized $a$-invariants and the Castelnuovo–Mumford regularity of the symbolic powers converge. In addition, we give a sufficient condition for the equality of the ordinary and symbolic powers of this family of ideals and relate it to Conforti–Cornuéjols conjecture. Finally, we interpret this condition in the context of linear optimization.

scholarly journals Rees algebras of square-free Veronese ideals and their a-invariants

2005 ◽  
Vol 302 (1-3) ◽  
pp. 7-21 ◽  
Author(s):  
Adrián Alcántar
Keyword(s):  
Rees Algebras

scholarly journals Symbolic Powers of Certain Cover Ideals of Graphs

2021 ◽  
Author(s):  
Arvind Kumar ◽  
Rajiv Kumar ◽  
Rajib Sarkar ◽  
S. Selvaraja
Keyword(s):  
Symbolic Powers

scholarly journals Sequentially Cohen–Macaulay Rees algebras

2017 ◽  
Vol 69 (1) ◽  
pp. 293-309
Author(s):  
Naoki TANIGUCHI ◽  
Tran Thi PHUONG ◽  
Nguyen Thi DUNG ◽  
Tran Nguyen AN
Keyword(s):  
Rees Algebras

scholarly journals A Note on the $k$-Buchsbaum Property of Symbolic Powers of Stanley-Reisner Ideals

2011 ◽  
Vol 34 (1) ◽  
pp. 221-227 ◽  
Author(s):  
Nguyên Công MINH ◽  
Yukio NAKAMURA
Keyword(s):  
Symbolic Powers

scholarly journals A note on Rees algebras of two dimensional local domains

1979 ◽  
Vol 19 (2) ◽  
pp. 327-333 ◽  
Author(s):  
Yasuhiro Shimoda
Keyword(s):  
Two Dimensional ◽  
Rees Algebras

scholarly journals Stanley depth and symbolic powers of monomial ideals

2017 ◽  
Vol 120 (1) ◽  
pp. 5 ◽  
Author(s):  
S. A. Seyed Fakhari
Keyword(s):  
Monomial Ideal ◽  
Monomial Ideals ◽  
Limit Behavior ◽  
Stanley Depth ◽  
Symbolic Powers

The aim of this paper is to study the Stanley depth of symbolic powers of a squarefree monomial ideal. We prove that for every squarefree monomial ideal $I$ and every pair of integers $k, s\geq 1$, the inequalities $\mathrm{sdepth} (S/I^{(ks)}) \leq \mathrm{sdepth} (S/I^{(s)})$ and $\mathrm{sdepth}(I^{(ks)}) \leq \mathrm{sdepth} (I^{(s)})$ hold. If moreover $I$ is unmixed of height $d$, then we show that for every integer $k\geq1$, $\mathrm{sdepth}(I^{(k+d)})\leq \mathrm{sdepth}(I^{{(k)}})$ and $\mathrm{sdepth}(S/I^{(k+d)})\leq \mathrm{sdepth}(S/I^{{(k)}})$. Finally, we consider the limit behavior of the Stanley depth of symbolic powers of a squarefree monomial ideal. We also introduce a method for comparing the Stanley depth of factors of monomial ideals.

scholarly journals Symbolic powers of cover ideals of graphs and Koszul property

2021 ◽  
pp. 1-17
Author(s):  
Yan Gu ◽  
Huy Tài Hà ◽  
Joseph W. Skelton
Keyword(s):  
Vertex Cover ◽  
Cycle Cover ◽  
Symbolic Powers ◽  
Koszul Property

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

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