M. Künzer A_{∞}-categories, WS 16/17

Sheet 14

Problem 32 Let R be an integral domain. Let p ∈ R^{×} ∖ U(R). Consider the grading category Z. Define a Z-graded module A by letting

\begin{aligned}A^{−1} & := & \{ (\begin{matrix}0 & 0 \\ c & 0 \end{matrix}) ∈ R^{2×2} : c ∈ R \} \\ A^{0} & := & \{ (\begin{matrix}a & 0 \\ 0 & d \end{matrix}) ∈ R^{2×2} : a, d ∈ R \} \\ A^{1} & := & \{ (\begin{matrix}0 & b \\ 0 & 0 \end{matrix}) ∈ R^{2×2} : b ∈ R \} \\ A^{i} & := & 0 for i ∈ Z ∖ \{−1, 0, 1\} \end{aligned}

Define the shift-graded linear map m_{1} : A → A of degree 1 as follows.

\begin{matrix}A^{−1} & ⟶^{m_{1}} & A^{0} \\ (\begin{matrix}0 & 0 \\ c & 0 \end{matrix}) & ↦ & p (\begin{matrix}c & 0 \\ 0 & c \end{matrix}) \\ A^{0} & ⟶^{m_{1}} & A^{1} \\ (\begin{matrix}a & 0 \\ 0 & d \end{matrix}) & ↦ & p (\begin{matrix}0 & a−d \\ 0 & 0 \end{matrix} ) \end{matrix}

Define the shift-graded linear map m_{2} : A^{⊗2} → A of degree 0 as follows. Suppose given i, j ∈ {−1, 0, 1}. Let

\begin{matrix}A^{i} & ⊗ & A^{j} & ⟶^{m_{2}} & A^{i+j} \\ S & ⊗ & T & ↦ & ST . \end{matrix}

(1) Show that A is a classical differential graded algebra.

(2) Calculate the Z-graded modules ZA, BA and HA.

(3) Show that there is no structure (m̃_{k})_{k} of a minimal A_{∞}-algebra on HA over Z such that there exists a quasiisomorphism from HA to A.

(4) Show that there is no structure (m̃_{k})_{k} of a minimal A_{∞}-algebra on HA over Z such that there exists a quasiisomorphism from A to HA.

(5) Let C be the complex having at positions 0 and 1 the differential R ⟶^{p} R, and zero objects elsewhere.

Show that A is strictly isomorphic to the regular differential graded category Ã on (C), which is in fact a classical differential graded algebra. By this we mean that there is a piecewise bijective shift-graded linear map g : A → Ã of degree 0 such that strict_{∞}(g) : A → Ã is a morphism of A_{∞}-algebras; cf. Problem 23.(4).

w5.mathematik.uni-stuttgart.de/fachbereich/Kuenzer/ai16/