Extremal Betti Numbers of t-Spread Strongly Stable Ideals
Let <i>K</i> be a field and let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>=</mo> <mi>K</mi> <mo>[</mo> <msub> <mi>x</mi> <mn>1</mn> </msub&g...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/8/695 |
| _version_ | 1819011503601221632 |
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| author | Luca Amata Marilena Crupi |
| author_facet | Luca Amata Marilena Crupi |
| author_sort | Luca Amata |
| collection | DOAJ |
| description | Let <i>K</i> be a field and let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>=</mo> <mi>K</mi> <mo>[</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> be a polynomial ring over <i>K</i>. We analyze the extremal Betti numbers of special squarefree monomial ideals of <i>S</i> known as the <i>t</i>-spread strongly stable ideals, where <i>t</i> is an integer <inline-formula> <math display="inline"> <semantics> <mrow> <mo>≥</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. A characterization of the extremal Betti numbers of such a class of ideals is given. Moreover, we determine the structure of the <i>t</i>-spread strongly stable ideals with the maximal number of extremal Betti numbers when <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>. |
| first_indexed | 2024-12-21T01:29:12Z |
| format | Article |
| id | doaj.art-62cbcbb346ad44c0a053e2041a936828 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-21T01:29:12Z |
| publishDate | 2019-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-62cbcbb346ad44c0a053e2041a9368282022-12-21T19:20:24ZengMDPI AGMathematics2227-73902019-08-017869510.3390/math7080695math7080695Extremal Betti Numbers of t-Spread Strongly Stable IdealsLuca Amata0Marilena Crupi1Department of Mathematics and Computer Sciences, Physics and Earth Sciences, University of Messina Viale Ferdinando Stagno d’Alcontres 31, 98166 Messina, ItalyDepartment of Mathematics and Computer Sciences, Physics and Earth Sciences, University of Messina Viale Ferdinando Stagno d’Alcontres 31, 98166 Messina, ItalyLet <i>K</i> be a field and let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>=</mo> <mi>K</mi> <mo>[</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> be a polynomial ring over <i>K</i>. We analyze the extremal Betti numbers of special squarefree monomial ideals of <i>S</i> known as the <i>t</i>-spread strongly stable ideals, where <i>t</i> is an integer <inline-formula> <math display="inline"> <semantics> <mrow> <mo>≥</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>. A characterization of the extremal Betti numbers of such a class of ideals is given. Moreover, we determine the structure of the <i>t</i>-spread strongly stable ideals with the maximal number of extremal Betti numbers when <inline-formula> <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>.https://www.mdpi.com/2227-7390/7/8/695graded idealssquarefree monomial idealsminimal graded resolutionsextremal Betti numberst-spread monomial ideals |
| spellingShingle | Luca Amata Marilena Crupi Extremal Betti Numbers of t-Spread Strongly Stable Ideals Mathematics graded ideals squarefree monomial ideals minimal graded resolutions extremal Betti numbers t-spread monomial ideals |
| title | Extremal Betti Numbers of t-Spread Strongly Stable Ideals |
| title_full | Extremal Betti Numbers of t-Spread Strongly Stable Ideals |
| title_fullStr | Extremal Betti Numbers of t-Spread Strongly Stable Ideals |
| title_full_unstemmed | Extremal Betti Numbers of t-Spread Strongly Stable Ideals |
| title_short | Extremal Betti Numbers of t-Spread Strongly Stable Ideals |
| title_sort | extremal betti numbers of t spread strongly stable ideals |
| topic | graded ideals squarefree monomial ideals minimal graded resolutions extremal Betti numbers t-spread monomial ideals |
| url | https://www.mdpi.com/2227-7390/7/8/695 |
| work_keys_str_mv | AT lucaamata extremalbettinumbersoftspreadstronglystableideals AT marilenacrupi extremalbettinumbersoftspreadstronglystableideals |