Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

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.


References [Enhancements On Off] (What's this?)

  • 1. A. Aramova, J. Herzog and T. Hibi, Squarefree lexsegment ideals, Math. Z. 228 (1998), 353 - 378. MR 1630500 (99h:13013)
  • 2. A. Aramova, J. Herzog and T. Hibi, Gotzmann theorems for exterior algebras and combinatorics, J. of Algebra 191 (1997), 174 - 211. MR 1444495 (98c:13025)
  • 3. A. Aramova, J. Herzog and T. Hibi, Shifting operations and graded Betti numbers, J. Algebraic Combin. 12 (2000), 202 - 222. MR 1803232 (2001k:13022)
  • 4. W. Bruns and J. Herzog, ``Cohen-Macaulay rings,'' Revised Edition, Cambridge University Press, 1996. MR 1251956 (95h:13020)
  • 5. P. Erdös, C. Ko and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. (2) 12 (1960), 313 - 320. MR 0140419 (25:3839)
  • 6. M. Green, Generic initial ideals, in ``Six Lectures on Commutative Algebra'' (J. Elias, J. M. Giral, R. M. Miró-Roig and S. Zarzuela, Eds.) Birkhäuser, 1998, pp. 119 - 186. MR 1648665 (99m:13040)
  • 7. J. Herzog, Generic initial ideals and graded Betti numbers, in ``Computational Commutative Algebra and Combinatorics'' (T. Hibi, Ed.), Advanced Studies in Pure Math., Volume 33, 2002, pp. 75 - 120. MR 1890097 (2003b:13021)
  • 8. G. Kalai, Algebraic shifting, in ``Computational Commutative Algebra and Combinatorics'' (T. Hibi, Ed.), Advanced Studies in Pure Math., Volume 33, 2002, pp. 121 - 163. MR 1890098 (2003e:52024)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13D02, 13F55

Retrieve articles in all journals with MSC (2000): 13D02, 13F55


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: https://doi.org/10.1090/S0002-9947-08-04707-7
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.

American Mathematical Society