# Hans-Bjørn Foxby's research while affiliated with University of Copenhagen and other places

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## Publications (27)

The new intersection theorem states that, over a Noetherian local ring R , for any non-exact complex concentrated in degrees n ,…,0 in the category P(length) of bounded complexes of finitely generated projective modules with finite-length homology, we must have n ≥ d = dim R .
One of the results in this paper is that the Grothendieck group of P(len...

Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory’s connections with relative homological algebra and with studies of local ring homomorphisms. It ends close to the starting point: with a characterization of Gorenstein...

The projective dimension of Cartan and Eilenberg and the Gorenstein dimension of Auslander and Bridger are two classical homological dimensions for the class of finite modules over commutative noetherian local rings. Recently, new homological dimensions have been defined: complete intersection dimension by Avramov, Gasharov and Peeva, Cohen-Macaula...

We prove that over a commutative noetherian ring the three approaches to introducing depth for complexes: via Koszul homology, via Ext modules, and via local cohomology, all yield the same invariant. Using this result, we establish a far reaching generalization of the classical Auslander-Buchsbaum formula for the depth of finitely generated modules...

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...

Numerical invariants which measure the Cohen–Macaulay character of homomorphismsϕ:R→Sof noetherian rings are introduced and studied. Comprehensive results are obtained for homomorphisms which are locally of finite flat dimension. They provide a point of view from which a variety of phenomena receive a unified treatment. The conceptual clarification...

this paper is to introduce the local stage of a new approach to the study of arbitrary homomorphisms of noetherian rings. If ' : R ! S is a local homomorphism, it involves breaking down the canonically associated homomorphism ` ' from R to the

Nontrivial relations between Bass numbers of local commutative rings are established in case there exists a local homomorphism ϕ:R → S which makes S into an R-module of finiteflat dimension. In particular, it is shown that an inequality μi + depth RR ≤ μi + depth SS holds for all i ϵ ℤ. This is a consequence of an equality involving the Bass series...

This paper explores various notions of projective, injective, and flat dimensions, arising from recent constructions of resolutions of unbounded complexes, proposed by N. Spaltenstein and by S. Halperin with the authors. The different versions of each dimension are compared to each other, and also to the classical concepts, whenever these may be de...

For a flat morphism $\varphi: A \rightarrow B$ of noetherian rings, the minimal injective resolution of the $B$-module $M \otimes_A B$ is described in terms of the minimal injective resolution of the finitely generated $A$-module $M$ and the minimal injective resolutions of the fibers of $\varphi$.

Since Bass' original paper [5] on injective resolutions and Gorenstein rings, there has been a number of papers following a similar theme. Perhaps the most notable is the paper of Peskine and Szpiro [33] which blends the algebraic theory begun by M. Auslander, Bass et al. together with the more geometric concepts of Serre and Grothendieck to solve...

It is proved that an injective module after a flat change of base is of pointwise finite injective dimension if and only if all the fibers at points associated to the injective module are Gorenstein rings (or trivial).

Let A be a noetherian commutative ring which is a homomorphic image of a Gorenstein ring. The first result in this note is that A is a Cohen-Macaulay ring if and only if every finitely generated A-module can be embedded in a finitely generated Amodule T which is pointwise of finite injective dimension (i.e. all localizations of T are of finite inje...

## Citations

... More generally, the Bass series of S is related to the Bass series for R and S/mS by the formula When ϕ is not flat, the properties in the previous paragraph can fail, e.g., for the natural surjection R → k when R is not regular, i.e., when pd R k is not finite. However, Avramov, Foxby, and Lescot [19,20,23] recognized that the full strength of flatness is not needed: (c) Assume further that the closed fibre S/mS is artinian, and either ϕ is not flat or S/mS is not a field. Then the following coefficient-wise inequality holds: ...

... The following classical result will be indispensable (cf. [Laz69], and [FF74] for the graded case): ...

... The relevant theory of derived functors for bounded complexes had been developed in the early 1960s by Grothendieck, Hartshorne, and Verdier, but had been applied only sporadically to questions in commutative algebra. In the short surveys [7,9] Foxby extended the basic numerical invariants of local algebra from finitely generated modules to bounded complexes with finitely generated homology. This is fairly straightforward for various homological dimensions and for depth, but not so for the notion of Krull dimension, which was suggested by work of Foxby and of Iversen. ...

Reference: To Hans-Bjørn Foxby

... Transfer of homological properties along ring homomorphisms is already a classical field of study, initiated in [31] and continued in the more recent series [7, 8, 9, 10, 11]. In this paper we investigate ascent properties of modules in the so-called Auslander categories of a commutative noetherian ring. ...

Reference: Ascent Properties of Auslander Categories

... Our final main result improves Bass' conjecture [7] as proved by Peskine, Szpiro, and Roberts [23,26,27]; see Theorem 5.4. It is an open question whether one can replace complete intersection Hom injective dimension with Gorenstein injective dimension in this result [10,Question 6]; see Takahashi and Yassemi [34,36] for progress on this question. ...

... People have done a lot of work to extend the rich theory of commutative Gorenstein rings to DG algebras (cf. [FHT1,AF,DGI,FJ,FIJ,MW2]). In this paper, we study the Gorensteinness of invariant DG subalgebras of some nice Gorenstein DG algebras. ...

... R. Fossum, H. Foxby, B. Iversen defined, for n ≥ 2, a Mennicke n-symbol U m n (R) wt → SK 1 R using the theory of acyclic based complexes. (We refer the reader to [20]; a copy of which can be got by making a request.) ...

... Grothendieck calls ϕ regular if it is flat and the closed fiber ring B ⊗ A K B/mB is geometrically regular (i.e., B/mB⊗ K F is regular for every finite extension F of K [12]). Avramov, Foxby and Herzog call ϕ weakly regular if it is flat and B/mB is regular [3]. Of course, if ϕ is regular, then it is weakly regular. ...

Reference: Basically regular local homomorphisms

... Second, we expand our perspective from modules and complexes over local rings to the setting of differential graded (DG) modules over commutative DG algebras. This point of view was developed by Avramov and his collaborators; see, e.g., [4,5,7,8,12,13,15,16,17,19,18,20]. It provides a construction whereby if one can solve a homological commutative algebra problem for finite dimensional algebras over a field, then one can sometimes solve the general problem by passing to an associated finite dimensional DG algebra over a field where one can solve a related problem. ...

Reference: Ascent Properties for Test Modules

... Theorem 59. [FFGR75,5.3] Let (R, m) be 1-dimensional local ring. Then R has a dualizing module iff the generic fibers of R →R are Gorenstein. ...