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Transactions of the American Mathematical Society
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
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
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
Email: hibi@math.sci.osaka-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04707-7
PII: S 0002-9947(08)04707-7
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.