\(K\)-theory and \(G\)-theory of dg-stacks (Q2906507)

This paper presents foundational definitions and main theorems about \(K\)-theory and \(G\)-theory of algebraic stacks (``so as to serve as a reference'') and, at the same time, extends these theories to dg-stacks, i.e.\ to algebraic stacks where the usual structure sheaf is replaced with a sheaf of commutative differential graded algebras.NEWLINENEWLINEInitial key results about dg-stacks have been established in the author's paper [Adv. Math. 209, No. 1, 1--68 (2007; Zbl 1117.14020)]. The paper is written in the framework of Waldhausen \(K\)-theory [\textit{F. Waldhausen}, in: Algebraic and geometric topology, Proc. Conf., New Brunswick/USA 1983, Lect. Notes Math. 1126, 318--419 (1985; Zbl 0579.18006)]. A large part of the originality of this paper is about choosing the definitions carefully, the proofs then often follow \textit{R. W. Thomason} and \textit{T. Trobaugh} [in: The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. III, Prog. Math. 88, 247--435. Appendix A: 398--408; appendix B: 409--417; appendix C: 418--423; appendix D: 424--426; appendix E: 427--430; appendix F: p. 431 (1990; Zbl 0731.14001)] (and ultimately \textit{D. Quillen} [in: Algebr. K--Theory I, Proc. Conf. Battelle Inst. 1972, Lect. Notes Math. 341, 85--147 (1973; Zbl 0292.18004)]).NEWLINENEWLINEThe paper starts with an exposition of fundamentals of \(K\)-theory and \(G\)-theory of Artin stacks and of dg-stacks. The main new result here gives several characterizations of perfect and of pseudo-coherent complexes on dg-stacks. The next section establishes the projective bundle theorem for dg-stacks. Then the devissage, localization and homotopy theorems for \(G\)-theory of dg-stacks are proved. Finally (co)homology theories on dg-stacks are discussed. The paper ends with an appendix about Waldhausen \(K\)-theory and with an appendix about injective resolutions of dg-modules.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14001].