Homological invariants of powers of fiber products

TL;DR

This paper investigates the homological properties of powers of fiber product ideals in polynomial rings, providing explicit formulas for depth and regularity under certain conditions, and extending known results on linearity defect.

Contribution

It offers new explicit formulas for depth and regularity of fiber product powers and extends the understanding of linearity defect in this context.

Findings

Depth of powers of fiber product is zero for all s ≥ 2.

Explicit formulas for regularity of fiber product powers when ideals are generated in a single degree.

Linearity defect of the fiber product equals the maximum of the linearity defects of the individual ideals.

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 R\"omer. The proofs exploit the so-called Betti splittings of powers of a fiber product.