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Questions tagged [virtual-motives]

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27 votes
4 answers

What is the status of the theory of motives?

It has been almost 60 years since Grothendieck conceived the conjectural theory of motives in order to grasp the common behavior of the most important (Weil) cohomology theories. But what is the ...
THC's user avatar
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4 votes
2 answers

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 ...
a_g's user avatar
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3 votes
1 answer

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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6 votes
1 answer

Virtual Motives Infinitely Divisible by Lefschetz Motive

Let $K_0(Var_k)$ be the grothendieck group of the category of $k$-varieties, and call its elements virtual motives. $\mathbb{L}:=[\mathbb{A}^1_k]$ is called the Lefschetz motive. I think that if a ...
user110215's user avatar
7 votes
1 answer

Conjecture of relation between residues of Feynman integrals and mixed Tate motives

In many articles (for example in articles given by M.Marcoli) there is statement that there is the following conjecture Residues of Feynman integrals in scalar field theories are always periods of ...
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8 votes
0 answers

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 ...
mikis's user avatar
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5 votes
0 answers

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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