M. Künzer A_{∞}-categories, WS 16/17

Sheet 11

Problem 25 Let Z be a grading category. Let A = (A, (m_{1}), (A^{⟨i⟩})_{i}) be a minimal eA_{1}-algebra over Z. Suppose that there exist shift-graded linear map d^{⟨i⟩} : A^{⟨i⟩} → A^{⟨i−1⟩} of degree 1 and shift-graded linear map e^{⟨i⟩} : A^{⟨i⟩} → A^{⩽i−2} of degree 1 for i ∈ Z_{⩾0} such that

ι^{⟨i⟩} · m_{1} = d^{⟨i⟩} · ι^{⟨i−1⟩} + e^{⟨i⟩} · ι^{⩽i−2} .

holds for i ∈ Z_{⩾0} .

(1) Express the Stasheff equation at 1 in terms of d^{⟨i⟩} and e^{⟨i⟩}, where i ∈ Z_{⩾0} .

(2) Show that A is diagonally resolving if and only if Kern(d^{⟨i⟩}) = Im(d^{⟨i+1⟩}) for i ∈ Z_{⩾1} .

Problem 26 Let Z be a grading category. Suppose given an eA_{∞}-algebra (A, (m_{k})_{k} , (A^{⟨i⟩})_{i}) over Z. Suppose that A^{⟨i⟩} = 0 for i ∈ Z∖[0, ℓ]. For which k ∈ Z_{⩾1} is the Schmid condition on m_{k} not void? For which k ∈ Z_{⩾1} is the strong Schmid condition on m_{k} not void?

(1) Consider the case ℓ = 1.

(2) Consider the case ℓ = 2.

(3) Consider the case ℓ = 3.

Problem 27 Let Z be a grading category. Suppose given an eA_{∞}-algebra (A, (m_{k})_{k}, (A^{⟨i⟩})_{i}) over Z. Let k ⩾ 1. Let (j_{1}, …, j_{k}) ∈ Z_{⩾0}^{×k}. What bound results from the Schmid condition for the image of A^{⟨j_{1}⟩} ⊗ … ⊗ A^{⟨j_{k}⟩} under a summand of the Stasheff equation at k?

Problem 28 Let X = (X, ⩽) be a poset. We call X artinian if it does not contain a strictly descending chain. We call X superartinian if X_{⩽ξ} is finite for all ξ. We call X discrete if (⩽) = (=). We call X narrow if each discrete subposet of X is finite. Suppose given k ∈ Z_{⩾1} and posets Y_{1} , …, Y_{k}.

(1) Show that X is artinian if and only if each nonempty subposet of X has a minimal element.

(2) If X is superartinian, show that X is artinian. Does the converse hold?

(3) Construct the product ∏_{i∈[1,k]} Y_{i} in Poset, which is to be equipped with monotone maps ∏_{i∈[1,k]} Y_{i} ⟶^{π_{j}} Y_{j} for j ∈ [1, k] such that for each poset T and each tuple (T ⟶^{t_{i}} Y_{i})_{i} of monotone maps, there exists a unique monotone map T ⟶^{t} ∏_{i∈[1,k]} Y_{i} such that t · π_{j} = t_{j} for j ∈ [1, k].

(4) If Y_{i} is artinian for i ∈ [1, k], show that ∏_{i∈[1,k]} Y_{i} is artinian.

(5) If Y_{i} is superartinian for i ∈ [1, k], show that ∏_{i∈[1,k]} Y_{i} is superartinian.

(6) Show that Z_{⩾0}^{×k} := ∏_{i∈[1,k]} Z_{⩾0} is superartinian and narrow.

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