The Stacks project

Exercise 111.66.1 (Definitions). Provide brief definitions of the italicized concepts.

1. a scheme $X$,

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

3. the Picard group of a scheme $X$,

4. give a morphism $f : X \to Y$ of schemes and an $\mathcal{O}_ X$-module $\mathcal{F}$, the pushforward $f_*\mathcal{F}$,

5. a closed immersion of schemes, and

6. a variety over a given field $k$.