Cohen-Macaulay and (S<sub>2</sub>) Properties of the Second Power of Squarefree Monomial Ideals
We show that Cohen-Macaulay and (S<inline-formula> <math display="inline"> <semantics> <msub> <mrow></mrow> <mn>2</mn> </msub> </semantics> </math> </inline-formula>) properties are equivalent for the second power of a...
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MDPI AG
2019-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/8/684 |
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| author | Do Trong Hoang Giancarlo Rinaldo Naoki Terai |
| author_facet | Do Trong Hoang Giancarlo Rinaldo Naoki Terai |
| author_sort | Do Trong Hoang |
| collection | DOAJ |
| description | We show that Cohen-Macaulay and (S<inline-formula> <math display="inline"> <semantics> <msub> <mrow></mrow> <mn>2</mn> </msub> </semantics> </math> </inline-formula>) properties are equivalent for the second power of an edge ideal. We give an example of a Gorenstein squarefree monomial ideal <i>I</i> such that <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>/</mo> <msup> <mi>I</mi> <mn>2</mn> </msup> </mrow> </semantics> </math> </inline-formula> satisfies the Serre condition (S<inline-formula> <math display="inline"> <semantics> <msub> <mrow></mrow> <mn>2</mn> </msub> </semantics> </math> </inline-formula>), but is not Cohen-Macaulay. |
| first_indexed | 2024-04-13T03:26:27Z |
| format | Article |
| id | doaj.art-9fc5732e046843e6b8791ff73719b299 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-13T03:26:27Z |
| publishDate | 2019-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-9fc5732e046843e6b8791ff73719b2992022-12-22T03:04:38ZengMDPI AGMathematics2227-73902019-07-017868410.3390/math7080684math7080684Cohen-Macaulay and (S<sub>2</sub>) Properties of the Second Power of Squarefree Monomial IdealsDo Trong Hoang0Giancarlo Rinaldo1Naoki Terai2Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi 10307, VietnamDepartment of Mathematics, University of Trento, via Sommarive, 14, 38123 Povo (Trento), ItalyFaculty of Education, Saga University, Saga 840-8502, JapanWe show that Cohen-Macaulay and (S<inline-formula> <math display="inline"> <semantics> <msub> <mrow></mrow> <mn>2</mn> </msub> </semantics> </math> </inline-formula>) properties are equivalent for the second power of an edge ideal. We give an example of a Gorenstein squarefree monomial ideal <i>I</i> such that <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>/</mo> <msup> <mi>I</mi> <mn>2</mn> </msup> </mrow> </semantics> </math> </inline-formula> satisfies the Serre condition (S<inline-formula> <math display="inline"> <semantics> <msub> <mrow></mrow> <mn>2</mn> </msub> </semantics> </math> </inline-formula>), but is not Cohen-Macaulay.https://www.mdpi.com/2227-7390/7/8/684Stanley-Reisner idealedge idealCohen-Macaulay(S2) condition |
| spellingShingle | Do Trong Hoang Giancarlo Rinaldo Naoki Terai Cohen-Macaulay and (S<sub>2</sub>) Properties of the Second Power of Squarefree Monomial Ideals Mathematics Stanley-Reisner ideal edge ideal Cohen-Macaulay (S2) condition |
| title | Cohen-Macaulay and (S<sub>2</sub>) Properties of the Second Power of Squarefree Monomial Ideals |
| title_full | Cohen-Macaulay and (S<sub>2</sub>) Properties of the Second Power of Squarefree Monomial Ideals |
| title_fullStr | Cohen-Macaulay and (S<sub>2</sub>) Properties of the Second Power of Squarefree Monomial Ideals |
| title_full_unstemmed | Cohen-Macaulay and (S<sub>2</sub>) Properties of the Second Power of Squarefree Monomial Ideals |
| title_short | Cohen-Macaulay and (S<sub>2</sub>) Properties of the Second Power of Squarefree Monomial Ideals |
| title_sort | cohen macaulay and s sub 2 sub properties of the second power of squarefree monomial ideals |
| topic | Stanley-Reisner ideal edge ideal Cohen-Macaulay (S2) condition |
| url | https://www.mdpi.com/2227-7390/7/8/684 |
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