Dualizing Differential Graded Modules and Gorenstein Differential Graded Algebras

Frankild, Anders; Iyengar, Srikanth B.; Jørgensen, Peter

doi:10.1112/s0024610703004496

Cited by 44 publications

(41 citation statements)

References 6 publications

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“…The first paper we are aware of that defined dualizing DG-modules over DG-rings was [11], where Hinich defined them over some specific DG-rings associated to a given local ring. In the generality we work in this paper, dualizing DG-modules were first defined by Frankild, Iyengar and Jorgensen in [8], where some basic properties of dualizing DG-modules were established. The recent paper [23] by Yekutieli generalized the definition further, and made a very detailed study of dualizing DG-modules in a very general commutative setting.…”

Section: Then There Is An Isomorphismmentioning

confidence: 99%

The twisted inverse image pseudofunctor over commutative DG rings and perfect base change

Shaul

2017

Advances in Mathematics

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ABSTRACT. Let K be a Gorenstein noetherian ring of finite Krull dimension, and consider the category of cohomologically noetherian commutative differential graded rings A over K, such that H 0 (A) is essentially of finite type over K, and A has finite flat dimension over K. We extend Grothendieck's twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup. We prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps, and prove a perfect base change result for f ! in this setting. As application, we deduce a perfect derived base change result for the twisted inverse image of a map between ordinary commutative noetherian rings. Our results generalize and solve some recent conjectures of Yekutieli.

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Section: Then There Is An Isomorphismmentioning

confidence: 99%

The twisted inverse image pseudofunctor over commutative DG rings and perfect base change

Shaul

2017

Advances in Mathematics

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“…Suppose that the ground ring A has a dualizing complex C and consider D = RHom A (R, C) which is sometimes a so-called dualizing DG module for R, see [6]. Since amp R < ∞ implies cmd…”

Section: Comments and Connectionsmentioning

confidence: 99%

Amplitude inequalities for Differential Graded modules

Jørgensen¹

2010

Forum Mathematicum

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Abstract. Differential Graded Algebras can be studied through their Differential Graded modules. Among these, the compact ones attract particular attention. This paper proves that over a suitable chain Differential Graded Algebra R, each compact Differential Graded module M satisfies amp M ≥ amp R, where amp denotes amplitude which is defined in a straightforward way in terms of the homology of a DG module.In other words, the homology of each compact DG module M is at least as long as the homology of R itself. Conversely, DG modules with shorter homology than R are not compact, and so in general, there exist DG modules with finitely generated homology which are not compact.Hence, in contrast to ring theory, it makes no sense to define finite global dimension of DGAs by the condition that each DG module with finitely generated homology must be compact. IntroductionDifferential Graded Algebras (DGAs) play an important role in both ring theory and algebraic topology. For instance, if M is a complex of modules, then the endomorphism complex Hom(M, M) is a DGA with multiplication given by composition of endomorphisms, and this can be used to prove ring theoretical results, see [12] and [13]. Another example is that over a commutative ring, the Koszul complex on a series of elements is a DGA, see [17, sec. 4.5], and again, ring theoretical results ensue, see [10].Likewise, DGAs occur naturally in algebraic topology, where the canonical example is the singular cochain complex C * (X) of a topological space X. Other constructions also give DGAs; for instance, if G is a topological monoid, then the singular chain complex C * (G) is a DGA whose multiplication is induced by the composition of G; see [4].Just as rings can be studied through their modules, DGAs can be studied through their Differential Graded modules (DG modules), and this is the subject of the present paper.The main results are a number of "amplitude inequalities" which give bounds on the amplitudes of various types of DG modules. Such results have been known for complexes of modules over rings since Iversen's paper [9], and it is natural to seek to extend them to DG modules.Another main point, implied by one of the amplitude inequalities, is that, in contrast to ring theory, it appears to make no sense to define finite global dimension of DGAs by the condition that each DG module with finitely generated homology must be compact. This is of interest since several people have been asking how one might define finite global dimension for DGAs.First main Theorem. To get to the first main Theorem of the paper, recall from [13] that if R is a DGA then a good setting for DG modules over R is the derived category of DG left-R-modules D(R).A DG left-R-module is called compact if it is in the smallest triangulated subcategory of D(R) containing R, or, to use the language of topologists, if it can be finitely built from R. The compact DG left-Rmodules form a triangulated subcategory D c (R) of D(R), and play the same important role as finitely presented modules of fi...

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“…Suppose that A is a quotient of R which is a Gorenstein local DGA, and let R → A be the quotient morphism. As R is concentrated in non-negative degrees, this clearly induces a surjection In [4,Thm. 4.3] it was proved that if R is a local DGA with residue class field , then (6) R is Gorenstein ⇔ dim Ext R ( , R) = 1.…”

Section: Theorem 22 Let a Be A Noetherian Local Commutative Ring Anmentioning

confidence: 94%

“…When R is a commutative DGA with H 0 R a noetherian ring, D(R) denotes the derived category of DG R-modules, and D f (R) denotes the full subcategory of DG modules M such that HM is a finitely generated module over H 0 R. (This is compatible with the use of the notation D f given before Definition 1.1.) For commutative DGAs, the definitions of Gorenstein DGAs and dualizing DG modules from [5] and [4] simplify as follows. Definition 1.3.…”

mentioning

confidence: 99%