Algebraic shifting and graded Betti numbers

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.