All Questions
21 questions
6
votes
0
answers
162
views
$K_0$ of arithmetic surfaces
In his paper "Algebraic K-Theory and classfield theory of arithmetic surfaces", Annals of Mathematics 114 (1981), Spencer Bloch proved the following result: if $A$ is a finitely generated ...
1
vote
0
answers
206
views
Motivic cohomology commutes with field extension
$\DeclareMathOperator\Cor{Cor}$Let $X$ be a smooth scheme over $k$ and $k \subset F$ a field extension. Let $X_F$ be the field extension of $X$. Then there is a map
$$\varinjlim_{k\subset E \subset F} ...
- 12.9k
5
votes
1
answer
222
views
Computation of the torsion of K-groups related to elliptic curves
Let $E$ be an elliptic curve over $\mathbb Q$. Let $F$ be the rational function field of $E$.
The $K_2$ group of $F$ may be described by elements in $F^\times ⊗_\mathbb{Z} F^\times$ quotiented by the ...
- 20.8k
37
votes
1
answer
3k
views
Morava on Shafarevich conjecture
$\DeclareMathOperator\Q{\mathbf{Q}}$Jack Morava has some interesting ideas stemming from stable homotopy theory and geometric topology on the Shafarevich Conjecture.
The Shafarevich Conjecture states: ...
- 63.9k
11
votes
2
answers
1k
views
Good reference for topological Hochschild homology
I want to start reading topological Hochschild homology(THH) as well as topological cyclic homology (TC).
I have read the Hochschild homology and cyclic homology from the book Cyclic homology by J. ...
- 63.9k
5
votes
0
answers
415
views
Modern context for hypercohomology spectra
In Thomason's paper Algebraic K-theory and étale cohomology, (Ann. ENS 1985, Numdam link) Thomason develops an elaborate theory of hypercohomology spectra, $\mathbb{H}(X,\mathcal{F})$ for presheafs of ...
- 254
12
votes
1
answer
2k
views
Why does K-theory need schemes to be Noetherian?
The definition of K-theory of a scheme $X$ is defined as
$G_i(X):=K_i(\mathrm{Coh}(X))$ or $K_i(X):=K_i(\mathrm{Vec}(X))$.
But usually the schemes are required to be (at least locally) Noetherian, and ...
- 16.5k
4
votes
0
answers
226
views
K-theoretic derivation of Bézout theorem
In the paper "$K$-Theory and Intersection Theory" by Henri Gillet in Handbook of K-theory, 2005 (link behind paywall at Springerlink), the author says:
"When the ground field $k = \mathbb C$, Bézout’...
- 63.9k
7
votes
1
answer
476
views
Equivalence between categories of coherent sheaf of codimension p
Let $X$ be a noetherian and separated scheme and $M(X)$ denote the abelian category of coherent sheaves on $X$. Let $M^{P}(X) = \lbrace \mathcal{F} \in M(X) \hspace{2mm} : Codim(sup(\mathcal{F}), X) \...
- 63.9k
13
votes
1
answer
700
views
Reference for the algebro-geometric proof of Matsumoto theorem
Matsumoto proved in his PhD thesis that if $F$ is a field then $$K_2(F)=(F^*\otimes F^*)/(x\otimes (1-x)).$$
The original Matsumoto proof as it is written in Milnor's book on algebraic K-theory looks ...
- 103
9
votes
1
answer
336
views
Positive cones in K-groups
Let $X$ be a topological space or a scheme, and let $K^0(X)$ be $K$-group of vector bundles of $X$. One may ask when an element $x$ of $K^0(X)$ is represented by an actual vector bundle, and not just ...
- 4,679
9
votes
2
answers
971
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 ...
- 342
11
votes
1
answer
1k
views
K-theory of an elliptic curve
Given an elliptic curve $E$ over $\mathbb{Q}$, I have read somewhere (But I can't remember exactly where) that the Beilinson conjecture asserts that: The rank of the albelian group $K_{2}(E)$ (the ...
- 17.4k
6
votes
2
answers
1k
views
K theory long exact sequence
(1) Suppose that $Z\subset X$ is a closed embedding, $U = X\setminus Z$ is the complement. If relevant, suppose that both $X, Z$ are smooth and even (if relevant) that the normal bundle of $Z\subset X$...
- 5,062
6
votes
0
answers
144
views
$K_0$ an $KH_0$ of a normal crossing variety
Let $k$ be a field (say, algebraically closed to fix the ideas) and let $X$ be a strict (aka simple) normal crossing variety over $k$, so that $X$ is union of regular varieties with intersection that ...
- 61
2
votes
0
answers
416
views
Do we have the following "devissage commutative diagram" in K-theory?
Let $X$ be a non-reduced Noetherian scheme. We define $K^0(X)$ to be the Grothendieck group of the derived category $Perf(X)$ and $K_0(X)$ to be the Grothendieck group of the derived category $D^b_{...
- 9,019
5
votes
1
answer
417
views
Could we extend the exact sequence $K^0(X)\to K_0(X)\to K_0(D_{sg}(X))\to 0$ to the left?
Let $X$ be a variety over a field $k$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category of perfect complex $Perf(X)$. It is clear that $Perf(X)$ is a ...
- 5,901
8
votes
1
answer
756
views
Equivariant algebraic K-theory of affine space
Unlike algebraic K-theory, equivariant K-theory of affine space (over a field $k$) can be quite nontrivial, depending on the action of the group in question. For example, if one takes the standard ...
- 805
1
vote
0
answers
94
views
Relative Gersten resolution for a flat projective morphism
I am reading two papers by Daniel Grayson: "Localization for flat modules in algebraic K-theory" and "Algebraic cycles and algebraic K-theory" and I am wondering if any recent advances in K-theory ...
2
votes
1
answer
237
views
For which local $R$ its K-theory mod l is isomorphic to the one of its residue field?
It is well-known (and was proved by Gabber?): if $R$ is a regular henselian local ring containing a field of characteristic prime to $l$, $k$ is its residue field, then $K_\ast(R,\mathbb{Z}/l)\cong ...
- 8,717
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 ...
- 4,909