# Questions tagged [grothendieck-rings]

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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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### 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 ...
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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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### 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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### 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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### 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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### 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 ...
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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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### 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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### $0$-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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### 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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### 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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### 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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### 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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1 vote
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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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### 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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### 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 ...
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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 ...
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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. ...
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