Questions tagged [derham-cohomology]

The cohomology of the complex of differential forms on a smooth manifold with differential given by exterior derivative.

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Nice proof that de Rham complex computes Lie algebra cohomology?

If $G$ is a nice enough group acting on a nice enough space $X$, then the relative de Rham complex $$\Omega^\bullet_{X/(X/G)}\ \simeq\ \mathcal{O}_X\otimes\text{Sym}\,\mathfrak{g}^*[-1]$$ is given by (...
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Existence of covering space with trivial pullback map on $H^1$

I have seen somewhere the following claim (but can't remember where): let $M$ be a connected orientable closed smooth manifold with $b_1(M)=1$, then there exists a connected covering space $p:\tilde{M}...
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Calculation of de Rham cohomology of abelian varieties/ jacobian varieties

It's known that for a elliptic curve like $E:y^2=x(x-1)(x-t)$ we have a basis $\frac{dx}{y}, \frac{xdx}{y}$ for $H_{dR}^1(E)$. But find such a basis is not an easy thing. I wonder for a general ...
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Calculation of Frobenius on de Rham cohomology of elliptic curves with good reduction

I'm reading "An introduction to the theory of $p$-adic representations" by Laurent Berger. In the page 14 it says: Suppose $F$ is an unramified finite extension of $\mathbb Q_p$ and $E$ is ...
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Algebraic de Rham cohomology with torus coefficients

Let $X$ be a smooth projective variety over $\mathbb{C}.$ On page 3 in this preprint of Simpson, it is stated that Notice first of all that the algebraic de Rham theory is not going to work well in ...
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Gysin isomorphism in de Rham cohomology using currents

I'd like to find a reference for the following fact. First, some background: we can define de Rham cohomology of a smooth manifold $X$ of dimension $d$ using the de Rham complex $$ \Omega^0_X\to \...
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Is anything known about de Rham $K(\pi,1)$'s?

Let $X$ be a connected qcqs scheme. We say that $X$ is a (étale) $K(\pi,1)$ if for every locally constant constructible abelian sheaf $\mathscr{F}$ on $X$ and every geometric point $\overline{x}$ the ...
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Clarification on smooth de Rham theorem

I am misunderstanding something in Theorem 2.1.9 in Dimca’s Sheaves in Topology: Let $X$ be a real smooth manifold. Then the natural morphism from the constant sheaf to the de Rham complex $$\mathbb{R}...
locally trivial's user avatar
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2 answers
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Exterior differentiation of foliations

Let $M$ be a differentiable manifold. Let $T^*M$ be the cotangent bundle of $M$. Consider the exterior differentiation $d: A^p(M)\longrightarrow A^{p+1}(M)$, where $A^p(M)=\Gamma(\...
Shiquan Ren's user avatar
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Monodromy group action on de Rham cohomology

Let $f : Y \longrightarrow X := \mathbb{P}^1\setminus\{0,1,\infty\}$ be the smooth proper morphism associated to the Legendre family, which is an elliptic fibration of the punctured line, with fibre ...
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Spencer complex and de Rham Complex

in those lectures notes written by Claude Sabbbah: https://perso.pages.math.cnrs.fr/users/claude.sabbah/livres/sabbah_nankai110705.pdf there is the proposition 1.4.4 where he says that there is a ...
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Compute de Rham-Witt sheaves

I am really new to this, but I am having a hard time understanding all the de Rham-Witt construction. It seems to be really difficult to compute anything with those beasts: like I cannot find any ...
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Cycles in algebraic de Rham cohomology

Let $F$ be a number field, $S$ a finite set of places, and $X$ a smooth projective $\mathscr{O}_{F,S}$-scheme with geometrically connected fibers. For each point $t\in \text{Spec}(\mathscr{O}_{F,S})$, ...
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Künneth formula for algebraic de Rham cohomology

Let $X$ and $Y$ be finite type schemes over a field $k$, and let $H^i(X/k)$ denote the $i$-th algebraic de Rham cohomology group of $X$ over $k$. I'm interested in the extent to which a Künneth ...
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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(\...
Alex's user avatar
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Can one make sense of de Rham cohomology for the complement of a (dense) irrational flow on the torus?

Recent work has led me to consider whether one could define consider the complement of a dense irrational flow on the torus $P_\alpha \subset T^2$ as some kind of generalized smooth space, and ...
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Modular forms and Petersson inner product via De Rham cohomology, Hodge filtration and cup products

I'm looking for an explanation on how and why you can define modular forms through De Rham cohomology via the Hodge filtration and especially how the Petersson inner product is related to the cup ...
Lukas Heger's user avatar
16 votes
4 answers
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Can one glue De Rham cohomology classes on a differential manifolds?

Let $M$ be a differential manifold and $\mathcal H^k$ the presheaf of real vector spaces associating to the open subset $U\subset M$ the $k$-th de Rham cohomology vector space: $\mathcal H^k(U)=H^k_{...
Georges Elencwajg's user avatar
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Interpreting the Higher-order Hodge-Laplace Operator

As an operator on functions, one intuitive way to think about the Laplacian seems to be as an operator that returns the average difference between a function's value at a point and the values of its ...
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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 ...
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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 ...
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17 votes
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De Rham via topoi

Étale cohomology of schemes $X$ is constructed as follows: one associates to $X$ the so-called étale topos of $X$, and then one just takes the sheaf cohomology of that topos. Is it possible to ...
user1022117's user avatar
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Hodge dual of de Rham cohomology and singular cohomology

We know that the de Rham cohomology is isomorphic to the singular cohomology, does the Hodge dual of differential forms induce a dual operation on de Rham cohomology, hence also on singular cohomology?...
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L^1 gradient bounds for potentials of weakly closed forms

Context: The Poincaré-lemma is a central statement in differential geometry. It shows that a k-form is closed iff it is exact. A special case is as follows: Let $\omega\in\Omega^k(U)$ with $\omega=\...
MrVolt16's user avatar
9 votes
1 answer
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Integrating hypercohomology classes

Let $X$ be a complex variety. By Poincare's lemma, its singular cohomology can be computed as hypercohomology of the holomorphic de Rham complex (viewing $X$ as a complex manifold) $$\text{H}^\cdot(X,\...
Pulcinella's user avatar
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de Rham cohomology of a specific ring

I'm running into a certain algebraic de Rham cohomology computation I could use some help with. Specifically, what is the algebraic de Rham cohomology of: $$ \mathbb{C}[x_1,\dots,x_n,y_1,\dots,y_n,(r^...
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Does Stokes theorem have anything to do with adjoint functors?

I notice some similarity between Stokes theorem in differential geometry and the definition of adjoint functors: in both cases, there is a 2-placed function (the $\operatorname{hom}$ functor, or the ...
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Artin vanishing for Stein manifolds and restriction maps

In the setting of complex Stein manifolds $X$ of complex dimension $d$, the theorem of Andreotti--Frankel implies the vanishing of the singular cohomology group $H^i(X,\mathbb Z)=0$ for $i>d$. With ...
Peter Scholze's user avatar
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de Rham Bloch-Ogus theory in positive characteristic

In their famous paper Gersten's conjecture and the homology of schemes, one of the results that Bloch and Ogus prove is that the second page of the coniveau spectral sequence for $X$ smooth over a ...
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Relationship between Dolbeault and de Rham cohomology on Riemann surface

A lecturer of mine once ``proved'' the existence of non-constant meromorphic functions on a compact Riemann surface $X$ by using analysis of the Laplacian to decompose the de Rham cohomology group as $...
Martin Skilleter's user avatar
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Where does this clever choice of differential come from? (calculating $\mathrm{H}^1_{\mathrm{dR}}(E/k))$

In these notes of Kedlaya, he calculates the de Rham cohomology of an affine part $X$ of an elliptic curve $E$ over a field $K$, given by $y^2 = P(x) = x^3 + ax + b$. He uses these relations: $0 = y^...
Somatic Custard's user avatar
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"Geometric" proof of Kunneth formula

The usual proof of the Kunneth formula (say for either the homology or cohomology of manifolds) is essentially pure homological algebra. I was wondering if there was a more geometric proof, i.e., one ...
Matt Larson's user avatar
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Berthelot-Ogus comparison isomorphism

On the link, page, $ 2 $, the Berthlot-Ogus isomorphism theorem is stated as follows, We have a canonical isomorphism, $$ \rho_{\mathrm{cris}} \ : \ H_{\mathrm{cris}}^{i} (X) \otimes_{K_ {0}} K \to H_{...
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Existence non-trivial parallel $p$-form implies non-triviality of $p$-th cohomology group using De Rham cohomology

Cross-post from MSE. Suppose $(M,g)$ be a closed Riemannian manifold. Because every parallel (nontrivial) $p$-form $\omega$ is harmonic so the $p$-th Betti number should be positive i.e. $b_p\geq 1$. ...
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The unit root subspace of a genus-2 odd degree hyperelliptic curve of semistable reduction

Let $K$ be a finite extension of $\mathbb{Q}_p$. If $A_K$ is a semistable Abelian variety over $K$, then we have a Frobenius endomorphism on $H_{dR}^1(A_K)$, whose definition depends on a choice of a ...
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how to see the Gysin map explicitly in an easy situation

Let $C$ be a smooth projective curve and let $U \subset C$ be an open affine subset, with closed complement $S$ consisting of a finite number of points. I am trying to see explicitly the Gysin map in ...
dR93's user avatar
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6 votes
2 answers
477 views

Künneth formula for de Rham cohomology with respect to an integrable connection

I am reading through https://stacks.math.columbia.edu/tag/0FM9 which proves that for $X,Y$ schemes over some base $S$ and $X \times _S Y \overset{p}{\rightarrow} X$ resp. $X \times _S Y \overset{q}{\...
Joachim's user avatar
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algebraic de Rham cohomology of toric varieties (reference request)

I haven't been able to find anything workable yet, but I'm looking for a reference on the de Rham cohomology of toric varieties, where as many as possible of the following conditions are handled: ...
Somatic Custard's user avatar
8 votes
2 answers
422 views

Degeneration twisted Hodge to de Rham spectral sequence

Let $X$ be a proper and smooth scheme over $\mathbf{C}$ and let $\mathbb{L}$ be a local system of finite dimensional $\mathbf{C}$-vector spaces. By the Riemann Hilbert correspondence, to $\mathbb{L}$ ...
franck's user avatar
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Understanding the Exercise 9.9.5 of Weibel homological algebra

The exercise 9.9.5 of Weibel's homological algebra states that $\textbf{Exercises 9.9.5}$ If $Z$ is a nilpotent ideal of $R$ and $k$ is a field of $char(k) = 0$, show that $H_{dR}^{\ast}(R) \cong H_{...
Sunny's user avatar
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12 votes
2 answers
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Different definitions for integral de Rham cohomology classes

Suppose that $S$ is a compact orientable surface. In this case, the top de Rham cohomology space $H^2(S)\cong \mathbb{R}$, with the isomorphism given by integration on $2$-forms along $S$. Now, one ...
G. Gallego's user avatar
4 votes
1 answer
432 views

Is the action of the absolute Frobenius on de Rham cohomology induced by an algebraic map?

Let $X\to \mathrm{Spec}\:\mathbb{Z}_{(p)}$ be a smooth proper morphism with a geometrically connected generic fiber. Assume that the special fiber has an $\mathbb{F}_p$-point. Via the isomorphism $H^{*...
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4 votes
1 answer
390 views

Can Hodge symmetry fail if there is a lift to $W_2$ and the crystalline cohomology is torsion-free?

Let $f:X\to \mathrm{Spec}\:\mathbb{F}_p$ be a smooth proper morphism with $p>\mathrm{dim}\:X$. Assume that $H^i_{\mathrm{crys}}(X/\mathbb{Z}_p)$ is torsion-free for all $i\geq 0$ and that there is ...
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6 votes
1 answer
761 views

De Rham's theorem for top-forms in manifolds with boundary

In page 79 of Bott-Tu, "Differential Forms in Algebraic Topology", they define the relative de Rham theory as follows: Let $f:S\to M$ be a smooth map. Define the complex $\Omega^*(f)$ by $$\...
Juan Margalef's user avatar
3 votes
1 answer
157 views

Models for computing cohomology of Lie groupoids

Given a Lie groupoid $\mathcal{G}=[\mathcal{G}_1\rightrightarrows \mathcal{G}_0]$, let $\mathcal{G}_\bullet$ be the associated simplicial manifold. Let $\Omega^\bullet(\mathcal{G}_\bullet)$ be the ...
Praphulla Koushik's user avatar
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0 answers
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When are the cotangent and tangent sheaves isomorphic?

Let $X$ be an $S$-scheme. Under what conditions, if any, is the cotangent sheaf $\Omega_{X/S}$ isomorphic to the tangent sheaf $\Theta_{X/S}$ as $\mathcal{O}_X$- modules? For example, given a ...
Plank's user avatar
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Geometric theory for cohomology groups $H^p(M;\mathbb{Z})$

An excerpt from the book Loop Spaces, Characteristic Classes and Geometric Quantization by Jean-Luc Brylinski is mentioned below: Characteristic classes are certain cohomology classes associated ...
Praphulla Koushik's user avatar
6 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 ...
Emily's user avatar
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2 votes
0 answers
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Relative de Rham Cohomology groups of k-algebra

Let $A$ be a commutative unital $k$-algebra. Then we have de Rham complex given as: $C_{\ast}(A)$ : $ 0 \rightarrow A \rightarrow \Omega_{A \lvert k}^{1} \rightarrow \Omega_{A \lvert k}^{2} \...
Sunny's user avatar
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1 vote
0 answers
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Find torsion classes using flat bundles

My question refers to a discussion from this older thread on Neron-Severi group of a Kähler manifold. In the comments below Ted Shifrin's answer there arose a discussion when the map $H^2(X,\mathbb{Z}...
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