All Questions
Tagged with kt.k-theory-and-homology ag.algebraic-geometry
149 questions
17
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 ...
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
3
votes
0
answers
912
views
Relations between rational algebraic K-theory and Chow groups
A consequence of Grothendieck's Riemann-Roch Theorem is the fact that the Chern character induces an isomorphism between algebraic
$ch: K_{0}(X) \otimes \mathbb{Q} \stackrel{\cong}{\rightarrow} C H^{*...
- 6,622
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
1
vote
1
answer
186
views
Confusion regarding a definition of cycles
For a projective variety $X$ over $\mathbb{C}$, let us denote by $CH_k(X)$ the Chow group of $k$-cycles of $X$, modulo rational equivalence. Also, let $CH_k(X)_{hom}$ denote $k$-cycles modulo ...
- 737
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
5
votes
1
answer
322
views
Is the motivic homotopy spectrum of Hermitian K-theory $\eta$-complete?
Theorem 1.2 of the paper "The motivic Hopf map solves the homotopy limit problem for K-theory" (see https://elibm.org/article/10011880) says that (under certain assumptions on the base field) the ...
- 16.9k
3
votes
1
answer
1k
views
Chern character of coherent sheaf on singular variety
Does there exist a definition of Chern character (or Chern classes) for a coherent sheaf $\mathscr{F}$ on a singular variety $X$? In this case I might not be able to find a projective resolution for $\...
- 635
4
votes
0
answers
226
views
K-theoretic derivation of Bézout theorem
In the paper "$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’...
5
votes
1
answer
366
views
K-theory for a (geometric) stack
There is a notion of $K$-theory for a manifold $M$.
Is there a notion of $K$-theory for a stack $\mathcal{D}\rightarrow \text{Man}$ that is representable by a Lie groupoid $\mathcal{G}$; that is $...
- 6,023
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 ...
- 129
4
votes
3
answers
532
views
The algebro-geometrical version of K-theory
It has been proved that the higher K groups of a (possible noncommutative, but here only comm. for convenience) ring $R$ are correctly defined by Q-construction or + construction.
Recently I'm ...
- 449
3
votes
1
answer
630
views
Description of higher chow groups
In the literature there are several descriptions of motivic cohomology groups, some of them rather explicit, but I don't always understand why they are equivalent. The simplest example I have in mind ...
- 31
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) \...
- 629
1
vote
0
answers
529
views
Algebraic K-theory of schemes and cohomology
Are there examples of:
two smooth projective schemes over a field having homotopy equivalent algebraic K-theory spectra and having different rational Voevodsky motives;
two smooth projective schemes ...
user138661
13
votes
1
answer
557
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
1
vote
0
answers
204
views
Fundamental group of the Grothendieck ring scheme
Let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties with $k$ a field. Let S$_k$ be its affine scheme. Is this a connected scheme for any field ? (I understand that this could be a very naive ...
- 4,547
10
votes
1
answer
346
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
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
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
4
votes
1
answer
236
views
Absoluteness of motivic cohomology and restriction of scalars
Maybe this is a question to naive for the MO community! For a projective smooth variety $X$ defined over a field $F$, and for simplicity let's assume $F$ is a number field. One way to define motivic ...
- 2,971
4
votes
0
answers
205
views
$\mathbf{A}^1$- contractibility
Suppose $U$ is an $\mathbf{A}^1$-contractible smooth scheme over a field $k$, that is, it is isomorphic to a point in the $\mathbf{A}^1$-homotopy category of smooth schemes over $k$.
Does motivic ...
user119470
7
votes
0
answers
245
views
What can be said about the topological K-theory of non-singular varieties of small codimension in projective space?
Working over $\mathbb{C}$, the Barth-Larsen results tell us a lot about the ordinary cohomology of non-singular varieties of small codimension in projective space. For example if $X \subseteq \mathbb{...
- 1,444
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 ...
user119470
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
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
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
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
0
votes
0
answers
171
views
Surjectivity of the Albanese map of the moduli space of stable vector bundles
I have a naive question:(I saw that it is related to relative K-theory of Hodge-Deligne and also Nadel-Chern-Weil theory )
Let $\mathcal M (r, d)$ be the moduli space of stable vector bundles of rank ...
- 189
-1
votes
1
answer
178
views
About n-tuple unimodular
Let ($\mathcal{O}$, $\mathcal{M}$, k) be an DVR and $F_{1},...,F_{n} \in \mathcal{O}[X_{1},...,X_{n}]$ such that detJF = 1 where JF is the matrix $(\frac{\partial F_{i}}{\partial X_{j}})$. Suppose ...
- 348
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 ...
9
votes
1
answer
667
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
3
votes
1
answer
285
views
Is there a notion of injective, projective, flat, dimension for a differential graded algebra?
Given a differential graded algebra $(A_\bullet,d)$, is there a well-defined notion of a K-injective, K-projective, K-flat dimension of a differential graded module, or even of the category of ...
- 1,716
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
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-...
- 528
4
votes
1
answer
182
views
Explicit description of Verdier quotient of effective motives
Let $DM^{eff}_{gm}(k)$ be the triangulated category of effective geometric motives over some field $k$ (in the sense of Voevodsky), and let $DM^{eff}_{gm}(k)(1)$ be its full triangulated subcategory ...
- 41
1
vote
0
answers
967
views
Trivial normal bundle
I would like to know if there is a theorem along those lines: let $V$ be a submanifold in $\mathbb{R}^n$ such that $V$ is the boundary of a submanifold with boundary $W$. Then, the normal bundle of $V$...
- 11
5
votes
0
answers
264
views
Do almost commutative flat degenerations induce equality in K-theory? (Or: Is the characteristic variety actually a support of a class in $K$-theory?)
I intentionally phrased the title to match a different question which is almost identical to the one i'm asking. However similar, the answer there, which uses commutative algebraic geometry, is not ...
- 7,789
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
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 $Z\subset X$...
- 7,952
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
7
votes
1
answer
485
views
Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2
Does every compact complex manifold of complex dimension greater than or equal two
possess a nontrivial rank 2 holomorphic vector bundle?
- 1,236
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 ...
- 9,019
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 ...
- 9,019
6
votes
0
answers
363
views
Why write GRR with the relative tangent sheaf?
The first published version of the Grothendieck-Riemann-Roch theorem, GRR for short, was written in the form
$$
\operatorname{ch}(f_!\alpha).\operatorname{Td}(Y)
=
f_*\left(\operatorname{ch}(\alpha).\...
- 2,011
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 ...
- 9,019
2
votes
0
answers
338
views
Algebraic K-theory of complex varieties
Maybe this question is trivial, but I was not able to find an answer. The question is this: Consider the algebraic K-theory of smooth complex projective varieties (such that the K-theory and the G-...
- 912
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
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