The Stacks project

22.14 Projective modules over algebras

In this section we discuss projective modules over algebras analogous to Algebra, Section 10.77. This section should probably be moved somewhere else.

Let $R$ be a ring and let $A$ be an $R$-algebra, see Section 22.2 for our conventions. It is clear that $A$ is a projective right $A$-module since $\mathop{\mathrm{Hom}}\nolimits _ A(A, M) = M$ for any right $A$-module $M$ (and thus $\mathop{\mathrm{Hom}}\nolimits _ A(A, -)$ is exact). Conversely, let $P$ be a projective right $A$-module. Then we can choose a surjection $\bigoplus _{i \in I} A \to P$ by choosing a set $\{ p_ i\} _{i \in I}$ of generators of $P$ over $A$. Since $P$ is projective there is a left inverse to the surjection, and we find that $P$ is isomorphic to a direct summand of a free module, exactly as in the commutative case (Algebra, Lemma 10.77.2).

We conclude

1. the category of $A$-modules has enough projectives,

2. $A$ is a projective $A$-module,

3. every $A$-module is a quotient of a direct sum of copies of $A$,

4. every projective $A$-module is a direct summand of a direct sum of copies of $A$.