Categorical Milnor squares and \(K\)-theory of algebraic stacks. (English) Zbl 1504.19004

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A Milnor square of rings (Section 2 of [J. W. Milnor, Introduction to algebraic \(K\)-theory. Princeton, NJ: Princeton University Press (1971; Zbl 0237.18005)]) is a cartesian square of rings \[ \begin{tikzcd} A \rar \dar & A/I \dar \\ B \rar & B/J. \end{tikzcd} \] The following result is due to Land and Tamme, building on work of Morrow, Geisser and Hesselholt, and Suslin (see Corollaries 2.10 and 2.33 in [M. Land and G. Tamme, Ann. Math. (2) 190, No. 3, 877–930 (2019; Zbl 1427.19002)], as well as [T. Geisser and L. Hesselholt, Invent. Math. 166, No. 2, 359–395 (2006; Zbl 1107.19002); M. Morrow, J. Reine Angew. Math. 736, 95–139 (2018; Zbl 1393.19003); G. Tamme, Compos. Math. 154, No. 9, 1801–1814 (2018; Zbl 1395.18013)] and [A. A. Suslin, in: Number theory, algebra and algebraic geometry. Collected papers. In honor of the seventieth birthday of Academician Igor Rostislavovich Shafarevich. Moscow: Maik Nauka/Interperiodica Publishing. 255–279 (1995; Zbl 0871.19002); translation from Tr. Mat. Inst. Steklova 208, 290–317 (1995)]): Given a Milnor square of rings in which \(\operatorname{Tor}_i^A(A/I, B) = 0\) for all \(i\) (Tor-independence), the associated square of \(K\)-theory spectra is cartesian.
The first purpose of this paper is to categorify the hypothesis in this theorem. Theorem A, part (i) says the following: Suppose \[ \begin{tikzcd} \mathcal A \rar{f^*} \dar{p^*} & \mathcal B \dar{q^*} \\ \mathcal{A}' \rar{g^*} & \mathcal{B}'\end{tikzcd} \] is a commutative square of presentable stable \(\infty\)-categories and colimit-preserving functors that furthermore satisfies the following:
the canonical functor \(\mathcal A \to \mathcal A' \times_{\mathcal B'} \mathcal B\) is fully faithful (an analogue of being cartesian)
the right adjoints of each of the functors \(f^*\), \(g^*\), \(p^*\) and \(q^*\) preserve filtered colimits (called compactness of the functor here) and these functors are also surjective, in that their images generate the codomains under colimits (these conditions are presumed to be analogues of the surjectivity in the Milnor square)
the base-change transformation \(f^*p_* \to q_*g^*\) is invertible (an analogue of the Tor condition above)
Then the associated square of \(K\)-theory spectra is cartesian.
This new theorem may be applied to the derived \(\infty\)-categories of left modules over rings, in which case the old theorem on Milnor squares is recovered. The virtues of the new formalism are twofold. First: the result now applies not only to \(K\)-theory but to any invariant of stable \(\infty\)-categories that sends exact sequences to exact triangles. Second: the result now can be applied in circumstances where one has a category resembling the derived category of modules over a ring. The examples of such categories considered here are the categories of perfect complexes in the derived category of an algebraic stack (with certain further good behaviour imposed: slightly weaker than affine diagonal and tameness in the sense of [D. Abramovich et al., Ann. Inst. Fourier 58, No. 4, 1057–1091 (2008; Zbl 1222.14004)]). The stacks in question are said to be “ANS” stacks.
With the application to stacks in mind, it seems, a variation on Theorem A, part (i) is also considered: Theorem A, part (ii) which is essentially the same as part (i), but applied to pro-objects.
The applications to stacks are two excision theorems: given a cartesian and cocartesian square of noetherian ANS stacks \[ \begin{tikzcd} Z' \rar \dar & X' \dar{f} \\ Z \rar{i} & X\end{tikzcd} \] in which \(i\) is a closed immersion, then there is an induced cartesian square of pro-spectra: \[ \begin{tikzcd} \{K(X)\} \rar \dar & \hat K(X^\wedge_Z) \dar \\ \{K(X')\} \rar & \hat K({X'}^\wedge_{Z'}) \end{tikzcd} \] where \(f\) is either an affine morphism (Theorem B) or a proper representable morphism that is an isomorphism away from \(Z\) (Theorem C). Here \(\hat K(X^\wedge_Z)\) denotes the formal completion of \(X\) along the closed substack \(Z\). In the case of Theorem B, if certain Tor-groups also vanish, the formal completions may be abandoned in favour of \(K(Z)\) and \(K(Z')\).
Theorem C allows the extension of the proof [M. Kerz et al., Invent. Math. 211, No. 2, 523–577 (2018; Zbl 1391.19007) ] of “Weibel’s conjecture” to finite dimensional ANS stacks. This says that the groups \(K_n(X)\) vanishes when \(n\) is less than the negative of the dimension ( [C. A. Weibel, Invent. Math. 61, 177–197 (1980; Zbl 0437.13009)] ). In these dimensions and also when \(n\) is equal to the negative of the dimension, the \(K\)-theory is homotopy invariant.
The main technical innovation in the proofs is an generalization of the \(\odot\) construction of [M. Land and G. Tamme, Ann. Math. (2) 190, No. 3, 877–930 (2019; Zbl 1427.19002)]. Here the construction is generalized from module categories to presentable stable \(\infty\)-categories.


19E08 \(K\)-theory of schemes
14A20 Generalizations (algebraic spaces, stacks)
14F42 Motivic cohomology; motivic homotopy theory
19D35 Negative \(K\)-theory, NK and Nil
19D55 \(K\)-theory and homology; cyclic homology and cohomology
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14D23 Stacks and moduli problems
Full Text: DOI arXiv


[1] Alper, J., Hall, J., Halpern-Leistner, D., Rydh, D.: Artin algebraization for pairs and applications to the local structure of stacks and Ferrand pushouts (in preparation)
[2] Alper, J., Hall, J., Rydh, D.: The étale local structure of algebraic stacks. arXiv e-prints, preprint arXiv:1912.06162 (2019)
[3] Artin, M.; Mazur, B., Etale Homotopy (1969), Cham: Springer, Cham · Zbl 0182.26001
[4] Abramovich, D.; Olsson, M.; Vistoli, A., Tame stacks in positive characteristic, Ann. Inst. Fourier, 58, 4, 1057-1091 (2008) · Zbl 1222.14004
[5] Artin, M., Algebraization of formal moduli. II: Existence of modifications, Ann. Math. (2), 91, 88-135 (1970) · Zbl 0177.49003
[6] Blumberg, AJ; Gepner, D.; Tabuada, G., A universal characterization of higher algebraic K-theory, Geom. Topol., 17, 2, 733-838 (2013) · Zbl 1267.19001
[7] Bondarko, M., Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-theory, 6, 3, 387-504 (2010) · Zbl 1303.18019
[8] Bondarko, M.; Sosnilo, V., Non-commutative localizations of additive categories and weight structures; applications to birational motives, J. Inst. Math. Jussieu, 17, 4, 785-821 (2016) · Zbl 1434.18006
[9] Cisinski, D.-C., Khan, A.A.: \({\bf A}^1\)-homotopy invariance in spectral algebraic geometry, preprint arXiv:1705.03340
[10] Elmanto, E., Hoyois, M., Iwasa, R., Kelly, S.: Milnor excision for motivic spectra, preprint arXiv:2004.12098 · Zbl 1477.14036
[11] Elmanto, E., Sosnilo, V.: On nilpotent extensions of \(\infty \)-categories and the cyclotomic trace, preprint arXiv:2010.09155
[12] Ferrand, D., Conducteur, descente et pincement, Bull. Soc. Math. Fr., 131, 4, 553-585 (2003) · Zbl 1058.14003
[13] Fontes, E.E.: Weight structures and the algebraic \(K\)-theory of stable \(\infty \)-categories, preprint arXiv:1812.09751 · JFM 26.0280.01
[14] Geisser, T.; Hesselholt, L., Bi-relative algebraic \(K\)-theory and topological cyclic homology, Invent. Math., 166, 2, 359-395 (2006) · Zbl 1107.19002
[15] Gaitsgory, D., Rozenblyum, N.: DG indschemes. Contemporary Mathematics 610, 139-251 (2014). doi:10.1090/conm/610/12080 · Zbl 1316.14006
[16] Gaitsgory, D.; Rozenblyum, N., A Study in Derived Algebraic Geometry. Volume I: Correspondences and Duality (2017), Providence, RI: American Mathematical Society (AMS), Providence, RI · Zbl 1409.14003
[17] Gross, P., Tensor generators on schemes and stacks, Algebra. Geom., 4, 501-522 (2017) · Zbl 1412.14002
[18] Hoyois, M.; Krishna, A., Vanishing theorems for the negative K-theory of stacks, Ann. K-Theory, 4, 3, 439-472 (2019) · Zbl 1469.19002
[19] Halpern-Leistner, D., Preygel, A.: Mapping stacks and categorical notions of properness, preprint arXiv:1402.3204
[20] Hoyois, M., The six operations in equivariant motivic homotopy theory, Adv. Math., 305, 197-279 (2017) · Zbl 1400.14065
[21] Hoyois, M.: K-theory of dualizable categories. http://www.mathematik.ur.de/hoyois/papers/efimov.pdf (2018) · Zbl 1403.18003
[22] Hall, J.; Rydh, D., Perfect complexes on algebraic stacks, Compos. Math., 153, 11, 2318-2367 (2017) · Zbl 1390.14057
[23] Kerz, M.: On negative algebraic \(K\)-groups. Proceedings of the International Congress of Mathematicians, ICM 2018, Rio de Janeiro, Brazil, August 1-9, 2018. Volume II. Invited lectures, pp. 163-172 (2018) · Zbl 1441.19006
[24] Khan, A.A.: Pro-systems of K-theory spectra. Lecture notes https://www.preschema.com/lecture-notes/kdescent/lect8.pdf (2017)
[25] Khan, AA, The Morel-Voevodsky localization theorem in spectral algebraic geometry, Geom. Topol., 23, 7, 3647-3685 (2019) · Zbl 1451.14068
[26] Khan, AA, Algebraic K-theory of quasi-smooth blow-ups and cdh descent, Ann. Henri Lebesgue, 3, 1091-1116 (2020) · Zbl 07272950
[27] Khan, AA, K-theory and G-theory of derived algebraic stacks, Jpn. J. Math., 17, 1-61 (2022)
[28] Krishna, A.; Østvær, PA, Nisnevich descent for \(K\)-theory of Deligne-Mumford stacks, J. K-Theory, 9, 2, 291-331 (2012) · Zbl 1284.19005
[29] Khan, A.A., Rydh, D.: Virtual Cartier divisors and blow-ups, preprint arXiv:1802.05702
[30] Krishna, A.; Ravi, C., Algebraic K-theory of quotient stacks, Ann. K-Theory, 3, 2, 207-233 (2018) · Zbl 1423.19007
[31] Khan, A.A., Ravi, C.: Generalized cohomology theories for algebraic stacks. arXiv e-prints, preprint arXiv:2106.15001 (2021)
[32] Kerz, M.; Strunk, F., On the vanishing of negative homotopy \(K\)-theory, J. Pure Appl. Algebra, 221, 7, 1641-1644 (2017) · Zbl 1372.19003
[33] Kerz, M.; Strunk, F.; Tamme, G., Algebraic K-theory and descent for blow-ups, Invent. Math., 211, 2, 523-577 (2017) · Zbl 1391.19007
[34] Laumon, G.; Moret-Bailly, L., Champs algébriques (2018), Berlin: Springer, Berlin · Zbl 0945.14005
[35] Land, M.; Tamme, G., On the K-theory of pullbacks, Ann. Math., 190, 3, 877-930 (2019) · Zbl 1427.19002
[36] Lurie, J., Higher Topos Theory (2009), Princeton: Princeton University Press, Princeton · Zbl 1175.18001
[37] Lurie, J.: Higher algebra, preprint www.math.harvard.edu/ lurie/papers/HigherAlgebra.pdf, version of 2017-09-18 (2017)
[38] Lurie, J.: Spectral algebraic geometry, preprint www.math.harvard.edu/ lurie/papers/SAG-rootfile.pdf, version of 2018-02-03 (2018)
[39] Milnor, JW, Introduction to Algebraic K-Theory (1971), Princeton: Princeton University Press, Princeton · Zbl 0237.18005
[40] Mathew, A.; Naumann, N.; Noel, J., Nilpotence and descent in equivariant stable homotopy theory, Adv. Math., 305, 994-1084 (2017) · Zbl 1420.55024
[41] Morrow, M., Pro unitality and pro excision in algebraic \(K\)-theory and cyclic homology, J. Reine Angew. Math., 736, 95-139 (2018) · Zbl 1393.19003
[42] Raoult, J-C, Compactification des espaces algébriques, C. R. Acad. Sci. Paris Sér. A, 278, 867-869 (1974) · Zbl 0278.14002
[43] Raynaud, M.; Gruson, L., Critères de platitude et de projectivité. Techniques de “platification” d’un module. (Criterial of flatness and projectivity. Technics of ”flatification of a module.), Invent. Math., 13, 1-89 (1971) · Zbl 0227.14010
[44] Rydh, D.: Compactification of tame Deligne-Mumford stacks, preprint https://people.kth.se/ dary/tamecompactification20110517.pdf (2011)
[45] Rydh, D.: Equivariant flatification, etalification and compactification (in preparation) (2017)
[46] Sosnilo, V., Theorem of the heart in negative K-theory for weight structures, Doc. Math., 24, 2137-2158 (2019) · Zbl 1441.18008
[47] Sosnilo, V., Regularity of spectral stacks and discreteness of weight-hearts, Q. J. Math. (2021) · Zbl 1496.18024
[48] Stapleton, J., Weibel’s conjecture for twisted K-theory, Ann. K-Theory, 5, 3, 621-637 (2020) · Zbl 1458.19001
[49] Suslin, A.A.: Excision in the integral algebraic \(K\)-theory. Number theory, algebra and algebraic geometry. Collected papers. In honor of the seventieth birthday of Academician Igor Rostislavovich Shafarevich, Maik Nauka/Interperiodica Publishing, Moscow, pp. 255-279 (1995) · Zbl 0863.00012
[50] Tamme, G., Excision in algebraic K-theory revisited, Compos. Math., 154, 9, 1801-1814 (2018) · Zbl 1395.18013
[51] The Stacks Project Authors: Stacks Project https://stacks.math.columbia.edu (2020)
[52] Thomason, R.W.: Algebraic K-theory of group scheme actions. Algebraic topology and algebraic K-theory, Proceedings Conference, Princeton, NJ (USA), Ann. Math. Stud. 113, 539-563 (1987) · Zbl 0701.19002
[53] Thomason, RW; Trobaugh, T., Higher Algebraic K-Theory of Schemes and of Derived Categories, 247-435 (2007), Berlin: Springer, Berlin
[54] Voevodsky, V., Homotopy theory of simplicial sheaves in completely decomposable topologies, J. Pure Appl. Algebra, 214, 8, 1384-1398 (2010) · Zbl 1194.55020
[55] Voevodsky, V., Unstable motivic homotopy categories in Nisnevich and cdh-topologies, J. Pure Appl. Algebra, 214, 8, 1399-1406 (2010) · Zbl 1187.14025
[56] Weibel, CA, K-theory and analytic isomorphisms, Invent. Math., 61, 177-197 (1980) · Zbl 0437.13009
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