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- W4301661602 abstract "Let $R$ and $S$ be polynomial rings of positive dimensions over a field $k$. Let $Isubseteq R, Jsubseteq S$ be non-zero homogeneous ideals none of which contains a linear form. Denote by $F$ the fiber product of $I$ and $J$ in $T=Rotimes_k S$. 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 $sge 2$ 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 $sge 1$, $text{reg}~ F^s=max_{iin [1,s]}{text{reg}~ I^i+s-i,text{reg}~ J^i+s-i}$. 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 Romer. The proofs exploit the so-called Betti splittings of powers of a fiber product." @default.
- W4301661602 created "2022-10-05" @default.
- W4301661602 creator A5010811164 @default.
- W4301661602 creator A5071238613 @default.
- W4301661602 date "2018-03-11" @default.
- W4301661602 modified "2023-10-16" @default.
- W4301661602 title "Homological invariants of powers of fiber products" @default.
- W4301661602 doi "https://doi.org/10.48550/arxiv.1803.04016" @default.
- W4301661602 hasPublicationYear "2018" @default.
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