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73 votes
1 answer
8k views

Derived Functors Versus Spectral Sequences

Let $A{\buildrel F\over\rightarrow}B{\buildrel G\over\rightarrow}C$ be additive functors between abelian categories. Hartshorne, in Proposition 5.4 of Residues and Duality, constructs the obvious ...
Steven Landsburg's user avatar
26 votes
1 answer
4k views

Voevodsky's counterexample to the existence of a motivic t-structure

I have been trying to unravel some of the known relationships between various ideas on mixed motives. I find the literature quite hard to follow -"from experts, for experts". Voevodsky in "...
plm's user avatar
  • 982
72 votes
3 answers
8k views

Where do all these projection formulas come from?

I have been intrigued for a long time by the formal similarity of results from different areas of mathematics. Here are some examples. Set theory Given a map $f:X\to Y$ and subsets $X' \subset X, Y'\...
Georges Elencwajg's user avatar
18 votes
1 answer
1k views

Which motivic cohomology groups of complex numbers are non-torsion?

I would like to know which motivic cohomology groups of complex numbers are non-zero and ("better") non-torsion, i.e., for which $(i,j)$ the $i$th cohomology of the complex ${\mathbb{Q}}(j)$ over $\...
Mikhail Bondarko's user avatar
14 votes
3 answers
2k views

A question on K_1 of an elliptic curve

Consider an elliptic curve $E/ \mathbb{Q}$, with a regular model $\mathcal{E} / \mathbb{Z}$. We have (Beilinson regulator) maps $$ K_1(\mathcal{E})^{(2)} \to K_1(E)^{(2)} \to H_D^3(E_{/ \mathbb{R}} , \...
Andreas Holmstrom's user avatar
14 votes
3 answers
3k views

References for equivariant K-theory

I want a good introduction to localization in equivariant $K$-theory. This introduction can be simple in several ways: I only care about torus actions. I only care about $K^0$. I only care about very ...
David E Speyer's user avatar
12 votes
0 answers
340 views

Homology of Gersten complex for singular schemes

It is one of the important facts in K-theory/motivic cohomology that the Gersten-type complexes (for Quillen K-theory, Milnor K-theory or more generally Rost's cycle modules) are exact for smooth ...
Matthias Wendt's user avatar
12 votes
3 answers
2k views

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 ...
user avatar
11 votes
1 answer
846 views

$K_0$ of a non-separated scheme

This question is on "computing" the Grothendieck group of the projective $n$-space with $m$ origins ($m\geq 1$). For any (noetherian) scheme $X$, let $K_0(X)$ be the Grothendieck group of coherent ...
Ariyan Javanpeykar's user avatar
8 votes
0 answers
440 views

Poincaré duality for topological $K$-theory

Let $n$ be an even number. Let $X$ be a $n$-dimensional complex projective manifold with $H^{2m+1}(X,\mathbb{Z})=0$, for all $0\leq m\leq n-1$. $H^{2m}(X,\mathbb{Z})$ is a free $\mathbb{Z}$-module ...
user avatar
7 votes
3 answers
966 views

If a colimit of distinguished triangles exists, is it also a distinguished triangle?

Consider the following situation in some triangulated category: We are given a collection of distinguished triangles $A_n \to B_n \to C_n \to A_n[1]$ indexed by the natural numbers, together with maps ...
Andreas Holmstrom's user avatar
6 votes
0 answers
137 views

Comparison of K-groups of (affine) singular schemes with K'=G-groups

It is well known that Quillen K-theory coincides with $K'=G$-theory for regular schemes, and can be distinct from it for singular ones. Are there any methods for studying this distinctions? In ...
Mikhail Bondarko's user avatar
3 votes
1 answer
374 views

Is $K^0(X)\to K_0(X)$ monomorphic for a noetherian scheme $X$?

This question is related to the MO questions What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes? and Does a fully faithful functor between triangulated categories induce ...
Zhaoting Wei's user avatar
  • 9,019
2 votes
0 answers
833 views

Applications of Jordan-Holder theorem in an abelian category

The Jordan-Holder theorem says that any chain of subobjects of a finite length object can be refined to a composition series, and that any composition series has the same length. This theorem holds ...
less's user avatar
  • 129
2 votes
0 answers
194 views

Varieties with Chow groups supported in positive codimension: examples and properties?

In their 1983 paper "Remarks on correspondences and algebraic cycles" Bloch and Srinivas proved several interesting properties of smooth proper varieties (over universal domains) whose Chow groups of ...
Mikhail Bondarko's user avatar