All Questions
Tagged with ag.algebraic-geometry kt.k-theory-and-homology
149 questions
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 ...
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 ...
- 23k
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'\...
- 47.5k
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 \...
- 21k
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: ...
- 2,734
30
votes
4
answers
2k
views
When can we cancel vector bundles from tensor products?
Let $E,F,G$ be algebraic vector bundles over $\mathbb P_{\mathbb C}^n$. My general question is:
Assume $E\otimes G \cong F\otimes G$, under what conditions can one conclude that $E\cong F$?
Some ...
- 30.5k
28
votes
0
answers
2k
views
derived category of equivariant coherent sheaves and fixed points
The K-group $K^T(X)$ of $T$(torus)-equivariant coherent sheaves on a variety $X$ is isomorphic to $K^T(X^T)$, that of the fixed point locus via the inclusion homomorphism, when we tensor the quotient ...
- 3,051
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 "...
- 982
25
votes
1
answer
839
views
Vector bundles on $\mathbb{A}^n / G$
Let $G$ be a finite group acting linearly on $\mathbb{A}^n$. Do we expect algebraic vector bundles on $X := \mathbb{A}^n/G$ to be trivial? Here by the quotient I mean the singular scheme, not the ...
- 2,300
25
votes
1
answer
1k
views
Comparison map between de Rham cohomology of analytic and formal neighborhoods of singularities
Suppose that $X$ is a complex algebraic (or complex analytic) variety, and $x \in X$ is a singular point. I am interested in two types of local differential forms at $x$: analytic and formal.
First, ...
- 536
18
votes
4
answers
2k
views
Why does one invert $G_m$ in the construction of the motivic stable homotopy category?
Morel and Voevodsky construct the motivic stable homotopy category, a category through which all cohomology theories factor and where they are representable, by starting with a category of schemes, ...
- 12.3k
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 $\...
- 16.9k
18
votes
1
answer
1k
views
Application of higher categories in algebra
Higher categories and derived algebraic geometry are relatively new areas and probably fewer people are working on them compared to the majority of topologists or geometers. I believe higher ...
15
votes
3
answers
3k
views
State of the art for Gersten's conjecture for K-theory?
Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence
$$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to \...
- 1,347
15
votes
1
answer
1k
views
Attaching maps for Grassmann manifolds
The Grassmannian $G_n(\mathbb{R}^k)$ of n-planes in $\mathbb{R}^k$ has a CW-complex structure coming from the Schubert cell decomposition.
What is known about the attaching maps in this CW-complex ...
- 865
14
votes
3
answers
3k
views
References for equivariant K-theory
I want a good introduction to localization in equivariant $K$-theory. This introduction can be simple in several ways:
I only care about torus actions.
I only care about $K^0$.
I only care about very ...
- 156k
14
votes
3
answers
2k
views
A question on K_1 of an elliptic curve
Consider an elliptic curve $E/ \mathbb{Q}$, with a regular model $\mathcal{E} / \mathbb{Z}$. We have (Beilinson regulator) maps
$$ K_1(\mathcal{E})^{(2)} \to K_1(E)^{(2)} \to H_D^3(E_{/ \mathbb{R}} , \...
- 5,637
14
votes
2
answers
1k
views
Is Deligne cohomology the motivic cohomology of analytic spaces?
Let $X$ be a smooth projective complex analytic space.
We can cook up a complex analytic version of Bloch's cycle complex by declaring
$z^n(X^{\rm an}, m)$
is the free abelian group on all ...
user92332
14
votes
1
answer
1k
views
Dirichlet's regulator vs Beilinson's regulator
Consider a number field $F$ with ring of integers $O_F$. The Beilinson regulator can in this particular setting be viewed as a map from $K_n(O_F)$ to a suitable real vector space. Here $n$ is any ...
- 5,637
13
votes
1
answer
558
views
Where does the $\hat A$ class get its name?
In K-theory we have the Todd class and the $\hat A$ class.
The Todd class is named after the Cambridge geometer John Arthur Todd.
Where does the name $\hat A$ come from? Does the A stand for Atiyah?...
- 683
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 ...
- 2,305
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 ...
- 155
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 ...
user95222
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 ...
- 449
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 ...
- 17.4k
11
votes
1
answer
520
views
Problems concerning subspaces of $M_{n}(\mathbb{Q}) $
Let $M_{n}(\mathbb{Q}) $ denote the $n$ times $n$ matrices over the rational number field. $N$ be a subspace of $M_{n}(\mathbb{Q}) $.Then if all the non-zero matrices in $N$ are invertible, what is ...
- 923
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. ...
- 629
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 ...
- 9,497
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 ...
- 233
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 ...
- 2,955
10
votes
1
answer
983
views
Friedlander Conjecture
I just read on the ALGTOP discussion list that Morel has announced a proof of the Friedlander conjecture. Question: Are there other applications besides the Milnor conjecture $H_*(G,F_p)=H_*(BG,F_p)$ ...
- 101
10
votes
1
answer
347
views
Is it possible for the Witt group of a scheme to have non-trivial odd torsion?
Let $F$ be a field of characteristic not $2$. A well-known theorem of Pfister asserts that the torsion of $W(F)$, the Witt group of $F$, is $2$-primary.
Baeza [B, V.6.3] extended this result to Witt ...
- 2,928
10
votes
1
answer
1k
views
What does the moduli stack of G-torsors over the multiplicative group look like?
I am an algebraic topologist and am trying to understand some computations related to p-adic complex K-theory and equivariant K-theory. However this has led me into the world of algebraic geometry ...
- 27.5k
10
votes
0
answers
340
views
Geometric vs combinatorial motives over Spec Z
Consider the category of reduced schemes of finite type over $\mathbb{Z}$. Take the Grothendieck group of this category, i.e. the free abelian group on isomorphism classes, modulo the usual "syzygy" ...
- 5,637
9
votes
5
answers
1k
views
Calculations of Pic^0, Pic, NS of surfaces
I'm looking for an example in the literature where $\mbox{Pic}^0(X)$, $\mbox{Pic}(X)$, and $NS(X)$ of a projective surface $X$ over a field are calculated. I want them for an example I'm trying to ...
- 183
9
votes
1
answer
3k
views
Why is the first chern class of a line bundle $c_1(L) = 1-L$ in complex K-theory?
I'm trying to understand why on earth the first chern class of a line bundle in K-theory $c_1(L) = 1-L$.
I understand that the first Chern class of the trivial bundle is zero, and that $H-1$ ...
- 3,539
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 ...
user19475
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 ...
- 16.9k
9
votes
1
answer
756
views
Does there exist a GRR-like generalization of the AS Index Theorem?
The Hirzebruch Riemann-Roch Theorem (HRR) expresses an analytic/algebraic invariant, namely the Euler-Poincaré characteristic of a vector bundle $V$ over a compact complex/algebraic manifold $X$, as ...
- 1,020
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 ...
- 1,274
9
votes
1
answer
669
views
Topological version of K-theory of coherent sheaves
My question is this: what is the topological analog of the Grothendieck group of coherent sheaves $G(X)$?
Background:
In Algebra/Algebraic Geometry there are two versions of the Grothendieck group ...
- 2,300
9
votes
1
answer
599
views
Numerical evidence of Beilinson's conjecture in local fields and function fields
The famous Beilinson's conjecture predicts a relationship between the regulator map in $K$-theory and special value of $L$-function generalizing the Dirichlet's theorem in number theory. Please see ...
- 91
9
votes
1
answer
820
views
Quantum equivariant $K$-theory and DAHA.
Theorem 3.2 of the paper "Quantum cohomology of the Springer resolution" by Braverman, Maulik and Okounkov relates equivariant quantum cohomology of the cotangent bundle of $G/B$ to the trigonometric ...
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 ...
- 2,356
9
votes
0
answers
463
views
Beilinson regulators and Bloch's mythological algebraic intermediate Jacobians
In the paper introducing his motivic cycle complexes, Bloch outlines a project he says he was going to return to in the future:
Towards the end of page 270, he says, given a smooth projective variety ...
user95222
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
8
votes
1
answer
691
views
Subspaces of $ A_{n}(\mathbb {Q})$ in which all nonzero matrices are invertible
Let $A_{n}(\mathbb{Q}) $ denote the $n$ times $n$ skew symmetric matrices over the rational number field. Let $N$ be a subspace of $A_{n}(\mathbb{Q}) $.
If all the non-zero matrices in $N$ are ...
- 923
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 ...
- 16.9k
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 ...
user39380
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 ...
- 5,637