Is grading unique?

Question

It must be a very basic question, but I just can't figure out...

Let $P$ be a graded $A$-module ($A$ is a commutative associative with unity). Can $P$ have two different direct decompositions, that is

$P=\bigoplus_{d\geq 0}M_d=\bigoplus_{l\geq 0}N_l$ with $M_d\ne N_l$?

Answer 1

Yes and no. Your question is not really clear anyway. I'll try to clarify your point of view.

Let's start with a small example: with $A=\mathbb{R}$ and $P=\mathbb{R}[X]$:

$$M_d=\mathbb{R}\cdot X^d,\qquad N_d=\mathbb{R}\cdot(X-1)^d.$$ We do have: $$P=\bigoplus_{m\in\mathbb{N}}M_m=\bigoplus_{n\in\mathbb{N}}N_n,$$ yet $$\forall m,n\in\mathbb{N}^*,\ M_m\neq N_n.$$ (note that $M_0=N_0$, but that could be artificially fixed). So the answer is yes.

Now let's have a look at the (pedantic) proper definitions of graded rings and modules (using $\mathbb{N}$ as the set of indices for the sake of simplicity):

A graded (Abelian) ring is a pair $\bigl(A,(A_n)_{n\in\mathbb{N}}\bigr)$ where $A$ is a (Abelian) ring and $(A_n)_{n\in\mathbb{N}}$ is a sequence of subgroups of $A$, such that $A=\bigoplus_{n\in\mathbb{N}}A_n$ and for all $m,n\in\mathbb{N}$, $A_mA_n\subset A_{m+n}$.

If $\bigl(A,(A_n)_{n\in\mathbb{N}}\bigr)$ is a graded (Abelian) ring, a graded $\bigl(A,(A_n)_{n\in\mathbb{N}}\bigr)$-module is a pair $\bigl(P,(P_n)_{n\in\mathbb{N}}\bigr)$ where $P$ is an $A$-module and $(P_n)_{n\in\mathbb{N}}$ is a sequence of subgroups of $P$, such that $P=\bigoplus_{n\in\mathbb{N}}P_n$ and for all $m,n\in\mathbb{N}$, $A_mP_n\subset P_{m+n}$.

As it's usually tedious to always specify both the ring/module and the sequence explicitly, and since we're (most of the time) only working with one sequence, we usually only mention the ring/module or the sequence, as this definition has some kind of redundancy. And this may bring some kind of confusion.

Now, formally, with the example I gave above: I'm talking about the graded ring

$$\bigl(\mathbb{R},(A_n)_{n\in\mathbb{N}}\bigr)$$ where $$A_0=\mathbb{R},\quad\forall n\geq1,\ A_n=\{0\},$$ and there I defined two distinct graded $\bigl(\mathbb{R},(A_n)_{n\in\mathbb{N}}\bigr)$-modules:

$$\bigl(P,(M_n)_{n\in\mathbb{N}}\bigr)\qquad\text{and}\qquad\bigl(P,(N_n)_{n\in\mathbb{N}}\bigr).$$ Even though $P$ is the same in both cases, as graded $\bigl(\mathbb{R},(A_n)_{n\in\mathbb{N}}\bigr)$-modules they are distinct. So the answer is no.