M. Künzer

A_{∞}-categories, WS 16/17

Sheet 12

Problem 29 Let A = (A, (m_{1})) be an A_{1}-algebra over the grading category Z. Suppose that A^{z} = 0 for z ∈ Z ∖ {0, 1}. Suppose given augmented projective resolutions

⋯ → Ã^{⟨3⟩,−2} ⟶^{d^{⟨3⟩,−2}} Ã^{⟨2⟩,−1} ⟶^{d^{⟨2⟩,−1}} Ã^{⟨1⟩,0} ⟶^{d^{⟨1⟩,0}} Ã^{⟨0⟩,1} ⟶^{ε^{1}} (HA)^{1}

and

⋯ → Ã^{⟨3⟩,−3} ⟶^{d^{⟨3⟩,−3}} Ã^{⟨2⟩,−2} ⟶^{d^{⟨2⟩,−2}} Ã^{⟨1⟩,−1} ⟶^{d^{⟨1⟩,−1}} Ã^{⟨0⟩,0} ⟶^{ε^{0}} (HA)^{0} .

(1) Construct a minimal eA_{1}-algebra Ã = (Ã, (m̃_{1}), (Ã^{⟨i⟩})_{i}) over Z and a quasiisomorphism (q_{1}) : Ã → A such that the following holds. For z ∈ Z, we have

(Ã^{−z} ⟶^{m̃_{1}^{−z}} Ã^{−z+1}) = (Ã^{⟨z+1⟩,−z} ⊕ Ã^{⟨z⟩,−z} \underrightarrow{(\begin{matrix}d^{⟨z+1⟩,−z} & e^{⟨z+1⟩,−z} \\ 0 & d^{⟨z⟩,−z+1} \end{matrix} )} Ã^{⟨z⟩,−z+1} ⊕ Ã^{⟨z−1⟩,−z+1})

for some linear maps e^{⟨z+1⟩,−z} : Ã^{⟨z+1⟩,−z} → Ã^{⟨z−1⟩,−z+1}. Cf. proof of Proposition 60.

(2) In the course of the construction in (1), write

(Ã^{−z} ⟶^{q_{1}^{−z}} A^{−z}) =: (Ã^{⟨z+1⟩,−z} ⊕ Ã^{⟨z⟩,−z} \underrightarrow{(\begin{matrix}q^{⟨z+1⟩,−z} \\ q^{⟨z⟩,−z} \end{matrix} )} A^{−z})

for z ∈ Z. Show that we may choose q^{⟨1⟩,0} = 0 if and only if the residue class map A^{1} → A^{1}/(A^{0})m_{1}^{0} is a retraction.

(3) Suppose that q^{⟨1⟩,0} = 0; cf. (2).

When is it possible to choose e^{⟨2⟩,−1} = 0?

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