Homotopy Type Theory graded module > history (changes)

Showing changes from revision #1 to #2: Added | Removed | Changed

Defintion

< graded module

\mathbb{N}-graded modules

Given a commutative ring RR, an \mathbb{N}-graded RR-module or just graded RR-module is an RR-module AA with a binary function ():A×A\langle - \rangle_{(-)}: A \times \mathbb{N} \to A called the grade projection operator such that

a:Aa= n:a n\prod_{a:A} a = \sum_{n:\mathbb{N}} \langle a \rangle_n
a:A b:A n:a+b n=a n+b n\prod_{a:A} \prod_{b:A} \prod_{n:\mathbb{N}} \langle a + b \rangle_n = \langle a \rangle_n + \langle b \rangle_n
a:A c:A n:(c=c 0)×(ca n=ca 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)
a:A n:a n n=a n\prod_{a:A} \prod_{n:\mathbb{N}} \langle \langle a \rangle_n \rangle_n = \langle a \rangle_n
a:A m: n:(mn)×(a m n=0)\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 AA are called multivectors.

For a natural number n:n:\mathbb{N}, the image of n\langle - \rangle_n under AA is called the nn-module and is denoted as A n\langle A \rangle_n.

A nim( n)\langle A \rangle_n \coloneqq \mathrm{im}(\langle - \rangle_n)

The terms of A n\langle A \rangle_n are called nn-vectors.

We define the filtration operator ():×AA\mathcal{F}_{(-)}: \mathbb{N} \times A \to A

n(v)= m=0 nv m\mathcal{F}_{n}(v) = \sum_{m = 0}^n \langle v \rangle_m

For a natural number n:n:\mathbb{N}, the image of n\mathcal{F}_{n} under AA is called the filtered nn-module and is denoted as n(A)\mathcal{F}_{n}(A).

n(A)im( n)\mathcal{F}_{n}(A) \coloneqq \mathrm{im}(\mathcal{F}_{n})

The terms of n(A)\mathcal{F}_{n}(A) are called nn-multivectors.

Every filtered RR-algebra is an \mathbb{N}-graded RR-module.

supermodules

See also

Last revised on June 14, 2022 at 21:18:41. See the history of this page for a list of all contributions to it.