
## 22.11 Projective modules over algebras

In this section we discuss projective modules over algebras and over graded algebras. Thus it is the analogue of Algebra, Section 10.76 in the setting of this chapter.

Algebras and modules. 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 10.76.2).

Graded algebras and modules. Let $R$ be a ring. Let $A$ be a graded algebra over $R$. Let $\text{Mod}_ A$ denote the category of graded right $A$-modules. For an integer $k$ let $A[k]$ denote the shift of $A$. For an graded right $A$-module we have

$\mathop{\mathrm{Hom}}\nolimits _{\text{Mod}_ A}(A[k], M) = M^{-k}$

As the functor $M \mapsto M^{-k}$ is exact on $\text{Mod}_ A$ we conclude that $A[k]$ is a projective object of $\text{Mod}_ A$. Conversely, suppose that $P$ is a projective object of $\text{Mod}_ A$. By choosing a set of homogeneous generators of $P$ as an $A$-module, we can find a surjection

$\bigoplus \nolimits _{i \in I} A[k_ i] \longrightarrow P$

Thus we conclude that a projective object of $\text{Mod}_ A$ is a direct summand of a direct sum of the shifts $A[k]$.

If $(A, \text{d})$ is a differential graded algebra and $P$ is an object of $\text{Mod}_{(A, \text{d})}$ then we say $P$ is projective as a graded $A$-module or sometimes $P$ is graded projective to mean that $P$ is a projective object of the abelian category $\text{Mod}_ A$ of graded $A$-modules.

Lemma 22.11.1. Let $(A, \text{d})$ be a differential graded algebra. Let $M \to P$ be a surjective homomorphism of differential graded $A$-modules. If $P$ is projective as a graded $A$-module, then $M \to P$ is an admissible epimorphism.

Proof. This is immediate from the definitions. $\square$

Lemma 22.11.2. Let $(A, d)$ be a differential graded algebra. Then we have

$\mathop{\mathrm{Hom}}\nolimits _{\text{Mod}_{(A, \text{d})}}(A[k], M) = \mathop{\mathrm{Ker}}(\text{d} : M^{-k} \to M^{-k + 1})$

and

$\mathop{\mathrm{Hom}}\nolimits _{K(\text{Mod}_{(A, \text{d})})}(A[k], M) = H^{-k}(M)$

for any differential graded $A$-module $M$.

Proof. This is clear from the discussion above. $\square$

Comment #2156 by shom on

In the Algebras and modules section, in the line " Then we can choose a surjection $\oplus_{i \in I} A \rightarrow M$ by choosing a set $\{m_{i}\}_{i \in I}$ of generators of $P$ over $A$. Since P is projective there is a left inverse to the surjection"

$M$ should be $P$ and perhaps the generating set should be $p_{i}$ instead of $m_{i}$ (to avoid confusion).

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