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24.4 Sheaves of graded modules

Please skip this section.

Definition 24.4.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of graded algebras on $(\mathcal{C}, \mathcal{O})$. A (right) graded $\mathcal{A}$-module or (right) graded module over $\mathcal{A}$ is given by a family $\mathcal{M}^ n$ indexed by $n \in \mathbf{Z}$ of $\mathcal{O}$-modules endowed with $\mathcal{O}$-bilinear maps

\[ \mathcal{M}^ n \times \mathcal{A}^ m \to \mathcal{M}^{n + m},\quad (x, a) \longmapsto xa \]

called the multiplication maps with the following properties

  1. multiplication satisfies $(xa)a' = x(aa')$,

  2. the identity section $1$ of $\mathcal{A}^0$ acts as the identity on $\mathcal{M}^ n$ for all $n$.

We often say “let $\mathcal{M}$ be a graded $\mathcal{A}$-module” to indicate this situation. A homomorphism of graded $\mathcal{A}$-modules $f : \mathcal{M} \to \mathcal{N}$ is a family of maps $f^ n : \mathcal{M}^ n \to \mathcal{N}^ n$ of $\mathcal{O}$-modules compatible with the multiplication maps. The category of (right) graded $\mathcal{A}$-modules is denoted $\textit{Mod}(\mathcal{A})$.

We can define left graded modules in exactly the same manner but our default in the chapter will be right modules.

Given a graded $\mathcal{A}$-module $\mathcal{M}$ and an object $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we use the notation

\[ \mathcal{M}(U) = \Gamma (U, \mathcal{M}) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{M}^ n(U) \]

This is a (right) graded $\mathcal{A}(U)$-module.

Lemma 24.4.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a graded $\mathcal{O}$-algebra. The category $\textit{Mod}(\mathcal{A})$ is an abelian category with the following properties

  1. $\textit{Mod}(\mathcal{A})$ has arbitrary direct sums,

  2. $\textit{Mod}(\mathcal{A})$ has arbitrary colimits,

  3. filtered colimit in $\textit{Mod}(\mathcal{A})$ are exact,

  4. $\textit{Mod}(\mathcal{A})$ has arbitrary products,

  5. $\textit{Mod}(\mathcal{A})$ has arbitrary limits.

The functor

\[ \textit{Mod}(\mathcal{A}) \longrightarrow \textit{Mod}(\mathcal{O}),\quad \mathcal{M} \longmapsto \mathcal{M}^ n \]

sending a graded $\mathcal{A}$-module to its $n$th term commutes with all limits and colimits.

The lemma says that we may take limits and colimits termwise. It also says (or implies if you like) that the forgetful functor

\[ \textit{Mod}(\mathcal{A}) \longrightarrow \text{graded }\mathcal{O}\text{-modules} \]

commutes with all limits and colimits.

Proof. Let us denote $\text{gr}^ n : \textit{Mod}(\mathcal{A}) \to \textit{Mod}(\mathcal{O})$ the functor in the statement of the lemma. Consider a homomorphism $f : \mathcal{M} \to \mathcal{N}$ of graded $\mathcal{A}$-modules. The kernel and cokernel of $f$ as maps of graded $\mathcal{O}$-modules are additionally endowed with multiplication maps as in Definition 24.4.1. Hence these are also the kernel and cokernel in $\textit{Mod}(\mathcal{A})$. Thus $\textit{Mod}(\mathcal{A})$ is an abelian category and taking kernels and cokernels commutes with $\text{gr}^ n$.

To prove the existence of limits and colimits it is sufficient to prove the existence of products and direct sums, see Categories, Lemmas 4.14.11 and 4.14.12. The same lemmas show that proving the commutation of limits and colimits with $\text{gr}^ n$ follows if $\text{gr}^ n$ commutes with direct sums and products.

Let $\mathcal{M}_ t$, $t \in T$ be a set of graded $\mathcal{A}$-modules. Then we can consider the graded $\mathcal{A}$-module whose degree $n$ term is $\bigoplus _{t \in T} \mathcal{M}_ t^ n$ (with obvious multiplication maps). The reader easily verifies that this is a direct sum in $\textit{Mod}(\mathcal{A})$. Similarly for products.

Observe that $\text{gr}^ n$ is an exact functor for all $n$ and that a complex $\mathcal{M}_1 \to \mathcal{M}_2 \to \mathcal{M}_3$ of $\textit{Mod}(\mathcal{A})$ is exact if and only if $\text{gr}^ n\mathcal{M}_1 \to \text{gr}^ n\mathcal{M}_2 \to \text{gr}^ n\mathcal{M}_3$ is exact in $\textit{Mod}(\mathcal{O})$ for all $n$. Hence we conclude that (3) holds as filtered colimits are exact in $\textit{Mod}(\mathcal{O})$; it is a Grothendieck abelian category, see Cohomology on Sites, Section 21.19. $\square$

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