# Michael: The Grothendieck Ring of Varieties (Tuesday, June 14)

:::{.remark}
Fix $S$ a Noetherian scheme and let $\Iso_S$ be the set isomorphism classes of finite type separated $S\dash$schemes.
For $S = \spec k$, finite-type means covered by finitely generated $k\dash$algebras, and separated is analogous to Hausdorff.
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:::{.definition title="Grothendieck ring of varieties"}
Define $\K_0(\Var\slice S) = \ZZ\freemod{\Iso_S}/\sim$ with relations $[X] = [Y] + [X\sm Y]$ where $Y$ is a closed subscheme of $X$.
This has a multiplication given by $[X][Y] \da [X\fiberprod{S} Y]$.
:::

:::{.definition title="Lefschetz motive"}
$\LL \da [\AA^1\slice S]$ is the **Lefschetz motive**.
:::

:::{.example title="?"}
Some explicit elements:

- $[\emptyset] = 0$ 
- $[S] = 1$ if $S = \spec k$.
- $[\PP^n\slice S] = 1+\LL + \LL^2 + \cdots + \LL^n$.
- $[X] = [X^\red]$
- Base change: $f:T\to S$ induces a map $\K_0(\Var\slice S) \mapstofrom^{f^*}_{f_!} \K_0(\var\slice T)$
  where $f^*[X] = [X\fiberprod{S} T]$ and $f_![Y] = [Y\slice S]$ regarding $Y$ as an $S\dash$scheme.

:::

:::{.definition title="Piecewise isomorphisms"}
$X$ and $Y$ are **piecewise isomorphic** iff there exist locally closed subvarieties $\ts{X_i}_{i\in I}$ and $\ts{Y_j}_{j\in J}$ with $X = \Union_i X_i$ and $Y = \Union_j Y_j$ a bijection $\sigma: I\to J$ inducing $X\con Y_{\sigma(i)}$.
Note that locally closed means open in its closure.
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:::{.remark}
Define $M_S\da \K_0(\Var\slice S)\invert{\LL}$.
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:::{.remark}
If $\Iso_S\to A$ for $A\in\CRing$ sends $\coprod$ to addition and fiber products to multiplication, then $f$ factors through $\K_0(\Var\slice S)$.
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:::{.example title="of invariants"}
Some examples of invariants:

- For $k=\FF_q$, point counting.
- For $k=\CC$, the Hodge characteristic.
- Any Euler characteristic of a scheme, particularly those coming from \(\ell\dash \)adic Galois representations.

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:::{.remark}
Recall that the Hasse-Weil zeta function for $X\slice{\FF_q}$ is defined as
\[
Z(X, t) = \sum_{n\geq 0} \abs{\Div_n(X)}t^n = \prod_{x\in \abs{X}}{1\over 1-t^{\deg(x)}}
\]
where $\Div_n(X)$ is the set of effective zero-dimensional divisors of degree $n$.
There is a motivic enhancement
\[
Z([X], t) = \sum_{n\geq 0}[\Div_n(X)] t^n
.\]
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:::{.definition title="?"}
Let $S = \spec k$ for $k$ a field (for the rest of the talk), fix $\dim X = d$, and define
\[
\mcl_n(X) \da \Hom(\spec k[t]/t^m, X)
.\]
Note that $\mcl_0 X = X$, $\mcl_1 X$ is the tangent bundle, and $\mcl_n$ is the $n$th jet bundle.
Define the **infinite jet bundle** $\mcl_\infty(X) = \cocolim \mcl L_n(X) \cong \Hom(\spec k\fps{t}, X)$, which will serve as a domain for integration.
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:::{.remark}
For $X$ smooth, there is map $\mcl_\infty(X) \mapsvia{\pi_n} \mcl_n(X)$, so for $C = \pi_n\inv(C_n)$ for $C_n$ constructible, define a measure
\[
\mu(C) = [\pi_n(C)]\LL^{-d(n+1)} \in M_k
.\]

If $f:X\to \AA^1\slice k$ then 
\[
Z_\mot(f, s) = \int_{\mcl_\infty(X)} \LL ^{-\ord_t f \cdot s} = \sum_{i\geq 0}\mu(\ord_t f\inv(i) ) \LL^{-is} \in M_k\fps{\LL^{-s}}
.\]
This recovers the original zeta function by setting $t = \LL^{-s}$.
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:::{.remark}
How bad is this ring?

- $\K_0(\Var\slice k)$ is always infinite,
- It's not Noetherian, even over a field (Linkind-Sebag?? 2010)
- It's not an integral domain (Poonen '02)
- $\K_0(\Var\slice k)/\LL \cong \ZZ[\SB]$, where $\SB$ denotes stably birational iso classes.
- $\LL$ is a zero divisor (Borisov).

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