In algebraic geometry, a motive (or sometimes motif, following French usage) denotes 'some essential part of an algebraic variety'. One has a satisfactory description of pure motives, whereas the mixed case is much more difficult. Pure motives are triples (X, p, m), where X is a smooth projective ...

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### Motivic cohomology is universal with respect to what (co)homology theories?

I have been told several times, at least implicitly, that motivic cohomology should be universal with respect to Bloch-Ogus cohomology theories. Is it proved somewhere or is it just some folk theorem?
...

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### holomorphic continuation of motivic $L$-functions

The question is rather easy to formulate: when is the $L$-function of a pure motive over $\mathbb{Q}$ expected to have a holomorphic (as opposed to simply meromorphic) continuation to the complex ...

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### Motive of a conic without points

Let $C\subset \mathbb{P}^2$ be a smooth conic without $k$-points.
Call the Chow $k$-motive in zero-dimensional if it is a sum of $M\mathbb{L}^n$ where $M$ is an Artin motive, i.e. a part of a motive ...

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### A question about the vanishing of motivic cohomology in negative Tate twist

Let $DM_{\text{gm}}$ be the category of Voevodsky´s geometric motives. Let $p,q\in \mathbb{Z}$ be integers with $p<0$.
Is it true that
$$\text{Hom}_{DM_{\text{gm}}}(M_{\text{gm}}(X),\mathbb{Z}(p)[...

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### A question on the slice filtration and the slice of the motive of the projective space

In the following $k$ is an algebraically closed field of characteristic $0$.
Consider the category $SH(k)$ (the Morel-Voevodsky stable motivic homotopy category).
By the work of Voevodsky (see for ...

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### Artin-Tate chow motives and graded Galois representations

Consider the category $\mathsf{Chow}_{\mathbb{Q}}$ of rational pure effective chow $k$-motives. The full subcategory of Artin motives (generated by (X,p,0), X smooth projective zero-dimensional, let $...

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### What is the best reference for motives?

I want to learn about homotopy theory on number fields, and I heard that the theory of motives made it possible, so I want to know what is a good textbook for motive theory.
To be honest, I don’t ...

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### Reference request: Tate's conjecture for L functions of motives

What's a good reference for the most general form of Tate's conjecture for the order of poles of the L function of a motive? Thanks!

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### Are exotic affine spaces motivic/whatever equivalent to affine space?

This question is inspired by this MO question; in turn by this MO; in turn by these MO, MO.
An exotic affine space is an affine variety $V$ whose $\mathbb{C}$-points are diffeomorphic to $\mathbb{R}^{...

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### Inequality in the Grothendieck ring of stacks

In the Grothendieck ring of varieties, there are ways of distinguishing classes of varieties, for example $\ell$-adic cohomology. The Grothendieck ring of stacks is a localization of the Grothendieck ...

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### Quotient of a motive by a finite group

Given a smooth scheme $X$ over a field $k$, we can consider its motive $M(X)$. It is an object in Voevodsky's triangulated category of motives $DM(k,\Lambda)$ where $\Lambda$ is the ring of ...

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### Reference request: Motivic Cohomology and Cycle class maps

For a smooth projective variety $X$ over any field $K$, Voevodsky showed in his paper ``Motivic Cohomology Groups Are Isomorphic to Higher Chow Groups in Any Characteristic" that the motivic ...

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### Poincare duality for mixed motives

Suppose $k$ is a field of characteristic zero (and we assume it is a number field if necessary). If $U$ is a smooth quasi-projective variety over $k$, then there is Poincare duality,
\begin{equation}
...

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### A question on Nekovar's paper Belinson's Conjectures

In Section 2 of Nevovar's paper "Beilinson's Conjectures", for a pure motive $M$ of the form $h^i(X)(n)$ where $X$ is a projective smooth variety over $\mathbb{Q}$ and $n$ is an integer such that the ...

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### Mixed motives and motivic cohomology

In Scholl's paper "Remarks on special values of $L$-functions", he defines that an object $M$ of $\textbf{MM}_{\mathbb{Q}}$ (the conjectured abelian category of mixed motives with coefficients $\...

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### Coniveau in étale motivic cohomology

Let $X$ be a smooth variety over a field.
Is there a spectral sequence:
$$E_1^{p,q} := \bigoplus_{x\in X^{(p)}}H^{q-p}(\kappa(x)_{\rm ét},\mathbf{Z}(n))\Rightarrow H^{q-p}(X_{\rm ét},\mathbf{Z}(n))$$...

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### Multiplicative structure on Deligne cohomology

Let $X$ be a smooth projective variety over the complex numbers, and $\mathbf{Z}(p)_{\mathcal{D}}$ the Deligne complex on $X$:
$$\mathbf{Z}(p)_{\mathcal{D}} : \ \ \mathbf{Z}(p)\to\mathcal{O}_X\to\...

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### Spectral sequence in Betti cohomology

Let $X$ be a smooth projective algebraic variety over the complex numbers, and let us name
$$f : X_{\rm an}\to X_{\rm Zar}$$
the morphism of sites induced by sending a Zariski open $U\subset X$ to $...

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### Torsion homologically trivial cycles

Is there an example of a smooth projective variety $X$ over the complex numbers, such that
$$\ker(\text{CH}^2(X)\to H^4(X,\mathbf{Z}(2))$$
is not torsion?

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### Filtrations and the Betti cycle map

Let $X$ be a smooth projective complex variety.
Calling $f : X_{\rm an}\to X_{\rm Zar}$ the "change-of-topology" morphism of sites induced by sending a Zariski open $U$ of $X$ to its analytification, ...

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### Integral coniveau spectral sequences in Hodge Theory

Let $X$ be a smooth projective complex analytic space.
We name $\mathcal{H}^*(\mathbf{Z}(n))$ the Zariski sheafification of Betti cohomology with $\mathbf{Z}(n)$ coefficients.
We have a "coniveau" ...

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### Cycles modulo homological equivalence

Let $\text{CH}^p(X)_{\rm hom}$ be the abelian group of codimension $p$ algebraic cycles on a smooth projective variety over a field $k$, modulo homological equivalence. Is $\text{CH}^p(X)_{\rm hom}$ ...

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### Absolute Hodge Cycles over $\mathbf{Q}$

In the 1986 notes by Milne "Hodge cycles on abelian varieties", Deligne defines the notion of absolute Hodge cycles.
For a smooth projective variety defined over $k\subset\mathbf{C}$ non ...

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### Blowup formula for motivic cohomology

If $X$ is a smooth projective variety over a field, $Z\subset X$ a smooth closed subvariety of codimension $d$, $X'\to X$ the blowup of $X$ along $Z$, there's the blowup formula
$$H^j(X'_{et},\mathbf{...

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### Two motivic complexes, compared

Bloch defines the motivic complexes $\mathbf{Z}(n)$ in his paper "Algebraic Cycles and Higher K-Theory" (1986).
Some references (that I currently am unable to track down) use $$\check{\mathbf{Z}}(n) :...

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### Is this Mayer-Vietoris sequence motivic?

Suppose $Y$ is a variety defined over $\mathbb{Q}$ and $pt$ is a rational point of $Y$. Let $\pi:X \rightarrow Y$ be the blow up of $Y$ at $pt$ and $D$ be the exceptional divisor. For simplicity let's ...

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### Tate Conjecture birational invariant?

Is the Tate Conjecture stable under birational equivalence?
In particular, is the Tate Conjecture for rational varieties known?

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### Locus of Hodge classes

Let $\pi: X\to S$ be a proper smooth morphism of complex analytic spaces, with connected smooth $X$ and $S$ over $\mathbf{C}$, projective fibers, and $$\mathscr{H}_{X/S}^p := R^p\pi_*\Omega^{\bullet}_{...

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### Absolute Hodge cycles

Let $X$ is a smooth projective variety defined over a finite extension $K/\mathbf{Q}$, $\sigma : K\to\mathbf{C}$ any of the finitely many field embeddings of $K$ into the complex numbers, and call $X^{...

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### Homotopy equivalence between two basepoints of the etale homotopy type of the one-torus

Let $T = \mathbb{G}_m$ be the torus, and let $\tilde{T}$ be its étale universal cover (a pro-object in schemes of finite type). Then both $T$ and $\tilde{T}$ have a well-defined étale homotopy type. ...

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### $\mathbf{A}^1$- contractibility

Suppose $U$ is an $\mathbf{A}^1$-contractible smooth scheme over a field $k$, that is, it is isomorphic to a point in the $\mathbf{A}^1$-homotopy category of smooth schemes over $k$.
Does motivic ...

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### How to think about infinite generatedness of motivic cohomology

In this question I previously asked how to think about the motivic complex $\mathbf{Z}(1)_{\mathcal{M}}$, whose Zariski hypercohomology should morally be the "singular cohomology" $H^*((-)\wedge S^{2n}...

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### How to think about $\mathbf{Z}(n)_{\mathcal{M}}$

One definition of motivic cohomology for smooth schemes $X$ over a field, is via Friedlander-Suslin complexes.
A refresher (you may skip to the question at the bottom)
One defines
(1) $z_n(X,d) :=$...

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### Structure of Deligne cohomology

It is a classical fact that for a smooth and proper complex-analytic space $X$, the Deligne cohomology $H^p_{\mathcal{D}}(X,\mathbf{Z}(q))$, defined as the hypercohomology of the complex
$$\mathbf{Z}(...

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### Precise formulation of conjectures on orders of vanishing?

Let $X$ be a smooth and proper scheme over $\text{Spec}(\mathbf{Z})$.
C. Soulé has conjectures about special values of the completed zeta function of $X$, $\zeta(X,s)$, which were first reformulated ...

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### Group completion of Chow varieties

Let $X$ be a quasi-projective variety over a perfect field $k$.
Given a projective embedding $j : X\to \mathbf{P}(\mathscr{E})$, the Chow variety $\text{Chow}_r(X, j)$ is a quasi-projective variety ...

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### Torsion in Deligne cohomology

Let $X$ be a smooth projective complex analytic space, $i,p\ge 0$ integers, $\mathbf{Z}(p)_{\mathcal{D}}$ the Deligne complex of $X$, $H^i_{\mathcal{D}}(X,\mathbf{Z}(p))$ its hypercohomology.
What ...

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### Semisimplicity conjecture

In this short note Ben Moonen proves that over fields of characteristic zero that are of finite type over their prime field, the Tate conjecture about surjectivity of cycle maps implies the semi-...

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### Sha finiteness vs $\ell$-primary torsion

Where do I find a proof of the fact that over global function fields of characteristic $p>0$, finiteness of the Tate-Shafarevich group of an abelian variety is equivalent to finiteness of its $\ell$...

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### Pull-back of algebraic cycles

Since today is the Chow-variety day, I'm going to ask my question here.
Suppose I have a smooth projective variety $X$ over a field of characteristic zero, and a smooth hyperplane $p: H\...

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### Codimension restrictions on intersections

This is a question I stumbled across earlier this week. I see a similar one has been asked here.
Suppose I have a smooth quasi projective variety $X$ over a field $K$, and I call $\text{Chow}^r(X\...

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### Effective cycles of codimension 1 and field extensions

Let $X$ be a smooth quasi-projective variety over a field $k$, with pure dimension $d$, $K/k$ an arbitrary field extension.
For any algebraic cycle $\eta$ of codimension $1$ on $X_K$ ($\eta\in Z^1(...

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### Around algebraic equivalence of cycles

Let $k$ be a finitely generated field, $X$ a smooth projective $k$-variety, $\ell$ a prime number, $\ell\in k^{\times}$, $r\ge 0$ an integer.
The Tate conjecture asserts surjectivity of the cycle ...

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### Vector bundles vs algebraic cycles

For integral schemes, the Picard group is isomorphic to the group of Cartier divisors modulo linear equivalence.
What is the correct analog of this isomorphism, for higher codimension Chow groups vs ...

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### Borel regulator and Bloch-Beilinson regulators

Is there any relation, either conjectural or known, between the Borel regulator for the Quillen $K$-theory of algebraic number rings, and the Bloch-Beilinson regulator from motivic cohomology to real ...

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### Beilinson regulators and Bloch's mythological algebraic intermediate Jacobians

In the paper introducing his motivic cycle complexes, Bloch outlines a project he says he was going to return to in the future:
Towards the end of page 270, he says, given a smooth projective variety ...

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### Is Deligne cohomology the motivic cohomology of analytic spaces?

Let $X$ be a smooth projective complex analytic space.
We can cook up a complex analytic version of Bloch's cycle complex by declaring
$z^n(X^{\rm an}, m)$
is the free abelian group on all ...

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### Motivic vs Deligne cohomology

Where can I find the construction of the cycle class map from motivic cohomology to Deligne cohomology of smooth projective varieties over the complex numbers?
It should be a construction by Bloch ...

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### Difference of Beilinson conjecture and equivariant Tamagawa number conjecture

As stated in the title, I am wondering the main difference between Beilinson conjecture and eTNC. If I read correctly, I can see that there are many literature treating both conjectures in the same ...

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### Number field analog of Artin-Tate $\Rightarrow$ BSD?

What is the difference between the alternating product of the Hasse-Weil $L$-functions of the generic fiber of an arithmetic scheme $X\to\text{Spec}(\mathbf{Z})$ and the zeta function of $X$? (each ...