Algebraic shifting and graded Betti numbers
| dc.creator | Hibi, Takayuki | |
| dc.creator | Murai, Satoshi | |
| dc.date | 2005-03-29 | |
| dc.date | 2008-02-12 | |
| dc.date.accessioned | 2026-07-07T09:19:59Z | |
| dc.date.available | 2026-07-07T09:19:59Z | |
| dc.description | Let $S = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $°x_i = 1$. Let $Δ$ be a simplicial complex on $[n] = \{1, ..., n \}$ and $I_Δ\subset S$ its Stanley--Reisner ideal. We write $Δ^e$ for the exterior algebraic shifted complex of $Δ$ and $Δ^c$ for a combinatorial shifted complex of $Δ$. Let $β_{ii+j}(I_Δ) = \dim_K \Tor_i(K, I_Δ)_{i+j}$ denote the graded Betti numbers of $I_Δ$. In the present paper it will be proved that (i) $β_{ii+j}(I_{Δ^e}) \leq β_{ii+j}(I_{Δ^c})$ for all $i$ and $j$, where the base field is infinite, and (ii) $β_{ii+j}(I_Δ) \leq β_{ii+j}(I_{Δ^c})$ for all $i$ and $j$, where the base field is arbitrary. Thus in particular one has $β_{ii+j}(I_Δ) \leq β_{ii+j}(I_{Δ^{lex}})$ for all $i$ and $j$, where $Δ^{lex}$ is the unique lexsegment simplicial complex with the same $f$-vector as $Δ$ and where the base field is arbitrary. | |
| dc.description | 14 pages; title changed, new section added. To appear in Trans. Amer. Math. Soc | |
| dc.identifier | https://arxiv.org/abs/math/0503685 | |
| dc.identifier | http://arxiv.org/abs/math/0503685 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/154591 | |
| dc.subject | Commutative Algebra | |
| dc.subject | 13F55; 13D02 | |
| dc.title | Algebraic shifting and graded Betti numbers | |
| dc.type | text |