The Stacks project

Exercise 111.62.1 (Definitions). Provide brief definitions of the italicized concepts.

1. a constructible subset of a Noetherian topological space,

2. the localization of an $R$-module $M$ at a prime $\mathfrak p$,

3. the length of a module over a Noetherian local ring $(A, \mathfrak m, \kappa )$,

4. a projective module over a ring $R$, and

5. a Cohen-Macaulay module over a Noetherian local ring $(A, \mathfrak m, \kappa )$.