Filtration of a module

A decreasing filtration of a module is a chain $M=M_0\supseteq M_1\supseteq \cdots$ of submodules.

The quotient filtration records images of the filtered pieces in a quotient module. Together with the subspace filtration, it gives exactness of associated graded modules for induced filtrations.

A submodule of a filtered module inherits the displayed filtration by intersection. It is also called the induced filtration.

An ascending filtration is an increasing chain of additive subgroups, or vector subspaces when a scalar field is fixed. It is exhaustive if their union is the whole object. For a filtered ring, multiplicative compatibility means $F_{i}R{F}_{j}R\subseteq F_{i+j}R$. For an ideal $I$, intersecting each level with $I$ gives the induced filtration. Its associated graded module has components $(I\cap F_{n}R)/(I\cap F_{n-1}R)$.

For a filtered algebra $A$ with finite-dimensional filtration pieces, a good filtration of a module on a finitely generated module is compatible with multiplication and has finitely generated associated graded module over the associated graded ring. Taking bounded-degree translates of a finite set of module generators supplies such a filtration of a module when the associated graded ring is Noetherian.

For a good filtration of a module over a fixed finite-dimensional filtration of a ring whose associated graded algebra is commutative and finitely generated, the cumulative Hilbert function is eventually a quasipolynomial by the Hilbert-Serre theorem. Its leading asymptotic degree is the dimension of the nonzero filtered module. Monotonicity forces the leading coefficients in its residue classes to agree. If the associated graded algebra is a standard graded algebra, the Hilbert function is eventually polynomial. Two good module filtrations bound one another after fixed shifts, so they have the same growth degree. Set $d(0)=-\infty$. For the Bernstein filtration this is the Bernstein growth dimension of a Weyl algebra module.

With the fixed algebra-filtration convention in dimension of a filtered module, the multiplicity is factorial times the common leading coefficient of the cumulative Hilbert function for $M\ne 0$, and is zero for $M=0$. It is a positive integer with a standard graded algebra normalization, and can be a positive rational number for a weighted grading. Fixed shifts between good filtrations of a module preserve its leading Hilbert coefficient. Rescaling the algebra filtration itself can change multiplicity, so its normalization must be specified.

The subspace filtration and quotient filtration in a short exact sequence are good when the associated graded algebra is Noetherian. Their cumulative Hilbert functions add in every degree. Positive leading coefficients make the degree of the sum the larger degree; when both degrees agree, factorial times the leading coefficients add. Thus equal-dimensional nonzero factors consume positive multiplicity of a filtered module. With a standard graded algebra normalization these multiplicities are integral.

Two decreasing filtrations $(M_n)$ and $(N_n)$ are equivalent when each contains a fixed shift of the other: $M_{n+a}\subseteq N_n$ and $N_{n+b}\subseteq M_n$ for fixed $a,b$ and every $n$.

The $I$-adic topology on a module $M$ has the powers $I^{n}M$ as a neighbourhood basis of zero. It is generated by the I-adic filtration; separatedness means ${\bigcap }_n{I}^{n}M=0$.

For an ideal $I$, the $I$-adic filtration of a module $M$ is $M\supseteq IM\supseteq I^{2}M\supseteq \cdots$.

If $R$ is a Noetherian local ring with maximal ideal $\mathfrak{m}$ and $I\subseteq \mathfrak{m}$, thenMore generally, the intersection of the powers acts trivially on every finitely generated module.

The $x$-adic filtration is the filtration by powers of the principal ideal $(x)$.

An $I$-filtration $(M_n)$ is stable when $I{M}_n=M_{n+1}$ for all sufficiently large $n$. Every stable $I$-filtration is equivalent to the $I$-adic filtration.

For an ideal $I\subseteq R$, the Rees ring is the graded subringIt packages all powers of $I$ into one algebra and turns stable filtrations into finitely generated graded modules.

The Rees module of an $R$-module $M$ with respect to an ideal $I$ is ${\mathcal{R}}_I(M)={⨁}_{j\ge 0}{I}^{j}M{t}^j$, a graded module over the Rees ring $\mathcal{R}(I)$. Generators of $M$ in degree zero generate this module. For $N\subseteq M$, the graded submodule ${⨁}_j(I^{j}M\cap N)t^j$ is finitely generated when $R$ is a Noetherian ring; a bound on its generators' degrees yields the Artin-Rees lemma.

If $R$ is Noetherian, $I$ is an ideal, and $N\subseteq M$ are finitely generated modules, then some $k$ satisfiesfor every $n\ge k$.

A filtered algebra has a nested family of vector subspaces whose index is compatible with multiplication.

For an ascending multiplicative filtration with $1\in F_{0}A$, its Rees ring contains the central element $h$. The quotient by $h$ is the associated graded ring, and the quotient by $h-1$ is the original algebra. If $[F_{i}A,F_{j}A]\subseteq F_{i+j-1}A$, commutators of lifts are divisible by $h$. Their first-order term defines the Poisson bracket from a central deformation parameter on the associated graded ring.

An almost commutative algebra over a field is an algebra with an exhaustive increasing nonnegative multiplicative filtration whose associated graded ring is commutative and finitely generated as an algebra over the field. A common presentation uses finitely many degree-one generators, scalar degree-zero part, and commutators that lower degree. The finite-generation requirement is essential: an infinitely generated commutative polynomial ring has commutative associated graded algebra but need not be Noetherian.

This is the degree-one convention for an almost commutative algebra: the exhaustive filtration is generated by a finite-dimensional first step containing $1$, the degree-zero part is $k$, and its associated graded algebra is commutative. The first step is a finite-dimensional Lie algebra under commutators. Its inclusion into $A$ extends by the universal property to a surjection from its universal enveloping algebra. The element $1$ of the first step is retained as a central Lie generator and its image is the associative identity; quotienting the first step by $k1$ before taking the bracket can lose scalar commutators. This convention is stronger than merely requiring a finitely generated commutative associated graded algebra for an arbitrary weighted filtration.

// Destination: noncommutative-algebra.bigb

A filtered ring is a ring with subgroups $(R_n)$ satisfying $R_m{R}_n\subseteq R_{m+n}$.

A filtration of a ring is an exhaustive increasing family of additive subgroups $(F_{i}R{)}_{i\in \mathbb{Z}}$, with $1\in F_{0}R$ and the displayed multiplicativity condition. Its associated graded ring is ${⨁}_i{F}_{i}R/F_{i-1}R$. It differs from a probabilistic filtration, which is a nested family of sigma-algebras.

A negative filtration of a ring has $F_{i}R=R$ for nonnegative indices. It is complete and separated when the displayed map to the inverse limit is an isomorphism. Products make $F_{-q}R$ two-sided ideals. Series whose terms eventually belong to every $F_{-q}R$ converge. In particular, for $x\in F_{-1}R$ and $r\in R$, the geometric series ${\sum }_{j\ge 0}(xr{)}^j$ inverts $1-xr$, so the unit criterion for the Jacobson radical gives $F_{-1}R\subseteq J(R)$.

Lift finite homogeneous generators of the associated graded module of a right ideal $I$ to $x_i\in I$ of degrees $d_i$. Successive leading-symbol cancellations write an element $x\in F_{d}I$ as finite partial sums ${\sum }_i{x}_i{a}_i^{(q)}$ with residual in $F_{d-q}R$. The coefficient increments lie in $F_{d-q-d_i}R$ and therefore converge in the complete negative filtration. Multiplication is continuous, so their limits express $x$ as a finite sum ${\sum }_i{x}_i{a}_i$. This generates the actual right ideal and does not presuppose that it is closed. The left-sided statement is analogous.

For an exhaustive nonnegative increasing multiplicative filtration, each left ideal $I$ induces a homogeneous ideal $\mathrm{gr}I$ in the associated graded ring. If that graded ring is a left Noetherian ring, lift a finite homogeneous generating set of $\mathrm{gr}I$ to elements of $I$. For any element of $I$, subtract a combination of these lifts with the same leading symbol; the remainder has strictly lower degree. Induction on the nonnegative degree proves the lifts generate $I$. There is no completion hypothesis because the cancellation terminates in finitely many steps. The analogous result holds for right ideals.

For a complete separated multiplicatively filtered ring with nonnegative integer valuations, a domain associated graded ring makes the original ring a domain: the leading symbols of two nonzero elements have nonzero product. If the graded ring is left Noetherian, lift finitely many homogeneous generators of the graded ideal of a left ideal. Successively cancelling leading terms leaves remainders of increasing valuation. Completeness makes the coefficient series converge, expressing every ideal element as a finite sum of the chosen generators with completed coefficients. The right-handed argument is identical. Completeness is necessary for this infinite cancellation proof.