Homological Invariants of Powers of Fiber Products

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 = max i∈ [1,s] 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. 2010 Mathematics Subject Classification. 13D02, 13C05, 13D05, 13H99. Proof. This follows from the proof of [46, Corollary 3.8]. 2.4. Elementary facts. Lemma 2.6. Let R, S be affine k-algebras. Let I, J be ideals of R, S, respectively. Then in T = R ⊗ k S, there is an equality I ∩ J = IJ. This standard fact follows from [48, Lemma 2.2(i)].Lemma 2.7. Let R, S be standard graded k-algebras, and M, N be finitely generated graded modules over R, S, respectively. Then for T = R ⊗ k S, there are equalitiesThis lemma is folklore. The first part follows from the description of local cohomology of tensor product due to Goto and Watanabe [29, Theorem 2.2.5]. For the remaining assertions, see [48, Lemma 2.3].Recall that a morphism of noetherian local rings (R, m) θ − → (S, n) is an algebra retract if there exists a local homomorphism (S, n) φ − → (R, m) such that φ • θ is the identity map of R. In that case, φ is called the retraction map of θ. Lemma 2.8 ([48, Lemma 2.4]). Let (R, m) θ − → (S, n) be an algebra retract of noetherian local rings with the retraction map S φ − → R. Let I ⊆ m be an ideal of R. Let J ⊆ n be an ideal containing θ(I)S such that φ(J)R = I. Then there are inequalities ld R (R/I) ≤ ld S (S/J), ld R I ≤ ld S J.

Powers of sums and their homological invariants

“…This paper is the second part, which contains Section 4 of [41] but goes significantly beyond that. A third part which contains the results in the last section of [41], among other things, is the preprint [44].…”

Local rings with quasi-decomposable maximal ideal

“…1 Furthermore, Nasseh and Takahashi [42, Theorem A] proved that the maximal ideal of a fiber product ring is always a direct summand of a direct sum of certain syzygies of finitely generated modules of infinite projective dimension. Several other properties and applications of these rings have also been studied in [1,16,18,20,26,31,37,41,44,54].…”

Geometric regularity of powers of two-dimensional squarefree monomial ideals

“…( 1) cannot be removed. But if assume further I is a squarefree monomial ideal, then the condition that d ≥ 3 could be dropped, as shown by [18,Corollary 5.6].…”