Anders Frankild's research while affiliated with Newcastle University and other places

Publications (20)

Article
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We investigate Buchbaum and Eisenbud's construction of the second symmetric power S^2_R(X) of a chain complex X of modules over a commutative ring R. We state and prove a number of results from the folklore of the subject for which we know of no good direct references. We also provide several explicit computations and examples. We use this construc...
Article
Consider a local chain Differential Graded algebra, such as the singular chain complex of a pathwise connected topological group.In two previous papers, a number of homological results were proved for such an algebra: An Amplitude Inequality, an Auslander–Buchsbaum Equality, and a Gap Theorem. These were inspired by homological ring theory.By the s...
Article
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We study the following question: Given two semidualizing complexes B and C over a commutative noetherian ring R, does the vanishing of Ext^n_R(B,C) for n>>0 imply that B is C-reflexive? This question is a natural generalization of one studied by Avramov, Buchweitz, and Sega. We begin by providing conditions equivalent to B being C-reflexive, each o...
Article
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Let $(R,\m)$ and $(S,\n)$ be commutative Noetherian local rings, and let $\phi:R\to S$ be a flat local homomorphism such that $\m S = \n$ and the induced map on residue fields $R/\m \to S/\n$ is an isomorphism. Given a finitely generated $R$-module $M$, we show that $M$ has an $S$-module structure compatible with the given $R$-module structure if a...
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Let R be a commutative, noetherian, local ring. Topological Q-vector spaces modelled on full subcategories of the derived category of R are constructed in order to study intersection multiplicities.
Article
The main result asserts that a local commutative noetherian ring is Gorenstein if it possesses a non-zero cyclic module of finite Gorenstein injec-tive dimension. From this follows a classical result by Peskine and Szpiro: A local ring is Gorenstein if it admits a non-zero cyclic module of finite (classical) injective dimension. The main result app...
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We show that the set S(R) of shift-isomorphism classes of semidualizing complexes over a local ring R admits a nontrivial metric. We investigate the interplay between the metric and several algebraic operations. Motivated by the dagger duality isometry, we prove the following: If K,L are homologically bounded below and degreewise finite R-complexes...
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Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let $R$ be a commutative Noetherian ring and $a$ an ideal in the Jacobson radical of $R$. Let $\hat{R}^a$ be the $a$-adic completion of $R$. If $M$ is a finitely generated $R$-module such that $\ext^i_R(\hat{R}^a,M)=0$ for all $i\neq 0$, then $M$ is $...
Article
In this paper we present a systematic study of the reflexivity properties of homologically finite complexes with respect to semidualizing complexes in the setting of nonlocal rings. One primary focus is the descent of these properties over ring homomorphisms of finite flat dimension, presented in terms of inequalities between generalized G-dimensio...
Article
Recently, Dwyer and Greenless established a Morita-like equivalence between categories consisting of complete modules and torsion modules. It turns out that these categories contain certain full subcategories which may be viewed as "perturbed" Auslander and Bass classes; Auslander and Bass classes are used in the study of so-called Gorenstein dimen...
Article
Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings.As opposed to their classical counterparts, these dimensions do not immediately come with practical and robust criteria for finiteness, not even over...
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We propose a definition of Gorenstein Differential Graded Algebra. In order to give examples, we introduce the technical notion of Gorenstein morphism. This enables us to prove the following: Theorem:Let A be a noetherian local commutative ring, let L be a bounded complex of finitely generated projective A-modules which is not homotopy equivalent...
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The paper explores dualizing differential graded (DG) modules over DG algebras. The focus is on DG algebras that are commutative local, and finite. One of the main results established is that, for this class of DG algebras, a finite DG module is dualizing precisely when its Bass number is 1. As a corollary, one obtains that the Avramov–Foxby notion...
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 We study the vanishing properties of local homology of complexes of modules without assuming that its homology is artinian. Using vanishing results for local homology and cohomology we prove new vanishing results for Ext- and Tor-modules.
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Taking the idea from classical Foxby equivalence, we develop an equivalence theory for derived categories over differential graded algebras. Both classical Foxby equivalence and the Morita equivalence for complete modules and torsion modules developed by Dwyer and Greenlees arise as special cases.
Article
The classical homological dimensions—the projective, flat, and injective ones—are usually defined in terms of resolutions and then proved to be computable in terms of vanishing of appropriate derived functors. In this paper we define restricted homological dimensions in terms of vanishing of the same derived functors but over classes of test module...
Article
For a large class of local homomorphisms φ: R → S, including those of finite G-dimension studied by Avramov and Foxby [Proc. London Math. Soc. 75 (1997), 241-270], we assign a new numerical invariant called the quasi Cohen-Macaulay defect of φ, and a local homomorphism is called quasi Cohen-Macaulay if it is of finite G-dimension and has trivial qu...

Citations

... The idea of reproving and extending known equivalences/dualities from commutative algebra via an abstract approach, like we do, is certainly not new. In fact, this is the main idea in, for example, [13,14] by Frankild and Jørgensen, however, these papers focus on the derived category setting, whereas we are interested in the the abelian category setting. ...
... We prove Theorems 2.3, 2.4, and 2.5 by means of dualizing DG modules (DG module being our abbreviation of Differential Graded module). These are the natural generalization of dualizing complexes from homological ring theory, and were made available in [13,14]. As any reader of the ring theoretic literature will know, dualizing complexes can be used to give nice proofs of homological identities; it is hence not surprising that dualizing DG modules enable us to prove homological identities for DGAs. ...
... Our second theorem is inspired by the Gap Theorem by Frankild and Jørgensen, see [7,Theorem 2.5]. The vanishing condition on the Hom-spaces can be viewed as a gap in the cohomology of the object X and it implies the splitting of truncation triangles. ...
... Projective, injective and flat dimensions are important and fundamental in classical homological algebra. As generalizations of the above homological dimensions, Christensen, Foxby and Frankild [2] introduced the notions of restricted homological dimensions; they use derived functors to define restricted projective, injective and flat dimensions of modules. Sharif and Yassemi [15] studied the behavior of the restricted flat dimension under change of rings, and generalized some classical results. ...
... (See Section 2 for definitions and background material.) The utility of these factorizations can be seen in their many applications; see, e.g., [3,4,6,7,8,13,22,28]. The main point of this construction is that it allows one to study a local ring homomorphism by replacing it with a surjective one; thus, one can assume that the target is finitely generated over the source, so one can apply finite homological algebra techniques. ...
... Let us now briefly explain where the full subcategories E l 1 , E l 2 , F l 1 , and F l 2 come from. The full subcategories E l 1 ⊂ C-comod and F l 1 ⊂ D-contra are our analogues of what are known as the Auslander and Bass classes in the literature [2,3,6,8,12]. So they are defined as the classes of all left C-comodules and left D-contramodules satisfying certain conditions with respect to the derived functors R Hom C (L • , −) and L • L D −, with the parameter l 1 meaning a certain (co)homological degree. ...
... We now recall the definition of dualizing DG-modules which are the appropriate generalization of dualizing complexes to the DG-setting. We refer the reader to [10,28] for more details. Fix a commutative noetherian DG-ring A. A DG-A-module R is a dualizing DG-module over A if it has the following properties: ...
... Consequently Q is a Gorenstein ring by [15,Theorem 4.5]. We plan to show that R ′ is a Cohen-Macaulay ring. ...
... Let a be an ideal of R. The right derived local cohomology functor with support in a is denoted by RΓ a (−). Its right adjoint, LΛ a (−), is the left derived local homology functor with support in a (see [8] for detail). ...
... The aim of this section is to generalize this result of Bass to the DG-setting. Such Gorenstein conditions on DG-rings were studied in detail in [8]. ...