Exercise 110.65.1 (Definitions). Provide brief definitions of the italicized concepts.

1. a scheme,

2. a quasi-coherent module on a scheme $X$,

3. a flat morphism of schemes $X \to Y$,

4. a finite morphism of schemes $X \to Y$,

5. a group scheme $G$ over a base scheme $S$,

6. a family of varieties over a base scheme $S$,

7. the degree of a closed point $x$ on a variety $X$ over the field $k$,

8. the usual logarithmic height of a point $p = (a_0 : \ldots : a_ n)$ in $\mathbf{P}^ n(\mathbf{Q})$, and

9. a $C_ i$ field.

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