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 | References | Similar Articles | Additional Information

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.