Timeline for Definition of a Grothendieck ring
Current License: CC BY-SA 3.0
3 events
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| Apr 23, 2019 at 12:16 | comment | added | user138661 | @DoriBejleri maybe the Grothendieck ring is the recipient of the universal Euler characteristic? | |
| Sep 4, 2011 at 0:58 | comment | added | Dori Bejleri | In this case the Grothendieck ring is a universal Euler characteristic. The Euler characteristic $\chi:\text{Var}_k \to \mathbb{Z}$ satisfies the relations, $\chi(X) = \chi(Y)$ if $X \cong Y$, $\chi(X) = \chi(X\setminus Y) + \chi(Y)$ for $Y$ a closed subsceme of $X$, and $\chi(X \times Y) = \chi(X)\chi(Y)$. So $\chi$ and any map from $\text{Var}_k$ to a commutative ring $R$ satisfying those relations factors uniquely as a ring homomorphism on $K_0(\text{Var}_k)$. Other interesting examples of such maps are the maps sending $X \to \text{#}X(\mathbb{F}_q)$ (the number of $\mathbb{F}_q$ points). | |
| Sep 3, 2011 at 13:54 | history | answered | Johan Öinert | CC BY-SA 3.0 |