All Questions
Tagged with derham-cohomology reference-request
9 questions
3
votes
1
answer
203
views
Cohomology of the complex of differential forms with Schwartz coefficients
Let $U$ be an open manifold (say an open subset of $\mathbb{R}^n$ for simplicity). Denote by $\mathscr{S}(U)$ the space of Schwartz functions on $U$. Schwartz functions are defined as usual to be ...
- 21.5k
3
votes
0
answers
131
views
Isomorphism between Weyl and Cartan models as Hom-Tensor Adjunction
Let $M$ be a manifold, $\Omega$ be the de Rham complex of $M$. Let $G$ is a compact Lie group acting on $M$, $\mathfrak g$ its Lie algebra and $W(\mathfrak g) = \Lambda(\mathfrak g^*) \otimes S(\...
- 131
3
votes
0
answers
118
views
Logarithmic differentials on an arithmetic surface, and Poincaré residue
Suppose that $X$ is an arithmetic surface, i.e. $\pi: X \to S$ flat and relative dimension 1 over a Dedekind scheme $S$, and assume $X$ smooth.
Let $Y \subset X$ be a horizontal effective Cartier ...
- 1,338
2
votes
0
answers
164
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Exact sequence for low-degree terms of relative de Rham cohomology
Let $\varphi:X\to Y$ be a morphism of schemes, smooth of relative dimension 1. The de Rham cohomology $H^\bullet(X/Y)$ is the result of applying the derived functor $R^\bullet\varphi_*$ to the de Rham ...
- 1,338
7
votes
2
answers
1k
views
On a mysterious reference of Grothendieck
These days I found a mysterious page on Google books describing a book entitled On the De Rham cohomology of schemes by Grothendieck, Coates, and Jussila.
At once I thought this was an error and ...
- 40.2k
8
votes
1
answer
773
views
A variant on characteristic $p$ de Rham cohomology
I was thinking about de Rham cohomology in characteristic $p$, and in particular the recent question about Poincare residues, and I came up with the following construction.
Let $k$ be a perfect field ...
- 156k
2
votes
0
answers
177
views
vanishing of higher algebraic de Rham cohomology and sheaves of differentials for singular curves
I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are:
$H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^...
- 1,142
3
votes
0
answers
217
views
Reference: Relative cohomology of a morphism
Let $f\colon Y \to X$ be a morphism of schemes, the inverse image in $K$-theory always fit into a long exact sequence
$$
\cdots \to K_i(f)\to K_i(X) \xrightarrow {f^*} K_i(Y)\to \cdots
$$
where the ...
- 2,871
5
votes
1
answer
540
views
Most general "finiteness of de Rham cohomology" statement for holonomic $D$-modules in the algebraic case?
Let $X$ be a nonsingular algebraic variety over a field $k$ of characteristic zero. (We may assume $k$ algebraically closed if need be, but I want to avoid specifically demanding $k = \mathbb{C}$.) ...
- 6,829