Algebraic shifting and graded Betti numbers

Abstract

Let $S = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$. Let $\Delta$ be a simplicial complex on $[n] = \{1, ..., n \}$ 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 $\beta_{ii+j}(I_{\Delta}) = \dim_K \Tor_i(K, I_\Delta)_{i+j}$ 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 $\Delta^{lex}$ is the unique lexsegment simplicial complex with the same $f$-vector as $\Delta$ and where the base field is arbitrary.