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Six operations on dg enhancements of derived categories of sheaves

Abstract

We lift Grothendieck–Verdier–Spaltenstein’s six functor formalism for derived categories of sheaves on ringed spaces over a field to differential graded enhancements. Our main tools come from enriched model category theory.

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Acknowledgements

We thank Valery Lunts for many inspiring discussions. He was hoping very much that a theory as presented in this work should exist. We thank Michael Mandell for discussions and Emily Riehl and Michael Shulman for useful correspondence concerning model categories. We thank Timothy Logvinenko, Hanno Becker, Alexander Efimov, James Gillespie, Greg Stevenson, Pierre-Yves Gaillard, Lorenzo Ramero and Amnon Neeman for useful discussions. Hanno Becker and Jan Weidner shared an observation which led to Lemma 4.4. Frédéric Déglise answered a question concerning Theorem 4.8. We thank the referee for very detailed comments, in particular for drawing our attention to set-theoretical problems concerning functor categories, and for suggesting a more intrinsic definition of the 2-multicategory \({{\text {ENH}}}_{\mathsf {k}}\) of dg enhancements. The author was supported by a postdoctoral fellowship of the DFG, and by SPP 1388 and SFB/TR 45 of the DFG.

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Correspondence to Olaf M. Schnürer.

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Schnürer, O.M. Six operations on dg enhancements of derived categories of sheaves. Sel. Math. New Ser. 24, 1805–1911 (2018). https://doi.org/10.1007/s00029-018-0392-4

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Mathematics Subject Classification

  • 14F05
  • 16E45
  • 18G10