Algebraic de Rham cohomology

Tags
- #AG #arithmetic-geometry
Refs:
- https://link.springer.com/content/pdf/10.1007%2F978-3-319-50926-6_3.pdf #resources/papers
Links:
- resolution of singularities
- differential forms
- For connections:
  - Unsorted/curvature form
  - Unsorted/torsion of a connection
- Unsorted/de Rham isomorphism

algebraic de Rham cohomology

Closed vs exact: attachments/Pasted%20image%2020220202222055.png

Definitions

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Properties

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Degeneration of the spectral sequence

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Logarithmic differentials

attachments/Pasted%20image%2020220209094009.png Comparison to Crystalline cohomology of the reduction mod \(p\): Frobenius action and Grothendieck-Lefschetz Trace Formula: # Examples

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Grothendieck’s Proof

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Derivations

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Motivations

From physics: attachments/Pasted%20image%2020220213222929.png

Exercises

Affine space:
Elliptic curves:

🗓️ Timeline

2021-09-24

Can you use the Unsorted/algebraic de Rham cohomology to get a volume form?

Links to this page

unresolved links output

Unsorted/Grothendieck-Lefschetz Trace Formula in Unsorted/Weil Conjectures, -Unsorted/algebraic de Rham cohomology
tangent bundle

de Rham cohomology
period

Algebraic de Rham
p-adic Hodge theory

de Rham cohomology
motive

de Rham cohomology
log differential forms

de Rham complex
formally unramified

\(\Omega_{X/S} = 0\) (i.e. the relative differentials vanish).
formally etale

The relative differentials \(\Omega_{X/S} = 0\) (the relative differentials vanish) and the conormal sheaf vanishes, \(\mathcal{C}_{X/S} = 0\).
etale comparison theorem

See de Rham.
deformation theory

relative differentials
crystalline cohomology

Relation to Weil cohomology for smooth and proper schemes, algebraic de Rham cohomology, and Witt vectors
cotangent complex

A nonabelian derived functor of relative differentials:

Recovers algebraic differentials in the smooth case.

See cocone and relative differentials:
connection

- Tags: - #geomtop/differential-geometry - Refs: - #todo/add-references - Links: - ASD - flat connection - Milnor fiber - Algebraic de Rham - Local systems biject with monodromy representations
arithmetic geometry

algebraic de Rham cohomology
Weil cohomology

See Algebraic de Rham, crystalline cohomology, l-adic cohomology
THH

For $R\in {}_{k} \mathsf{Alg} $, \(H_1(R,R) = \Omega_{R/k}\), the module of relative differentials . If \(Q\subseteq R\) then there is an algebraic decomposition of this homology analogous to the Hodge decomposition for complex manifolds.
Notes - The cotangent complex and derived de Rham cohomology

de Rham cohomology
Fundamental SESs in algebraic geometry

The Cartier divisor exact sequences: \begin{align*} 0\to \mathcal{O}_{X}(-D) \to \mathcal{O}_{X} \to \mathcal{O}_{D} \to 0 \end{align*} and for algebraic differentials: \begin{align*} 0\to \Omega^{k}_{X}(-D)\to \Omega^{k}_{X}\to \Omega^{k}_{D} \to 0 \end{align*}
Arpon Raksit - Hochschild homology and the derived de Rham complex revisited

The Unsorted/algebraic de Rham cohomology is used to define derived de Rham cohomology :

#AG #arithmetic-geometry #resources/papers