# Tagged Questions

**4**

votes

**0**answers

78 views

### The 'most general' papers on rational Borel-Moore motivic homology and K'-theory?

There are two ways to define Borel-Moore motivic homology (of schemes) with rational coefficients: one should either consider certain complexes of algebraic cycles, or the $\gamma$-filtrations of ...

**3**

votes

**1**answer

171 views

### the graded pieces of the gamma-filtration of Quillen K-theory and Chow groups of a regular scheme

Let $X$ be a regular scheme and consider Grothendieck's $\gamma$-filtration $F^nK(X)$ on $K(X)$. For the graded pieces, one has $Gr^0K(X) = CH^0(X)$ and $Gr^1K(X) = \mathrm{Pic}(X) = CH^1(X)$. Does ...

**8**

votes

**1**answer

471 views

### Motivic cohomology and cohomology of Milnor K-theory sheaf

Let $X$ be a smooth variety over a field $k$. (Assume $k$ has characteristic 0 if it helps; in fact I'd be happy to assume that $k$ is a finite extension of either $\mathbf{Q}$ or $\mathbf{Q}_p$).
...

**5**

votes

**2**answers

388 views

### When do the $\gamma$-filtration and codimension filtration of K-theory agree?

Let $X$ be a smooth quasiprojective algebraic variety over a field $k$. Then the $K$-groups $K_m(X)$ are defined, and there are two standard filtrations on them: the "codimension filtration" given by
...

**5**

votes

**2**answers

782 views

### Relation between motivic cohomology and Quillen K-theory

What's the relation between Voevodsky's motivic cohomology and Quillen K-theory of a scheme?

**10**

votes

**0**answers

865 views

### Are there analogues of Beilinson's conjectures for motives with coefficients?

There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My ...

**33**

votes

**6**answers

3k views

### Why is Milnor K-theory not ad hoc?

When Milnor introduced in "Algebraic K-Theory and Quadratic Forms" the Milnor K-groups he said that his definition is motivated by Matsumoto's presentation of algebraic $K_2(k)$ for a field $k$ but is ...