**Graded modules, part I**

This page is about graded modules over the polynomial algebra R = K[T] in 1 variable.
Many of the remarks we make here apply more generally. The K-algebra R is *graded*.
What does this mean? It just means that we have a direct sum decomposition

R = ⊕ R_d

of K-vector spaces where R_d are the elements of R of degree d (and zero). In the special case at hand we think of T as sitting in degree 1 and constants sitting in degree 0. Thus we have

- R_0 = K
- R_1 = KT = polynomials of the form λT with λ ∈ K
- R_2 = KT^2 = polynomials of the form λT^2 with λ ∈ K
- and so on

With these definitions it is clear that we have the direct sum decomposition as above: This just means that every polynomial in T can be uniquely written as λ_0 + λ_1T + … + λ_dT^d for some d and λ_i ∈ K. Note that we have

R_d ⋅ R_{d'} ⊂ R_{d + d'}.

for d, d' integers.

As usual in this project we do not define the most general notion of a graded module.
We just define what we need. A *graded module* M over the ring R above is an R-module M
which comes equipped with a direct sum
decomposition

M = ⊕ M_n

where now the integers n are allowed to be negative as well. The condition imposed on this decomposition is that

R_d ⋅ M_n ⊂ M_{d + n}.

That is all there is to it!

**Example:**
Let M = K[T, T^{-1}] = ⊕_{n ∈ Z} KT^n. There is an obvious R-module structure and with
this M becomes a graded R-module.

Note that M_n is nonzero for every integer n. However, it turns out that this doesn't happen
for finitely generated R-modules. Before we formulate the corresponding exercise we need to
introduce some more terminology. We say an element x ∈ M is *homogeneous* if x ∈ M_n for
some integer n. In this case we say that x is *homogeneous of degree n*. The key to many
arguments about graded modules is to find suitable homogeneous elements, such as in the
following two exercises.

**Exercise 28:**
Show that if M is a graded R-module which is finitely generated as a (plain) R-module,
then there exist finitely many homogeneous elements of M which generate M as an R-module.

Exercise 28 means there is no confusion if we just say “Let M be a finitely generated graded R-module”.

**Exercise 29:**
Prove: If M is a finitely generated graded R-module, then (a) M_n = 0 for
all n < < 0 and (b) dim_K(M_n) < ∞ for all n.

A very important invariant of a finitely generated graded R-module M is the
*Hilbert function* which is the map

H_M : Z —> Z_{≥ 0}, n |—> dim_K(M_n)

There is a lot you can say about these functions, and we'll need some of it. Before we develop more theory, try to answer the following question in the current case R = K[T].

**Exercise 30:**
What are all possible Hilbert functions of finitely generated graded R-modules?

Continue reading about graded modules. Back to the start page.