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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 $\\text{depth}(T/F^s)=0$. 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Nguyen, Thanh Vu","submitted_at":"2018-03-11T19:10:20Z","abstract_excerpt":"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 $\\text{depth}(T/F^s)=0$. 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