Filtration (abstract algebra)
Encyclopedia
In mathematics
, a filtration is an indexed set Si of subobject
s of a given algebraic structure
S, with the index i running over some index set
I that is a totally ordered set, subject to the condition that if i ≤ j in I then Si ⊆ Sj. The concept dual
to a filtration is called a cofiltration.
Sometimes, as in a filtered algebra
, there is instead the requirement that the
be subobject
s with respect to certain operations (say, vector addition), but with respect to other operations (say, multiplication), they instead satisfy
, where here the index set is the natural numbers; this is by analogy with a graded algebra
.
Sometimes, filtrations are supposed to satisfy the additional requirement that the union of the
be the whole 
, or (in more general cases, when the notion of union does not make sense) that the canonical homomorphism from the direct limit of the 
to 
is an isomorphism. Whether this requirement is assumed or not usually depends on the author of the text and is often explicitly stated. We are not going to impose this requirement in this article.
There is also the notion of a descending filtration, which is required to satisfy
in lieu of 
(and, occasionally, 
instead of 
). Again, it depends on the context how exactly the word "filtration" is to be understood. Descending filtrations are not to be confused with cofiltrations (which consist of quotient
objects rather than subobjects).
Filtrations are widely used in abstract algebra
, homological algebra
(where they are related in an important way to spectral sequence
s), and in measure theory and probability theory
for nested sequences of σ-algebras. In functional analysis
and numerical analysis
, other terminology is usually used, such as scale of spaces or nested spaces.
s of G (that is, for any n we have Gn+1 ⊆ Gn). Note that this use of the word "filtration" corresponds to our "descending filtration".
Given a group G and a filtration Gn, there is a natural way to define a topology on G, said to be associated to the filtration. A basis for this topology is the set of all translates of subgroups appearing in the filtration, that is, a subset of G is defined to be open if it is a union of sets of the form aGn, where a∈G and n is a natural number.
The topology associated to a filtration on a group G makes G into a topological group
.
The topology associated to a filtration Gn on a group G is Hausdorff
if and only if ∩Gn = {1}.
If two filtrations Gn and G′n are defined on a group G, then the identity map from G to G, where the first copy of G is given the Gn-topology and the second the G′n-topology, is continuous if and only if for any n there is an m such that Gm ⊆G′n, that is, if and only if the identity map is continuous at 1. In particular, the two filtrations define the same topology if and only if for any subgroup appearing in one there is a smaller or equal one appearing in the other.
An important special case is known as the I-adic topology (or J-adic, etc.). Let R be a commutative ring, and I an ideal of R.
Given an R-module M, the sequence InM of submodules of M forms a filtration of M. The I-adic topology on M is then the topology associated to this filtration. If M is just the ring R itself, we have defined the I-adic topology on R.
When R is given the I-adic topology, R becomes a topological ring
. If an R-module M is then given the I-adic topology, it becomes a topological R-module
, relative to the topology given on R.
) of the set. For instance, the filtration
corresponds to the ordering 
. From the point of view of the field with one element
, an ordering on a set corresponds to a maximal flag (a filtration on a vector space), considering a set to be a vector space over the field with one element.
es, a filtration is an increasing sequence of σ-algebras on a measurable space. That is, given a measurable space
, a filtration is a sequence of σ-algebras 
with 
for each t and
The exact range of the "times" t will usually depend on context: the set of values for t might be discrete or continuous, bounded
or unbounded. For example,
Similarly, a filtered probability space (also known as a stochastic basis) is a probability space
with a filtration of its σ-algebra.
It is also useful (in the case of an unbounded index set) to define
as the σ-algebra generated by the infinite union of the 
's, which is contained in 
:
A σ-algebra defines the set of events that can be measured, which in a probability
context is equivalent to events that can be discriminated, or "questions that can be answered at time t". Therefore a filtration is often used to represent the change in the set of events that can be measured, through gain or loss of information
. A typical example is in mathematical finance
, where a filtration represents the information available up to and including each time t, and is more and more precise (the set of measurable events is staying the same or increasing) as more information from the evolution of the stock price becomes available.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a filtration is an indexed set Si of subobject
Subobject
In category theory, a branch of mathematics, a subobject is, roughly speaking, an object which sits inside another object in the same category. The notion is a generalization of the older concepts of subset from set theory and subgroup from group theory...
s of a given algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
S, with the index i running over some index set
Index set
In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set...
I that is a totally ordered set, subject to the condition that if i ≤ j in I then Si ⊆ Sj. The concept dual
Dual (category theory)
In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...
to a filtration is called a cofiltration.
Sometimes, as in a filtered algebra
Filtered algebra
In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory....
, there is instead the requirement that the
be subobjectSubobject
In category theory, a branch of mathematics, a subobject is, roughly speaking, an object which sits inside another object in the same category. The notion is a generalization of the older concepts of subset from set theory and subgroup from group theory...
s with respect to certain operations (say, vector addition), but with respect to other operations (say, multiplication), they instead satisfy
, where here the index set is the natural numbers; this is by analogy with a graded algebraGraded algebra
In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
.
Sometimes, filtrations are supposed to satisfy the additional requirement that the union of the
be the whole , or (in more general cases, when the notion of union does not make sense) that the canonical homomorphism from the direct limit of the to is an isomorphism. Whether this requirement is assumed or not usually depends on the author of the text and is often explicitly stated. We are not going to impose this requirement in this article.There is also the notion of a descending filtration, which is required to satisfy
in lieu of (and, occasionally, instead of ). Again, it depends on the context how exactly the word "filtration" is to be understood. Descending filtrations are not to be confused with cofiltrations (which consist of quotientQuotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
objects rather than subobjects).
Filtrations are widely used in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
(where they are related in an important way to spectral sequence
Spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations...
s), and in measure theory and probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
for nested sequences of σ-algebras. In functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
and numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
, other terminology is usually used, such as scale of spaces or nested spaces.
Groups
In algebra, filtrations are ordinarily indexed by N, the set of natural numbers. A filtration of a group G, is then a nested sequence Gn of normal subgroupNormal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
s of G (that is, for any n we have Gn+1 ⊆ Gn). Note that this use of the word "filtration" corresponds to our "descending filtration".
Given a group G and a filtration Gn, there is a natural way to define a topology on G, said to be associated to the filtration. A basis for this topology is the set of all translates of subgroups appearing in the filtration, that is, a subset of G is defined to be open if it is a union of sets of the form aGn, where a∈G and n is a natural number.
The topology associated to a filtration on a group G makes G into a topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
.
The topology associated to a filtration Gn on a group G is Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
if and only if ∩Gn = {1}.
If two filtrations Gn and G′n are defined on a group G, then the identity map from G to G, where the first copy of G is given the Gn-topology and the second the G′n-topology, is continuous if and only if for any n there is an m such that Gm ⊆G′n, that is, if and only if the identity map is continuous at 1. In particular, the two filtrations define the same topology if and only if for any subgroup appearing in one there is a smaller or equal one appearing in the other.
Rings and modules: descending filtrations
Given a ring R and an R-module M, a descending filtration of M is a decreasing sequence of submodules Mn. This is therefore a special case of the notion for groups, with the additional condition that the subgroups be submodules. The associated topology is defined as for groups.An important special case is known as the I-adic topology (or J-adic, etc.). Let R be a commutative ring, and I an ideal of R.
Given an R-module M, the sequence InM of submodules of M forms a filtration of M. The I-adic topology on M is then the topology associated to this filtration. If M is just the ring R itself, we have defined the I-adic topology on R.
When R is given the I-adic topology, R becomes a topological ring
Topological ring
In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as mapswhere R × R carries the product topology.- General comments :...
. If an R-module M is then given the I-adic topology, it becomes a topological R-module
Topological module
In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.- Examples :A topological vector space is a topological module over a topological field....
, relative to the topology given on R.
Rings and modules: ascending filtrations
Given a ring R and an R-module M, an ascending filtration of M is an increasing sequence of submodules Mn. In particular, if R is a field, then an ascending filtration of the R-vector space M is an increasing sequence of vector subspaces of M. Flags are one important class of such filtrations.Sets
A maximal filtration of a set is equivalent to an ordering (a permutationPermutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
) of the set. For instance, the filtration
corresponds to the ordering . From the point of view of the field with one elementField with one element
In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French-English pun, Fun...
, an ordering on a set corresponds to a maximal flag (a filtration on a vector space), considering a set to be a vector space over the field with one element.
Measure theory
In measure theory, in particular in martingale theory and the theory of stochastic processStochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
es, a filtration is an increasing sequence of σ-algebras on a measurable space. That is, given a measurable space
, a filtration is a sequence of σ-algebras with for each t andThe exact range of the "times" t will usually depend on context: the set of values for t might be discrete or continuous, bounded
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
or unbounded. For example,
Similarly, a filtered probability space (also known as a stochastic basis) is a probability space
Probability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
with a filtration of its σ-algebra.
It is also useful (in the case of an unbounded index set) to define
as the σ-algebra generated by the infinite union of the 's, which is contained in :A σ-algebra defines the set of events that can be measured, which in a probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
context is equivalent to events that can be discriminated, or "questions that can be answered at time t". Therefore a filtration is often used to represent the change in the set of events that can be measured, through gain or loss of information
Information
Information in its most restricted technical sense is a message or collection of messages that consists of an ordered sequence of symbols, or it is the meaning that can be interpreted from such a message or collection of messages. Information can be recorded or transmitted. It can be recorded as...
. A typical example is in mathematical finance
Mathematical finance
Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical...
, where a filtration represents the information available up to and including each time t, and is more and more precise (the set of measurable events is staying the same or increasing) as more information from the evolution of the stock price becomes available.