Abstract
For a smooth surface S, Porta–Sala defined a categorical Hall algebra generalizing previous work in K-theory of Zhao and Kapranov–Vasserot. We construct semi-orthogonal decompositions for categorical Hall algebras of points on S. We refine these decompositions in K-theory for a topological K-theoretic Hall algebra.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ballard, M., Favero, D., Katzarkov, L.: A category of kernels for equivariant factorizations and its implications for Hodge theory. Publ. Math. Inst. Hautes Études Sci. 120, 1–111 (2014)
Ben-Zvi, D., Francis, J., Nadler, D.: Integral transforms and Drinfeld centers in derived algebraic geometry. J. Am. Math. Soc. 23(4), 909–966 (2010)
Blanc, A.: Topological K-theory of complex noncommutative spaces. Compos. Math. 152(3), 489–555 (2016)
Corti, A., Hanamura, M.: Motivic decomposition and intersection Chow groups II. Pure Appl. Math. Q. 3(1, Special Issue: In honor of Robert D. MacPherson. Part 3), 181–203 (2007)
Davison, B.: Purity and 2-Calabi-Yau categories. https://arxiv.org/pdf/2106.07692.pdf
Davison, B.: BPS Lie algebra and the less perverse filtration on the preprojective CoHA. https://arxiv.org/pdf/2007.03289.pdf
Davison, B.: The integrality conjecture and the cohomology of preprojective stacks. https://arxiv.org/pdf/1602.02110.pdf
Efimov, A., Positselski, L.: Coherent analogues of matrix factorizations and relative singularity categories. Algebra Number Theory 9(5), 1159–1292 (2015)
Gaitsgory, D.: Ind-coherent sheaves. Mosc. Math. J. 13(3), 399–528, 553 (2013)
Halpern-Leistner, D., Pomerleano, D.: Equivariant Hodge theory and noncommutative geometry. Geom. Topol. 24(5), 2361–2433 (2020)
Halpern-Leistner, D.: Derived \(\Theta \)-stratifications and the \(D\)-equivalence conjecture. http://pi.math.cornell.edu/~danielhl/dcts_2020_09_22.pdf
Halpern-Leistner, D., Preygel, A.: Mapping stacks and categorical notions of properness. https://arxiv.org/pdf/1402.3204.pdf
Isik, M.U.: Equivalence of the derived category of a variety with a singularity category. Int. Math. Res. Not. 12, 2787–2808 (2013)
Kapranov, M., Vasserot, E.: The cohomological Hall algebra of a surface and factorization cohomology. https://arxiv.org/pdf/1901.07641.pdf
Keller, B.: On Differential Graded Categories. International Congress of Mathematicians, vol. II, pp. 151–190. European Mathematical Society, Zürich (2006)
Meinhardt, S., Reineke, M.: Donaldson–Thomas invariants versus intersection cohomology of quiver moduli. J. Reine Angew. Math. 754, 143–178 (2019)
Minets, A.: Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces. Sel. Math. (N.S.) 26(2), 67, Paper No. 30 (2020)
Neguţ, A.: Shuffle algebras associated to surfaces. Sel. Math. (N.S.) 25(3), 57, Paper No. 36, (2019)
Okounkov, A., Smirnov, A.: Quantum difference equation for Nakajima varieties. https://arxiv.org/pdf/1602.09007.pdf
Pădurariu, T.: Generators for K-theoretic Hall algebras of quivers with potential. https://arxiv.org/pdf/2108.07919.pdf
Pădurariu, T.: Non-commutative resolutions and intersection cohomology of quotient singularities. https://arxiv.org/pdf/2103.06215.pdf
Porta, M., Sala, F.: Two-dimensional categorified Hall algebras. https://arxiv.org/pdf/1903.07253.pdf
Preygel, A.: Thom-Sebastiani and Duality for Matrix Factorizations. https://arxiv.org/pdf/1101.5834.pdf
Sala, F., Schiffmann, O.: Cohomological Hall algebra of Higgs sheaves on a curve. Algebr. Geom. 7(3), 346–376 (2020)
Schiffmann, O., Vasserot, E.: The elliptic Hall algebra and the K-theory of the Hilbert scheme of \({\mathbb{A} }^2\). Duke Math. J. 162(2), 279–366 (2013)
Toda, Y.: Moduli stacks of semistable sheaves and representations of Ext-quivers. Geom. Topol. 22(5), 3083–3144 (2018)
Toda, Y.: Categorical Donaldson–Thomas invariants for local surfaces. arxiv:1907.09076
Toda, Y.: Hall-type algebras for categorical Donaldson–Thomas theories on local surfaces. Sel. Math. (N.S.) 26(4), 72, Paper No. 62 (2020)
Toda, Y.: Semiorthogonal decompositions for categorical Donaldson–Thomas theory via \(\Theta \)-stratifications (preprint)
Toda, Y.: Categorical Donaldson–Thomas theory for local surfaces: \({\mathbb{Z}}/2\)-periodic version (preprint)
Varagnolo, M., Vasserot, E.: K-theoretic Hall algebras, quantum groups and super quantum groups. https://arxiv.org/pdf/2011.01203.pdf
Zhao, Y.: On the K-theoretic Hall algebra of a surface. Int. Math. Res. Not. 6, 4445–4486 (2021)
Acknowledgements
I thank Francesco Sala, Yukinobu Toda, and the referee for useful comments and suggestions. I thank the Institute of Advanced Studies for support during the preparation of the paper. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1926686.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Pădurariu, T. Generators for Hall algebras of surfaces. Math. Z. 303, 4 (2023). https://doi.org/10.1007/s00209-022-03158-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00209-022-03158-6