 Filtration (mathematics)

In mathematics, a filtration is an indexed set S_{i} of subobjects 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 S_{i} ⊆ S_{j}. The concept dual to a filtration is called a cofiltration.
Sometimes, as in a filtered algebra, there is instead the requirement that the S_{i} be subobjects^{[clarification needed]} 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 S_{i} be the whole S, or (in more general cases, when the notion of union does not make sense) that the canonical homomorphism from the direct limit of the S_{i} to S 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 sequences), and in measure theory and probability theory for nested sequences of σalgebras. In functional analysis and numerical analysis, other terminology is usually used, such as scale of spaces or nested spaces.
Contents
Examples
Algebra
See also: Filtered algebraGroups
See also: Length functionIn algebra, filtrations are ordinarily indexed by N, the set of natural numbers. A filtration of a group G, is then a nested sequence G_{n} of normal subgroups of G (that is, for any n we have G_{n+1} ⊆ G_{n}). Note that this use of the word "filtration" corresponds to our "descending filtration".
Given a group G and a filtration G_{n}, 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 aG_{n}, 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 G_{n} on a group G is Hausdorff if and only if ∩G_{n} = {1}.
If two filtrations G_{n} 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 G_{n}topology and the second the G′_{n}topology, is continuous if and only if for any n there is an m such that G_{m} ⊆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 Rmodule M, a descending filtration of M is a decreasing sequence of submodules M_{n}. 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 Iadic topology (or Jadic, etc.). Let R be a commutative ring, and I an ideal of R.
Given an Rmodule M, the sequence I^{n}M of submodules of M forms a filtration of M. The Iadic topology on M is then the topology associated to this filtration. If M is just the ring R itself, we have defined the Iadic topology on R.
When R is given the Iadic topology, R becomes a topological ring. If an Rmodule M is then given the Iadic topology, it becomes a topological Rmodule, relative to the topology given on R.
Rings and modules: ascending filtrations
Given a ring R and an Rmodule M, an ascending filtration of M is an increasing sequence of submodules M_{n}. In particular, if R is a field, then an ascending filtration of the Rvector 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 permutation) of the set. For instance, the filtration corresponds to the ordering (0,1,2). 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.
Measure theory
In measure theory, in particular in martingale theory and the theory of stochastic processes, a filtration is an increasing sequence of σalgebras on a measurable space. That is, given a measurable space , a filtration is a sequence of σalgebras 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 σalgebra.
It is also useful (in the case of an unbounded index set) to define as the σalgebra generated by the infinite union of the 's, which is contained in :
A σalgebra 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.
See also
References
 Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. Berlin: Springer. ISBN 3540047581.
Categories: Algebra
 Measure theory
 Stochastic processes
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