22.15 Projective modules over graded algebras

In this section we discuss projective graded modules over graded algebras analogous to Algebra, Section 10.77.

Let $R$ be a ring. Let $A$ be a $\mathbf{Z}$-graded algebra over $R$. Section 22.2 for our conventions. 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 a 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 $A[k]$.

We conclude

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

2. $A[k]$ is a projective $A$-module for every $k \in \mathbf{Z}$,

3. every graded $A$-module is a quotient of a direct sum of copies of the modules $A[k]$ for varying $k$,

4. every projective $A$-module is a direct summand of a direct sum of copies of the modules $A[k]$ for varying $k$.

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