Questions tagged [grothendieck-rings]
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49
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Are there orthogonality relations for the Burnside ring / table of marks known?
I would like to ask the following.
Are there orthogonality relations for the Burnside ring / table of marks known?
There are formulæ known for idempotent elements, but I am searching for something ...
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107
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$K_0$ of finite graphs
We have two operations on finite graphs, first the disjoint union and the categorical product. I want to use these operations to associate a r(i)ng $R$ to finite graphs. An element of that ring is an ...
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Generators of triangulated category and Grothendieck groups
Let $\mathcal{T}$ be a triangulated category that is generated by one object, say $A$ in the sense that the smallest triangulated subcategory containing $A$ and closed under coproducts and ...
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142
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Grothendieck ring of varieties in positive characteristic, away from the characteristic
In "The universal Euler characteristic for varieties of characteristic zero", Bittner shows that over a field $k$ of characteristic zero, the Grothendieck ring $K_{0}(Var_{k})$ of varieties ...
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Grothendieck group of finite cyclic groups
Let $\mathcal{C}$ be the set of isomorphic classes of all finite cyclic groups $[C_n]$, with $C_n \cong \mathbb{Z}/n\mathbb{Z}$, $n \in \mathbb{N}_0$.
Define relations on $\mathcal{C}$ as follows: if $...
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190
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Relationship between vector bundles and modules
THE GROTHENDIECK RING IN GEOMETRY AND TOPOLOGY - M.F. ATIYAH
§1. The Grothendieck ring in homotopy theory
I am going to be talking about vector bundles, i.e. fibre bundles with
fibre a vector space ...
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154
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Class of finite quotient affine space in Grothendieck ring of varieties
Let $G$ be a finite group acting linearly on affine space $\mathbb{A}^n_k$ over $k = \mathbb{C}$. Since the action is linear, it can be extended to an action of $\operatorname{GL}_n(\mathbb{C})$, ...
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Representation-induced relations in the Grothendieck of varieties
Let $X$ be a variety over $k = \mathbb{C}$ with an action of a finite group $G$. According to this paper (Section 4), the induced action of $G$ on the cohomology of $X$ respects the mixed Hodge ...
- 197
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208
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Signed number of pieces in a decomposition in the Grothendieck ring of varieties
Let $X/k$ be a (geometrically integral and connected) variety over $k$ either a field of characteristic $0$ or a finite field. Let $[X] = \sum_{i\in I}[Y_i] - \sum_{j\in J}[Z_j]$ be a decomposition ...
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550
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Dimension is an invariant in the Grothendieck ring of algebraic varieties
$\DeclareMathOperator\Var{Var}$Let $k$ be any field and let $K_0(\Var_k)$ be the Grothendieck ring of $k$-algebraic varieties (i.e. algebraic varieties up to cut-and-paste relations). Given an ...
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1
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Grothendieck rings and the Tannakian formalism
I understand that the Tannakian formalism (which I only "know" extremely superficially) is very important for the theory of motives. I guess "the" conjectural category of motives ...
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votes
2
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1k
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Grothendieck group generated by classes of invertible sheaves
Given a smooth, projective (complex) varieties $X$, is it true that the grothendieck group $K_0(X)$ of equivalence classes of coherent sheaves on $X$, is generated by clases of invertible sheaves i.e.,...
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2
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400
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Piecewise isomorphism versus equivalence in Grothendieck ring
$\DeclareMathOperator\Var{Var}$Let $K_{0}(\Var_{\mathbb{C}})$ be the Grothendieck ring of varieties over $\mathbb{C}$. The class of a variety, $X$, in $K_{0}$ is denoted $[\,X\,]$. If $X$ and $Y$ are ...
user108998
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Quotients by algebraic group actions at the level of the Grothendieck ring
$\DeclareMathOperator\SGro{SGro}\DeclareMathOperator\Gro{Gro}\DeclareMathOperator\GL{GL}$For an algebraically closed field $K$, the Grothendieck semiring of $K$ consists of, say, quasi-projective $K$-...
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90
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Cone of morphism in families
I am working in derived category $D^b(X)$ of coherent sheaves on a smooth projective varitey.
Let $E,F$ be two sheaves on $X$, with $\mathrm{R}Hom(E,F)=k\oplus k[-1]$, I consider the following ...
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146
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$0$-Dimensional $k$-varieties in the Grothendieck ring $K_0(V_k)$
Let $k$ be a field, and let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties. Let $K_0(V_k^0)$ be the Grothendieck ring of $0$-dimensional $k$-varieties. I also assume that $k$ is perfect.
(1) I ...
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288
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Why the scissor relations in Grothendieck rings?
Let $k$ be a field, and let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties. One type of relation which defines $K_0(V_k)$ is the following: if $A$ is a $k$-variety and $C$ a closed subset of $A$,...
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1
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346
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Field extensions in Grothendieck rings
Let $k$ be a field, and consider the Grothendieck ring of $k$-varieties, $K_0(V_k)$. Let $K/k$ and $K'/k$ be field extensions of $k$. We view $\mathrm{Spec}(K)$ and $\mathrm{Spec}(K')$ as $k$-schemes ...
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2
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490
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Double points in the Grothendieck ring
Let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties, and consider the scheme $X = \mathrm{Spec}(k[x]/(x^2))$.
I understand that this scheme has one point, but I am missing the fact that in $K_0(...
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Restriction to the maximal torus
$\DeclareMathOperator\ad{ad}\DeclareMathOperator\ind{ind}\DeclareMathOperator\res{res}\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Hom{Hom}$Let me say that I am kind of sure that all the things I ...
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The Grothendieck ring of varieties with classical Zariski
Consider $\mathcal{V}_k$, the category of $k$-varieties over the finite field $k = \mathbb{F}_q$ with $q$ elements. We see varieties in the old "classical" sense of the word, foreseen with the old ...
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132
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Same class in the Grothendieck ring, different $\pi_2$
Let $k$ be an algebraically closed field. Are there two connected smooth projective $k$-schemes that have the same class in the Grothendieck ring and have non-isomorphic etale $\pi_2$ groups?
user138661
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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 ...
- 4,533
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1
answer
595
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An inverse problem for Grothendieck rings of varieties
Suppose $A$ is a given commutative ring, and suppose that one knows that $A$ is isomorphic to the Grothendieck ring of $k$-varieties for some unknown field $k$.
Can $k$ be recovered from $A$ ? If ...
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157
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What is the interest of Grothendieck ring in Model theory
I've hear of the notion of Grothendieck ring in model theory.
Could someone tell me what are their interest and what are the relevant questions about them ?
I'm sorry for this quite vague question.
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Do we expect abelian varieties (and “Artin motives”) to generate the Grothendieck ring of varieties over a finite field?
The Tate conjecture implies that the category of motives over a finite field is generated (as tensor category) by the motives of abelian varieties and Artin motives. See [1] for details.
Let $K(\...
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On isomorphisms of closed subsets in Grothendieck rings of varieties
Consider the Grothendieck ring $K_0(\mathcal{V}_k)$ of $k$-varieties. Then for any $k$-variety $X$, and closed subset $C$ of $X$, we have the relation $$[X] = [X \setminus C] + [C],$$
where $[X]$ ...
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answer
526
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Quadrics in the Grothendieck ring
Let $\mathcal{Q}$ be an irreducible quadric in $\mathbb{P}^n(k)$, with $n \geq 2$ and $k$ a finite field. Let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties. It is well known (it appears) that ...
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What is $K_1(\mathrm{Var}_\Bbbk)$? [duplicate]
Ok, this is a very naive question, and not seriously motivated. But I was just wondering: did anybody define any (interesting) higher K-theory Grothendieck group of varieties $K_n(\mathrm{Var}_{\Bbbk})...
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Grothendieck group of limit of categories
I am in the following situation. I have a stable presentable $\infty$-category $\cal{C}$, and a sequence of full stable subcategories $\dots\subset\cal{C}_{-2}\subset\cal{C}_{-1}\subset\cal{C}_0\...
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What is the meaning of rationality for these series?
Let me start with a couple of examples of rationality.
Let $X$ be a nonsingular, projective Calabi-Yau threefold. Let $\beta\in H_2(X)$ be a homology class. The rationality of the reduced Donaldson-...
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Virtual mixed Tate motives
Let $\mathbf{Sch}_k$ be the category of $k$-schemes of finite type, and let $K_0(\mathbf{Sch}_k)$ be the Grothendieck ring of $k$-schemes. Let $\mathbb{Z}[\mathbb{L}]$ the subring generated by the ...
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What is the algebro-geometric or measure-theoretic "content" of Dhillon and Mináč's motivic Artin symbols over an arbitrary ground field?
1. Short version. In this text, Dhillon and Mináč define motivic Artin symbols. Having fixed a ground field $k$ and a smooth projective curve $Y$ over $k$ equipped with the action of a finite group $G$...
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Constructible sets II (Grothendieck rings)
Here is my second question on constructible sets, now on Grothendieck rings. Let $K_0(Sch_k)$ be the Grothendieck ring of schemes over $k$. I have read that if $S$ is a constructible set in a ...
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3
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What does it mean that $[X]+[Y]=0$ in the Grothendieck ring of varieties?
This is a really basic question. If I have two non-isomorphic varieties $X$ and $Y$, is it possible that $[X]+[Y]=0$ in the Grothendieck ring?
If so, what does this mean geometrically? Obviously $[\...
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2
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Some questions about the map $K_0(\text{Var})\to K_0(\text{Mot})$
Let $k$ be a field. The naive Grothendieck ring of varieties $K_0(\text{Var})$ is generated by isomorphism classes of varieties over $k$ with the scissors relation $[X]=[X-Y]+[Y]$ for $Y$ a closed ...
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What is the motivic class of a quotient stack?
The Grothendieck ring of complex varieties $K(Var_\mathbb C)$ is the free abelian group generated by isomorphism classes $[X]$ of $\mathbb C$-varieties, modulo the scissor relation $[X]=[Z]+[X\...
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Properties of Grothendieck ring for field of characterictic $p$
In this article there is a proof that for field $k$ of characteristic zero Grothendieck ring $K(\mathbf{Var}_k)$ is not an integral domain. In many articles I found statement that similar theorem for ...
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Polynomially countable varieties and virtual mixed Tate motives
Let $K_0(Var_k)$ be the Grothendieck ring of $k$-varieties for a field $k$. Let $\mathbb{L}$ denote the class of the affine line over $k$. Let $S$ be a $k$-variety and $[S] \in \mathbb{Z}[\mathbb{L}]$,...
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951
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Reference for the Hodge polynomial or the Hodge Characteristic
What is the first work that studies, refers to, or mentions the Hodge characteristic?
The Hodge polynomial is the unique ring homomorphism
$$
P_{hdg}:K_0(\mathbf{Var}/\mathbb{C)}\to \mathbb{Z}[u,v,u^{...
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answer
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Is there a motivic Cauchy integral formula?
Let $R$ be a complete dvr with fraction field $K$ and residue field $k$, and let $X, Y$ be two smooth projective $R$-schemes with isomorphic generic fibers.
Is it true that $[X_k]=[Y_k]$ in $K_0(\...
- 22.9k
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2
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715
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Should the Grothendieck ring of varieties be K_0 of numerical motives?
Assuming the Standard Conjectures, should the Grothendieck ring of varieties be the $K_0$ of the abelian category of numerical motives?
- 15.6k
8
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Etale local fibrations in the Grothendieck ring of varieties
Let $k$ be a field and $K_0(Var_k)$ the Grothendieck ring of varieties over $k$. This is the ring generated by isomorphism classes of varieties over $k$ with multiplication given by
$$
[X \times_k Y] ...
- 3,210
2
votes
1
answer
436
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On a property of the Grothendieck group of a smooth projective curve
Let $K$ be a complete DVR of characteristic $0$, $X$ a smooth projective curve over $K$. Denote by $K^0(X)$ the Grothendieck group of locally free sheaves on $X$ and by $\mbox{det}$ the natural group ...
- 1,979
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votes
1
answer
2k
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Is there a higher Grothendieck ring of varieties?
Fix a field $k$. The Grothendieck ring $K_0(\mathrm{Var}_k)$ of varieties over $k$ is defined as the quotient of the free abelian group on isomorphism classes of algebraic varieties by the scissor ...
- 293
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1
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432
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Virtual Lefschetz motive
Hi there,
I have a question which popped up while reading papers on motives.
Let $V_k$ be the category of (projective) k-varieties, and let $K_0(V_k)$ be the Grothendieck ring of $V_k$; then $\...
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2
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1k
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Grothendieck ring of "varieties carrying a function"
Fix a base ring $R$, and consider pairs $(X,f)$ where $X$ is a scheme
of finite type over $R$ and $f:X\to R$ is an $R$-valued algebraic (not
constructible!) function on $X$.
I want to consider a ...
- 27.8k
6
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0
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421
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Higher K-theory of Orlik-Solomon algebras (and possible generalizations?)
This topic of this question is a bit outside my comfort zone, and I should say that my end goal is to really understand how much "graph theory" is captured by contraction-deletion relations. ...
- 85.3k
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1
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When are representation rings special lambda-rings? (variations of an old question)
Status: Questions 2 and 4 answered in the negative. Questions 1 and 3 ARE STILL UNANSWERED, despite previous claims.
On the third page of Wolfang K. Seiler's paper "lambda-rings and Adams ...
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