All Questions
Tagged with derham-cohomology ag.algebraic-geometry
43 questions
21
votes
1
answer
2k
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 ...
- 211
18
votes
1
answer
682
views
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 ...
- 714
18
votes
0
answers
2k
views
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 $t\in \text{Spec}(\mathscr{O}_{F,S})$, ...
- 23k
17
votes
3
answers
1k
views
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 $\mathbb C$...
- 47.5k
15
votes
3
answers
3k
views
algebraic de Rham cohomology of singular varieties
Hi,
Is there a simple example of an (affine) algebraic variety $X$ over $\mathbb C$ where
the $H^*_{dR}(X/\mathbb C) = H^*(\Omega^\bullet_{A/\mathbb C})$ differs from the singular cohomology $H^*_{...
- 2,842
15
votes
1
answer
911
views
Monsky's proof of the finiteness of de Rham cohomology
I'd really like to understand the proof that Paul Monsky wrote about the finiteness of the de Rham cohomology of algebraic varieties. I'd like it very much because it seems to explain in concrete ...
- 1,044
10
votes
1
answer
1k
views
algebraic de rham cohomology of a curve
Let $X$ be a smooth projective curve over a field $k$ of characteristic zero. The algebraic de Rham cohomology of $X$ is, by definition, the hypercohomology of the complex of Kähler differentials for ...
- 101
10
votes
1
answer
395
views
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 ...
- 101
9
votes
1
answer
392
views
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,\...
- 5,701
8
votes
2
answers
515
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}$ ...
- 273
8
votes
1
answer
766
views
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 \...
- 708
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
8
votes
0
answers
333
views
Triple comparison of cohomology in algebraic geometry
Let $X$ be a smooth proper variety over $\mathbb{Q}$ and $p$ a prime number. For an integer $k$, we have:
a finitely-generated abelian group $H^k(X^{\mathrm{an}}(\mathbb{C});\mathbb{Z})$
a finitely-...
- 15.4k
8
votes
0
answers
680
views
Hard Lefschetz in De Rham cohomology
I'm looking for a reference for Hard Lefschetz theorem in algebraic De Rham cohomology. By this I mean the statement that
If $i: Y \hookrightarrow X$ is a smooth hyperplane section of a smooth ...
- 81
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 ...
- 11.8k
7
votes
1
answer
440
views
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 ...
- 711
6
votes
3
answers
1k
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 ...
- 61
6
votes
2
answers
533
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}{\...
- 61
5
votes
2
answers
758
views
Remove denominators in de Rham cohomology
Let $\omega = \mathrm d \eta$ be an exact rational $n$-form on $\Bbb P^n$.
It may happen that the polar locus of $\eta$ is not included in the polar locus of $\omega$. But is it true that $\omega = \...
- 1,044
5
votes
2
answers
649
views
If a $d \log$ form is exact, is it zero?
Let $T = \mathrm{Spec}\ \mathbb{C}[x_1^{\pm 1}, x_2^{\pm 1}, \ldots, x_n^{\pm 1}]$ be an algebraic torus and $X$ a closed subvariety. Let $\eta$ be a differential form on $T$ of the form
$$\sum_I ...
- 156k
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 $k = \mathbb{C}$.) ...
- 409
5
votes
0
answers
144
views
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 (...
- 5,701
5
votes
0
answers
248
views
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 ...
- 123
5
votes
0
answers
659
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
$cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$ ...
- 806
4
votes
1
answer
447
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^{*...
user158636
4
votes
1
answer
405
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 ...
user158636
4
votes
0
answers
262
views
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 ...
- 2,044
4
votes
0
answers
441
views
Finite Field Varieties and the de Rham Complex of Kähler Differentials
In an answer to a previous question I asked, where the Kahler differentials of a variety over a finite field were discussed, it was stated that:
You can certainly define de Rham cohomology using ...
- 3,399
3
votes
0
answers
171
views
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_{...
- 629
3
votes
0
answers
494
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 ...
- 39
3
votes
0
answers
185
views
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 ...
- 7,377
2
votes
1
answer
291
views
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^...
- 1,142
2
votes
1
answer
657
views
algebraic de Rham cohomology functoriality
Suppose that $Y/k$ is a an algebraic variety over a field $k$ of characteristic zero and
that $Y\subseteq X$ is a closed embedding into a smooth variety over $k$. Then the completion
of the de Rham ...
- 2,842
2
votes
1
answer
487
views
Algebraic de Rham cohomology - open subvariety and normal crossing
I need to do some explicit computation with algebraic de Rham cohomology on some projective varieties, also some open subsets of them.
I don't know the theory well, but I just search some notes to ...
- 1,109
2
votes
0
answers
241
views
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 ...
- 2,907
2
votes
0
answers
253
views
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 ...
- 333
2
votes
0
answers
173
views
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:
$$
\mathbb{C}[x_1,\dots,x_n,y_1,\dots,y_n,(r^...
- 1,077
2
votes
0
answers
546
views
Gauss Manin connection in algebraic geometry and DG setting
E.Geltzler defined Gauss Manin connection on periodic cyclic homology of smooth proper DG algebra.I just began to learn his definition.But it seems that his construction is very different to the Gauss ...
- 1,982
1
vote
0
answers
167
views
How does the cohomology theory on de Rham (pre)stack compute de Rham cohomology?
Recently when I'm reading PTVV, Shifted Symplectic Structures, in Sec. 2.1 Mapping stacks, the authors use the identification $H^*(X,E)\simeq H^{*}_{dR}(Y/k,\mathcal{E})$ to show $X=Y_{dR}$ admits an ...
- 83
1
vote
0
answers
181
views
Calculation of de Rham cohomology of abelian varieties/ jacobian varieties
It's known that for a elliptic curve like $E:y^2=x(x-1)(x-t)$ we have a basis $\frac{dx}{y}, \frac{xdx}{y}$ for $H_{dR}^1(E)$. But find such a basis is not an easy thing. I wonder for a general ...
- 775
1
vote
0
answers
303
views
Berthelot-Ogus comparison isomorphism
On the link, page, $ 2 $, the Berthlot-Ogus isomorphism theorem is stated as follows,
We have a canonical isomorphism, $$ \rho_{\mathrm{cris}} \ : \ H_{\mathrm{cris}}^{i} (X) \otimes_{K_ {0}} K \to H_{...
- 595
1
vote
0
answers
348
views
Vanishing of cohomology groups
Given a smooth hypersurface $X \subset \mathbb{P}^{2p+1}_{\mathbb{C}}$ of degree $d \ge 5$ (with $p\ge 2$), when can we say that
$H^{p+1}(\mathcal{O}_X)=H^{p+1}(\Omega^1_X)=...=H^{p+1}(\Omega^{p-2}...
- 1,040
0
votes
0
answers
413
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 ...
- 327