Sequences for complexes II

Christensen, Lars Winther

doi:10.7146/math.scand.a-14384

Cited by 7 publications

(5 citation statements)

References 6 publications

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“…. We note that [8,9] uses homological notation, making the definitions look slightly different. The support of a DG-module was defined in Definition 1.10.…”

Section: Regular Sequences and The Derived Bass Conjecturementioning

confidence: 99%

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The Cohen-Macaulay property in derived commutative algebra

Shaul

2020

Trans. Amer. Math. Soc.

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By extending some basic results of Grothendieck and Foxby about local cohomology to commutative DG-rings, we prove new amplitude inequalities about finite DGmodules of finite injective dimension over commutative local DG-rings, complementing results of Jørgensen and resolving a recent conjecture of Minamoto. When these inequalities are equalities, we arrive to the notion of a local-Cohen-Macaulay DG-ring. We make a detailed study of this notion, showing that much of the classical theory of Cohen-Macaulay rings and modules can be generalized to the derived setting, and that there are many natural examples of local-Cohen-Macaulay DG-rings. In particular, local Gorenstein DG-rings are local-Cohen-Macaulay. Our work is in a non-positive cohomological situation, allowing the Cohen-Macaulay condition to be introduced to derived algebraic geometry, but we also discuss extensions of it to non-negative DG-rings, which could lead to the concept of Cohen-Macaulayness in topology.

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“…. We note that [8,9] uses homological notation, making the definitions look slightly different. The support of a DG-module was defined in Definition 1.10.…”

Section: Regular Sequences and The Derived Bass Conjecturementioning

confidence: 99%

“…The next definitions are DG-versions of [8, Definition 2.3] and [9, Definition 2.1]. We note that [8,9] uses homological notation, making the definitions look slightly different. The support of a DG-module was defined in Definition 1.10.…”

Section: Regular Sequences and The Derived Bass Conjecturementioning

confidence: 99%

The Cohen-Macaulay property in derived commutative algebra

Shaul

2020

Trans. Amer. Math. Soc.

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“…Regular sequences and associated primes over commutative DG-rings. Following [14, Section 3.2], [17, Section 5], as well as the earlier [6,7], given a commutative noetherian local DG-ring (A, m), and given M ∈ D + (A), we say that an element…”

Section: Preliminariesmentioning

confidence: 99%

Koszul complexes over Cohen-Macaulay rings

Shaul

2020

Preprint

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We prove a Cohen-Macaulay version of a result by Avramov-Golod and Frankild-Jørgensen about Gorenstein rings, showing that if a noetherian ring A is Cohen-Macaulay, and a 1 , . . . , an is any sequence of elements in A, then the Koszul complex K(A; a 1 , . . . , an) is a Cohen-Macaulay DG-ring. We further generalize this result, showing that it also holds for commutative DG-rings. In the process of proving this, we develop a new technique to study the dimension theory of a noetherian ring A, by finding a Cohen-Macaulay DG-ring B such that H 0 (B) = A, and using the Cohen-Macaulay structure of B to deduce results about A. As application, we prove that if f : X → Y is a morphism of schemes, where X is Cohen-Macaulay and Y is nonsingular, then the homotopy fiber of f at every point is Cohen-Macaulay. As another application, we generalize the miracle flatness theorem. Generalizations of these applications to derived algebraic geometry are also given.

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“…The purpose of this section is to fix notation and recall some definitions and results of hyperhomological algebra relevant to this article. As a general reference, we mention [2] and references therein. For more details, see also [3], [4], [8] and [9].…”

Section: Preliminariesmentioning

confidence: 99%

Canonical modules of complexes

Akhavin

Hyry

2016

Journal of Pure and Applied Algebra

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We define the notion of the canonical module of a complex. We then consider Serre's conditions for a complex and study their relationship to the local cohomology of the canonical module and its ring of endomorphisms. M : . . .The derived category is triangulated, the suspension functor Σ being defined by the formulas (ΣM) n = M n−1 and d ΣM n = −d n−1 . The symbol "≃" is reserved for isomorphisms in D(R). We use the subscripts "b", "+" and "−" to denote the homological boundness, the homological boundness from below and the homological boundness from above, respectively. The superscript "f " denotes the homological finiteness. So the full subcategory of D(R) consisting of complexes with finitely generated homology modules is denoted by D f (R). As usual, we identify the category of R-modules as the full subcategory of D(R) of complexes M satisfying H i (M) = 0 for i = 0. For a complex M ∈ D(R), by sup M and inf M, we mean its homological supremum and infimum. Let M

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Sequences for complexes II

Cited by 7 publications

References 6 publications

The Cohen-Macaulay property in derived commutative algebra

The Cohen-Macaulay property in derived commutative algebra

Koszul complexes over Cohen-Macaulay rings

Canonical modules of complexes

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