# 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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### Image of first group of Cech cohomology in the second group of De Rham cohomology

Let $M$ be a smooth manifold and $\mathcal{U}$ an open good cover of $M$. If $\underline{\mathbb{Z}}$ denotes sheaf of locally constant functions, $C^{\infty}(U) := C^{\infty}(U, \mathbb{R})$ and $\...

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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_{...

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437 views

### 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 ...

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308 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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**1**answer

255 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 ...

**4**

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279 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
$$\...

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64 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 ...

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227 views

### 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 ...

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132 views

### 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
...

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937 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 ...

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66 views

### 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} \...

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125 views

### 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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577 views

### Direct proof that Chern-Weil theory yields integral classes

Suppose $E$ is a complex vector bundle of rank $n$ on a compact oriented manifold (both assumed smooth). Let $h$ be a Hermitian metric on $E$, and let $A$ be a Hermitian connection on $E$ and $F_A$ ...

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86 views

### Multidifferential operators with vanishing integrals

(Moved from math.stackexchange.)
Is the following proposition true?
Given a multidifferential operator $D$ on $\mathbb{R}^n$ with constant coefficients, i.e. for all functions $f_1,\dots,f_k \in C^...

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504 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 ...

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113 views

### When are automorphisms of the cohomology ring realized by isometries?

Let $(M,g)$ be a closed smooth Riemannian manifold, and denote by $G$ a closed subgroup of its isometry group. By considering the maps $g^*$ induced by elements $g\in G$ in the (de Rham) cohomology $H^...

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225 views

### Easier ways to compute homology/cohomology by adding extra structure

Suppose $X$ is a topological space and I want to talk about its “homology”.
There is this notion of singular homology obtained from singular chain complex. This is not very easy to compute.
Suppose ...

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72 views

### Generalized de Rham cohomology on product bundle giving specified cohomology

Given a compact, smooth manifold $M$ and a real vector bundle $E \to M$ (in general not flat). There already have been numerous questions about how to equip the space $\bigoplus_k \Gamma(\Lambda^k T^* ...

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### 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^...

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226 views

### De Rham cohomology of Lie groupoid

Let $G$ be a Lie group acting on a manifold $M$.
Consider the transformation groupoid $\mathcal{G}=(G\times M\rightrightarrows M)$. We have the notion of de Rham cohomology of a Lie groupoid by ...

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325 views

### Suppose that two cohomologous forms agree on every restriction. Do they agree?

Let $\eta$, $\omega$ be two $(1,1)$-forms on $\mathbb{C}^m \times Y$, where $Y$ is a compact Kahler manifold with vanishing first Chern class, i.e., a Calabi-Yau manifold. Suppose that for all $z \in \...

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269 views

### Leafwise de Rham cohomology (A true definition of differential forms along leaves)

For a foliated space $(M, \mathcal{F})$, one associate a leafwise de Rham cohomology. This cohomology and trace-class operators on this cohomology and trace interpretations for closed orbits of ...

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196 views

### De Rham cohomology and extension of scalars

Let $K$ be a field of characteristic zero and let $X$ be a smooth variety over $K$. Given a field extension $L/K$, I denote by $X_L$ the variety $X \times_{Spec(K)} Spec(L)$.
What is the easiest way ...

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**1**answer

124 views

### Vanishing product of a closed and coclosed form on a Riemannian manifold

For a (compact) Riemannian manifold $(M,g)$, can it happen that for a non-zero form $\text{d}^*\omega$, and a smooth function $f$ such that $\text{d}f \neq 0$, we can have
$$
\text{d}f \wedge \text{d}^...

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81 views

### Alternating property of H_2(T, Z)

Let us consider the torus $T = S^1_X \times S^1_Y$, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. We have an alternate property $dX \wedge dY = - dY \wedge dX \...

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267 views

### Formality of surfaces

The de Rham dg algebra $\Omega(F)$
of a closed orientable surface $F$
is formal
(that is, weakly equivalent to its cohomology algebra).
This is a special case of the fact of formality of Kähler ...

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926 views

### Classification of line bundles by second cohomology of a manifold

In the book Loop spaces, Characteristic classes and geometric quantization by Brylinski I see following result when trying to motivate geometric description of $H^3(M,\mathbb{Z})$.
$H^2(M,\mathbb{Z}...

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### Definition of algebraic de Rham cohomology of non-smooth affine variety

$\newcommand{\Hdr}{H_{\mathrm{dRh}}}$
$\newcommand{\spec}[1]{\mathrm{spec}(#1)}$
$\require{amsmath}$
Let $A = k[x_1,\ldots,x_n]$ the polynomial ring over a field $k$ of characteristic zero and $I \...

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518 views

### Sheaf / de Rham cohomology of a stack with values in a complex of abelian sheaves

I am reading Differentiable Stacks and Gerbes to understand about (hyper) cohomology groups of a stack $\mathcal{X}$ with values in a complex $\mathcal{M}$ of abelian sheaves over $\mathcal{X}$.
...

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162 views

### Can every De Rham cohomology class be represented by a closed form $\alpha$ with $L_X \alpha=0$

Assume that $M$ is a manifold and $X$ is a vector field on $M$.
Is it true to say that every closed form is De Rham-cohomologue to a closed form $\alpha$ with $L_X \alpha =0$?

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332 views

### Comparison for cycle class maps for de Rham and etale cohomology via p-adic Hodge theory

Let $K$ be a p-adic local field, $X$ a smooth projective variety over $Spec(K)$, $CH^k(X)$ the Chow group of pure codimension $k$. Then there are cycle maps
$cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$ ...

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208 views

### Which compact (finite dimensional) Lie groups have $H^1_{DR}(G)\neq 0$

In particular, I am wondering if $H^1_{DR}(G)\neq 0$ implies that the group can written as a semidirect product of $\mathbb{S^1}$ and something else, with the $\mathbb{S^1}$ factor being responsible ...

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647 views

### What is the scope of validity of Kunneth formula for de Rham?

In books like Bott-Tu or all pdf texts I have found on internet, the Kunneth formula for manifolds $M$ and $N$ and their de Rham cohomology
$$ H^{\bullet}_{dR}(M \times N) \simeq H^{\bullet}_{dR}(M) \...

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506 views

### Counterexample showing that G-invariant de Rham cohomology different from cohomology of G-invariant sub-complex?

If $G$ is a discrete or a Lie Group acting smoothly on a manifold $M$, we can define the algebra of $G$-invariant de Rham classes, $H(M)^G$, and we can also consider the cohomology of the sub-complex ...

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407 views

### How does one introduce characteristic classes [closed]

How does one introduce, or how were you introduced to characteristic classes?
You can assume that the student is comfortable with principal bundles and connections on principal bundles.
I am not ...

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113 views

### Question on de Rham complex with distributional coefficients

Let $X$ be a smooth manifold (usually assumed to be paracompact). Let us denote by $\underline{\Omega}^{p,-\infty}_X$ the sheaf of real valued $p$-forms with distributional coefficients in the ...

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### Lagrangian up to Hamiltonian in cotangent bundle

I want to understand the folklore conjecture that, in a CY manifold, Lagrangians up to Hamiltonian isotopies are represented by special Lagrangians by examining cotangent bundle and Hodge theory.
...

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### How does one compute the space of algebraic global differential forms $\Omega^i(X)$ on an affine complex scheme $X$?

In 1963 Grothendieck introduced the algebraic de Rham cohomolog in a letter to Atiyah, later published in the Publications Mathématiques de l'IHES, N°29.
If $X$ is an algebraic scheme over $\mathbb C$...

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### Countability assumption for good covers in Bott-Tu

In chapter II of their text Differential Forms in Algebraic Topology, Bott and Tu construct the Čech-De Rham complex with regards to an open covering indexed by some ordered and countable indexing set....

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### What is known about this type of generalisation of de Rham cohomology?

I will describe a certain generalisation of de Rham cohomology; things could be generalised further but I will stick to a concrete example. A $0$-double-form is a function on the complex plane $\...

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**1**answer

1k views

### naive de Rham cohomology fails for singular varieties

Let $X$ be a variety over a field $k$ of characteristic zero. If $X$ is smooth, algebraic de Rham cohomology defined as
$$
H^n_{dR}(X / k)=\mathbb{H}^n(X, \Omega^\bullet_{X/k})\qquad (\star)
$$ is a ...

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### Cohomology classes functorial under etale morphisms

Consider the full subcategory $\mathcal C$ in the big etale site of a field $k$ consisting of smooth schemes, namely the category of smooth varieties over a field $k$ with etale maps as morphisms.
I ...

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### Generalization of de Rham cohomology, or cohomology for non-smooth case

Let $\Omega\subseteq \mathbb{R}^{3}$ be a contractible region and $f:\Omega\to \mathbb{R}^{3}$ be a vector field on $f$. If $f$ is smooth vector field and $\nabla\cdot f=0$, then $f=\nabla\times g$ ...

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478 views

### Proof of Lefschetz-Hopf Fixpoint Theorem with de Rham cohomology?

Looking for a proof of the Lefschetz-Hopf Fixpoint Theorem with the de Rham Cohomology.
(I´m more interestet in the Formula then just the simple statement that if the Lefschetz number is not zero ...

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### 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 ...

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982 views

### Integration currents VS Poincaré Dual

Let $M$ be a complex manifold of dimension $n$ and $S \subset M$ a closed complex submanifold of complex codimension $r$. Let $[S] \in H_{2r}(S)$ be the fundamental class of $S$.
We have the ...

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**1**answer

210 views

### Some clarifications regarding Deligne's paper on $\ell$-adic representations arising from modular forms

I've posted this question few days ago on math.stackexchange because it seems quite superficial. However, since I've got no responses at all, I'm posting it here. If the question is not suitable, ...

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445 views

### Gauss Manin connection in algebraic geometry and DG setting

E.Geltzler defined Gauss Manin connection on periodic cyclic homology of smooth proper DG algebra.I just began to learn his definition.But it seems that his construction is very different to the Gauss ...

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765 views

### Steenrod operations in algebraic geometry

What are some applications of Steenrod operations (or similar constructions) in algebraic geometry?
I am dimly aware of the the use of these Voevodsky's work on motivic cohomology, and would be ...

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### 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 $k = \mathbb{C}$.) ...