M. Künzer A_{∞}-categories, WS 16/17

Sheet 4

Problem 9 Let q ∈ Z_{⩾1} . Consider the cyclic group C_{q} = ⟨ c : c^{q} ⟩. Abbreviate K := K(RC_{q} -Mod).

(1) Construct a projective resolution P of the trivial RC_{q}-module R that is periodic of period length 2.

(2) Calculate _{K}(P, Conc(R)^{[i]}) for i ∈ Z.

(3) Calculate _{K}(P, P^{[i]}) for i ∈ Z.

(4) Calculate the composition map

K(P, P^{[i]}) ⊗ _{K}(P^{[i]}, P^{[i+j]}) → _{K}(P, P^{[i+j]})

for i, j ∈ Z.

Problem 10 Let Z and Z̃ be grading categories. Let Z ^{\overline{}^{F}} Z̃ be a morphism of grading categories; cf. Problem 5.

(1) Construct an isomorphism F_{&}(⨂_{i∈[1,n]} M_{i}) ⟶_{∼}^{σ_{\underline{M}}} ⨂_{i∈[1,n]} F_{&}M_{i} in Z̃-grad.

(2) Show that the following quadrangle commutes.

\xymatrix{F_{&}(⨂_{i∈[1,n]} M_{i}) \ar[r]^{σ_{\underline{M}}} \ar[d]_{F_{&}(⨂_{i∈[1,n]}(f_{i} ,k_{i}))} & ⨂_{i∈[1,n]} F_{&}M_{i} \ar[d]^{⨂_{i∈[1,n]} F_{&}(f_{i} ,k_{i})} \\ F_{&}(⨂_{i∈[1,n]} M′_{i}) \ar[r]^{σ_{\underline{M}′}} & ⨂_{i∈[1,n]} F_{&}M′_{i}}

Problem 11 Let Z be a grading category. Let A be a Z-graded module. Suppose given shift-graded maps m_{1} : A → A of degree 1 and m_{2} : A^{⊗2} → A of degree 0. For n ∈ Z_{⩾3} , we let m_{n} := 0, as shift-graded linear map from A^{⊗n} to A of degree 2 − n. Suppose that (m_{n})_{n∈Z_{⩾1}} satisfies the Stasheff equations for k ∈ [1, 3]. Suppose that for each X ∈ Ob(Z), there exists an element 1_{X} ∈ A^{id^{X}} such that for z, w ∈ Mor(Z) such that zt_{Z} = X = ws_{Z} and for a ∈ A^{z} and b ∈ A^{w}, we have (a ⊗ 1_{X})m_{2} = a and (1_{X} ⊗ b)m_{2} = b. Show that (A, (m_{n})_{n∈Z_{⩾1}}) is a differential graded algebra over Z.

Problem 12 Suppose given a grading category Z. Suppose given A_{∞}-algebras Ã and A. Suppose given a shift-graded linear map f_{1} : Ã → A of degree 0.

Suppose that f_{1}^{⊗k} · m_{k}^{A} = mÃ ; k · f_{1} for k ∈ Z_{⩾1} .

Let f_{k} = 0 for k ∈ Z_{⩾2} , as shift-graded linear map from Ã^{⊗k} to A of degree 1 − k. Show that (f_{k})_{k∈Z_{⩾1}} is a morphism of A_{∞}-algebras from Ã to A.

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