Algebraic \(K\)-theory of Gorenstein projective modules. (English) Zbl 1381.16007
Summary: We introduce the Gorenstein algebraic \(K\)-theory space and the Gorenstein algebraic \(K\)-group of a ring, and show the relation with the classical algebraic \(K\)-theory space, and also show the ’resolution theorem’ in this context due to Quillen. We characterize the Gorenstein algebraic \(K\)-groups by two different algebraic \(K\)-groups and by the idempotent completeness of the Gorenstein singularity category of the ring. We compute the Gorenstein algebraic \(K\)-groups along a recollement of the bounded Gorenstein derived categories of CM-finite Gorenstein algebras.
MSC:
| 16E20 | Grothendieck groups, \(K\)-theory, etc. |
| 16E65 | Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) |
| 16D40 | Free, projective, and flat modules and ideals in associative algebras |
| 19D50 | Computations of higher \(K\)-theory of rings |
| 18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |
Keywords:
Frobenius pair; Gorenstein projective module; Gorenstein algebraic \(K\)-group; idempotent complete category; recollementReferences:
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