Negative K-theory of derived categories

By M. Schlichting. 2006

204 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/ae2dcbf20c2103bc7d60142a9191a2142dc9f1b6

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Citations in the corpus

Chow motives versus non-commutative motives (2011, background): “Thanks to the work of Blumberg–Mandell, Keller, Schlichting, and Thomason–Trobaugh (see [8], [30], [29], [42], [47]), all the mentioned invariants satisfy localization3 and so give rise to localizing invariants.”
A Guided Tour Through the Garden of Noncommutative Motives (2011, background): “3) satisfy additivity, and hence are additive invariant of dg categories; see [6, 29, 40, 53].”
Relative singularity categories I: Auslander resolutions (2012, unknown): “Firstly, ∆R(A) is idempotent complete: using Schlichting’s negative K-Theory for triangulated categories [82] this may be deduced from the idempotent completeness of MCM(R) (see [23, Theorem 3.”
Higher K-theory via universal invariants (2007, background): “This shows us that Waldhausen’s K-theory is completely characterized by its additive property and ‘intuitively’ it is the universal construction with values in a stable context which satisfies additivity.”
The Local Structure of Algebraic K-Theory (2012, background): “7 [301] and [245]) Let C and D be categories with cofibrations and weak equivalences, where the weak equivalences have the twoout-of-three property, and suppose that every morphism in C may be factored as a cofibration followed by a weak equivalence.”
Topological K-theory of complex noncommutative spaces (2012, methodology): “on we give of topological K-theory starts with algebraic K-theory and proceeds in two steps. If Tis a C-dg-category, we denote by K(T) the nonconnective algebraic K-theory of T (dened by Schlichting [Sch06]). We also have a presheaf of spectra on the site of complex ane C-schemes, K(T) : Spec(A) 7! ”
A universal characterization of higher algebraic K-theory (2010, methodology): “Schlichting’s axiomatic framework implies that this construction agrees with his when both are defined, and therefore we deduce from his comparison results [42, §8] that the non-connective K-theory spectrum of Definition 8.5 agrees with the various classical constructions of non-connective K-theory ”
Reducing Homological Conjectures by n-Recollements (2014, methodology): “Here, we focus on the recollements of derived categories of algebras which are the generalization of derived equivalences and provide a useful reduction technique for some homological properties such as the finiteness of global dimension [32, 21, 1], the finiteness of finitistic dimension [17, 9] an”
Invariance de la K-théorie par équivalences dérivées (2010, unknown): “…(voir le th´eor`eme 2.9), ce qui permet de se ramener au th´eor`eme d’approximation de Waldhausen, dont nous donnons une nouvelle d´emonstration (laquelle a aussi ´et´e d´egag´ee, en substance, par Marco Schlichting dans [Sch06]), qui n’utilise pas l’existence de factorisations fonctorielles.”
Symmetric monoidal structure on non-commutative motives (2010, unknown): “Thanks to the work of Keller [31, 32], Schlichting [41], Thomason-Trobaugh [47], and Blumberg-Mandell [2], all the mentioned invariants satisfy localization(1), and so give rise to localizing invariants.”
Semi-derived and derived Hall algebras for stable categories (2014, background): “p)M,N). The latter being ﬁnite, so is the former.6 MIKHAIL GORSKY It follows that the classical Hall algebra H(C) of the exact category F satisfying conditions (C1) and (C2) is well-deﬁned. Following [Schl06], we deﬁne a map between Frobenius categories to be an exact functor between them sending pr”
On differential graded categories (2006, background): “More generally, every localization pair [82] (=Frobenius pair [134]) gives rise to a dg category.”