## Defintion ## ### $\mathbb{N}$-graded modules ### Given a [[commutative ring]] $R$, an **$\mathbb{N}$-graded $R$-module** or just **graded $R$-module** is an $R$-[[module]] $A$ with a binary function $\langle - \rangle_{(-)}: A \times \mathbb{N} \to A$ called the **grade projection operator** such that $$\prod_{a:A} a = \sum_{n:\mathbb{N}} \langle a \rangle_n$$ $$\prod_{a:A} \prod_{b:A} \prod_{n:\mathbb{N}} \langle a + b \rangle_n = \langle a \rangle_n + \langle b \rangle_n$$ $$\prod_{a:A} \prod_{c:A} \prod_{n:\mathbb{N}} (c = \langle c \rangle_0) \times (\langle c a \rangle_n = c \langle a \rangle_n)$$ $$\prod_{a:A} \prod_{n:\mathbb{N}} \langle \langle a \rangle_n \rangle_n = \langle a \rangle_n$$ $$\prod_{a:A} \prod_{m:\mathbb{N}} \prod_{n:\mathbb{N}} (m \neq n) \times (\langle \langle a \rangle_m \rangle_n = 0)$$ Terms of $A$ are called **multivectors**. For a natural number $n:\mathbb{N}$, the [[image]] of $\langle - \rangle_n$ under $A$ is called the **$n$-module** and is denoted as $\langle A \rangle_n$. $$\langle A \rangle_n \coloneqq \mathrm{im}(\langle - \rangle_n)$$ The terms of $\langle A \rangle_n$ are called **$n$-vectors**. We define the **filtration operator** $\mathcal{F}_{(-)}: \mathbb{N} \times A \to A$ $$\mathcal{F}_{n}(v) = \sum_{m = 0}^n \langle v \rangle_m$$ For a natural number $n:\mathbb{N}$, the [[image]] of $\mathcal{F}_{n}$ under $A$ is called the **filtered $n$-module** and is denoted as $\mathcal{F}_{n}(A)$. $$\mathcal{F}_{n}(A) \coloneqq \mathrm{im}(\mathcal{F}_{n})$$ The terms of $\mathcal{F}_{n}(A)$ are called **$n$-multivectors**. Every filtered $R$-algebra is an $\mathbb{N}$-graded $R$-module. ### supermodules ### ... ## See also ## * [[filtered algebra]] * [[geometric algebra]] * [[module]]