M. Künzer A_{∞}-categories, WS 16/17

Sheet 6

Problem 15 Let Z be a grading category. Let L and M be Z-graded modules.

Let L ⟶^{f} M be a shift-graded linear map of degree a ∈ Z.

(1) Let K be the Z-graded module defined by K^{z} := Kern(L^{z} ⟶^{f} M^{z[a]}) for z ∈ Mor(Z). Let K ⟶^{i} L be the shift-graded linear map given by inclusion, which is of degree 0.

Suppose given a Z-graded module T and a shift-graded linear map T ⟶^{t} L of degree d such that tf = 0.

Show that there exist a unique shift-graded linear map T ⟶^{ť} K of degree d such that ťi = t.

(2) Suppose R to be a field. Suppose f to be piecewise surjective.

Show that there exists a piecewise injective shift-graded linear map L ⟵^{g} M of degree −a such that gf = id_{M} .

Problem 16 Let Z be a grading category. Let n ∈ Z_{⩾1} . Let K_{i}, L_{i} and M_{i} be Z-graded modules for i ∈ [1, n]. Let K_{i} ⟶^{u_{i}} L_{i} be a shift-graded linear map of degree c_{i} ∈ Z for i ∈ [1, n].

Let L_{i} ⟶^{f_{i}} M_{i} be a piecewise surjective shift-graded linear map of degree a_{i} ∈ Z for i ∈ [1, n].

(1) Show that ⨂_{i∈[1,n]} f_{i} is piecewise surjective.

(2) For i ∈ [1, n], suppose that K_{i} ⟶^{u_{i}} L_{i} ⟶^{f_{i}} M_{i} to be exact at L_{i}, i.e. suppose K_{i}^{z[−c_{i}]} ⟶^{u_{i}} L_{i}^{z} ⟶^{f_{i}} M_{i}^{z[a_{i}]} to be exact at L_{i}^{z} for z ∈ Mor(Z).

Suppose given a Z-graded module T and a shift-graded linear map ⨂_{i∈[1,n]} L_{i} ⟶^{t} T of degree d.

Suppose that (id^{⊗j−1} ⊗u_{j} ⊗ id^{⊗n−j})t = 0 for j ∈ [1, n].

Show that there exists a unique shift-graded linear map ⨂_{i∈[1,n]} M_{i} ⟶^{t̃} T of degree d − ∑_{i∈[1,n]} a_{i} such that (⨂_{i∈[1,n]} f_{i})t̃= t.

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