Holmstrom Algebraic de Rham cohomology

Schemes of finite type over a field $\ldots \to H^{i-1}(U) \to H^{i-2}(Y) \to H^i(X) \to H^i(U) \to \ldots chunk77760940categorychunk --- ## Algebraic de Rham cohomology An [introduction by de Jong](http://www.math.columbia.edu/~dejong/note_on_algebraic_de_Rham_cohomology.pdf) [Grothendieck article](http://www.numdam.org/item?id=PMIHES_1966__k$ of characteristic zero

category: Domain Category [private]

Algebraic de Rham cohomology

Hartshorne: On the de Rham cohomology of algebraic varieties.

Hartshorne: Algebraic de Rham cohomology

category: Paper References

Algebraic de Rham cohomology

$H_{dR}^*(X) = \mathbf{H}^*(X, \Omega^*_{X/k})$ (hypercohomology of the de Rham complex)

category: Definition

Algebraic de Rham cohomology

Easy to compute cohomology of affine and projective space (see de Jong).

category: Computations and Examples

Algebraic de Rham cohomology

Hodge to de Rham spectral sequence

For any $X$ of finite type over $k$ , there is a spectral sequence

E_1^{p,q} = H^q( X, \Omega^p_{X/k} ) \implies H^{p+q}_{dR}(X)

In particular, when $X$ is affine, the de Rham cohomology is just the cohomology of the global sections of the de Rham complex.

Mayer-Vietoris spectral sequence

See de Jong.

category: Spectral Sequences [private]

Algebraic de Rham cohomology

de Rham cohomology is a Weil cohomology with coefficients $K=k$ on the category of smooth projective varieties over $k$ .

Cup product comes from wedge product on the de Rham complex.

Functoriality: Given a morphism of schemes $f:X \to Y$ , we have a map of complexes $f^{-1} \Omega^*_Y \to \Omega^*_X$ , which induces a map on cohomology.

Homotopy invariance: de Jong writes: Let $X, T, Y$ be quasiprojective $k$ -varieties, and let $f: X \times T \to Y$ , with $T$ smooth and connected. Then for any two $k$ -points of $T$ , the inclusion of the associated fibers induce the same map on cohomology.

category: Properties

Algebraic de Rham cohomology

We have the first Chern class $c_1: Pic(X) \to H^2(X)(1)$ , at least for $X$ quasi-projective. Also, Chern classes $c_i(E) \in H^{2i}(X)(i)$ for any finite locally free sheaf $E$ , up to $i = rank(E)$ . Lots of properties, see de Jong.

Chern character of a coherent sheaf: A homomorphism $ch: K^0(X) \to \oplus H^{2i}(X)(i)$ . Various properties of this map.

category: Extra Structure [private]

Algebraic de Rham cohomology

An affine bundle induces isomorphism on cohomology.

Kunneth decomposition.

Trace map: Let $f: X \to Y$ be a finite morphism, with $X,Y$ of the same dimension. Then we have a map $Tr: f_* \mathcal{O}_X \to \mathcal{O}_Y$ . For properties, see de Jong.

Poincaré duality and Tate twists, see de Jong.

Projective bundle theorem.

Let $Y$ be a nonsingular divisor on a smooth projective variety $X$ , and $U$ the open complement. Then, using the logarithmic de Rham complex, one gets a long exact sequence

  \ldots \to H^{i-1}(U) \to H^{i-2}(Y) \to H^i(X) \to H^i(U) \to \ldots

category: Standard theorems

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## Algebraic de Rham cohomology

An [introduction by de Jong](http://www.math.columbia.edu/~dejong/note_on_algebraic_de_Rham_cohomology.pdf)

[Grothendieck article](http://www.numdam.org/item?id=PMIHES_1966__29__95_0)

Some [notes by Clark](http://www.math.mcgill.ca/goren/SeminarOnCohomology/derham4.pdf)

[Friedrich](http://www.math.uni-leipzig.de/MI/huber/preprints/periods.pdf) on periods and algebraic de Rham cohomology

category: Online References

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## Algebraic de Rham cohomology

See Lecomte and Wach for representability in DM by an ind-object.

category: Representability [private]

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## Algebraic de Rham cohomology

The Hodge to de Rham spectral sequence implies the finiteness and vanishing axioms of a Weil cohomology.

<http://mathoverflow.net/questions/75961/is-de-rham-cohomology-of-affine-schemes-over-discrete-valuation-rings-finitely-ge>

category: Finiteness properties [private]

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## Algebraic de Rham cohomology

[arXiv:1207.6193](http://front.math.ucdavis.edu/1207.6193) Completions and derived de Rham cohomology
fra arXiv Front: math.AG av Bhargav Bhatt
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.

[arXiv:1009.3108](http://front.math.ucdavis.edu/1009.3108) Sur le topos infinitésimal p-adique d'un schéma lisse I
from arXiv Front: math.AG by Alberto Dario Arabia, Zoghman Mebkhout
In order to have cohomological operations for de Rham p-adic cohomology with
coefficients as manageable as possible, the main purpose of this paper is to
solve intrinsically and from a cohomological point of view the lifting problem
of smooth schemes and their morphisms from characteristic p > 0 to
characteristic zero which has been one of the fundamental difficulties in the
theory of de Rham cohomology of algebraic schemes in positive characteristic
since the beginning. We show that although smooth schemes and morphisms fail to
lift geometrically, it is as if this was the case within the cohomological
point of view, which is consistent with the theory of Grothendieck Motives. We
deduce the p-adic factorization of the Zeta function of a smooth algebraic
variety, possibly open, over a finite field, which is a key testing result of
our methods.


Toen: Algebres simplicicales etc, file Toen web prepr rhamloop.pdf. Comparison between functions on derived loop spaces and de Rham theory. Take a smooth k-algebra, k aof char zero. Then (roughly) the de Rham algebra of A and the simplical algebra $S^1 \otimes A$ determine each other (functorial equivalence). Consequence: For a smooth k-scheme $X$, the algebraic de Rham cohomology is identified with $S^1$-equivariant functions on the derived loop space of $X$. Conjecturally this should follow from a more general comparison between functions on the derived loop space and cyclic homology. Also functorial and multiplicative versions of HKR type thms on decompositions of Hochschild cohomology, for any separated k-scheme.

category: Some Research Articles

nLab page on [[nlab:Algebraic de Rham cohomology]]

Created on June 10, 2014 at 21:14:54 by Andreas Holmström