Homological invariants of powers of fiber products

Abstract

Let $R$ and $S$ be polynomial rings of positive dimensions over a field $k$. Let $I\subseteq R,J\subseteq S$ be non-zero homogeneous ideals none of which contains a linear form. Denote by $F$ the fiber product of $I$ and $J$ in $T=R{\otimes }_{k}S$. We compute homological invariants of the powers of $F$ using the data of $I$ and $J$. Under the assumption that either $\text{char}k=0$ or $I$ and $J$ are monomial ideals, we provide explicit formulas for the depth and regularity of powers of $F$. In particular, we establish for all $s\ge 2$ the intriguing formula $\text{depth}(T/F^s)=0$. If moreover each of the ideals $I$ and $J$ is generated in a single degree, we show that for all $s\ge 1$, $\text{reg}{F}^s={\max }_{i\in [1,s]}\{\text{reg}{I}^i+s-i,\text{reg}{J}^i+s-i\}$. Finally, we prove that the linearity defect of $F$ is the maximum of the linearity defects of $I$ and $J$, extending previous work of Conca and Römer. The proofs exploit the so-called Betti splittings of powers of a fiber product.

An abridged version of this paper (without the appendix) is to be published in the Acta Mathematica Vietnamica. In this arXiv version, we make a minor correction to a small error in Lemma 5.3(ii) (which unfortunately is present in the online-first AMV version). Nevertheless, this correction has no influence on other parts of the paper. See also the new Remark 5.4