The motives tag has no wiki summary.

**4**

votes

**1**answer

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### Is the scalar extension functor for Chow motives conservative?

Denote $CHM(F)$ to be the category of Chow motives over a field $F$.
Let's consider an algebraic exension $E/F$, then
there is a natural extension of scalars functor $CHM(F) \to CHM(E)$.
I was ...

**8**

votes

**6**answers

1k views

### Kunneth formula for motivic cohomology

I was wondering when the Kunneth formula holds for motivic cohomology:
$$
H^p(X,A(\alpha)) = \bigoplus_{i+j=p;\beta+\gamma = \alpha} H^j(X,A(\beta)) \otimes H^i(X,A(\gamma))
$$
where ...

**6**

votes

**2**answers

545 views

### Néron theory for motives of arbitrary weight

SGA 7, tome 1, exp. IX, contains in its introduction and in section 13.4 remarks about ideas and conjectures of Deligne on a “théorie de Néron pour motifs de poids quelconque”. Would someone please ...

**57**

votes

**7**answers

5k views

### What is the field with one element?

I've heard of this many times, but I don't know anything about it.
What I do know is that it is supposed to solve the problem of the fact that the final object in the category of schemes is ...

**8**

votes

**3**answers

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### ubiquitous quantum cohomology

Manin stressed that every projective scheme should have a quantum-cohomology structure. I'd like to know more about that. And since the varieties considered in texts about monodromy resp. vanishing ...

**44**

votes

**3**answers

6k views

### What's the “Yoga of Motives”?

There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new ...

**5**

votes

**3**answers

512 views

### Solving “a, b, a+b have given divisors” problem

I've read an interesting article, math.NT/0409456 where you're just trying to solve a simple problem:
For a given (finite) set of primes S find all solutions to an equation ...

**8**

votes

**0**answers

598 views

### Kato's log motives

What are they and what are their intended uses? Does anyone have notes/slides of this talk?
I am curious about "log motives" because there seems to exist a "log motivic yoga" among experts in ...

**18**

votes

**4**answers

1k views

### Spectrum of the Grothendieck ring of varieties

Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my ...