Homological invariants of powers of fiber products

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 $\text{depth}(T\mathrm{/}{F}^s)=0$. 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.