By Thomas W Hungerford

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Then the lower sequence is exact whenever the upper one is exact. Proof. Let f : Z → X ⊗ X be a morphism such that ∆ (τa ⊗ τb )a+b=n ◦ f = 0. Then (τa ⊗ τb ) ◦ f = 0, for every a + b = n. (16) Since ξn ⊗ X is a monomorphism, we get the exact sequence ξn ⊗X τ ⊗X n 0 → Xn ⊗ X −→ X ⊗ X −→ X ⊗ X. Xn Since X0 = 0, we have (16) (τn ⊗ X) ◦ f = (τn ⊗ τ0 ) ◦ f = 0. By the universal property of the kernel, there exists a unique morphism f : Z → Xn ⊗ X such that (ξn ⊗ X) ◦ f = f. (17) Since Xn ⊗ ξn is a monomorphism, we have an exact sequence Xn ⊗ξn X ⊗τ n n Xn ⊗ 0 → Xn ⊗ Xn −→ Xn ⊗ X −→ X .

By hI (u) := min t ∈ N : ut ∈ kX ut +I . (7) With these conventions, we are now able to state the main result of this subsection. 14. [Kh]. Keep the notation above. Then BI := B ({1 + I} , [SI ]c + I, <, hI ) is a PBW-basis of H = T (V )/I. 14 are used later. See [Kh] for proofs. 15. A word u belongs to GI if and only if the corresponding hyperletter [u]c is not a linear combination, modulo I, of greater hyperwords of the same degree as u and of hyperwords of lower degree, where all the hyperwords belong to BI .

We ﬁx the following assumptions. • M is a monoidal category which is abelian with additive tensor functors. • ((Xi )i∈N , (ξij )i,j∈N ) is a direct system in M where, for i ≤ j, ξij : Xi → Xj . • (ξi : Xi → X)i∈N is a compatible family of morphisms with respect to the given direct system. , there exists a morphism λii+1 : Xi+1 → Xi such that λii+1 ◦ ξii+1 = IdXi . • X0 = 0, • ξn : Xn → X is a monomorphism. ξn ξn ⊗ X are monomorphisms for every • ξn ⊗ X, X ⊗ ξn , ξn ⊗ XXn , XXn ⊗ ξn , X a b 0 ≤ a, b ≤ n.