Abstract
We study smooth maps that arise in derived algebraic geometry. Given a map \(A \rightarrow B\) between non-positive commutative noetherian DG-rings which is of flat dimension 0, we show that it is smooth in the sense of Toën–Vezzosi if and only if it is homologically smooth in the sense of Kontsevich. We then show that B, being a perfect DG-module over \(B\otimes ^{{\mathrm {L}}}_A B\) has, locally, an explicit semi-free resolution as a Koszul complex. As an application we show that a strong form of Van den Bergh duality between (derived) Hochschild homology and cohomology holds in this setting.
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References
Avramov, L.L., Iyengar, S.B.: Homological criteria for regular homomorphisms and for locally complete intersection homomorphisms, algebra, arithmetic and geometry (Mumbai, 2000). In: T.I.F.R. Studies in Math., vol. 16, pp. 97–122. New Delhi, Narosa (2002)
Avramov, L., Iyengar, S.B., Lipman, J.: Reflexivity and rigidity for complexes, I: commutative rings. Algebra Number Theory 4(1), 47–86 (2010)
Avramov, L.L., Iyengar, S.B., Lipman, J., Nayak, S.: Reduction of derived Hochschild functors over commutative algebras and schemes. Adv. Math. 223(2), 735–772 (2010)
Gaitsgory, D., Rozenblyum, N.: A study in derived algebraic geometry, vol. I. Correspondences and duality. In: Mathematical Surveys and Monographs (2017). https://doi.org/10.1090/surv/221.1
Jørgensen, P.: Amplitude inequalities for differential graded modules. Forum Mathematicum 22(5), 941–948 (2010)
Kontsevich, M., Soibelman, Y.: Notes on \(A^{\infty }\)-algebras, \(A^{\infty }\)-categories and non-commutative geometry. In: Homological Mirror Symmetry, pp. 153–219. Springer, Berlin (2008)
Lurie, J.: Higher algebra. Version from September 2017. https://www.math.ias.edu/~lurie/papers/HA.pdf (2017)
Minamoto, H.: Resolutions and homological dimensions of DG-modules. arXiv preprint arXiv:1802.01994v4 (2018)
Minamoto, H.: Homological identities and dualizing complexes of commutative differential graded algebras. Isr. J. Math. (2021). https://doi.org/10.1007/s11856-021-2095-3
Rodicio, A.G.: Smooth algebras and vanishing of Hochschild homology. Commentarii Mathematici Helvetici 65(1), 474–477 (1990)
Shaul, L.: The twisted inverse image pseudofunctor over commutative DG rings and perfect base change. Adv. Math. 320, 279–328 (2017)
Shaul, L.: Koszul complexes over Cohen–Macaulay rings. arXiv preprint arXiv:2005.10764 (2020)
The Stacks Project. In: de Jong, J.A. (ed.) http://stacks.math.columbia.edu
Toën, B., Vezzosi, G.: Homotopical Algebraic Geometry II: Geometric Stacks and Applications. Memoirs of the AMS, vol. 193 (2008). https://doi.org/10.1090/memo/0902
Van den Bergh, M.: A relation between Hochschild homology and cohomology for Gorenstein rings. Proc. Am. Math. Soc. 126(5), 1345–1348 (1998)
Yekutieli, A.: Duality and tilting for commutative DG rings. arXiv preprint arXiv:1312.6411v4 (2016)
Yekutieli, A.: The squaring operation for commutative DG rings. J. Algebra 449, 50–107 (2016)
Yekutieli, A.: Derived Categories (Cambridge Studies in Advanced Mathematics). Cambridge University Press, Cambridge (2019)
Yekutieli, A., Zhang, J.: Rigid complexes via DG algebras. Trans. Am. Math. Soc. 360(6), 3211–3248 (2008)
Acknowledgements
The author thanks Amnon Yekutieli for helpful discussions. The author is thankful to an anonymous referee for several corrections that helped significantly improving this manuscript. This work has been supported by the Charles University Research Centre program No.UNCE/SCI/022, and by the Grant GA ČR 20-02760Y from the Czech Science Foundation.
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