M. Künzer A_{∞}-categories, WS 16/17

Sheet 7

Problem 17 Let p > 0 be a prime. Let P ∈ Ob C(F_{p}C_{p} -Mod) be the projective resolution of the trivial F_{p}C_{p}-module as found in Problem 9.(1). Let \underline{X} := (P), so that \underline{X} has P as its only tuple entry. Let A := Hom_{F_{p}C_{p}}(\underline{X}) be the regular differential graded category, i.e. differential graded algebra over Z = Z × [1, 1]^{×2}. Recall from Problem 9 and Problem 14 that we have calculated the Z-graded module HA, i.e. that we know F_{p}-linear generators for its graded pieces. Find a minimal A_{3}-structure (m̃_{1}, m̃_{2}, m̃_{3}) on HA and a quasiisomorphism (q_{1}, q_{2}, q_{3}) : HA → A of A_{3}-algebras.

Problem 18 Suppose R to be a field. Let Z be a grading category. Let n ∈ [1, ∞]. Let A be a unital A_{n}-algebra over Z. Consider the shift-graded linear residue class map ZA ⟶^{ρ} HA of degree 0. Show that there exists a shift graded linear map ZA ⟵^{σ} HA of degree 0 such that σρ = id_{HA} and such that (1_{X}ρ)σ = 1_{X} for X ∈ Ob(Z).

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