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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 L ass A = ass(A)x , where x∈M Fx (A) . ass(A)x = S Fy (A) y<x 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.