Sosnilo, Vladimir

Theorem of the Heart in Negative K-Theory for Weight Structures

Doc. Math. 24, 2137-2158 (2019)
DOI: 10.25537/dm.2019v24.2137-2158


We construct the strong weight complex functor for a stable infinity-category \(\mathcal{C}\) equipped with a bounded weight structure \(w\). Along the way we prove that \(\mathcal{C}\) is determined by the infinity-categorical heart of \(w\). This allows us to compare the K-theory of \(\mathcal{C}\) and the K-theory of \(Hw\), the classical heart of \(w. In\) particular, we prove that \(K_n(\mathcal{C}) \to K_n(Hw)\) are isomorphisms for \(n \le 0\).

Mathematics Subject Classification

18E05, 18E30, 14F42


homological algebra, stable homotopy theory, infinity-categories, K-theory, weight structures, Voevodsky motives


  • [AGH18]. Benjamin Antieau, David Gepner, Jeremiah Heller, K-theoretic obstructions to bounded t-structures, Inventiones mathematicae 216(1), 2019, 241-300. DOI 10.1007/s00222-018-00847-0; zbl 07051048; MR3935042; arxiv 610.07207.
  • [Bar15]. Clark Barwick, On exact infinity-categories and the Theorem of the Heart, Compositio Mathematica 151, 2015, 2160-2186. DOI 10.1112/S0010437X15007447; zbl 1333.19003; MR3427577; arxiv 1212.5232.
  • [Bar16]. Clark Barwick, On the algebraic K-theory of higher categories, Journal of Topology 9(1), 2016, 245-347. DOI 10.1112/jtopol/jtv042; zbl 1364.19001; MR3465850; arxiv 1204.3607.
  • [Bass68]. Hyman Bass, Algebraic K-Theory, Benjamin, New York, 1968. zbl 0174.30302; MR0249491.
  • [BeVo08]. Alexander Beilinson, Vadim Vologodsky, A DG guide to Voevodsky's motives, Geometric and Functional Analysis 17(6), 2008, 1709-1787. DOI 10.1007/s00039-007-0644-5; zbl 1144.14014; MR2399083; arxiv math/0604004.
  • [BGT13]. Andrew J. Blumberg, David Gepner, Goncalo Tabuada, A universal characterization of higher algebraic K-theory, Geom. Topol. 17, 2013, 733-838. DOI 10.2140/gt.2013.17.733; zbl 1267.19001; MR3070515; arxiv 1001.2282.
  • [BM14]. Andrew Blumberg, Michael A. Mandell, The homotopy groups of the algebraic K-theory of the sphere spectrum, Geom. Topol. 23(1), 2019, 101-134. DOI 10.2140/gt.2019.23.101; zbl 1412.19001; MR3921317; arxiv 1408.0133.
  • [Bon09]. Mikhail Bondarko, Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura, J. of the Inst. of Math. of Jussieu 8(1), 2009, 39-97. DOI 10.1017/S147474800800011X; zbl 1161.14014; MR2461902; arxiv math/0601713.
  • [Bon10a]. Mikhail Bondarko, Weight structures vs. \(t\)-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. of K-Theory 6(3), 2010, 387-504. DOI 10.1017/is010012005jkt083; zbl 1303.18019; MR2746283; arxiv 0704.4003.
  • [Bon10b]. Mikhail Bondarko, Motivically functorial coniveau spectral sequences; direct summands of cohomology of function fields, Doc. Math., Extra volume: Andrei Suslin's Sixtieth Birthday, 2010, 33-117.; zbl 1210.14023; MR2804250; arxiv 0812.2672.
  • [Bon11]. Mikhail Bondarko, Z[1/p]-motivic resolution of singularities, Compos. Math. 147(5), 2011, 1434-1446. DOI 10.1112/S0010437X11005410; zbl 1230.14013; MR2834727; arxiv 1002.2651.
  • [Bon19]. Mikhail Bondarko, On weight complexes, pure functors, and detecting weights, preprint, 2019. arxiv 1812.11952.
  • [Bon18]. Mikhail Bondarko, Gersten weight structures for motivic homotopy categories; retracts of cohomology of function fields, motivic dimensions, and coniveau spectral sequences, preprint, 2018. arxiv 1803.01432.
  • [BoK17]. Mikhail Bondarko, David Kumallagov, On Chow weight structures without projectivity and resolution of singularities, (Russian), translated from Algebra i Analiz 30(5), 2018, 57-83, St. Petersbg. Math. J. 30(5), 2019, 803-819. DOI 10.1090/spmj/1570; zbl 07090624; MR3856101; arxiv 1711.08454.
  • [BoS18]. Mikhail Bondarko, Vladimir Sosnilo, On constructing weight structures and extending them to idempotent extensions, Homology, Homotopy and Appl. 20(1), 2018, 37-57. DOI 10.4310/HHA.2018.v20.n1.a3; zbl 1394.14006; MR3775347; arxiv 1605.08372.
  • [Fon18]. Ernest E. Fontes, Weight structures and algebraic K-theory of stable \(\infty \)-categories, preprint, 2018. arxiv 1812.09751.
  • [Hel19]. Aron Heleodoro, Determinant map for the prestack of Tate objects, preprint, 2019. arxiv 1907.00384v1.
  • [Joy04]. Andre Joyal, Notes on quasi-categories, 2008.
  • [KeSa17]. Shane Kelly, Shuji Saito, Weight homology of motives, Int. Math. Res. Notices 13, 2017, 3938-3984. DOI 10.1093/imrn/rnw111; zbl 1405.14052; MR3671508; arxiv 1411.5831.
  • [Lur17a]. Jacob Lurie, Higher Topos Theory, Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton, NJ, 2009. DOI 10.1515/9781400830558; zbl 1175.18001; MR2522659; arxiv math/0608040.
  • [Lur17b]. Jacob Lurie, Higher Algebra, September 2017.
  • [Lur18]. Jacob Lurie, Spectral Algebraic Geometry, February 2018.
  • [MSS07]. F. Muro, S. Schwede and N. Strickland, Triangulated categories without models, Invent. Math. 170(2), 2007, 231-241. DOI 10.1007/s00222-007-0061-2; zbl 1125.18009; MR2342636; arxiv 0704.1378.
  • [Pau08]. David Pauksztello, Compact corigid objects in triangulated categories and co-t-structures, Central European Journal of Mathematics 6(1), 2008, 25-42. DOI 10.2478/s11533-008-0003-2; zbl 1152.18009; MR2379950; arxiv 0705.0102.
  • [Rog03]. John Rognes, The smooth Whitehead spectrum of a point at odd regular primes, Geom. Topol. 7, 2003, 155-184. DOI 10.2140/gt.2003.7.155; zbl 1130.19300; MR1988283; arxiv math/0304384.
  • [Sch02]. Marco Schlichting, A note on K-theory and triangulated categories, Invent. Math. 150(1), 2002, 111-116. DOI 10.1007/s00222-002-0231-1; zbl 1037.18007; MR1930883.
  • [Sch06]. Marco Schlichting, Negative K-theory of derived categories, Math. Z. 253(1), 2006, 97-134. DOI 10.1007/s00209-005-0889-3; zbl 1090.19002; MR2206639.
  • [ScSh03]. Stefan Schwede, Brooke Shipley, Stable model categories are categories of modules, Topology 42, 2003, 103-153. DOI 10.1016/S0040-9383(02)00006-X; zbl 1013.55005; MR1928647.
  • [TT90]. Robert W. Thomason, Thomas Trobaugh, Higher algebraic K-theory of schemes and of derived categories, Progress in Mathematics, vol. 88, 1990, 247-435. zbl 0731.14001; MR1106918.
  • [Qui73]. Daniel Quillen, Higher algebraic K-theory. I. Algebraic \(K\)-theory. In Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, WA, 1972), Lecture Notes in Math., vol. 341, Springer, Berlin, 1973, 85-147. DOI 10.1007/BFb0067053; zbl 0292.18004; MR0338129.
  • [Sch11]. Olaf Schnürer, Homotopy categories and idempotent completeness, weight structures and weight complex functors, preprint, 2011.
  • [Voe95]. Vladimir Voevodsky, A nilpotence theorem for cycles algebraically equivalent to zero, International Mathematics Research Notices 1995(4), 1995, 187-199. DOI 10.1155/S1073792895000158; zbl 0861.14006; MR1326064.
  • [Voe00]. Vladimir Voevodsky, Triangulated categories of motives over a field, in: Voevodsky V., Suslin A., and Friedlander E., Cycles, transfers and motivic homology theories, Annals of Mathematical Studies, vol. 143, Princeton University Press, Princeton, NJ, 2000, 188-238. zbl 1019.14009; MR1764202.
  • [Wal85]. Friedhelm Waldhausen, Algebraic K-theory of spaces, in Algebraic and geometric topology (New Brunswick, NJ, 1983), Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, 318-419. zbl 0579.18006; MR0802796.


Sosnilo, Vladimir
Laboratory of Modern Algebra and Applications, Saint Petersburg State University, 14th Line V.O., 29B, Saint Petersburg 199178, Russia and St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka, 27, Saint Petersburg 191011, Russia