Foxby equivalence over associative rings

By Henrik Holm, D. White. 2006

207 citations recorded by the source corpus.

We extend the definition of a semidualizing module to general associative rings. This enables us to define and study Auslander and Bass classes with respect to a semidualizing bimodule C. We then study the classes of C-flats, C-projectives, and C-injectives, and use them to provide a characterization of the modules in the Auslander and Bass classes. We extend Foxby equivalence to this new setting. This paper contains a few results which are new even in the commutative, noetherian setting.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/16e4a35e04293f6ece4e4411aecf1dc3f25a4ce4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Tirdad Sharif
2. Zhaoyong Huang
3. Sean Sather-Wagstaff
4. D. White
5. Zengfeng Liu
6. Aimin Xu
7. Henrik Holm

In Geometry and Topology

1. Tirdad Sharif
2. Zhaoyong Huang
3. Sean Sather-Wagstaff
4. D. White
5. Zengfeng Liu
6. Aimin Xu
7. Henrik Holm

Citations in the corpus

• Proper Resolutions and Gorenstein Categories (2012, background): “Gorenstein projective and injective modules and some related generalized versions have been studied by many authors, see [1,4,6–8,10–21], and the literatures listed in them.”
• Tate cohomology with respect to semidualizing modules (2009, background): “H. Holm and D. White, Foxby equivalence over associative rings, J. Math.”
• Comparison of relative cohomology theories with respect to semidualizing modules (2007, background): “Foxby [6] when C is dualizing, and by Christensen [11] for arbitrary C. (Note that these works (and others) use the notation AC and BC for certain categories of complexes, while our categories consist precisely of the modules in these other categories.) In the non-noetherian setting, these definitio”
• Gorenstein Projective Dimension Relative to a Semidualizing Bimodule (2013, background): “Recall from [12] that a semidualizing bimodule SCR is called faithfully semidualizing if it satisfies the following conditions for all modules SN and MR: (1) If HomS C N = 0, then N = 0; (2) If HomRop C M = 0, then M = 0.”
• Stability of Gorenstein categories (2007, background): “) In the non-noetherian setting, these definitions are taken from [16, 20].”