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3 votes
2 answers
342 views

Reference Request: Beilinson-Bloch conjecture in terms of Beilinson regulator isomorphism

I'm looking for a reference that provides a concise statement of the Beilinson-Bloch conjecture, specifically formulated in terms of an isomorphism under the Beilinson regulator map. More precisely, I'...
kindasorta's user avatar
  • 2,907
13 votes
1 answer
2k views

Roadmap for Algebraic Geometry/Homotopy Theory/Algebraic $K$-Theory intersection

I’m afraid that this is quite a general question but I am hoping some experts can weigh in. I am a student generally interested in learning more about the intersection of algebraic geometry, algebraic ...
Song Ye's user avatar
  • 155
2 votes
0 answers
129 views

Flag variety type Beilinson resolution

The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of ...
fool rabbit's user avatar
9 votes
0 answers
1k views

Some questions about Clausen's third IHES lecture on Efimov K-theory

I have some questions about the last theorem stated by Clausen at https://youtu.be/2xNG4rHUC6U?si=yw9eYiygLegH0nQK&t=4319. I'm not very familar with the definitions, so please correct me about any ...
Andy Jiang's user avatar
  • 2,356
55 votes
8 answers
7k views

Applications of Grothendieck-Riemann-Roch?

I am currently trying to learn a bit about Grothendieck-Riemann-Roch... To try to get a better feeling for it, I am looking for examples of nice applications of GRR applied to a proper morphism $X \...
Kevin H. Lin's user avatar
80 votes
2 answers
7k views

Vladimir Voevodsky's works

Vladimir Voevodsky has made several contributions in abstract algebraic geometry, focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and ...
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
2 votes
1 answer
267 views

Semi-orthogonal decompositions over singular schemes

Where can I find any more or less explicit semi-orthogonal decompositions of derived categories of perfect complexes or of bounded derived categories for singular schemes that are proper over a ring R?...
Mikhail Bondarko's user avatar
1 vote
0 answers
68 views

Metric and connection on virtual bundles

Let $E=[E^+]-[E^-]$ be an element of the Grothendieck group $K(X)$ of a compact Kahler manifold $X$. Does it exist a way to define more "geometric" structures on $E\in K(X)$ such that a ...
BinAcker's user avatar
  • 789
2 votes
0 answers
181 views

Is there a degeneration formula for Gromov-Witten K-theoretic invariants?

By Gromov-Witten K-theoretic invariants (call them KGW) I mean the invariants defined by Givental and Lee. I expect the formula expresses the KGW of the generic fiber of a given degeneration in terms ...
jimmy's user avatar
  • 21
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 ...
cll's user avatar
  • 2,305
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
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
4 votes
1 answer
564 views

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 ...
user avatar
3 votes
1 answer
203 views

"Direct" calculation of $K_0$ for surfaces, 3-folds

I apologize in advance for the vagueness of my question, but I am looking for sources (if they exist) where $K_0$ (the Grothendieck group of coherent sheaves) is computed "by hand" for some low-...
A. S.'s user avatar
  • 528
10 votes
2 answers
1k views

Grothendieck group for projective space over the dual numbers

Fix a field $k$. For a singular variety $X$, I understand that the Grothendieck group $K^0(X)$ of vector bundles on $X$ is not necessarily isomorphic to the Grothendieck group $K_0(X)$ of coherent ...
Daniel Erman's user avatar
  • 2,955
8 votes
1 answer
569 views

Continuity of l-adic cohomology: is the cohomology of the generic point isomorphic to the completion of the limit of cohomology of open subvarieties?

Let $X$ be a variety over an algebraically closed field $k$. Denote by $\eta$ its generic point; it is the inverse limit of the open subvarieties $X_i$ of $X$. It is well known that the etale ...
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
9 votes
1 answer
932 views

The vanishing of non-connective K-theory in negative degrees

In the works of Cisinski, Tabuada, and Schlichting certain non-connective K-theory groups for a differential graded category $C$ are defined. As far as I understand, $K_i(C)$ is not necessarily zero ...
Mikhail Bondarko's user avatar
5 votes
2 answers
1k views

Survey of Algebraic K-Theory Since 1980?

I just came across Charles Weibel's Development of Algebraic K-Theory until 1980, and found it really helpful. Is there been anything analogous which surveys the developments in the last 30 years? I'...
Jesse Wolfson's user avatar
1 vote
0 answers
203 views

Is there any computation of $K^0(X)$ and $K_0(X)$ for a singular curve $X$?

Let $X$ be a projective curve over an algebraic closed field $k$ which characteristic zero. Define $K^0(X)$ as the Grothendieck group of the derived category $Perf(X)$ and $K_0(X)$ as the Grothendieck ...
Zhaoting Wei's user avatar
  • 9,019
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
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
2 votes
1 answer
387 views

Zero-cycles on an arithmetic surface

Could anyone give a reference for the following statement, which I believe is true. "Let X be a regular scheme, flat over $Spec( \mathbb{Z}) $, with fiber dimension $1$. Then the Chow group $CH^2(X)$ ...
Andreas Holmstrom's user avatar
4 votes
0 answers
324 views

The restriction of the Gersten resolution (for etale cohomology) onto a closed subvariety.

There is a very important result of Bloch and Ogus: for any smooth variety $X$ and fixed $r\in \mathbb{Z}$, $r\ge 0$, $l$ is prime to the residue field characteristic, the Zariski sheafification of ...
Mikhail Bondarko's user avatar