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Homotopy invariants of singularity categories. (English) Zbl 1469.16011

Summary: We present a method for computing \(\mathbb{A}^1\)-homotopy invariants of singularity categories of rings admitting suitable gradings. Using this we describe any such invariant, e.g. homotopy \(K\)-theory, for the stable categories of self-injective algebras admitting a connected grading. A remark is also made concerning the vanishing of all such invariants for cluster categories of type \(A_{2n}\) quivers.

MSC:

16E35 Derived categories and associative algebras
18G80 Derived categories, triangulated categories
16E20 Grothendieck groups, \(K\)-theory, etc.
16S37 Quadratic and Koszul algebras
19D55 \(K\)-theory and homology; cyclic homology and cohomology
18G35 Chain complexes (category-theoretic aspects), dg categories
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References:

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