The Stacks project

22.25 Graded categories

Just some definitions.

Definition 22.25.1. Let $R$ be a ring. A graded category $\mathcal{A}$ over $R$ is a category where every morphism set is given the structure of a graded $R$-module and where for $x, y, z \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ composition is $R$-bilinear and induces a homomorphism

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(y, z) \otimes _ R \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, z) \]

of graded $R$-modules (i.e., preserving degrees).

In this situation we denote $\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}^ i(x, y)$ the degree $i$ part of the graded object $\mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y)$, so that

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y) = \bigoplus \nolimits _{i \in \mathbf{Z}} \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}^ i(x, y) \]

is the direct sum decomposition into graded parts.

Definition 22.25.2. Let $R$ be a ring. A functor of graded categories over $R$, or a graded functor is a functor $F : \mathcal{A} \to \mathcal{B}$ where for all objects $x, y$ of $\mathcal{A}$ the map $F : \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(F(x), F(y))$ is a homomorphism of graded $R$-modules.

Given a graded category we are often interested in the corresponding “usual” category of maps of degree $0$. Here is a formal definition.

Definition 22.25.3. Let $R$ be a ring. Let $\mathcal{A}$ be a graded category over $R$. We let $\mathcal{A}^0$ be the category with the same objects as $\mathcal{A}$ and with

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{A}^0}(x, y) = \mathop{\mathrm{Hom}}\nolimits ^0_\mathcal {A}(x, y) \]

the degree $0$ graded piece of the graded module of morphisms of $\mathcal{A}$.

Definition 22.25.4. Let $R$ be a ring. Let $\mathcal{A}$ be a graded category over $R$. A direct sum $(x, y, z, i, j, p, q)$ in $\mathcal{A}$ (notation as in Homology, Remark 12.3.6) is a graded direct sum if $i, j, p, q$ are homogeneous of degree $0$.

Example 22.25.5 (Graded category of graded objects). Let $\mathcal{B}$ be an additive category. Recall that we have defined the category $\text{Gr}(\mathcal{B})$ of graded objects of $\mathcal{B}$ in Homology, Definition 12.16.1. In this example, we will construct a graded category $\text{Gr}^{gr}(\mathcal{B})$ over $R = \mathbf{Z}$ whose associated category $\text{Gr}^{gr}(\mathcal{B})^0$ recovers $\text{Gr}(\mathcal{B})$. As objects of $\text{Gr}^{gr}(\mathcal{B})$ we take graded objects of $\mathcal{B}$. Then, given graded objects $A = (A^ i)$ and $B = (B^ i)$ of $\mathcal{B}$ we set

\[ \mathop{\mathrm{Hom}}\nolimits _{\text{Gr}^{gr}(\mathcal{B})}(A, B) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathop{\mathrm{Hom}}\nolimits ^ n(A, B) \]

where the graded piece of degree $n$ is the abelian group of homogeneous maps of degree $n$ from $A$ to $B$. Explicitly we have

\[ \mathop{\mathrm{Hom}}\nolimits ^ n(A, B) = \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits _\mathcal {B}(A^{-q}, B^ p) \]

(observe reversal of indices and observe that we have a product here and not a direct sum). In other words, a degree $n$ morphism $f$ from $A$ to $B$ can be seen as a system $f = (f_{p, q})$ where $p, q \in \mathbf{Z}$, $p + q = n$ with $f_{p, q} : A^{-q} \to B^ p$ a morphism of $\mathcal{B}$. Given graded objects $A$, $B$, $C$ of $\mathcal{B}$ composition of morphisms in $\text{Gr}^{gr}(\mathcal{B})$ is defined via the maps

\[ \mathop{\mathrm{Hom}}\nolimits ^ m(B, C) \times \mathop{\mathrm{Hom}}\nolimits ^ n(A, B) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^{n + m}(A, C) \]

by simple composition $(g, f) \mapsto g \circ f$ of homogeneous maps of graded objects. In terms of components we have

\[ (g \circ f)_{p, r} = g_{p, q} \circ f_{-q, r} \]

where $q$ is such that $p + q = m$ and $-q + r = n$.

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

\[ \mathop{\mathrm{Hom}}\nolimits _{\text{Mod}^{gr}_ A}(L, M) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathop{\mathrm{Hom}}\nolimits ^ n(L, M) \]

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

\[ \mathop{\mathrm{Hom}}\nolimits ^ n(L, M) \subset \prod \nolimits _{p + q = n} \mathop{\mathrm{Hom}}\nolimits _ R(L^{-q}, M^ p) \]

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

\[ f_{p, q}(m a) = f_{p - i, q + i}(m)a \]

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

\[ \mathop{\mathrm{Hom}}\nolimits ^ m(L, M) \times \mathop{\mathrm{Hom}}\nolimits ^ n(K, L) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^{n + m}(K, M) \]

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

Remark 22.25.7. Let $R$ be a ring. Let $\mathcal{D}$ be an $R$-linear category endowed with a collection of $R$-linear functors $[n] : \mathcal{D} \to \mathcal{D}$, $x \mapsto x[n]$ indexed by $n \in \mathbf{Z}$ such that $[n] \circ [m] = [n + m]$ and $[0] = \text{id}_\mathcal {D}$ (equality as functors). This allows us to construct a graded category $\mathcal{D}^{gr}$ over $R$ with the same objects of $\mathcal{D}$ setting

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}^{gr}}(x, y) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(x, y[n]) \]

for $x, y$ in $\mathcal{D}$. Observe that $(\mathcal{D}^{gr})^0 = \mathcal{D}$ (see Definition 22.25.3). Moreover, the graded category $\mathcal{D}^{gr}$ inherits $R$-linear graded functors $[n]$ satisfying $[n] \circ [m] = [n + m]$ and $[0] = \text{id}_{\mathcal{D}^{gr}}$ with the property that

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}^{gr}}(x, y[n]) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{D}^{gr}}(x, y)[n] \]

as graded $R$-modules compatible with composition of morphisms.

Conversely, suppose given a graded category $\mathcal{A}$ over $R$ endowed with a collection of $R$-linear graded functors $[n]$ satisfying $[n] \circ [m] = [n + m]$ and $[0] = \text{id}_\mathcal {A}$ which are moreover equipped with isomorphisms

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y[n]) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(x, y)[n] \]

as graded $R$-modules compatible with composition of morphisms. Then the reader easily shows that $\mathcal{A} = (\mathcal{A}^0)^{gr}$.

Here are two examples of the relationship $\mathcal{D} \leftrightarrow \mathcal{A}$ we established above:

  1. Let $\mathcal{B}$ be an additive category. If $\mathcal{D} = \text{Gr}(\mathcal{B})$, then $\mathcal{A} = \text{Gr}^{gr}(\mathcal{B})$ as in Example 22.25.5.

  2. If $A$ is a graded ring and $\mathcal{D} = \text{Mod}_ A$ is the category of graded right $A$-modules, then $\mathcal{A} = \text{Mod}^{gr}_ A$, see Example 22.25.6.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09L1. Beware of the difference between the letter 'O' and the digit '0'.