I lately been introduced to monoid filtrations and I have a couple of questions.
Let $(\mathfrak{M},\star,1_\mathfrak{M})$ be a monoid with total order, $(A,+)$ the additive subgroup and $(R,+,\cdot,1_R)$ a ring. Then an $\mathfrak{M}$-filtration of $R$ is a family $(F_m(A))_{m\in\mathfrak{M}}$ of subgroups of $A$ such that $$x<y\Rightarrow F_x(A)\subseteq F_y(A).$$
If we want to filter a ring $R$ we just add that it also fulfills
- that it is a filtration of $(R,+),$
- $1_R\in F_1(r),$
- $a\in F_x(R) \land b\in f_y(R)\Rightarrow a\cdot b\in F_{x\star y}(R).$
Fnally we define $F_{<x}(A)= \bigcup_{y\in\mathfrak{M},y<x}F_y(A)$ and the graded assiciated object $gr(A)_m$is defined as
$$gr(A)_m:=\frac{F_m(A)}{F_{<m}(A)},\quad gr(A):=\bigoplus_{\mathfrak{M}}gr(A)_m.$$
Now, I would like to understand these structures a bit better. Let $R=k[x_1,\ldots,x_n]$ and let $\mathfrak{M}$ be graded by $<_{lex}$. Now let me ask a couple of questions.
- we have $F_{<x_1}(R)=\{0_R\}$ is this correct?
- how does $F_1=F_{x_1}(R)$ look like? how does $F_2=F_{<x_1}(R)$ look like? and what about $gr(R)_{x_1}$. It seems that both $F_1,F_2$ are the whole ring so $gr(R)_{x_1}=0_r$? Are these examples too simple?
- say $f=x_1^3x_2^4x_3^3$. Then $f\in F_{x_1^2x_2^4x_3^4}$ right? How does $F_{<x_1^2x_2^4x_3}$ looks like? And $gr(R)_{x_1^2x_2^4x_3}$?
Thanks for the help