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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Algebraic shifting and graded Betti numbers


Authors: Satoshi Murai and Takayuki Hibi
Journal: Trans. Amer. Math. Soc. 361 (2009), 1853-1865
MSC (2000): Primary 13D02; Secondary 13F55
DOI: https://doi.org/10.1090/S0002-9947-08-04707-7
Published electronically: October 20, 2008
MathSciNet review: 2465820
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Abstract: Let $S = K[x_1, \ldots , x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$. Let $\Delta$ be a simplicial complex on $[n] = \{ 1, \ldots , n \}$ and $I_\Delta \subset S$ its Stanley–Reisner ideal. We write $\Delta ^e$ for the exterior algebraic shifted complex of $\Delta$ and $\Delta ^c$ for a combinatorial shifted complex of $\Delta$. Let $\beta _{ii+j}(I_{\Delta }) = \dim _K \mathrm {Tor}_i(K, I_\Delta )_{i+j}$ denote the graded Betti numbers of $I_\Delta$. In the present paper it will be proved that (i) $\beta _{ii+j}(I_{\Delta ^e}) \leq \beta _{ii+j}(I_{\Delta ^c})$ for all $i$ and $j$, where the base field is infinite, and (ii) $\beta _{ii+j}(I_{\Delta }) \leq \beta _{ii+j}(I_{\Delta ^c})$ for all $i$ and $j$, where the base field is arbitrary. Thus in particular one has $\beta _{ii+j}(I_\Delta ) \leq \beta _{ii+j}(I_{\Delta ^{lex}})$ for all $i$ and $j$, where $\Delta ^{\operatorname {lex}}$ is the unique lexsegment simplicial complex with the same $f$-vector as $\Delta$ and where the base field is arbitrary.


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Additional Information

Satoshi Murai
Affiliation: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan
MR Author ID: 800440
Email: s-murai@ist.osaka-u.ac.jp

Takayuki Hibi
Affiliation: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan
MR Author ID: 219759
Email: hibi@math.sci.osaka-u.ac.jp

Received by editor(s): March 2, 2007
Published electronically: October 20, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.