K-THEORY AND DERIVED EQUIVALENCES

By Daniel Dugger, B. Shipley. 2002

107 citations recorded by the source corpus.

We show that if two rings have equivalent derived categories then they have the same algebraic K-theory. Similar results are given for G-theory, and for a large class of abelian categories.

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

View Paper: https://www.semanticscholar.org/paper/ed64a24caf494708f8a3c5b96c35d5d501521969

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In Algebra and Number Theory

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Citations in the corpus

A remark on K-theory and S-categories (2002, methodology): “Indeed, in [4] the authors only deal with very particular type of Waldhausen categories, the categories of complexes over some rings, which from a homotopical point of view behave in a very particular way (see Remarks 2.5 and 6.8 of [ 4 ]).”
Topological equivalences for differential graded algebras (2006, methodology): “The equivalence of (1) and (2) was established by Rickard [Ri], and the equivalence with (3) was explicitly noted in [ DS1 , 2.6] (this reference only discusses pointed Quillen equivalences, but that restriction can be removed using [D2, 5.5(b)]: if ChR and ChS are Quillen equivalent, then they are ”
Finiteness of finitistic dimension is invariant under derived equivalences (2009, background): “Introduction Derived equivalence preserves many invariants of algebras, for example, the number of irreducible representations, the finiteness of global dimension, and the algebraic K-theory and G-theory (see [1,2,4,6–8]).”
On differential graded categories (2006, background): “Shipley show in [39] that Ktheory is even preserved under topological Morita equivalence.”
Morita theory in abelian, derived and stable model categories (2003, background): “Such invariants include the algebraic K-theory [ DuSh ], topological Hochschild homology and topological cyclic homology.”
Invariance de la K-théorie par équivalences dérivées (2010, unknown): “D’autre part, Dugger et Shipley [DS04] ont ensuite d´emontr´e un cas particulier de ce th´eor`eme (lorsque A et A ′ sont des sous-cat´egories de Waldhausen de cat´egories mod`eles et lorsque Φ est la restriction d’une ´equivalence de Quillen d´eﬁnie au niveau des cat´egories de mod`eles…”