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
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.