Commutative Algebra, Notes 4: Graduations and Filtrations


Definition 1.  Let M be a monoid.  An Mgraded ring is a ring R where

\displaystyle R=\bigoplus_{x\in M}A_x

and A_xA_y\subseteq A_{xy} where A_\alpha are abelian groups.  A graded ring will mean an \mathbb{N}-graded ring.  A is an Mgraded Ralgebra if it is M-graded as a ring.  If A is a graded R-algebra and R is graded, then we say A is graded with respect to R if R_iA_j\subseteq A_{i+j} and A_iR_j\subseteq A_{i+j}.

Example 2.  The tensor algebra T(M) of a module is trivially graded with its concatenation product.  A group ring R[G] is a G-graded ring, which can be seen with its decomposition R[G]=\oplus_{g\in G}Rg.  A \mathbb{Z}_2-graded ring (algebra) is called a super ring (super algebra).  Note that \mathbb{Z}-graded rings and graded rings induce a \mathbb{Z}_2 grading as follows

\displaystyle R=R_0\oplus R_1=\bigoplus_{2n}R_n\oplus\bigoplus_{2n+1}R_n.

Note we could slightly generalize the above definition of A being graded with respect to R as follows:  suppose R is M-graded and A is N-graded for monoids M,N and that there exists a monoid homomorphism \varphi:M\to N.  Then we can say that A is an (M,N)graded Ralgebra if

\displaystyle R_mA_n\subseteq A_{\varphi(m)+n}


\displaystyle A_nR_m\subseteq A_{n+\varphi(m)}.

A similar definition exists for an (M,N)-graded R-module.  Note an N-graded R-algebra has an (M,N)-grading where M is the trvial monoid grading R and \varphi is of course the trivial morphism.  An element of an (M,N)-graded R-algebra is called homogeneous (of order n) if it has the form (\cdots,0,a_n,0,\cdots) for some n\in N.  Let A,B be (M,N)-graded R-algebras and f:A\to B be an R-algebra homomorphism.  Then f is called a graded homomorphism if f(A_n)\subseteq B_n.  In more generality, we could have B be an (M,N')-graded R-algebra together with a module homomorphism \varphi:N\to N' and require f(A_n)\subseteq B_{\varphi(n)}.

Definition 3.  A subset S\subseteq A is homogeneous if for every element of S, the component elements (which are homogeneous) are in S.  If S is an ideal, then we call it a homogeneous (or graded) ideal.  Note it has a grading as it can be written as a direct sum of the ideals generated by the homogeneous elements.

Proposition 4.  Let A be an (M,N)-graded R-algebra and \mathfrak{a} be a homogeneous ideal in A.  Then A/\mathfrak{a} is an (M,N)-graded R-algebra.

Proof.  Since \mathfrak{a} is homogeneous, it is graded (since elements give component elements).  So we have

\displaystyle\mathfrak{a}=\bigoplus_{n\in N}\mathfrak{a_n}=\bigoplus_{n\in N}(\mathfrak{a}\cap A_n)

Cosets of \mathfrak{a} in the quotient thus have the form

\displaystyle A+\mathfrak{a}=\bigoplus_{n\in N}A_n+\bigoplus\mathfrak{a}_n=\bigoplus_{n\in N}\left(A_n+\mathfrak{a}_n\right),

which gives us a decomposition.  Hence

R_m(A/\mathfrak{a})_n=R_m(A_n/\mathfrak{a}_n)\subseteq A_{\varphi(m)+n}/\mathfrak{a}_{\varphi(m)+n}=(A/\mathfrak{a})_{\varphi(m)+n}.


Definition 5.  Let (P,\leq) be a poset.  A subset F is called a filter if the following hold

  1. x,y\in F\Rightarrow\exists z\in F with z\leq x and z\leq y.
  2. If x\in F, y\in P, and x\leq y, then y\in F.

Definition 6.  Let A be a structure and F be a filter.  An descending (ascending) filtration on A with respect to F is a collection \{A_x\}_{x\in F} of substructures of A such that x\leq y\Rightarrow A_x\supseteq A_y (A_x\subseteq A_y).  A filtration on A will mean a filtration with respect to the filter (\mathbb{N},\leq) (either ascending/descending).  We can similarly define filtrations of modules and filtrations of modules that respect the filtration of their ring.

Definition 7.  If \mathfrak{a} is an ideal in R and E is an R-module with a descending filtration, then the filtration is called an \mathfrak{a}filtration if \mathfrak{a}E_n\subseteq E_{n+1} for all n. It is called \mathfrak{a}stable if \mathfrak{a}E_n=E_{n+1} for all n\geq m for some m.

Hence we can view multiplication by \mathfrak{a} as increasing/decreasing the degree (depending upon preferred terminology) of elements in E_n.

Induced Graduations:

Let \mathfrak{a} be an ideal of R.  Then R has an \mathfrak{a}-filtration:


We can define the Rees algebra (which Lang calls the “first associated graded ring”) as

\displaystyle R[\mathfrak{a}t]=\bigoplus_{n=0}^\infty\mathfrak{a}^nt^n.

This is clearly a graded R-algebra.  We could also consider

\displaystyle gr_{\mathfrak{a}}(R)=\bigoplus_{n=0}^\infty\mathfrak{a}^n/\mathfrak{a}^{n+1}.

This is easily verified as a graded R-algebra with a product defined componentwise:


Definition 8.  Let E be a graded R-module with grading E=\oplus_n E_n.  We define the Hilbert polynomial by H_E(n)=\dim_R(E_n).  We define the \textbf{Poincar\'{e} series} of E as

\displaystyle P_E(t)=\sum_{n=0}^\infty H_E(n)t^n.

[1] Lang, Serge.  Algebra.  Revised Third Edition.  Springer-Verlag.  2000.

[2] Dummit, David and Richard Foote.  Abstract Algebra.  Third Edition.  John Wiley and Sons, Inc.  2004.


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