TITLE: Grothendieck ring of varieties
QUESTION [5 upvotes]: 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 ?

REPLY [2 votes]: 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.