Title: A topological proof of a theorem of Larsen and Lunts
Speaker: Inna Zakharevich
Speaker Info: University of Chicago
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Abstract:
The Grothendieck ring of varieties $K_0(\mathrm{V})$ is defined to be the free abelian group generated by varieties, modulo the relation that for any variety $X$ and any subvariety $Y$, $[X] = [Y] + [X \backslash Y]$. Multiplication is defined by $[X] [Y] = [X \times Y]$. This is the universal additive invariant, in the sense that if any function $\chi$ on varieties satisfies the formula $\chi(X) = \chi(Y) + \chi(X \backslash Y)$ (such as point counting or Euler characteristic) factors through this ring. When Kapranov introduced his motivic zeta function he conjectured that it would always be rational. However, in their 2002 paper Larsen and Lunts showed that this is in fact generally not the case. The main step in their proof was a computation of the quotient ring $K_0(V)/([\mathbb{A}^1])$. In this talk we give an alternate proof of this computation by replacing the ring $K_0(V)$ by an $E_\infty$ ring spectrum $K(V)$ and computing $\pi_0$ of the cofiber of the map $K(V) \to K(V)$ induced by multiplication by $\mathbb{A}^1$.Date: Saturday, February 06, 2016