Grothendieck ring of varieties

Question

In the context of Grothendieck ring of varieties there is often used notion of variety over variety (for example here -2.2.1). I always used only varietes over field. My question is : how is it define ? (do anyone known any introductory articles in this subject ?) What is the most general definition of variety over something ?

Answer 1

A variety $X$ over a base variety $S$ is just a variety $X$ together with any morphism $f : X \to S$.

A very simple example would be that you can consider varieties over $\operatorname{Spec} \mathbb{C}$ or $\operatorname{Spec} \overline{\mathbb{F}_p}$, which are of course just single points.

This is important if you want to form the Cartesian product of varieties, for instance. $\mathbb{C} \otimes_\mathbb{Z} \mathbb{C}$ is a very different ring from $\mathbb{C} \otimes_\mathbb{C} \mathbb{C} \cong \mathbb{C}$. So if you want to form the product of two complex varieties, it's not enough just to have the varieties themselves; you need to know that they're varieties over $\mathbb{C}$ and not over something else. That's something that the "variety over (whatever)" structure gives you.

(Note: sometimes a variety over $\operatorname{Spec} k$ is referred to as a "variety over $k$.")

If $S$ has more than one point, you can also think of $X$ as being a family of varieties, one for each point $s \in S$, where the varieties in the family are the fibers $f^{-1}(\{s\})$.

Of course a variety over a base is the same thing as a morphism, so why the redundant term? Well, generally when we talk about a variety over $S$ the idea is to suggest that we have an element of the category of all varieties over $S$. Morphisms in this category need to preserve the morphisms to $S$, and the product is a fiber product over $S$.

Answer 2

It seems to me that the first page of Sahasrabudhe's thesis was included to make it a complete document and has been taken from Bittner's paper, see here.

• F. Bittner. The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140, 1011–1032 (2004).

Defining a variety over a variety $S$ is a slight generalization of defining it over an algebraically closed field $k$ (more precisely on $\text{Spec}\,k$). Every variety comes with a map. In the case of affine varieties over $\text{Spec}\,k$, $k$ maps into the coordinate ring of $X$, say $C(X)$. And when we talk of morphisms between two varieties $X$ and $Y$ over $\text{Spec}\,k$ then they form a commutative diagram respecting this embedding of $k$. This can be generalized over any $S$ and morphisms between $S$-schemes $X$ and $Y$ have to respect this embedding. Since every ring admits a map from $\mathbb{Z}$ (in case of $k$-algebras, a map from $k$), hence any variety is a variety over $\text{Spec}\,\mathbb{Z}$ ($\text{Spec}\,k$ respectively). Isomorphism of varieties over $S$, are morphisms with inverses, and this is an equivalence relation so one can talk of isomorphism classes.