M. Künzer A_{∞}-categories, WS 16/17

Sheet 5

Problem 13 Let B be an algebra.

Suppose given a diagram X′ ⟶^{i} X ⟶^{r} X′′ in C(B -Mod) such that X^{′k} ⟶^{i^{k}} X^{k} ⟶^{r^{k}} X^{′′k} is short exact for k ∈ Z. Such a diagram is called a short exact sequence of complexes in B.

(1) Suppose given T ⟶^{f} X in C(B -Mod) such that fr = 0. Show that there exists a unique morphism T ⟶^{f′} X′ such that f′i = f.

(2) Suppose given X ⟶^{g} T in C(B -Mod) such that ig = 0. Show that there exists a unique morphism X′′ ⟶^{g′′} T such that rg′′ = g.

(3) A Z-graded B-module M is a tuple M = (M^{z})_{z∈Z} of B-modules M^{z}. A graded B-linear map f : L → M between Z-graded B-modules is a tuple f = (f^{z})_{z∈Z} of B-linear maps f^{z}. Write B-Z-grad for the category of Z-graded B-modules and graded B-linear maps. Construct an additive functor H : C(B -Mod) → B-Z-grad having

(HX)^{k} = Kern(d^{k})/ Im(d^{k−1})

for a complex X with differential d = (X^{k} ⟶^{d^{k}} X^{k+1})_{k}.

For Y ⟶^{f} Z in C(B -Mod), we often write ((HY)^{k} ⟶^{(Hf)^{k}} (HZ)^{k}) =: (H^{k}Y ⟶^{H^{k}f} H^{k}Z).

(4) Construct a B-linear map H^{k}X′′ ⟶^{γ_{(i,r)}^{k}} H^{k+1}X′ for k ∈ Z, called connector of the given short exact sequence X′ ⟶^{i} X ⟶^{r} X′′, subject to the following conditions (i, ii).

(i) The sequence

… → H^{k}X′ ⟶^{H^{k}i} H^{k}X ⟶^{H^{k}r} H^{k}X′′ ⟶^{γ_{(i,r)}^{k}} H^{k+1}X′ ⟶^{H^{k+1}i} H^{k+1}X ⟶^{H^{k+1}r} H^{k+1}X′′ → …

is exact at each position. (ii) Given a morphism of short exact sequences, i.e. a commutative diagram

\xymatrix{X′ \ar[r]^{i} \ar[d]^{f′} & X \ar[r]^{r} \ar[d]^{f} & X′′ \ar[d]^{f′′} \\ Y′ \ar[r]^{j} & Y \ar[r]^{s} & Y′′}

in C(B -Mod) with (i, r) and (j, s) short exact, we get, for k ∈ Z, the commutative quadrangle

\xymatrix{H^{k}X′′ \ar[r]^{γ_{(i,r)}^{k}} \ar[d]^{H^{k}f′′} & H^{k+1}X′ \ar[d]^{H^{k+1}f′} \\ H^{k}Y′′ \ar[r]^{γ_{(j,s)}^{k}} & H^{k+1}Y′ .}

p.t.o.

Problem 14 Suppose given an algebra B. Suppose given n ⩾ 1. Suppose given X_{s} ∈ Ob C(B -Mod) for s ∈ [1, n]. Abbreviate \underline{X} := (X_{s})_{s∈[1,n]} . Abbreviate Z := Z × [1, n]^{×2}, C := C(B -Mod) and K := K(B -Mod). Consider the Z-graded module Z Hom_{B}(\underline{X}) having, for z ∈ Mor(Z),

(Z Hom_{B}(\underline{X}))^{z} := Kern((m_{1}^{Hom_{B}(\underline{X})})^{z}) .

Consider the Z-graded module H Hom_{B}(\underline{X}), having, for z ∈ Mor(Z),

(H Hom_{B}(\underline{X}))^{z} := Kern((m_{1}^{Hom_{B}(\underline{X})})^{z})/ Im((m_{1}^{Hom_{B}(\underline{X})})^{z[−1]}) .

(1) Show that (Z Hom_{B}(\underline{X}))^{(j,(s,t))} = _{C}(X_{s}, X_{t}^{[j]}) for (j, (s, t)) ∈ Mor(Z).

(2) Show that (H Hom_{B}(\underline{X}))^{(j,(s,t))} = _{K}(X_{s}, X_{t}^{[j]}) for (j, (s, t)) ∈ Mor(Z).

(3) Show that m_{2}^{Hom_{B}(\underline{X})} induces a map m_{2}^{H Hom_{B}(\underline{X}))} : H Hom_{B}(\underline{X})^{⊗2} → H Hom_{B}(\underline{X}) that maps [f] ⊗ [g] to [f · g] for each composable pair of morphisms (f, g) in C, where we use brackets to denote residue classes of morphisms of C in K.

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