*K*-theoretic obstructions to bounded *t*-structures

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## Abstract

Schlichting conjectured that the negative *K*-groups of small abelian categories vanish and proved this for noetherian abelian categories and for all abelian categories in degree \(-1\). The main results of this paper are that \(\mathrm {K}_{-1}(E)\) vanishes when *E* is a small stable \(\infty \)-category with a bounded *t*-structure and that \(\mathrm {K}_{-n}(E)\) vanishes for all \(n\geqslant 1\) when additionally the heart of *E* is noetherian. It follows that Barwick’s theorem of the heart holds for nonconnective *K*-theory spectra when the heart is noetherian. We give several applications, to non-existence results for bounded *t*-structures and stability conditions, to possible *K*-theoretic obstructions to the existence of the motivic *t*-structure, and to vanishing results for the negative *K*-groups of a large class of dg algebras and ring spectra.

## Mathematics Subject Classification

Primary: 16E45 18E30 19D35 Secondary: 16P40 18E10 55P43## Notes

### Acknowledgements

BA and JH thank the Hausdorff Institute for Mathematics in Bonn and DG thanks the Max Planck Institute for Mathematics: these were our hosts during the summer of 2015, when this project was conceived. BA thanks Akhil Mathew for several conversations that summer at HIM, especially about bounded *t*-structures for compact modules over cochain algebras. BA thanks John Calabrese, Denis-Charles Cisinski, Michael Gröchenig, Jacob Lurie, Matthew Morrow, Marco Schlichting, Jesse Wolfson, and Matthew Woolf for conversations and emails about material related to this paper. DG thanks Andrew Blumberg and Markus Spitzweck for conversations about material related to this paper. Both BA and DG would especially like to thank Benjamin Hennion for explaining Tate objects and the subtleties behind excisive squares. We all are very grateful for detailed, helpful comments from an anonymous referee. We also thank the UIC Visitors’ Fund, Purdue University, UIUC, and Lars Hesselholt for supporting collaborative visits in 2016.

## References

- 1.Abramovich, D., Polishchuk, A.: Sheaves of t-structures and valuative criteria for stable complexes. J. Reine Angew. Math.
**590**, 89–130 (2006)MathSciNetzbMATHGoogle Scholar - 2.Antieau, B., Gepner, D.: Brauer groups and étale cohomology in derived algebraic geometry. Geom. Topol.
**18**(2), 1149–1244 (2014)MathSciNetzbMATHGoogle Scholar - 3.Balmer, P., Schlichting, M.: Idempotent completion of triangulated categories. J. Algebra
**236**(2), 819–834 (2001)MathSciNetzbMATHGoogle Scholar - 4.Barwick, C.: On exact \(\infty \)-categories and the Theorem of the Heart. Compos. Math.
**151**(11), 2160–2186 (2015)MathSciNetzbMATHGoogle Scholar - 5.Barwick, C.: On the algebraic K-theory of higher categories. J. Topol.
**9**(1), 245–347 (2016)MathSciNetzbMATHGoogle Scholar - 6.Barwick, C., Lawson, T.: Regularity of structured ring spectra and localization in K-theory. ArXiv e-prints arXiv:1402.6038 (2014)
- 7.Bass, H.: Some problems in “classical” algebraic K-theory, Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic. In: Proceedings of the Conference Held at the Seattle Research Center of Battelle Memorial Institute, Seattle, Washington, 1972. Lecture Notes in Mathematics, vol. 342, pp. 3–73. Springer, Berlin (1973)Google Scholar
- 8.Bayer, A., Macrì, E., Toda, Y.: Bridgeland stability conditions on threefolds I: Bogomolov–Gieseker type inequalities. J. Algebr. Geom.
**23**(1), 117–163 (2014)MathSciNetzbMATHGoogle Scholar - 9.Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100, pp. 5–171. Soc. Math. France, Paris (1982)Google Scholar
- 10.Blumberg, A.J., Gepner, D., Tabuada, G.: A universal characterization of higher algebraic K-theory. Geom. Topol.
**17**(2), 733–838 (2013)MathSciNetzbMATHGoogle Scholar - 11.Blumberg, A.J., Gepner, D., Tabuada, G.: K-theory of endomorphisms via noncommutative motives. Trans. Am. Math. Soc.
**368**(2), 1435–1465 (2016)MathSciNetzbMATHGoogle Scholar - 12.Blumberg, A.J., Mandell, M.A.: The localization sequence for the algebraic K-theory of topological K-theory. Acta Math.
**200**(2), 155–179 (2008)MathSciNetzbMATHGoogle Scholar - 13.Braunling, O., Groechenig, M., Wolfson, J.: Tate objects in exact categories. Mosc. Math. J.
**16**(3), 433–504 (2016).**(With an appendix by Jan Šťovíček and Jan Trlifaj)**MathSciNetzbMATHGoogle Scholar - 14.Bridgeland, T.: Stability conditions on triangulated categories. Ann. Math. (2)
**166**(2), 317–345 (2007)MathSciNetzbMATHGoogle Scholar - 15.Buchweitz, R.-O.: Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings. http://hdl.handle.net/1807/16682 (1986). Accessed 20 Dec 2018
- 16.Curtis, C.W., Reiner, I.: Representation Theory of Finite Groups and Associative Algebras. Pure and Applied Mathematics, vol. XI. Interscience Publishers, New York (1962)zbMATHGoogle Scholar
- 17.Dwyer, W.G., Greenlees, J.P.C., Iyengar, S.: Duality in algebra and topology. Adv. Math.
**200**(2), 357–402 (2006)MathSciNetzbMATHGoogle Scholar - 18.Farrell, F.T., Jones, L.E.: The lower algebraic K-theory of virtually infinite cyclic groups. K-Theory
**9**(1), 13–30 (1995)MathSciNetzbMATHGoogle Scholar - 19.Glaz, S.: Commutative Coherent Rings. Lecture Notes in Mathematics, vol. 1371. Springer, Berlin (1989)zbMATHGoogle Scholar
- 20.Happel, D.: On the derived category of a finite-dimensional algebra. Comment. Math. Helv.
**62**(3), 339–389 (1987)MathSciNetzbMATHGoogle Scholar - 21.Heller, A.: The loop-space functor in homological algebra. Trans. Am. Math. Soc.
**96**, 382–394 (1960)MathSciNetzbMATHGoogle Scholar - 22.Hennion, B.: Tate objects in stable (\(\infty, 1\))-categories. Homol. Homotopy Appl.
**19**(2), 373–395 (2017)MathSciNetzbMATHGoogle Scholar - 23.Hennion, B., Porta, M., Vezzosi, G.: Formal glueing for non-linear flags. ArXiv e-prints arXiv:1607.04503 (2016)
- 24.Hovey, M.: Model Categories. Mathematical Surveys and Monographs, vol. 63. American Mathematical Society, Providence (1999)Google Scholar
- 25.Huybrechts, D.: Fourier-Mukai Transforms in Algebraic Geometry. Oxford Mathematical Monographs. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
- 26.Kahn, B.: Algebraic K-Theory, Algebraic Cycles and Arithmetic Geometry. Handbook of K-Theory, vol. 1, 2, pp. 351–428. Springer, Berlin (2005)zbMATHGoogle Scholar
- 27.Keller, B., Nicolás, P.: Weight structures and simple dg modules for positive dg algebras. Int. Math. Res. Not. IMRN
**5**, 1028–1078 (2013)MathSciNetzbMATHGoogle Scholar - 28.Krause, H.: Deriving Auslander’s formula. Doc. Math.
**20**, 669–688 (2015)MathSciNetzbMATHGoogle Scholar - 29.Lück, W., Reich, H.: The Baum-Connes and the Farrell-Jones Conjectures in K- and L-Theory. Handbook of K-Theory, vol. 1, 2, pp. 703–842. Springer, Berlin (2005)zbMATHGoogle Scholar
- 30.Lurie, J.: Higher Topos Theory. Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009)zbMATHGoogle Scholar
- 31.Lurie, J.: Higher algebra 2012. http://www.math.harvard.edu/-lurie/. Version dated 10 Mar 2016
- 32.Lurie, J.: Spectral algebraic geometry. http://www.math.harvard.edu/-lurie/. Version dated 13 Oct 2016
- 33.Mathew, A.: The Galois group of a stable homotopy theory. Adv. Math.
**291**, 403–541 (2016)MathSciNetzbMATHGoogle Scholar - 34.McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings (Revised edition). Graduate Studies in Mathematics, vol. 30. American Mathematical Society, Providence (2001).
**(With the cooperation of L. W. Small)**Google Scholar - 35.Quillen, D.: Higher algebraic K-theory I. Algebraic K-theory I: higher K-theories. In: Proceedings of the Conference Held at the Seattle Research Center of Battelle Memorial Institute, Seattle, Washington, 1972). Lecture Notes in Mathematics, vol. 341, pp. 85–147. Springer, Berlin (1973)Google Scholar
- 36.Renault, G.: Sur les anneaux de groupes. C. R. Acad. Sci. Paris Sér. A-B
**273**, A84–A87 (1971)MathSciNetzbMATHGoogle Scholar - 37.Saito, S.: On Previdi’s delooping conjecture for K-theory. Algebra Number Theory
**9**(1), 1–11 (2015)MathSciNetzbMATHGoogle Scholar - 38.Schlichting, M.: A note on K-theory and triangulated categories. Invent. Math.
**150**(1), 111–116 (2002)MathSciNetzbMATHGoogle Scholar - 39.Schlichting, M.: Negative K-theory of derived categories. Math. Z.
**253**(1), 97–134 (2006)MathSciNetzbMATHGoogle Scholar - 40.Sosnilo, V.: Theorem of the heart in negative K-theory for weight structures. ArXiv e-prints arXiv:1705.07995 (2017)
- 41.Swan, R.G.: Algebraic K-Theory. Lecture Notes in Mathematics, vol. 76. Springer, Berlin (1968)Google Scholar
- 42.Thomason, R.W.: The classification of triangulated subcategories. Compos. Math.
**105**(1), 1–27 (1997)MathSciNetzbMATHGoogle Scholar - 43.Thomason, R.W., Trobaugh, T.: Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift, vol. III. Progress in Mathematics, vol. 88, pp. 247–435. Birkhäuser Boston, Boston (1990)Google Scholar
- 44.Toën, B., Vezzosi, G.: A remark on K-theory and S-categories. Topology
**43**(4), 765–791 (2004)MathSciNetzbMATHGoogle Scholar - 45.Verdier, J.-L.: Des catégories dérivées des catégories abéliennes, Astérisque 239 (1996), pp. xii+253 (1997) (With a preface by Luc Illusie; Edited and with a note by Georges Maltsiniotis)Google Scholar
- 46.Voevodsky, V.: A nilpotence theorem for cycles algebraically equivalent to zero. Int. Math. Res. Not.
**4**, 187–198 (1995)MathSciNetzbMATHGoogle Scholar - 47.Voevodsky, V.: Triangulated categories of motives over a field: cycles, transfers, and motivic homology theories. In: Annals of Mathematics Studies, vol. 143, pp. 188–238. Princeton University Press, Princeton (2000)Google Scholar
- 48.Waldhausen, F.: Algebraic K-theory of generalized free products: I, II. Ann. Math. (2)
**108**(1), 135–204 (1978)MathSciNetzbMATHGoogle Scholar - 49.Weibel, C.A.: Negative K-theory of varieties with isolated singularities. In: Proceedings of the Luminy Conference on Algebraic K-Theory (Luminy, 1983), pp. 331–342 (1984)Google Scholar
- 50.Weibel, C.A.: The K-Book. Graduate Studies in Mathematics, vol. 145. American Mathematical Society, Providence (2013).
**(An introduction to algebraic K-theory)**Google Scholar