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Acta Mathematica Vietnamica

Homological Invariants of Powers of Fiber Products

icon-email Hop D. Nguyen , Thanh Vu

Abstract

Let R and S be polynomial rings of positive dimensions over a field k. Let IR, JS be non-zero homogeneous ideals none of which contains a linear form. Denote by F the fiber product of I and J in T=RkS. We compute homological invariants of the powers of F using the data of I and J. Under the assumption that either chark=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 s2 the intriguing formula depth(T/Fs)=0. If moreover each of the ideals I and J is generated in a single degree, we show that for all s1, regFs=maxi[1,s]{regIi+si,regJi+si}. 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.