TY - JOUR
T1 - Smooth flat maps over commutative DG-rings
AU - Shaul, Liran
N1 - Funding Information:
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.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2021/12/1
Y1 - 2021/12/1
N2 - We study smooth maps that arise in derived algebraic geometry. Given a map A→ 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⊗ALB 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.
AB - We study smooth maps that arise in derived algebraic geometry. Given a map A→ 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⊗ALB 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.
UR - http://www.scopus.com/inward/record.url?scp=85104890896&partnerID=8YFLogxK
U2 - 10.1007/s00209-021-02748-0
DO - 10.1007/s00209-021-02748-0
M3 - Article
AN - SCOPUS:85104890896
SN - 0025-5874
VL - 299
SP - 1673
EP - 1688
JO - Mathematische Zeitschrift
JF - Mathematische Zeitschrift
IS - 3-4
ER -