# Questions tagged [grothendieck-rings]

The grothendieck-rings tag has no usage guidance.

47
questions

0
votes

1
answer

149
views

### 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 ...

3
votes

0
answers

120
views

### 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 ...

1
vote

0
answers

82
views

### 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 $...

0
votes

0
answers

104
views

### 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 ...

3
votes

0
answers

135
views

### 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 $\operatorname{GL}_n(\mathbb{C})$, ...

2
votes

1
answer

193
views

### 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 ...

4
votes

0
answers

197
views

### 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 ...

4
votes

2
answers

517
views

### 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 ...

1
vote

1
answer

330
views

### 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 ...

7
votes

2
answers

909
views

### 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.,...

3
votes

2
answers

367
views

### 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 ...

10
votes

0
answers

248
views

### 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 $K$-...

2
votes

0
answers

89
views

### 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 ...

0
votes

0
answers

145
views

### $0$-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 ...

3
votes

0
answers

262
views

### 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 $A$,...

3
votes

1
answer

333
views

### 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 ...

3
votes

1
answer

238
views

### 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(...

2
votes

0
answers

144
views

### 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 ...

0
votes

0
answers

171
views

### 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 ...

2
votes

0
answers

131
views

### 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?

1
vote

0
answers

199
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 ...

12
votes

1
answer

583
views

### 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 ...

2
votes

0
answers

155
views

### 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.

15
votes

0
answers

434
views

### 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(\...

2
votes

0
answers

113
views

### 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 $[X]$ ...

10
votes

1
answer

502
views

### 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 ...

1
vote

0
answers

84
views

### 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})...

4
votes

0
answers

147
views

### 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\...

3
votes

0
answers

181
views

### 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-...

6
votes

1
answer

379
views

### 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 ...

5
votes

0
answers

515
views

### 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 $G$...

2
votes

1
answer

269
views

### 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 ...

11
votes

3
answers

907
views

### 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 $[\...

8
votes

2
answers

716
views

### 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 ...

7
votes

1
answer

470
views

### 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\...

8
votes

0
answers

275
views

### 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 ...

5
votes

0
answers

166
views

### 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 $[S] \in \mathbb{Z}[\mathbb{L}]$,...

3
votes

0
answers

824
views

### 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^{...

22
votes

1
answer

1k
views

### 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(\...

6
votes

2
answers

705
views

### 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?

8
votes

3
answers

758
views

### 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] ...

2
votes

1
answer

433
views

### 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 ...

28
votes

1
answer

2k
views

### 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 ...

3
votes

1
answer

413
views

### 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 $\...

18
votes

2
answers

1k
views

### 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 ...

6
votes

0
answers

408
views

### 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. ...

11
votes

1
answer

1k
views

### 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 operations ...