M. Künzer

A_{∞}-categories, WS 16/17

Sheet 3

Problem 7 Let Z = (Z, S, deg) be a grading category. Suppose given 1 ⩽ ℓ ⩽ n and Z-shift-graded linear maps L_{i} ⟶^{(f_{i},k_{i})} M_{i} for i ∈ [1, n]. Suppose given Z-shift-graded linear maps L ⟶^{(f,k)} M and L̃ ⟶^{(f̃,k̃)} M̃.

(1) Show that

(M_{1} ⊗ … ⊗ M_{ℓ}) ⊗ (M_{ℓ+1} ⊗ … ⊗ M_{n}) = M_{1} ⊗ … ⊗ M_{n} .

(2) Show that ((f_{1}, k_{1}) ⊗ … ⊗ (f_{ℓ}, k_{ℓ})) ⊗ ((f_{ℓ+1}, k_{ℓ+1}) ⊗ … ⊗ (f_{n}, k_{n})) = (f_{1}, k_{1}) ⊗ … ⊗ (f_{n}, k_{n}) .

(3) Construct a Z-graded module Ṙ such that (f, k) ⊗ (idṘ , 0) = (f, k) and (idṘ , 0) ⊗ (f, k) = (f, k).

(4) Construct an isomorphism L ⊗ L̃ ⟶_{∼}^{τ_{L,L̃}} L̃ ⊗ L in Z-grad, and likewise τ_{M,M̃} , such that the

following quadrangle commutes.

\xymatrix{L ⊗ L̃ \ar[r]_{∼}^{τ_{L,L̃}} \ar[d]_{(f,k)⊗(f̃,k̃)} & L̃ ⊗ L \ar[d]^{(−1)^{kk̃}(f̃,k̃)⊗(f,k)} \\ M ⊗ M̃ \ar[r]_{∼}^{τ_{M,M̃}} & M̃ ⊗ M}

Problem 8 Let B be an algebra.

(1) Let A be a linear additive category. Let N ⊆ A be a full additive subcategory. Write Null_{A,N} (X, Y) := { X ⟶^{f} Y : there exists N ∈ Ob(N) and morphisms X ⟶^{u} N ⟶^{v} Y such that f = uv } . Let A/N be the category that has

Ob(A/N) := Ob(A)
A/N(X, Y) := _{A}(X, Y)/ Null_{A,N} (X, Y) for X, Y ∈ Ob(A/N) .

For X ⟶^{f} Y ⟶^{g} Z in A, we define composition of the respective residue classes in A/N by

(f + Null_{A,N} (X, Y)) · (g + Null_{A,N} (Y, Z)) = f · g + Null_{A,N} (X, Z) .

Show that A/N is a linear additive category. Show that A ⟶^{R} A/N is a linear functor with RN ≃ 0 for N ∈ Ob(N).

We often write f̄:= f + Null_{A,N} (X, Y).

Given a linear additive category B and a linear functor A ⟶^{F} B with F N ≃ 0 for N ∈ Ob(N), show that there exists a unique linear functor A/N ⟶^{F̄} B such that F = F̄ ∘ R.

(2) Let A := C(B -Mod) be the category of complexes of B-modules. Let the differential of a complex X ∈ Ob(A) be denoted by d = d_{X} . Let N ⊆ A be the full additive subcategory of split acyclic complexes, i.e. those isomorphic to a complex of the form

⋯ → U^{i−1} ⊕ U^{i} \underrightarrow{(\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix})} U^{i} ⊕ U^{i+1} → …, where U^{i} ∈ Ob A for i ∈ Z.

Show that Null_{A,N} (X, Y) consists of those morphisms of complexes X ⟶^{f} Y for which

there exists a tuple of morphisms (X^{i} ⟶^{h^{i}} Y^{i−1})_{i∈Z} such that

f^{i} = h^{i}d_{Y}^{i−1} + d_{X}^{i}h^{i+1} for i ∈ Z.

Define K(B -Mod) := A/N to be the homotopy category of complexes of B-modules. Write shorthand _{K}(X, Y) := _{K(B -Mod)}(X, Y) for X, Y ∈ Ob(K(B -Mod)) = Ob(C(B -Mod)).

(3) Let M be a B-module. Let P be a projective resolution of M with augmentation ε : P_{0} → M. Let Conc(M) ∈ Ob(C(B -Mod)) have M at position 0, and 0 elsewhere. Let ε̂: P → Conc(M) be the morphism of complexes having entry ε at position 0.

Let Q be a complex consisting of projective B-modules, bounded above. Show that K(Q, ¯ ε̂) : _{K}(Q, P) → _{K}(Q, Conc(M)) is an isomorphism.

(4) Using the universal property from (1), construct a shift functor S on K(B -Mod) such that (SX)^{i} = X^{i+1} and such that d_{SX}^{i} = −d_{X}^{i+1} for i ∈ Z. Show that S is an automorphism.

We also write S^{k} =: (−)^{[k]} for k ∈ Z.

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