{"title":"平衡简单配合物的分级Betti数","authors":"Martina Juhnke-Kubitzke, Lorenzo Venturello","doi":"10.1007/s40306-021-00449-8","DOIUrl":null,"url":null,"abstract":"<div><p>We prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings <i>S</i>/<i>I</i>, where <i>S</i> is a polynomial ring and <span>\\(I\\subseteq S\\)</span> is a homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner rings of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40306-021-00449-8.pdf","citationCount":"2","resultStr":"{\"title\":\"Graded Betti Numbers of Balanced Simplicial Complexes\",\"authors\":\"Martina Juhnke-Kubitzke, Lorenzo Venturello\",\"doi\":\"10.1007/s40306-021-00449-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings <i>S</i>/<i>I</i>, where <i>S</i> is a polynomial ring and <span>\\\\(I\\\\subseteq S\\\\)</span> is a homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner rings of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type.</p></div>\",\"PeriodicalId\":45527,\"journal\":{\"name\":\"Acta Mathematica Vietnamica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2021-10-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s40306-021-00449-8.pdf\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Vietnamica\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40306-021-00449-8\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Vietnamica","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40306-021-00449-8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Graded Betti Numbers of Balanced Simplicial Complexes
We prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings S/I, where S is a polynomial ring and \(I\subseteq S\) is a homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner rings of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type.
期刊介绍:
Acta Mathematica Vietnamica is a peer-reviewed mathematical journal. The journal publishes original papers of high quality in all branches of Mathematics with strong focus on Algebraic Geometry and Commutative Algebra, Algebraic Topology, Complex Analysis, Dynamical Systems, Optimization and Partial Differential Equations.