Filtered algebra generally refers to structures in algebra where there is a filtration, which is a systematic way to impose a structure or hierarchy on the elements of the algebra. More formally, a filtered algebra is an algebra equipped with a filtration, which is an ascending chain of subalgebras indexed by a directed set. This can be useful in various branches of mathematics, particularly in homological algebra, algebraic topology, and representation theory. ### Definitions and Properties 1.
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A filtered algebra over a field has increasing vector subspaces with . In the usual exhaustive nonnegative convention, , , and . Multiplication on the associated graded ring is induced by multiplying representatives and retaining their highest filtered degree.
For a polynomial ring, polynomials of degree at most give this filtration. For a Weyl algebra, the Bernstein filtration assigns degree one to coordinates and derivatives. Its defining commutators have lower degree, so the associated graded algebra becomes commutative. A filtration of a module compatible with the algebra filtration similarly gives an associated graded module.