Completions and derived de Rham cohomology
By B. Bhatt. 2012
46 citations recorded by the source corpus.
We show that Illusie's derived de Rham cohomology (Hodge-completed) coincides with Hartshorne's algebraic de Rham cohomology for a finite type map of noetherian schemes in characteristic 0; the case of lci morphisms was a result of Illusie. In particular, the E_1-differentials in the derived Hodge-to-de Rham spectral sequence for singular varieties are often non-zero. Another consequence is a completely elementary description of Hartshorne's algebraic de Rham cohomology: it is computed by the completed Amitsur complex for any variety in characteristic 0.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/fb65448b82ec1145673c86ff7a65d8ca738fd59a
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Citations in the corpus
- Derived algebraic geometry (2014, background): “in terms of derived de Rham theory of X relative to Y . It is a special case of a more general theorem relating derived de Rham cohomology and algebraic de Rham cohomology in characteristic zero (see [Bhat1]), which itself possesses p-adic and finite characteristic versions (see [Bhat2]). 3.3 Tangent”
- p-adic derived de Rham cohomology (2012, methodology): “ives meaning to the resulting theory as it specialises to classical de Rham cohomologyfor smooth maps. This is also the variant used in [Bei], but is insufficient for the p-adic applications of§10. In [Bhaa], we show that this Hodge-completed theoryalways coincides with Hartshorne’s algebraic de Rham”
- Hochschild homology and the derived de Rham complex revisited (2020, methodology): “We first recall the definition of the former (going back to work of Illusie [Ill72], and studied in particular by Bhatt [Bha12]).”
- Shifted Poisson structures and deformation quantization (2015, background): “t conditions 1, 2, and 3 above are indeed satisfied for D(A) (for 2, recall that A→Ared being an epimorphism, we have π0(LA red/A) = 0). The fact that D is fully faithful is essentially the content of [Bh], stating that the relative derived de Rham cohomology of any closed immersion is the correspond”
- Topological Hochschild homology and integral p$p$-adic Hodge theory (2018, methodology): “homology In this section, we prove flat descent for topological Hochschild homology via reduction to the case of (exterior powers of) the cotangent complex, which was first proved by the first author in [Bha12a]. Theorem 3.1. Fix a base ring R. For each i≥0, the functor A→∧i ALA/R is an fpqc sheaf with”