Porta, Mauro; Sala, Francesco Two-dimensional categorified Hall algebras. (English) Zbl 1524.14005 J. Eur. Math. Soc. (JEMS) 25, No. 3, 1113-1205 (2023). For a small abelian category \(\mathcal{A}\) of finite cohomological dimension and with finite \(\mathrm{Hom}\)- and \(\mathrm{Ext}^i\)-groups, one classically defines its Hall algebra as an associative algebra whose vector space basis is given by isomorphism classes of objects and equipped with structure constants defined in terms of counting extensions. More flexible approaches and generalizations are in place with important examples including quantum groups, Yangians, algebras of BPS-states in physics, cohomological Hall algebras of quivers etc. A cohomological Hall algebra attached to \(\mathcal{A}\) is defined via a diagram \(\mathcal{M}_{\mathcal{A}}\times\mathcal{M}_{\mathcal{A}}\stackrel{p}\leftarrow\mathcal{M}_{\mathcal{A}}^{\mathrm{ext}}\stackrel{q}\rightarrow\mathcal{M}_{\mathcal{A}}\) where \(\mathcal{M}_{\mathcal{A}}\) and \(\mathcal{M}_{\mathcal{A}}^{\mathrm{ext}}\) are moduli stacks of objects and of extensions in \(\mathcal{A}\) and \(p,q\) are natural projections. The multiplication is then defined on Borel-Moore homology classes as a pull-push (convolution) product \(q_*\circ p^*\colon H_*^{\mathrm{BM}}(\mathcal{M}_{\mathcal{A}})\otimes H_*^{\mathrm{BM}}(\mathcal{M}_{\mathcal{A}})\to H_*^{\mathrm{BM}}(\mathcal{M}_{\mathcal{A}})\). This construction works under some regularity assumptions on \(p\) which hold if \(\mathcal{A}\) has cohomological dimension \(1\) but may fail if \(\mathcal{A}\) has cohomological dimension \(2\). A uniform satisfactory approach in dimension \(2\) to the construction of Hall multiplication is proposed in the article under review by replacing moduli stacks \(\mathcal{M}_{\mathcal{A}}, \mathcal{M}_{\mathcal{A}}^{\mathrm{ext}}\) by certain derived enhancements \(\mathbb{R}\mathcal{M}_{\mathcal{A}}, \mathbb{R}\mathcal{M}_{\mathcal{A}}^{\mathrm{ext}}\). In fact, a categorification of Hall multiplication is constructed: instead of working with homological or K-theoretic classes, the convolution formula gives a tensor product \(q_*\circ p^*\colon\mathrm{Coh}^{\mathrm{b}}(\mathbb{R}\mathcal{M}_{\mathcal{A}})\otimes\mathrm{Coh}^{\mathrm{b}}(\mathbb{R}\mathcal{M}_{\mathcal{A}})\to\mathrm{Coh}^{\mathrm{b}}(\mathbb{R}\mathcal{M}_{\mathcal{A}})\) of an \(\mathbb{E}_1\)-monoidal structure on the dg-category of coherent sheaves on \(\mathbb{R}\mathcal{M}_{\mathcal{A}}\) with bounded cohomology. Coherences for this \(\mathbb{E}_1\)-monoidal structure would be probably too hard to construct using deformation theory in terms of triangulated categories; throughout, the setup of stable \(\infty\)-categories is used instead.Given a smooth and proper scheme \(S\), authors construct a derived moduli stack \(\mathbf{Coh}(S)\) of coherent sheaves on \(S\) and its categorical Waldhausen S-construction \(\mathcal{S}_\bullet\mathbf{Coh}(S)\), a simplicial object in stable \(\infty\)-category of derived stacks satisfying 2-Segal condition, where \(\mathcal{S}_1\mathbf{Coh}(S)=\mathbf{Coh}(S)\) and \(\mathcal{S}_2\mathbf{Coh}(S)=\mathbf{Coh}^{\mathrm{ext}}(S)\) is the derived moduli stack of extensions of coherent sheaves. The corresponding convolution diagram is \(\mathbf{Coh}(S)\times\mathbf{Coh}(S)\stackrel{(\partial_o,\partial_1)}\longleftarrow\mathbf{Coh}^{\mathrm{ext}}(S)\stackrel{\partial_1}\rightarrow\mathbf{Coh}(S)\). The construction in the article builds on an insight of Dyckerhoff and Kapranov how 2-Segal simplicial objects induce Hall type structures [I. Gálvez-Carrillo et al., Adv. Math. 333, 1242–1292 (2018; Zbl 1403.18016)].Hall-type \(\mathbb{E}_1\)-monoidal structure on the stable \(\infty\)-category \(\mathrm{Coh}^{\mathrm{b}}_{\mathrm{pro}}(\mathbf{Coh}(X))\) is constructed if \(X\) is a complex scheme of dimension \(1\) or \(2\) or the Betti, de Rham or Dolbeault stack of a smooth projective curve. Derived stacks of coherent sheaves on the latter stacks (Simpson’s shapes) enhance classical stacks of local systems, flat vector bundles and Higgs sheaves on \(X\), respectively. This enables introducing the Hall monoidal products for the latter. An analytic version of the formalism is developed as well and a cohomological Hall algebra version of the derived Riemann-Hilbert correspondence is proven as an equivalence of stable \(\mathbf{E}_1\)-monoidal \(\infty\)-categories, relating the de Rham and Betti side in analytic setup. A version of non-abelian Hodge correspondence at the categorified Hall algebra level is demonstrated as well, in terms of Deligne shape interpolating between the de Rham and Dolbeault shape. These constructions are introduced with rich motivation from previously known cohomological Hall algebra structures in dimensions \(1\) and \(2\). Paths to decategorifications like \(K\)-theoretic Hall algebras are studied to make connections to the earlier known Hall algebras and missing conjectured cases. One of the motivations was to relate different categorifications of quantum groups, which have Hall algebra interpretations.The technical part of the paper are numerous auxiliary results in derived algebraic geometry, including methods of deformation theory in construction of derived moduli stacks. These are refining and adapting known techniques (including some from earlier papers of the authors) to the present needs. Such needs are concisely but explicitly articulated. The wealth of new and background material is presented precisely, clearly and sufficiently motivated even for non-specialists. Reviewer: Zoran Škoda (Zadar) Cited in 9 Documents MSC: 14A20 Generalizations (algebraic spaces, stacks) 17B37 Quantum groups (quantized enveloping algebras) and related deformations 55P99 Homotopy theory Keywords:Hall algebras; derived algebraic geometry; Higgs bundles; flat bundles; local systems; categorification; stable \(\infty\)-categories Citations:Zbl 1403.18016 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Arinkin, D., Gaitsgory, D.: Singular support of coherent sheaves and the geometric Langlands conjecture. Selecta Math. (N.S.) 21, 1-199 (2015) Zbl 1423.14085 MR 3300415 · Zbl 1423.14085 [2] Bartocci, C., Bruzzo, U., Hernández Ruipérez, D.: Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics. Progr. Math. 276, Birkhäuser Boston, Boston, MA (2009) Zbl 1186.14001 MR 2511017 · Zbl 1186.14001 [3] Bernstein, J.: Algebraic theory of D-modules. Lecture notes, available as ps file at this link (1983) [4] Burban, I., Schiffmann, O.: On the Hall algebra of an elliptic curve, I. Duke Math. J. 161, 1171-1231 (2012) Zbl 1286.16029 MR 2922373 · Zbl 1286.16029 [5] Calaque, D.: Three lectures on derived symplectic geometry and topological field theories. Indag. Math. (N.S.) 25, 926-947 (2014) Zbl 1298.81345 MR 3264781 · Zbl 1298.81345 [6] Conrad, B.: Grothendieck Duality and Base Change. Lecture Notes in Math. 1750, Springer, Berlin (2000) Zbl 0992.14001 MR 1804902 · Zbl 0992.14001 [7] Davison, B.: The critical CoHA of a quiver with potential. Quart. J. Math. 68, 635-703 (2017) Zbl 1390.14056 MR 3667216 · Zbl 1390.14056 [8] Davison, B., Meinhardt, S.: Cohomological Donaldson-Thomas theory of a quiver with poten-tial and quantum enveloping algebras. Invent. Math. 221, 777-871 (2020) Zbl 1462.14020 MR 4132957 · Zbl 1462.14020 [9] Diaconescu, D.-E., Porta, M., Sala, F.: McKay correspondence, cohomological Hall algebras and categorification. arXiv:2004.13685 (2020) [10] Douady, A.: Le problème des modules pour les sous-espaces analytiques compacts d’un espace analytique donné. Ph.D. thesis, Paris (1966) · Zbl 0146.31103 [11] Dyckerhoff, T., Kapranov, M.: Higher Segal Spaces. Lecture Notes in Math. 2244, Springer, Cham (2019) Zbl 1459.18001 MR 3970975 · Zbl 1459.18001 [12] Gaitsgory, D., Rozenblyum, N.: A Study in Derived Algebraic Geometry. Vol. I. Correspond-ences and Duality. Math. Surveys Monogr. 221, Amer. Math. Soc., Providence, RI (2017) Zbl 1409.14003 MR 3701352 · Zbl 1408.14001 [13] Gaitsgory, D., Rozenblyum, N.: A Study in Derived Algebraic Geometry. Vol. II. Deform-ations, Lie Theory and Formal Geometry. Math. Surveys Monogr. 221, Amer. Math. Soc., Providence, RI (2017) MR 3701353 · Zbl 1409.14003 [14] Gepner, D., Haugseng, R.: Enriched 1-categories via non-symmetric 1-operads. Adv. Math. 279, 575-716 (2015) Zbl 1342.18009 MR 3345192 · Zbl 1342.18009 [15] Ginzburg, V., Rozenblyum, N.: Gaiotto’s Lagrangian subvarieties via derived symplectic geo-metry. Algebras Represent. Theory 21, 1003-1015 (2018) Zbl 1398.53084 MR 3855670 · Zbl 1398.53084 [16] Gothen, P. B., King, A. D.: Homological algebra of twisted quiver bundles. J. London Math. Soc. (2) 71, 85-99 (2005) Zbl 1095.14012 MR 2108248 · Zbl 1095.14012 [17] Grojnowski, I.: Affinizing quantum algebras: From D-modules to K-theory. Unpublished manuscript, available at the author’s webpage (1994) [18] Halpern-Leistner, D.: Derived ‚-stratifications and the D-equivalence conjecture. arXiv:2010.01127 (2020) [19] Halpern-Leistner, D., Preygel, A.: Mapping stacks and categorical notions of properness. arXiv:1402.3204 (2014) [20] Harvey, J. A., Moore, G.: On the algebras of BPS states. Comm. Math. Phys. 197, 489-519 (1998) Zbl 1055.81616 MR 1652775 · Zbl 1055.81616 [21] Hinich, V.: Yoneda lemma for enriched 1-categories. Adv. Math. 367, 107129, 119 (2020) Zbl 1454.18003 MR 4080581 · Zbl 1454.18003 [22] Holstein, J., Porta, M.: Analytification of mapping stacks. arXiv:1812.09300 (2018) [23] Hotta, R., Takeuchi, K., Tanisaki, T.: D-Modules, Perverse Sheaves, and Representation Theory. Progr. Math. 236, Birkhäuser Boston, Boston, MA (2008) Zbl 1136.14009 MR 2357361 · Zbl 1136.14009 [24] Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. 2nd ed., Cambridge Math. Library, Cambridge Univ. Press, Cambridge (2010) Zbl 1206.14027 MR 2665168 · Zbl 1206.14027 [25] Kapranov, M., Vasserot, E.: The cohomological Hall algebra of a surface and factorization cohomology. J. Eur. Math. Soc. (online, 2022) [26] Khovanov, M., Lauda, A. D.: A diagrammatic approach to categorification of quantum groups. I. Represent. Theory 13, 309-347 (2009) Zbl 1188.81117 MR 2525917 · Zbl 1188.81117 [27] Khovanov, M., Lauda, A. D.: A categorification of quantum sl.n/. Quantum Topol. 1, 1-92 (2010) Zbl 1206.17015 MR 2628852 [28] Khovanov, M., Lauda, A. D.: A diagrammatic approach to categorification of quantum groups II. Trans. Amer. Math. Soc. 363, 2685-2700 (2011) Zbl 1214.81113 MR 2763732 · Zbl 1214.81113 [29] Kontsevich, M., Soibelman, Y.: Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants. Comm. Number Theory Phys. 5, 231-352 (2011) Zbl 1248.14060 MR 2851153 · Zbl 1248.14060 [30] Lauda, A. D.: A categorification of quantum sl.2/. Adv. Math. 225, 3327-3424 (2010) Zbl 1219.17012 MR 2729010 · Zbl 1219.17012 [31] Laumon, G., Moret-Bailly, L.: Champs algébriques. Ergeb. Math. Grenzgeb. (3) 39, Springer, Berlin (2000) Zbl 0945.14005 MR 1771927 · Zbl 0945.14005 [32] Lurie, J.: Higher Topos Theory. Ann. of Math. Stud. 170, Princeton Univ. Press, Princeton, NJ (2009) Zbl 1175.18001 MR 2522659 · Zbl 1175.18001 [33] Lurie, J.: Derived algebraic geometry V: Structured spaces. Available at J. Lurie’s webpage (2011) [34] Lurie, J.: Derived algebraic geometry IX: Closed immersions. Available at J. Lurie’s webpage (2011) [35] Lurie, J.: Higher algebra. Available at J. Lurie’s webpage (2017) [36] Lurie, J.: Spectral algebraic geometry. Available at J. Lurie’s webpage (2018) [37] Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc. 3, 447-498 (1990) Zbl 0703.17008 MR 1035415 · Zbl 0703.17008 [38] Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. Amer. Math. Soc. 4, 365-421 (1991) Zbl 0738.17011 MR 1088333 · Zbl 0738.17011 [39] Macpherson, A. W.: A bivariant Yoneda lemma and .1; 2/-categories of correspondences. arXiv:2005.10496 (2020) [40] Minets, A.: Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces. Selecta Math. (N.S.) 26, art. 30, 67 pp. (2020) Zbl 1444.14079 MR 4090584 · Zbl 1444.14079 [41] Neguţ, A.: Exts and the AGT relations. Lett. Math. Phys. 106, 1265-1316 (2016) Zbl 1348.14030 MR 3533570 · Zbl 1348.14030 [42] Neguţ, A.: The q-AGT-W relations via shuffle algebras. Comm. Math. Phys. 358, 101-170 (2018) Zbl 1407.16029 MR 3772034 · Zbl 1407.16029 [43] Neguţ, A.: Hecke correspondences for smooth moduli spaces of sheaves. Publ. Math. Inst. Hautes Études Sci. 135, 337-418 (2022) Zbl 07531906 MR 4426742 · Zbl 1506.14029 [44] Neguţ, A.: Shuffle algebras associated to surfaces. Selecta Math. (N.S.) 25 (2019), no. 3, art. 36, 57 pp. Zbl 1427.14023 MR 3950703 · Zbl 1427.14023 [45] Nitsure, N.: Moduli space of semistable pairs on a curve. Proc. London Math. Soc. (3) 62, 275-300 (1991) Zbl 0733.14005 MR 1085642 · Zbl 0733.14005 [46] Pȃdurariu, T.: K-theoretic Hall algebras for quivers with potential. arXiv:1911.05526 (2019) [47] Pantev, T., Toën, B.: Poisson geometry of the moduli of local systems on smooth varieties. Publ. RIMS Kyoto Univ. 57, 959-991 (2021) Zbl 07445182 MR 4322004 · Zbl 1496.14035 [48] Porta, M.: GAGA theorems in derived complex geometry. J. Algebraic Geom. 28, 519-565 (2019) Zbl 1453.14004 MR 3959070 · Zbl 1453.14004 [49] Porta, M.: The derived Riemann-Hilbert correspondence. arXiv:1703.03907 (2017) [50] Porta, M., Sala, F.: Simpson’s shapes of schemes and stacks. Available at this link [51] Porta, M., Yu, T. Y.: Higher analytic stacks and GAGA theorems. Adv. Math. 302, 351-409 (2016) Zbl 1388.14016 MR 3545934 · Zbl 1388.14016 [52] Porta, M., Yu, T. Y.: Representability theorem in derived analytic geometry. J. Eur. Math. Soc. 22, 3867-3951 (2020) Zbl 1456.14018 MR 4176782 · Zbl 1456.14018 [53] Porta, M., Yu, T. Y.: Derived Hom spaces in rigid analytic geometry. Publ. RIMS Kyoto Univ. 57, 921-958 (2021) Zbl 1487.14060 MR 4322003 · Zbl 1487.14060 [54] Porta, M., Yu, T. Y.: Non-archimedean quantum K-invariants. arXiv:2001.05515 (2020) [55] Rapčák, M., Soibelman, Y., Yang, Y., Zhao, G.: Cohomological Hall algebras, vertex algebras and instantons. Comm. Math. Phys. 376, 1803-1873 (2020) Zbl 07207197 MR 4104538 · Zbl 1508.81977 [56] Ren, J., Soibelman, Y.: Cohomological Hall algebras, semicanonical bases and Donaldson-Thomas invariants for 2-dimensional Calabi-Yau categories (with an appendix by Ben Davison). In: Algebra, Geometry, and Physics in the 21st Century, Progr. Math. 324, Birkhäuser/Springer, Cham, 261-293 (2017) Zbl 1385.16012 MR 3727563 · Zbl 1385.16012 [57] Rouquier, R.: 2-Kac-Moody algebras. arXiv:0812.5023 (2008) [58] Rouquier, R.: Quiver Hecke algebras and 2-Lie algebras. Algebra Colloq. 19, 359-410 (2012) Zbl 1247.20002 MR 2908731 · Zbl 1247.20002 [59] Safronov, P.: Quasi-Hamiltonian reduction via classical Chern-Simons theory. Adv. Math. 287, 733-773 (2016) Zbl 1440.53096 MR 3422691 · Zbl 1440.53096 [60] Sala, F., Schiffmann, O.: Cohomological Hall algebra of Higgs sheaves on a curve. Algebr. Geom. 7, 346-376 (2020) Zbl 1467.14034 MR 4087863 · Zbl 1467.14034 [61] Schiffmann, O.: Canonical bases and moduli spaces of sheaves on curves. Invent. Math. 165, 453-524 (2006) Zbl 1142.17004 MR 2242625 · Zbl 1142.17004 [62] Schiffmann, O.: Lectures on canonical and crystal bases of Hall algebras. In: Geometric Meth-ods in Representation Theory. II, Sémin. Congr. 24, Soc. Math. France, Paris, 143-259 (2012) Zbl 1356.17015 MR 3202708 · Zbl 1356.17015 [63] Schiffmann, O., Vasserot, E.: Hall algebras of curves, commuting varieties and Langlands duality. Math. Ann. 353, 1399-1451 (2012) Zbl 1252.14012 MR 2944034 · Zbl 1252.14012 [64] Schiffmann, O., Vasserot, E.: Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A 2 . Publ. Math. Inst. Hautes Études Sci. 118, 213-342 (2013) Zbl 1284.14008 MR 3150250 · Zbl 1284.14008 [65] Schiffmann, O., Vasserot, E.: The elliptic Hall algebra and the K-theory of the Hilbert scheme of A 2 . Duke Math. J. 162, 279-366 (2013) Zbl 1290.19001 MR 3018956 · Zbl 1290.19001 [66] Schiffmann, O., Vasserot, E.: On cohomological Hall algebras of quivers: Yangians. arXiv:1705.07491 (2017) [67] Schiffmann, O., Vasserot, E.: On cohomological Hall algebras of quivers: generators. J. Reine Angew. Math. 760, 59-132 (2020) Zbl 1452.16017 MR 4069884 · Zbl 1452.16017 [68] Schlichting, M.: A note on K-theory and triangulated categories. Invent. Math. 150, 111-116 (2002) Zbl 1037.18007 MR 1930883 · Zbl 1037.18007 [69] Schürg, T., Toën, B., Vezzosi, G.: Derived algebraic geometry, determinants of perfect com-plexes, and applications to obstruction theories for maps and complexes. J. Reine Angew. Math. 702, 1-40 (2015) Zbl 1320.14033 MR 3341464 · Zbl 1320.14033 [70] Shan, P., Varagnolo, M., Vasserot, E.: Coherent categorification of quantum loop algebras: the SL.2/ case. J. Reine Angew. Math. 792, 1-59 (2022) Zbl 07612786 MR 4504090 · Zbl 1515.81139 [71] Simpson, C. T.: Moduli of representations of the fundamental group of a smooth projective variety. I. Inst. Hautes Études Sci. Publ. Math. 79, 47-129 (1994) Zbl 0891.14005 MR 1307297 · Zbl 0891.14005 [72] Simpson, C. T.: Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. 80, 5-79 (1994) Zbl 0891.14006 MR 1320603 · Zbl 0891.14006 [73] Simpson, C.: The Hodge filtration on nonabelian cohomology. In: Algebraic Geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, RI, 217-281 (1997) Zbl 0914.14003 MR 1492538 · Zbl 0914.14003 [74] Simpson, C.: Geometricity of the Hodge filtration on the 1-stack of perfect complexes over X DR . Moscow Math. J. 9, 665-721 (2009) Zbl 1189.14020 MR 2562796 · Zbl 1189.14020 [75] The Stacks Project Authors: Stacks Project. http://stacks.math.columbia.edu [76] Thomason, R. W.: The classification of triangulated subcategories. Compos. Math. 105, 1-27 (1997) Zbl 0873.18003 MR 1436741 · Zbl 0873.18003 [77] Toën, B.: The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167, 615-667 (2007) Zbl 1118.18010 MR 2276263 · Zbl 1118.18010 [78] Toën, B.: Proper local complete intersection morphisms preserve perfect complexes. arXiv:1210.2827 (2012) [79] Toën, B., Vaquié, M.: Moduli of objects in dg-categories. Ann. Sci. École Norm. Sup. (4) 40, 387-444 (2007) Zbl 1140.18005 MR 2493386 · Zbl 1140.18005 [80] Toën, B., Vezzosi, G.: A remark on K-theory and S -categories. Topology 43, 765-791 (2004) Zbl 1054.55004 MR 2061207 · Zbl 1054.55004 [81] Varagnolo, M., Vasserot, E.: Canonical bases and KLR-algebras. J. Reine Angew. Math. 659, 67-100 (2011) Zbl 1229.17019 MR 2837011 · Zbl 1229.17019 [82] Yang, Y., Zhao, G.: The cohomological Hall algebra of a preprojective algebra. Proc. London Math. Soc. (3) 116, 1029-1074 (2018) Zbl 1431.17013 MR 3805051 · Zbl 1431.17013 [83] Yang, Y., Zhao, G.: Cohomological Hall algebras and affine quantum groups. Selecta Math. (N.S.) 24, 1093-1119 (2018) Zbl 1431.17012 MR 3782418 · Zbl 1431.17012 [84] Yang, Y., Zhao, G.: On two cohomological Hall algebras. Proc. Roy. Soc. Edinburgh Sect. A 150, 1581-1607 (2020) Zbl 1446.14036 MR 4091073 · Zbl 1446.14036 [85] Zhao, Y.: On the K-theoretic Hall algebra of a surface. Int. Math. Res. Notices 2021, 4445-4486 Zbl 1475.19005 MR 4230402 · Zbl 1475.19005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.