Example 22.25.6 (Graded category of graded modules). Let $A$ be a $\mathbf{Z}$-graded algebra over a ring $R$. We will construct a graded category $\text{Mod}^{gr}_ A$ over $R$ whose associated category $(\text{Mod}^{gr}_ A)^0$ is the category of graded $A$-modules. As objects of $\text{Mod}^{gr}_ A$ we take right graded $A$-modules (see Section 22.14). Given graded $A$-modules $L$ and $M$ we set

where $\mathop{\mathrm{Hom}}\nolimits ^ n(L, M)$ is the set of right $A$-module maps $L \to M$ which are homogeneous of degree $n$, i.e., $f(L^ i) \subset M^{i + n}$ for all $i \in \mathbf{Z}$. In terms of components, we have that

(observe reversal of indices) is the subset consisting of those $f = (f_{p, q})$ such that

for $a \in A^ i$ and $m \in L^{-q - i}$. For graded $A$-modules $K$, $L$, $M$ we define composition in $\text{Mod}^{gr}_ A$ via the maps

by simple composition of right $A$-module maps: $(g, f) \mapsto g \circ f$.

## Comments (0)