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Peter Jorgensen

Publications and source records attributed to Peter Jorgensen.

83 records · Page 5Linked to original sources
arXiv2004

Tate cohomology over fairly general rings

Tate cohomology was originally defined over finite groups. More recently, Avramov and Martsinkovsky showed how to extend the definition so that it now works well over Gorenstein rings. This paper improves the theory further by giving a new definition that works over more general rings, specifically, those with a dualizing complex. The new definition of Tate cohomology retains the desirable properties established by Avramov and Martsinkovsky. Notably, there is a long exact sequence connecting it to ordinary Ext groups.

math.RA↗
arXiv2003

Finite flat and projective dimension

Over a right-noetherian algebra admitting a dualizing complex, any left-module with finite flat dimension also has finite projective dimension.

math.RA↗
arXiv2003

The homotopy category of complexes of projective modules

The homotopy category of complexes of projective left-modules over any reasonably nice ring is proved to be a compactly generated triangulated category, and a duality is given between its subcategory of compact objects and the finite derived category of right-modules.

math.RA↗
arXiv2003

Auslander-Reiten sequences on schemes

Let X be a smooth projective scheme of dimension d >= 1 over the field k, and let C be an indecomposable coherent sheaf on X. Then there is an Auslander-Reiten sequence in the category of quasi-coherent sheaves on X, with C as right hand end term. The left hand end term of the sequence can be computed. It is the (d-1)'st syzygy in a minimal injective resolution of C, tensor the dualizing sheaf of X.

math.AG↗
arXiv2003

Ext vanishing and infinite Auslander-Buchsbaum

A vanishing theorem is proved for Ext groups over non-commutative graded algebras. Along the way, an "infinite" version is proved of the non-commutative Auslander-Buchsbaum theorem.

math.RA↗
arXiv2003

Auslander-Reiten triangles and quivers over topological spaces

In this paper, Auslander-Reiten triangles are introduced into algebraic topology, and it is proved that their existence characterizes Poincare duality spaces. Invariants in the form of quivers are also introduced, and Auslander-Reiten triangles and quivers over spheres are computed. The quiver over the d-dimensional sphere turns out to consist of d-1 components, each isomorphic to ZA_{\infty}. So quivers are sufficiently sensitive invariants to tell spheres of different dimension apart.

math.AT↗
arXiv2003

The Auslander-Reiten quiver of a Poincare duality space

In a previous paper, math.AT/0304079, Auslander-Reiten triangles and quivers were introduced into algebraic topology. This paper shows that over a Poincare duality space, each component of the Auslander-Reiten quiver is isomorphic to ZA_{\infty}.

math.AT↗
arXiv2003

A non-commutative BGG correspondence

A non-commutative version of the Bernstein-Gel'fand-Gel'fand (BGG) correspondence is set up, and a sample application is given to periodic injective resolutions.

math.AG↗
arXiv2003

Recognizing dualizing complexes

Let A be a noetherian local commutative ring and let M be a suitable complex of A-modules. This paper proves that M is a dualizing complex for A if and only if the trivial extension A \ltimes M is a Gorenstein Differential Graded Algebra. As a corollary follows that A has a dualizing complex if and only if it is a quotient of a Gorenstein local Differential Graded Algebra.

math.AC↗