17
votes
Accepted
What exactly do the standard conjectures in characteristic zero refer to?
To prove that the standard conjectures are true for any Weil cohomology over a given field $k$, it suffices to prove them for an arbitrary chosen Weil cohomology over each subfield of finite type of $...
- 13.6k
15
votes
Accepted
Why can Euler systems constructed from algebraic cycles only be anticyclotomic?
Let me explain a bit more what that footnote was supposed to mean.
As I'm sure you know, an Euler system for a Galois representation $V$ over a number field $K$ consists of a bunch of classes in $H^1(...
- 37k
14
votes
Accepted
How to think about infinite generatedness of motivic cohomology
While waiting for someone more competent than me to answer, let me turn the question right back to you. Why should motivic cohomology be finitely generated?
The answer is, of course, that there's no ...
- 16.5k
12
votes
Chow Groups of varieties over number fields
The statement you want follows fairly straightforwardly from Bass' conjecture -- sufficiently straightforwardly that it may well not have a separate name of its own.
If $\Sigma$ is a sufficiently ...
- 37k
11
votes
Accepted
Algebraic cycles, Chow spaces and Hilbert-Chow morphisms
Mathoverflow answer
In my thesis [R4], I gave an ad hoc definition of a Chow functor ($\mathrm{Chow}_r$ above) that was meaningful also in characteristic p and close to Barlet's and Angéniol's ...
- 5,039
11
votes
Accepted
How to think about $\mathbf{Z}(n)_{\mathcal{M}}$
[All cohomology will be reduced cohomology for ease of notation].
There is no analog for classical homotopy theory. This is related to the fact that the Picard group of the category of spectra is $\...
- 16.5k
11
votes
Accepted
Does Poincaré duality preserve algebraic cycles?
A positive answer to your question* would imply one of Grothendieck's standard conjectures (conjecture A) for complex varieties. See his note: Standard conjectures on algebraic cycles. It's safe to ...
- 35.2k
10
votes
Accepted
Subrings of Chow rings
Plenty!
$R$ is generated, as a ring, by $F$. So its structure as a ring is going to be $\mathbb Q(\alpha)/f(\alpha)$, where $f$ is the minimal polynomial of $F$. Because you are using homological ...
- 148k
9
votes
Accepted
Does a conservativity conjecture imply the standard conjectures?
Here is a partial answer: In characteristic 0 it is known that conservativity + algebraicity (edit: modulo rational equivalence) of the Künneth projectors implies the remaining standard conjectures.
...
- 5,484
9
votes
What exactly do the standard conjectures in characteristic zero refer to?
Any characteristic zero field is an inductive limit of fields that can be embedded into complex numbers (i.e., those of characteristic 0 and of cardinality at most continuum). Hence the assumption ...
- 16.9k
8
votes
Hodge standard conjecture for étale cohomology
Another way to phrase the main obstacle in positive characteristic is the following:
Although we can formulate the Hodge standard conjecture purely cycle-theoretically, in practice the only way we ...
- 28.7k
8
votes
Accepted
Matrix obtained by recursive multiplication and a cyclic permutation
Denote the matrix $A$, and index all $a_i$, and all rows and columns starting from $0$ for convenience. If, say, $a_0 = 0$, then $\det A = (-1)^{\lfloor (n - 1) / 2 \rfloor} \prod_{i = 1}^{n - 1} a_i^...
- 5,590
7
votes
A question on heights and Northcott's Theorem
If you don't require any further properties, then I don't think you get a particularly interesting class of functions. For example, $X(K)$ is countable, list them as $P_1,P_2,\ldots$. Define $h(P_n)=n$...
- 47.4k
7
votes
Chow Groups of varieties over number fields
See Conjecture 5.0 (attributed to Swnnerton-Dyer) in the paper "Height pairing between algebraic cycles" by Beilinson. The paper by Swinnerton-Dyer that Beilinson refers to is "The conjectures of ...
- 71
7
votes
Does Poincaré duality preserve algebraic cycles?
I don't think the question makes sense as stated. What is the definition of the integration of $\int_{W_i} \colon H^{2n-2k}(X,\mathbb C) \to \mathbb C$ for $W_i$ a variety of codimension $n-k$? This ...
- 148k
6
votes
Accepted
Analytic cycles on complex-analytic spaces
Unless I misunderstand your definition, wouldn't $CH^p(X)$ coincide with the usual Chow group when $X$ is smooth projective, by GAGA? And of course Deligne cohomology would be the same. So the answers ...
- 35.2k
5
votes
Accepted
Effective cycles of codimension 1 and field extensions
The answer to the both questions is no. Consider $X=\mathbb{A}^1$, $k=\mathbb{Q}$, and $K=\mathbb{C}$ (any transcendental extension will do here). Let $\eta=\{\pi\}$. The cycles on $X_K$ of the form $\...
- 9,061
5
votes
Around algebraic equivalence of cycles
A summary of some of these results is given in §19.3 of Fulton's Intersection theory [Ful98]. To address for example the questions you ask:
If $A \stackrel f\twoheadrightarrow B \stackrel g \to C$ ...
- 28.7k
5
votes
Accepted
Absolute Hodge cycles
A couple of comments. First of all, the question of absoluteness of Hodge cycles is only interesting if there is more than one embedding if your field of definition into $\mathbb{C}$. So you really ...
- 35.2k
4
votes
Accepted
Properties of codimension under pull back
This is false as stated: taking for $f$ the embedding of a closed subscheme $X\subset Y$, it would mean that $\operatorname{codim}(X\cap Z,X)\leq \operatorname{codim}(Z,Y) $. There are well-known ...
- 38k
4
votes
Chow Groups of varieties over number fields
Just adding to Lucifer's answer: This is Conjecture 5 in Beilinson's paper
https://mathscinet.ams.org/mathscinet-getitem?mr=902590
For a smooth projective variety over a number field and a fixed ...
- 528
4
votes
Accepted
Higher Chow cycles
There is a rather concrete construction in Landsburg's 1991 paper "Relative Chow groups" which gives an explicit isomorphism from $CH^k(X, 1)$ to the degree 1 homology of the Gersten complex,...
- 37k
4
votes
Accepted
Finite flat pullback of the diagonal
If $f$ is finite flat of degree $d$, then $f \times f \colon X \times X \to Y \times Y$ has degree $d^2$, but $\Delta_f \colon \Delta_X \to \Delta_Y$ has degree $d$. So equality cannot hold scheme-...
- 28.7k
3
votes
Blow-ups, pullbacks and proper transforms
I guess you are right... :)
It seems to me that this indeed fails quite often.
Let $d=\mathrm{codim}_XZ$ and $P=f^{-1}Z\subseteq \widetilde X$ the exceptional divisor. By the assumptions $P\to Z$ is ...
- 42.9k
3
votes
Accepted
Functoriality of crystalline cohomology
Let's first figure out why the definition given in Berthelot-Ogus coincides with the one from the Stacks project.
Unraveling the definition 5.8.3 we see that for a sheaf $G$ on $(Y/W)_{cris}$ the ...
- 7,367
3
votes
The Ogus conjecture for crystalline cohomology
Look at Yves André's book "Une introduction aux motifs", subchapter 7.4.
- 532
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