M. Künzer A_{∞}-categories, WS 16/17

Sheet 13

Problem 30 Suppose given a grading category Z. Suppose given n ∈ Z_{⩾0}. Suppose given a piecewise projective Z-graded module M_{i} for i ∈ [1, n]. Show that ⨂_{i∈[1,n]} M_{i} is piecewise projective.

Problem 31 Suppose we are in the setup of §2.4. In particular, we consider the poset 𝔄 of admissible triples. A subposet Y of a poset X is called a lower subposet if for y ∈ Y and x ∈ X with x ⩽ y, we may conclude that x ∈ Y.

(1) Suppose given a poset X and a subposet Y ⊆ X. Show that Y ∩ min(X) ⊆ min(Y).

If Y is a lower subposet in X, show that Y ∩ min(X) = min(Y).

(2) Show that for each totally ordered subposet T of 𝔄, there exists a unique element s ∈ 𝔄 such that t ⩽ s for t ∈ T and such that whenever given s′ ∈ 𝔄 such that t ⩽ s′ for t ∈ T, then s ⩽ s′. Write s =: sup T.

(3) Using the Lemma of Zorn and Lemma 64, show that for each element x = (L, M, Q) ∈ 𝔄 there exists an element x′ = (L′, M′, Q′) ∈ 𝔄 such that x ⩽ x′ and such that L′ = Z_{⩾0}^{×n}.

(4) Show the assertion of (3) again. Use Lemma 64 to do so. But do not use the Lemma of Zorn. Rather, form an ascending chain x = x_{0} ⩽ x_{1} ⩽ … such that, writing x_{k} = (L_{k}, M_{k}, Q_{k}) for k ∈ Z_{⩾0} , we have L_{k+1} = L_{k} ∪ min(Z_{⩾0} ∖ L_{k}). Then take x′ = sup{ x_{k} : k ∈ Z_{⩾0} }.

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