All Questions
Tagged with ag.algebraic-geometry kt.k-theory-and-homology
149 questions
3
votes
1
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294
views
A class in the motivic cohomology group $H^{0,1}(\operatorname{Spec}k;\mathbb{Z}/p)$
In the following paper
N. Yagita, Examples for the Mod p Motivic Cohomology of Classifying Spaces,
on the first page, below display (1.1), it says "It is known that there is an element $\tau\in H^...
- 935
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
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
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
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
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
3
votes
1
answer
394
views
K/G-theory of affine bundles
Setting: $f : C \to D$ is a morphism of Artin stacks over $X$ which is a torsor for a vector bundle $T \to X$: étale-locally in $X$, we have $C \simeq D \times_X T$. I want to conclude that $f^*: G(D)...
- 1,094
3
votes
0
answers
89
views
Is there a reasonable K-grroup of Behrend’s absolutely convergent complexes?
Let $\mathfrak X$ be an algebraic stack over $\mathbb F_q$ and let $D_{\mathrm{abs}}(\mathfrak X)$ be the derived category of complexes of $\overline{\mathbb Q}_\ell$-sheaves which are absolutely ...
- 137
3
votes
0
answers
152
views
Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts are categories of modules?
I am interested in $P$ that is smooth and proper over a field and such that the derived category of coherent sheaves $D^b(P)$ possesses a $t$-structure whose heart is the category of finitely ...
- 16.9k
3
votes
0
answers
224
views
Loop spaces of motivic Eilenberg-Mac Lane spaces
Consider the unstable $\mathbb{A}^1$-homotopy category (say over $\mathbb{C}$). By the loop space $\Omega X$ of an object $X$, we mean the homotopy fiber of $pt\to X$.
For an abelian group A and the ...
- 935
3
votes
0
answers
171
views
Understanding the Exercise 9.9.5 of Weibel homological algebra
The exercise 9.9.5 of Weibel's homological algebra states that
$\textbf{Exercises 9.9.5}$ If $Z$ is a nilpotent ideal of $R$ and $k$ is a field of $char(k) = 0$, show that $H_{dR}^{\ast}(R) \cong H_{...
- 629
3
votes
0
answers
913
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
3
votes
0
answers
118
views
Extending cohomology classes to compactifications of Kuga varieties
I am trying to understand the proof of lemma 3 in the paper "Algebraic cycles and the Hodge structure of a Kuga fiber variety" by B. Brent Gordon,
available at http://www.ams.org/journals/tran/1993-...
2
votes
1
answer
307
views
Difference between $K(1)$-local K theory and l-adic completion of etale $K$ theory
Let $X$ be an scheme. Fix a prime $l$ which is invertible in $X$. Consider the $K(1)$-localization at prime $l$ of algebraic K theory $L_1K(X)$ and $l$-adic completion of etale K theory $K^{et}(X)$.
...
- 127
2
votes
1
answer
434
views
Calculating topological $K(X)$ for complex projective manifolds
In the introduction to the book Vector bundles and K-theory
http://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html
two approaches to classification of (topological) vector bundles are discussed - the ...
- 14.3k
2
votes
1
answer
400
views
${\rm SL}_2(\mathbb C)$-equivariant K-theory of $\mathbb C P^1$
Consider the action of ${\rm SL}_2(\mathbb C)$ on $\mathbb C P^1$ induced by the action of 2×2 matrices on 2-vectors. Is it true that $K^{{\rm SL}_2(\mathbb C)}(\mathbb C P^1)$ is $\mathbb C[t,t^{-1}]$...
- 2,964
2
votes
1
answer
103
views
K-ring of splittable bundles
Is there something non-trivial to be said about the subring of $K(X)$ spanned by one-dimensional bundles?
If I am not mistaken, it is still a functor into commutative rings from the category of ...
- 1,374
2
votes
1
answer
510
views
Does the Grothendieck group detect the Picard group?
Let $C$ be a curve (=smooth projective curve) of genus $g$ over an algebraic closed field $\mathbb{k}$.
It is well known that the Grothendieck group $K_0(\operatorname{coh} C)$ of the category of ...
- 405
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 $CH^2(X)$ ...
- 5,637
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?...
- 16.9k
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 ...
- 16.9k
2
votes
0
answers
162
views
Equivariant Künneth formula for partial flag variety
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$. Let $P$ be a parabolic subgroup of $G$, $\mathscr{F}:=G/P$ the partial flag variety associated to $P$. For a $G$-variety $X$, ...
- 653
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 ...
- 653
2
votes
0
answers
136
views
Is there the specialisation map of etale K theory?
Take a smooth proper morphism of schemes $X\to S$. Fix a point $t\in S$ and a point $s\in \overline\{s\}$. For a prime $l$ which is invertible in $S$, is there the natural specialization map of etale ...
- 519
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 ...
- 21
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
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
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
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 ...
- 16.9k
1
vote
1
answer
94
views
Problem concerning about an $n$-subspace of $ A_{n}(F) $
Let $A_{n}(F) $ denote the $n \times n$ skew symmetric matrices over a finite field $F$. Suppose $n$ be even and $N$ be a subspace of $A_{n}(F) $. Now if all the non-zero matrices in $N$ are ...
- 923
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
1
vote
0
answers
129
views
Is $K_0(\mathrm{Vect}(X))\to K_0'(X)$ injective for a proper variety $X$?
Let $X$ be an integral scheme, proper over an algebraically closed field $k$. Let $\mathrm{Vect}(X)$ be the exact category of finite locally free $O_X$-modules. Let $K_0(\mathrm{Vect}(X))$ be its ...
- 615
1
vote
0
answers
124
views
K-theory of l-adic sheaves of a curve
I am trying to understand if there is a good notion of a $K$-theory attached to the etale topology on a nice scheme $X$ (say smooth projective goem connected curve over a finite field is enough for me)...
- 11
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 ...
- 789
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
$$\varinjlim_{k\subset E \subset F} ...
- 1,168
1
vote
0
answers
89
views
Commutative square of module of differential is cartesian?
Is it true that the following square is Cartesian? $\require{AMScd}$
\begin{CD}
R @>{d}>> \Omega^{1}_{R} \\
@VVV @VVV\\
\widehat{R} @>{ \widehat{d}}>> \Omega^{1}_{\widehat{R}}
\end{...
- 629
1
vote
0
answers
529
views
Algebraic K-theory of schemes and cohomology
Are there examples of:
two smooth projective schemes over a field having homotopy equivalent algebraic K-theory spectra and having different rational Voevodsky motives;
two smooth projective schemes ...
user138661
1
vote
0
answers
204
views
Fundamental group of the Grothendieck ring scheme
Let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties with $k$ a field. Let S$_k$ be its affine scheme. Is this a connected scheme for any field ? (I understand that this could be a very naive ...
- 4,547
1
vote
0
answers
967
views
Trivial normal bundle
I would like to know if there is a theorem along those lines: let $V$ be a submanifold in $\mathbb{R}^n$ such that $V$ is the boundary of a submanifold with boundary $W$. Then, the normal bundle of $V$...
- 11
1
vote
0
answers
203
views
Is there any computation of $K^0(X)$ and $K_0(X)$ for a singular curve $X$?
Let $X$ be a projective curve over an algebraic closed field $k$ which characteristic zero. Define $K^0(X)$ as the Grothendieck group of the derived category $Perf(X)$ and $K_0(X)$ as the Grothendieck ...
- 9,019
1
vote
0
answers
261
views
Two questions on canonical line bundle over $\mathbb{C}P^{n}$
The canonical line bundle over $\mathbb{C}P^{n}$ is denoted by $\ell_{n}$. It is well known that: $$(\ell_{1}\otimes \ell_{1})\oplus1 \simeq\ell_{1}\oplus\ell_{1}$$ Both sides are isomorphic ...
- 356
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 ...
1
vote
0
answers
140
views
The right (not the left 'Suslin complex' one) adjoint to the embedding of $ DM^{eff} $ into $D(ShvTr) $?
Inside the derived category of Nisnevich sheaves with transfers there is the category $DM^{eff} $ of Voevodsky's effective motivic complexes (actually, Voevodsky only considered bounded above ...
- 16.9k
1
vote
0
answers
243
views
Special properties of (the $\gamma$-filtration of) $K$-theory of affine varieties.
Let $A$ be a smooth affine variety of dimension $n$. Are there any facts known on $K_{\ast}(A)$ and its $\gamma$-filtration which do not hold for $K_*(V)$ for an arbitrary smooth $V$ (of the same ...
- 16.9k
1
vote
0
answers
96
views
When the class of a complex is necessarily equi-dimensional
Let $P$ be a smooth projective variety. For an object $X$ of $D_{perf}(P)$ (i.e. a bounded perfect complex of $\mathfrak{O}_P$-module sheaves) we consider its class $[X]$ in $K_0(P)\otimes \mathbb{Q}(\...
- 16.9k
0
votes
1
answer
204
views
Equivariant sheaves on $\mathbb P^1$
Let $K(\mathbb P^1)$ be the Grothendieck group of sheaves on $\mathbb P^1$. I want to show that the map $K^{{\rm PGL}(2)\times \{\pm 1\}}(\mathbb P^1) \to K(\mathbb P^1)$ is not onto. I read somewhere ...
- 2,964
0
votes
0
answers
86
views
Projectivity of equivariant K-theory of toric variety
I'm looking at Vezzosi and Vistoli's paper: Higher algebraic K-theory for actions of diagonalizable groups.
In Theorem 6.9, they prove that the $T$-equivariant K-theory of a smooth projective toric ...
- 959
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