# Questions tagged [chow-groups]

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103
questions

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### Locus where a family of cycles is rationally trivial is countable union of closed subvarieties?

Following up on this question which received a negative answer, I wonder if something weaker is true.
We work in the same set-up as the previous question. Let $B$ be a smooth quasi-projective variety ...

1
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0
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132
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### Locus where a family of cycles is rationally trivial is closed?

Let $B$ be a smooth quasi-projective variety over a field of characteristic zero.
Let $\pi\colon \mathcal{X} \rightarrow B$ be a smooth and projective morphism with geometrically integral fibres. Let $...

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138
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### Bloch's higher Chow group as relative ordinary Chow group

If X is a variety and $Y\subset X$ is a closed subscheme then one can define relative Chow group. The definition is follows: there is subcomplex $\psi_Y\colon z^r_Y(X,*)\hookrightarrow z^r(X,*)$ of ...

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### Universal properties for Bloch's higher Chow groups

I work in the category of varieties over some field of characteristic zero. Assume that for any variety I can define the group $\widetilde{CH}^r(X,n)$ which behave like classical Bloch's higher Chow ...

2
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### Tate's conjecture for arithmetic schemes

Tate's conjecture is about a map from Chow groups of a smooth projective variety $X$ to the $l$-adic cohomology i.e. $CH^n(X)\rightarrow (H^{2n}(\bar{X}, \mathbb{Q}_l(n)))^G$ where $G$ is the Galois ...

1
vote

2
answers

229
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### Correspondences acting on cohomology groups $H^*(X)$ & splittings

Let $X$ be a smooth connected proper scheme over field $k$. It is known that
correspondences $\alpha \subset X \times X$ regarded as
objects in Chow groups $\text{CH}^*(X \times X)$
act on cohomology $...

2
votes

0
answers

55
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### Composition of correspondences pulled back to $\mathrm{CH}_0$

Let $X,Y,Z$ be varieties. Given two correspondences $\Gamma_1 \subset X \times Y$ and $\Gamma_2 \subset Y \times Z$ there is a composition,
$$ [\Gamma_1] \circ [\Gamma_2] = \pi_{13 *} (\pi_{12}^* [\...

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### Interpretation of Tate conjecture using motivic homotopy

For a smooth projective variety $X$ over a field $k$ the Tate conjecture says that the cycle class maps
$$CH^i(X)\otimes \mathbb{Q}_l \to H^{2i}(X_{\bar{k}},\mathbb{Q}_l(i))^{G_k}$$
are surjective. To ...

3
votes

0
answers

226
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### Do Weil cohomology theories for schemes over arbitrary rings exist, and do the standard theorems (Lefschetz fixed point, Tr. Formula etc.) still hold?

A Weil cohomology theory is a functor that assigns to a smooth projective variety $X$ of dimension $d$ over a field $k$ a graded ring of cohomology groups with values in a field $K$ of characteristic $...

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233
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### Pushforward and pullback on the level of Chow varieties

Let $X$ and $Y$ be complex projective varieties. Let's assume we have a finite flat morphism $f:X\rightarrow Y$ of degree $k$. We know that it is possible to pullback and also pushforward algebraic
...

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242
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### A question on the Chow group on stacks

Let $X$ be a separated Deligne-Mumford stack finite type over the ground field. Then there is a Chow group $A_*(X)$ of $X$ which is well-behaved under flat pull-back, defined as follows.
Let $\...

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75
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### General fiber and the symmetric product of an ample hypersurface

Let $Sym^m(X)$ be the $m$th symmetric product of a smooth projective variety $X$, $n=\dim(X)$, $Y_1$ an ample hypersurface of $X$, and $CH_0(X)_{hom}$ the Chow groups of $0$-cycles of degree $0$....

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1
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186
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### Meaning of torsion points in a Roitman's theorem

I am having some problems to understand the meaning of the following theorem due to Roitmann. I found this theorem in Voisin's book: Hodge Theory and Complex Algebraic Geometry, Volume II, page ...

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### About finite dimensionality of Chow groups of zero cycles

Let $S$ be a connected smooth complex projective surface.
Let $Sym^{d}(S)$, $d\in \mathbb{Z}^+_0$, be the $d$-th symmetric product of $S$ parametrizing $0$-cycles of degree $d$.
Let $Sym^{d,d}(S)=...

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votes

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102
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### Chow group of different reductions of a smooth projective variety

Let $(R,\mathfrak{m},k)$ be a discrete valuation ring with fraction field $K$. Let $X/K$ be a smooth projective variety.
Let $\mathcal{X},\mathcal{X}'$ be smooth projective models of $X$ over $R$. Let ...

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102
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### About Definition 2 in Roĭtman's Paper

Let $A(X)$ be the Chow group of $0$-cycles on a nonsingular irreducible projective variety $X$ over an uncountable algebraically closed field of characteristic zero.
In Definition 2 of Roĭtman's paper ...

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173
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### Higher Chow group of complex field

It is well-known fact that there is an isomorphism
$$
K_i(\mathbb{C})\simeq \left\{
\begin{array}{ll}
\mathbb{Q}/\mathbb{Z} & \text{if } i:odd \\
0 & \text{if }i:even
\end{array}
\right.
$$
My ...

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0
answers

201
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### Algebraic correspondence as morphisms in Betti cohomology

$\newcommand{\sing}{\mathrm{sing}}$Take a commutative ring $R$ and smooth projective complex varieties $X$ and $Y$. An element $\alpha\in CH^*(X\times Y)_R$ induces the algebraic correspondence for ...

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1
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314
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### Singularities of Chow varieties

Let $X$ be a smooth complex projective variety. The Chow variety of degree $d$, $r$ dimensional subvarieties is denoted by $C_{d,r}(X)$. The Chow variety can have many topologically connected ...

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69
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### Non-torsion infinitely divisible elements in the Chow group

It was shown in "Clemens, Herbert, Homological equivalence, modulo algebraic equivalence, is not finitely generated.", that the Chow group mod algebraic equivalence of smooth complex ...

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### Chow countability argument

I would like to know what the "Chow countability argument or HIlbert schemes countability argument" says in order to finish an exercise. Any reference will also be very useful :)!

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votes

1
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### Notation on a Mumford's paper

I am reading the paper " D. Mumford. Rational equivalence of $0$-cycles on surfaces. J. Math. Kyoto Univ. 9 (1969) 195 - 204" and I do not understand a notation in bottom of page 197. It ...

2
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### Motivic complexes associated to adequate equivalence relations

Motivic cohomology sheaves $\mathbb{Z}(n)$ are homotopy invariant sheaves with transfers (under finite maps) and they satisfy excision long exact sequence when everything is smooth. The motivic ...

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### Are cubical higher Chow groups of field $CH^{n-1}(F,n)$ generated by linear cycles?

In the paper "The linearization of higher Chow cycles of dimension one" W. Gerdes proved that Higher Chow homology group $CH^{n-1}(F,n)=H^{n}(z^{n-1}(F,*))$ are generated by linear cycles.
...

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155
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### Pullback of algebraic $K$-theory along the surjection of abelian varieties

Given a surjective homomorphism of abelian varieties $f:A\rightarrow B$ where $\text{dim}(A)>\text{dim}(B)$, does $f^*$ induce a rational injection of algebraic $K$-theory? According to the ...

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245
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### Generic rank of proper pushforward of the trivial line bundle

Given a proper surjective morphism $f:X\rightarrow Y$ where $X$ and $Y$ are smooth projective varieties. The proper pushforward $f_!$ is the homomorphism that sends the class of a coherent sheaf $M$ ...

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1
answer

417
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### The Ogus conjecture for crystalline cohomology

How is the Ogus conjecture explicitly stated, which is a variant of the Hodge and the Tate conjectures for crystalline cohomology ?
How do we build its class cycle map, and how do we formulate its ...

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1
answer

372
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### How to show that, $ \mathrm{CH}_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} \simeq \Omega_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} $?

Let $ X $ be a $ n $ - dimentional oriented closed real manifold ( i.e : compact and without boundary ).
Can you tell me how to show that,
$$ \mathrm{CH}_k (X) \otimes_{ \mathbb{Z} } \mathbb{Q} \simeq ...

2
votes

0
answers

87
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### Homological and rational adequate equivalences for product of curves

There is a variant of Standard Conjecture D for projective varieties over finite fields. It claims that rational and homological equivalences are equivalent on cycles after tensoring with $\mathbb{Q}$....

5
votes

1
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245
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### Moduli of rational equivalent classes of 0-cycles

Let $X$ be a smooth variety over a field $k$ and I'd like to think about $CH_0(X)$ the 0-Chow group i.e. the group of rational equivalent classes of 0-cycles. I'm wondering if there is any reasonable ...

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### Injectivity of pushforward of rational Chow groups

I'd like to know whether there is a known counter-example to the following statement. Let $X$ be a smooth projective variety over a finite field. Let $Z$ be a codimension $2$ smooth subvariety which ...

2
votes

1
answer

390
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### Invariance of Chow groups of projective bundles under automorphisms of bundles

Suppose $X$ is a smooth scheme, $E=O_X^{\oplus n}$ and $\varphi\in SL_n(E)$, i.e. $\varphi$ has trivial determinant and is an isomorphism. Is the morphism
$$\mathbb{P}(\varphi)^*:CH^{\bullet}(\mathbb{...

5
votes

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### Applications of Chow rings of classifying spaces in algebraic geometry

For an algebraic group $G$, the Chow ring of its classifying space $BG$, in the sense of
Totaro, The Chow ring of a classifying space
has been computed in many cases. Is there any interesting ...

3
votes

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242
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### Descent and Chow groups

One of the features of the $\mathbb{A}^1$-homotopy theory is the existence of the motivic Eilenberg-MacLane space $K(\mathbb{Z}(n),2n)$ such that for $k$-schemes $X$, we have
$$[X,K(\mathbb{Z}(n),2n)]\...

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### Localization of Chow groups and flat base change

For any flat morphism $f:X\rightarrow Y$, we have a flat pullback of Chow groups
$$Ch^i(Y)\rightarrow Ch^i(X).$$
A particular example of this is of course an open immersion $U\rightarrow X$. In that ...

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87
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### What can be said about the Chow rings of classifying spaces of semi-direct products of groups?

For instance, what can we say about the Chow ring of the classifying space of a semi-direct product $CH^*(B(G\ltimes H))$, in terms of the Chow rings of $CH^*(BG)$, $CH^*(BH)$, and the singular ...

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### Which field extensions do not affect Chow groups?

Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively?
...

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2
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413
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### Are Chow groups invariant under universal homeomorphisms?

Let $f:X\to Y$ be a universal homeomorphism of schemes, $R$ a coefficient ring. Which assumptions on $f$ and $R$ are suffient to ensure that the pullback map $f^*$ of $R$-linear Chow groups is ...

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votes

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671
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### Relations between rational algebraic K-theory and Chow groups

A consequence of Grothendieck's Riemann-Roch Theorem is the fact that the Chern character induces an isomorphism between algebraic
$ch: K_{0}(X) \otimes \mathbb{Q} \stackrel{\cong}{\rightarrow} C H^{*...

2
votes

1
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### Looking for examples of not injective maps and not surjective maps of the form $ A_{k} (X) \to H_{2k} ( X , \mathbb{Z} ) $

Here: https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002-c9.pdf, on pages: $ 1 $ and $ 2 $, we find the following paragraph:
For any scheme of finite type over a ground field ...

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### Chow ring of $E_7$ varieties

Consider a split algebraic group $G$ of type $E_7$ over a field of characteristic zero.
It is known that some subgroups $P_i$ of $G$, which are called parabolic, have the property that the object $G/...

4
votes

1
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232
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### Question regarding intersection product in Chow group of $\mathbb{P}^n\times\mathbb{P}^m $

Assume we are working over complex numbers. Let $\pi:\mathbb{P}^n\times\mathbb{P}^m\rightarrow \mathbb{P}^n (m,n>0)$ be the first projection. Suppose we are given a vector bundle $E$ over $\mathbb{...

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### Numerical and rational equivalences on intersection of divisors

Let $X$ be a smooth projective variety over a finite field. Since $Pic^0(X)$ is finite and $Pic^0(X)$ can be identified with numerically equivalent to zero divisors this implies that for divisors on $...

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votes

1
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490
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### Question regarding Chow group of a blow-up

Let $X$ be a smooth complex projective variety, and $Y\hookrightarrow X$ be a smooth projective subvariety. Let $\pi:\tilde{X}\rightarrow X$ be the blow-up along $Y$, and let $j:E\hookrightarrow \...

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### Few questions about the algebraic cycles and the conjectures of Beilinson and Tate

I have three slightly related questions about algebraic cycles which I am just going to list them. I'd really appreciate any answers:
1) Is there any example of a smooth projective variety $X$ over a ...

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195
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### Computing Chow group of a variety which is almost a blow-up of another variety

Let $X$ be a normal complex projective variety (not necessarily smooth), and let $Y$ be a smooth complex projective variety. Let $Z\subset X$ be a smooth closed subvariety. I have a morphism which is ...

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### Vanishing of Chow groups in high codimension

Let $X$ be a smooth affine variety of dimension $n>2$ over $\mathbb{C}$. From the examples I have seen (admittedly very little) it seems to me that these varieties don't have torsion classes a ...

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### Chow group of a pair

In a paper by S. Landsburg the (higher) Chow groups of a pair $(X,Y)$ are defined when $Y$ is a smooth closed subvariety of a smooth variety $X$ as follows.
We consider the sub-complex $z^{*}(X;.)_{Y}...

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### Calculations of residue homomorphisms in cycle modules

In the proof of Proposition 2.2 and Theorem 2.3 in Chow groups with coefficients https://eudml.org/doc/233731 written by M. Rost, he wrote
$\mathbb{A}^{1}={\rm Spec}F[u], \mathbb{A}^{2}={\rm Spec}F[...

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212
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### Proper locally trivial bundle is injective on Chow groups

If $X\to Y$ is a map of varieties that is Zariski-locally isomorphic to a projection $U\times P\to U$ with $P$ (smooth) proper, I think the pullback $A_{\bullet}(Y)\to A_{\bullet}(X)$ is supposed to ...