Gorenstein projective dimension for complexes

By Oana Veliche. 2004

120 citations recorded by the source corpus.

We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cf9649c729de3a2c55c12e71e3621249f420d416

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Beyond totally reflexive modules and back (2008, methodology): “It is developed in detail by Asadollahi and Salarian [4], Christensen, Frankild, and Holm [31] and [32], Christensen and Sather-Wagstaff [35], and by Veliche [110].”
On Gorenstein projective, injective and flat dimensions—A functorial description with applications (2004, background): “However, in [44] Veliche introduces a Gorenstein projective dimension for unbounded complexes.”
Existence of Gorenstein projective resolutions and Tate cohomology (2007, background): “2) coincide with the previously defined Tate cohomology groups from [3] and[26] (see Remark3.”
Tate (co)homology via pinched complexes (2011, methodology): “In this section we recall some definitions and facts from works of Iacob [9] and Veliche [14], and we establish some auxiliary results for later use.”
Acyclicity versus total acyclicity for complexes over noetherian rings (2005, background): “Indeed, Avramov and Veliche [3, (3.3.3)] prove that for an arbitrary commutative local ring R with residue field k, if the Vogel cohomology with X = k = Y has finite rank even in a single degree, then R is Gorenstein.”
Vanishing of Tate homology and depth formulas over local rings (2011, background): “This definition is independent of the choice of complete resolution; see [30] for details.”
Semi-dualizing modules and related Gorenstein homological dimensions (2004, background): “A C ] 2 7 M ay We show how each such module gives rise to three new homological dimensions which we call C–Gorenstein projective, C–Gorenstein injective, and C–Gorenstein flat dimension, and investigate the properties of these dimensions.”