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 |
|---|---|
| Pages (from-to) | 1853-1865 |
| Number of pages | 13 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 361 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 2009 Apr 1 |
| Externally published | Yes |
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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 journal › Article
}
TY - JOUR
T1 - Algebraic shifting and graded betti numbers
AU - Murai, Satoshi
AU - Hibi, Takayuki
PY - 2009/4/1
Y1 - 2009/4/1
N2 - 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.
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.
UR - http://www.scopus.com/inward/record.url?scp=77950632770&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=77950632770&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-08-04707-7
DO - 10.1090/S0002-9947-08-04707-7
M3 - Article
AN - SCOPUS:77950632770
VL - 361
SP - 1853
EP - 1865
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
SN - 0002-9947
IS - 4
ER -