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Questions tagged [grothendieck-rings]

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6
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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 ...
2
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0answers
135 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.
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253 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
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0answers
92 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
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1answer
385 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
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0answers
62 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})...
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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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0answers
153 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
1answer
332 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
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0answers
425 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
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1answer
220 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 ...
8
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3answers
734 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 $[\...
7
votes
2answers
479 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 ...
6
votes
1answer
298 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
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0answers
256 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 ...
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0answers
138 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
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0answers
405 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^{...
19
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1answer
747 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
2answers
566 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?
7
votes
3answers
469 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
1answer
357 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 ...
26
votes
1answer
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
1answer
321 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
2answers
940 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
0answers
362 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. It seems ...
11
votes
1answer
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 ...