Commutative Algebra, Notes 4: Graduations and Filtrations

Definition 1.  Let $M$ be a monoid.  An $M$graded 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 $M$graded $R$algebra 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 $R$algebra if

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

and

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

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

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

Filtrations:

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 $E_n.$

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

$R=\mathfrak{a}^0R\supseteq\mathfrak{a}R\supseteq\cdots\supseteq\mathfrak{a}^nR\supseteq\cdots.$

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:

$(ab)_{m+n}=a_mb_n.$

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.