# Algebraic shifting and graded betti numbers

Satoshi Murai, Takayuki Hibi

Research output: Contribution to journalArticle

2 Citations (Scopus)

### Abstract

Let S = K[x1, ⋯,xn] denote the polynomial ring in n variables over a field K with each deg xi = 1. Let Δ be a simplicial complex on [n] = {1, ⋯,n} and I. Δ S its Stanley-Reisner ideal. We write.e for the exterior algebraic shifted complex of Δ and.c for a combinatorial shifted complex of. Let ßii+j (I.) = dimK Tori(K, I.)i+j denote the graded Betti numbers of I. In the present paper it will be proved that (i) βii%+j (I.e) = βii%+j (I.c) for all i and j, where the base field is infinite, and (ii) βii%+j (I.) = βii%+j (I.c) for all i and j, where the base field is arbitrary. Thus in particular one has βii%+j (I.) = βii%+j (I.lex) for all i and j, where.lex is the unique lexsegment simplicial complex with the same f-vector as Δ and where the base field is arbitrary.

Original language English 1853-1865 13 Transactions of the American Mathematical Society 361 4 https://doi.org/10.1090/S0002-9947-08-04707-7 Published - 2009 Apr 1 Yes

### Fingerprint

Polynomials
Simplicial Complex
Denote
F-vector
Si
Arbitrary
Polynomial ring
Torus

### ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics

### Cite this

Algebraic shifting and graded betti numbers. / Murai, Satoshi; Hibi, Takayuki.

In: Transactions of the American Mathematical Society, Vol. 361, No. 4, 01.04.2009, p. 1853-1865.

Research output: Contribution to journalArticle

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abstract = "Let S = K[x1, ⋯,xn] denote the polynomial ring in n variables over a field K with each deg xi = 1. Let Δ be a simplicial complex on [n] = {1, ⋯,n} and I. Δ S its Stanley-Reisner ideal. We write.e for the exterior algebraic shifted complex of Δ and.c for a combinatorial shifted complex of. Let {\ss}ii+j (I.) = dimK Tori(K, I.)i+j denote the graded Betti numbers of I. In the present paper it will be proved that (i) βii{\%}+j (I.e) = βii{\%}+j (I.c) for all i and j, where the base field is infinite, and (ii) βii{\%}+j (I.) = βii{\%}+j (I.c) for all i and j, where the base field is arbitrary. Thus in particular one has βii{\%}+j (I.) = βii{\%}+j (I.lex) for all i and j, where.lex is the unique lexsegment simplicial complex with the same f-vector as Δ and where the base field is arbitrary.",
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AB - Let S = K[x1, ⋯,xn] denote the polynomial ring in n variables over a field K with each deg xi = 1. Let Δ be a simplicial complex on [n] = {1, ⋯,n} and I. Δ S its Stanley-Reisner ideal. We write.e for the exterior algebraic shifted complex of Δ and.c for a combinatorial shifted complex of. Let ßii+j (I.) = dimK Tori(K, I.)i+j denote the graded Betti numbers of I. In the present paper it will be proved that (i) βii%+j (I.e) = βii%+j (I.c) for all i and j, where the base field is infinite, and (ii) βii%+j (I.) = βii%+j (I.c) for all i and j, where the base field is arbitrary. Thus in particular one has βii%+j (I.) = βii%+j (I.lex) for all i and j, where.lex is the unique lexsegment simplicial complex with the same f-vector as Δ and where the base field is arbitrary.

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