Homological Invariants of Powers of Fiber Products
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Homological Invariants of Powers of Fiber Products Hop D. Nguyen1 · Thanh Vu2 Dedicated to Professor Lˆe Tuˆa´n Hoa on the occasion of his sixtieth birthday Received: 17 March 2018 / Revised: 6 November 2018 / Accepted: 12 November 2018 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019
Abstract Let R and S be polynomial rings of positive dimensions over a field k. Let I ⊆ R, J ⊆ 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 ⊗k S. We compute homological invariants of the powers of F using the data of I and J . Under the assumption that either 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 ≥ 2 the intriguing formula 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 ≥ 1, reg F s = maxi∈[1,s] {reg I i + s − i, 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. Keywords Powers of ideals · Fiber product · Depth · Regularity · Castelnuovo–Mumford regularity · Linearity defect Mathematics Subject Classification (2010) 13D02 · 13C05 · 13D05 · 13H99
1 Introduction In this paper, we are concerned with the theory of powers of ideals. There is many work on the asymptotic properties of powers of ideals (see, e.g., [9, 12, 19, 22, 23, 33, 35, 39]). The Hop D. Nguyen
[email protected] Thanh Vu [email protected] 1
Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Hanoi, Vietnam
2
Hanoi University of Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam
H.D. Nguyen, T. Vu
reader is referred to a recent survey of Chardin [13] for an overview of work on regularity of powers of ideals. The novel feature of the present work is its focus on exact formulas for homological invariants of all the powers, under some fairly general circumstances. Let R, S be standard graded polynomial rings over a field k. Denote by m and n the corresponding graded maximal ideals of R and S, and T = R ⊗k S. Let I ⊆ m2 , J ⊆ n2 be homogeneous ideals. By abuse of notation, we also denote by I and J the extensions of these ideals to T . In [30] and [47], formulas for the depth and regularity of powers of I + J were provided. One motivation for both papers is that the sum I + J defines a fundamental operation on the k-algebras R/I and S/J , namely their tensor product. Among others, fiber product is also a fundamental operation on k-algebras. Indeed, fiber products of algebras (and more generally, of rings) were investigated by many authors (see, for example, [4, 16, 20, 40, 43, 44]). Now the fiber product of the k-algebras R/I and S/J is defined by F = I +J +mn, which accordingly is called
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