[Merged by Bors] - feat: the mapping cone of a monomorphism, up to a quasi-isomorphism #13675
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PR summaryImport changesNo significant changes to the import graph
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This was referenced Jun 11, 2024
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…13675) In this PR, given a short exact sequence `S` in the category of cochain complexes in an abelian category, we compare the homology sequence of `S`, and the homology sequence of the homological functor on the homotopy category applied to the distinguished triangle of the mapping cone of the monomorphism `S.f`. In particular, up to a quasi-isomorphism, the mapping cone of `S.f` identifies to `S.X₃`. In future PRs, this shall be used in order to attach a distinguished triangle in the derived category to `S` (#13742), and compare the homology sequence of `S` and the homology sequence attached to this distinguished triangle. Co-authored-by: Joël Riou <[email protected]>
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…13675) In this PR, given a short exact sequence `S` in the category of cochain complexes in an abelian category, we compare the homology sequence of `S`, and the homology sequence of the homological functor on the homotopy category applied to the distinguished triangle of the mapping cone of the monomorphism `S.f`. In particular, up to a quasi-isomorphism, the mapping cone of `S.f` identifies to `S.X₃`. In future PRs, this shall be used in order to attach a distinguished triangle in the derived category to `S` (#13742), and compare the homology sequence of `S` and the homology sequence attached to this distinguished triangle. Co-authored-by: Joël Riou <[email protected]>
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…13675) In this PR, given a short exact sequence `S` in the category of cochain complexes in an abelian category, we compare the homology sequence of `S`, and the homology sequence of the homological functor on the homotopy category applied to the distinguished triangle of the mapping cone of the monomorphism `S.f`. In particular, up to a quasi-isomorphism, the mapping cone of `S.f` identifies to `S.X₃`. In future PRs, this shall be used in order to attach a distinguished triangle in the derived category to `S` (#13742), and compare the homology sequence of `S` and the homology sequence attached to this distinguished triangle. Co-authored-by: Joël Riou <[email protected]>
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…13675) In this PR, given a short exact sequence `S` in the category of cochain complexes in an abelian category, we compare the homology sequence of `S`, and the homology sequence of the homological functor on the homotopy category applied to the distinguished triangle of the mapping cone of the monomorphism `S.f`. In particular, up to a quasi-isomorphism, the mapping cone of `S.f` identifies to `S.X₃`. In future PRs, this shall be used in order to attach a distinguished triangle in the derived category to `S` (#13742), and compare the homology sequence of `S` and the homology sequence attached to this distinguished triangle. Co-authored-by: Joël Riou <[email protected]>
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In this PR, given a short exact sequence
Sin the category of cochain complexes in an abelian category, we compare the homology sequence ofS, and the homology sequence of the homological functor on the homotopy category applied to the distinguished triangle of the mapping cone of the monomorphismS.f. In particular, up to a quasi-isomorphism, the mapping cone ofS.fidentifies toS.X₃.In future PRs, this shall be used in order to attach a distinguished triangle in the derived category to
S(#13742), and compare the homology sequence ofSand the homology sequence attached to this distinguished triangle.