## 12.16 Graded objects

We make the following definition.

Definition 12.16.1. Let $\mathcal{A}$ be an additive category. The *category of graded objects of $\mathcal{A}$*, denoted $\text{Gr}(\mathcal{A})$, is the category with

objects $A = (A^ i)$ are families of objects $A^ i$, $i \in \mathbf{Z}$ of objects of $\mathcal{A}$, and

morphisms $f : A = (A^ i) \to B = (B^ i)$ are families of morphisms $f^ i : A^ i \to B^ i$ of $\mathcal{A}$.

If $\mathcal{A}$ has countable direct sums, then we can associate to an object $A = (A^ i)$ of $\text{Gr}(\mathcal{A})$ the object

\[ A = \bigoplus \nolimits _{i \in \mathbf{Z}} A^ i \]

and set $k^ iA = A^ i$. In this case $\text{Gr}(\mathcal{A})$ is equivalent to the category of pairs $(A, k)$ consisting of an object $A$ of $\mathcal{A}$ and a direct sum decomposition

\[ A = \bigoplus \nolimits _{i \in \mathbf{Z}} k^ iA \]

by direct summands indexed by $\mathbf{Z}$ and a morphism $(A, k) \to (B, k)$ of such objects is given by a morphism $\varphi : A \to B$ of $\mathcal{A}$ such that $\varphi (k^ iA) \subset k^ iB$ for all $i \in \mathbf{Z}$. Whenever our additive category $\mathcal{A}$ has countable direct sums we will use this equivalence without further mention.

However, with our definitions an additive or abelian category does not necessarily have all (countable) direct sums. In this case our definition still makes sense. For example, if $\mathcal{A} = \text{Vect}_ k$ is the category of finite dimensional vector spaces over a field $k$, then $\text{Gr}(\text{Vect}_ k)$ is the category of vector spaces with a given gradation all of whose graded pieces are finite dimensional, and not the category of finite dimensional vector spaces with a given graduation.

Lemma 12.16.2. Let $\mathcal{A}$ be an abelian category. The category of graded objects $\text{Gr}(\mathcal{A})$ is abelian.

**Proof.**
Let $f : A = (A^ i) \to B = (B^ i)$ be a morphism of graded objects of $\mathcal{A}$ given by morphisms $f^ i : A^ i \to B^ i$ of $\mathcal{A}$. Then we have $\mathop{\mathrm{Ker}}(f) = (\mathop{\mathrm{Ker}}(f^ i))$ and $\mathop{\mathrm{Coker}}(f) = (\mathop{\mathrm{Coker}}(f^ i))$ in the category $\text{Gr}(\mathcal{A})$. Since we have $\mathop{\mathrm{Im}}= \mathop{\mathrm{Coim}}$ in $\mathcal{A}$ we see the same thing holds in $\text{Gr}(\mathcal{A})$.
$\square$

Definition 12.16.4. Let $\mathcal{A}$ be an additive category. If $A = (A^ i)$ is a graded object, then the $k$th *shift* $A[k]$ is the graded object with $A[k]^ i = A^{k + i}$.

If $A$ and $B$ are graded objects of $\mathcal{A}$, then we have

12.16.4.1
\begin{equation} \label{homology-equation-hom-into-shift} \mathop{\mathrm{Hom}}\nolimits _{\text{Gr}(\mathcal{A})}(A, B[k]) = \mathop{\mathrm{Hom}}\nolimits _{\text{Gr}(\mathcal{A})}(A[-k], B) \end{equation}

and an element of this group is sometimes called a map of graded objects *homogeneous of degree $k$*.

Given any set $G$ we can define $G$-graded objects of $\mathcal{A}$ as the category whose objects are $A = (A^ g)_{g \in G}$ families of objects parametrized by elements of $G$. Morphisms $f : A \to B$ are defined as families of maps $f^ g : A^ g \to B^ g$ where $g$ runs over the elements of $G$. If $G$ is an abelian group, then we can (unambiguously) define shift functors $[g]$ on the category of $G$-graded objects by the rule $(A[g])^{g_0} = A^{g + g_0}$. A particular case of this type of construction is when $G = \mathbf{Z} \times \mathbf{Z}$. In this case the objects of the category are called *bigraded* objects of $\mathcal{A}$. The $(p, q)$ component of a bigraded object $A$ is usually denoted $A^{p, q}$. For $(a, b) \in \mathbf{Z} \times \mathbf{Z}$ we write $A[a, b]$ in stead of $A[(a, b)]$. A morphism $A \to A[a, b]$ is sometimes called a *map of bidegree $(a, b)$*.

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