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Filtered and graded associated objects.
Let k be a field.
Let M be an ordered monoid. I.e., M is equipped with an
associative multiplication with a neutral element 1 (sometimes
written as addition and 0, respectively), and a total order <
on M, such that 1 ≤ x and (x < y =⇒ xz < yz & zx < zy)
for all x, y, z ∈ M. (Mostly, < is a well-ordering of M.)
Let A be an abelian group. (Mostly, A is a k (vector) space.)
An increasing (decreasing) filtration F (A) of A is a family
Fx (A) x∈M
of subgroups of A, such that x < y =⇒ Fx (A) ⊆ Fy (A)
(x < y =⇒ Fx (A) ⊇ Fy (A), respectively.)
The graded associated object associated to A is the direct sum
ass A =
ass(A)x , where
Fx (A)
ass(A)x = S
Fy (A)
If A has extra structure, we assume it to be compatible with
the filtration. In particular:
If A is a k space, then the Fx (A) should be subspaces of A.
If A is a ring, a ∈ Fx (A), and b ∈ Fy (B), then ab ∈ Fxy (A).
If A is a right module over a filtered ring R, r ∈ Fx (R), and
a ∈ Fy (A), then ra ∈ Fxy (A).
If f : A −→ B is a homomorphism between filtered groups,
then we expect f (Fx (A)) ⊆ Fx (B), ∀ x ∈ M.
If so, then we speak of filtered k spaces, rings, homomorphisms
et cetera; and the graded associated counterparts will respect
the same structures.