Notes on the linearity defect and applications

Journal of Pure and Applied Algebra

2019

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Let R and S be standard graded algebras over a field k, and I ⊆ R and J ⊆ S homogeneous ideals. Denote by P the sum of the extensions of I and J to R ⊗ k S. We investigate several important homological invariants of powers of P based on the information about I and J, with focus on finding the exact formulas for these invariants. Our investigation exploits certain Tor vanishing property of natural inclusion maps between consecutive powers of I and J. As a consequence, we provide fairly complete information about the depth and regularity of powers of P given that R and S are polynomial rings and either char k = 0 or I and J are generated by monomials.2010 Mathematics Subject Classification. 13D02, 13C05, 13D05, 13H99.

Section: 2mentioning

confidence: 83%

Section: Maps Of Tor and Algebraic Invariantsmentioning

confidence: 99%

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Powers of sums and their homological invariants

Journal of Pure and Applied Algebra

2019

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“…Lemma 4.11 (Nguyen,[29]). Let (R, m) be a noetherian local ring, and M φ − −→ P be a morphism of finitely generated R-modules.…”

Section: Morphisms Which Induce Trivial Maps Of Tormentioning

confidence: 99%

Linearity defect of edge ideals and Fröberg’s theorem

2016

J Algebr Comb

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Abstract. Fröberg's classical theorem about edge ideals with 2-linear resolution can be regarded as a classification of graphs whose edge ideals have linearity defect zero. Extending his theorem, we classify all graphs whose edge ideals have linearity defect at most 1. Our characterization is independent of the characteristic of the base field: the graphs in question are exactly weakly chordal graphs with induced matching number at most 2. The proof uses the theory of Betti splittings of monomial ideals due to Francisco, Hà, and Van Tuyl and the structure of weakly chordal graphs. Along the way, we compute the linearity defect of edge ideals of cycles and weakly chordal graphs. We are also able to recover and generalize previous results due to Dochtermann-Engström, Kimura and Woodroofe on the projective dimension and Castelnuovo-Mumford regularity of edge ideals. IntroductionLet (R, m) be a standard graded algebra over a field k with the graded maximal ideal m. Let M be a finitely generated graded R-module. For integers i, j,The regularity is an important invariant of graded modules over R. When R is a polynomial ring and M is a monomial ideal of R, its regularity exposes many combinatorial flavors. This fact has been exploited and proved to be very useful for studying the regularity; for recent surveys, see [13], [28], [38]. A classical and instructive example is Fröberg's theorem. Recall that if G is a graph on the vertex set {x 1 , . . . , x n } (where n ≥ 1), and by abuse of notation, R is the polynomial ring k[x 1 , . . . , x n ], then the edge ideal of G is I(G) = (x i x j : {x i , x j } is an edge of G). Unless otherwise stated, whenever we talk about an invariant of I(G) (including the regularity and the linearity defect, to be defined below), it is understood that the base ring is the polynomial ring R. For m ≥ 3, the cycle C m is the graph on vertices x 1 , . . . , x m with edges x 1 x 2 , . . . , x m−1 x m , x m x 1 . We say that a graph G is weakly chordal (or weakly triangulated) if for every m ≥ 5, neither G, nor its complement contains C m as an induced subgraph. G is chordal if for any m ≥ 4, C m is not an induced subgraph of G. Fröberg's theorem [10] says that I(G) has regularity 2 if and only if the complement graph of G is chordal. It is of interest to find generalizations of this important result; see, for instance, [7], [8]. Theorem 4.1] extended Fröberg's theorem by providing a combinatorial characterization of connected bipartite graphs G such that reg I(G) = 3. In general, the regularity 3 condition on edge ideals is not 2010 Mathematics Subject Classification. 05C25, 13D02, 13H99.

“…Nguyen,[46, Proposition 2.5]). Let (R, m) be a noetherian local ring with the residue field k. Consider an exact sequence 0 −→ M −→ P −→ N −→ 0 of finitely generated R-modules.…”

mentioning

confidence: 99%

Homological Invariants of Powers of Fiber Products

2019

Acta Math Vietnam

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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.