Maps in cohomology of shifted morphisms of complexes of modules, on classes #
A morphism f : K ⟶ L⟦n⟧ of cochain complexes induces maps K.homology a ⟶ L.homology a' for
n + a = a' (Functor.shiftMap for the shift sequence of the cohomology functor). For complexes
of modules, this map sends the class of a cycle x of K to the class of f x, read in L in
degree a' (CochainComplex.shiftMap_homologyπ_moduleCatCyclesMk).
References #
- M. Kashiwara, P. Schapira, Categories and Sheaves, Chapter 11.
theorem
CochainComplex.shiftMap_homologyπ_moduleCatCyclesMk
{R : Type u_1}
[Ring R]
{K L : CochainComplex (ModuleCat R) ℤ}
{n : ℤ}
(f : K ⟶ L⟦n⟧)
(a a' : ℤ)
(h : n + a = a')
(x : ↑(K.X a))
(hx : (ModuleCat.Hom.hom (K.d a (a + 1))) x = 0)
(y : ↑(L.X a'))
(hy : y = (ModuleCat.Hom.hom (L.shiftFunctorObjXIso n a a' ⋯).hom) ((ModuleCat.Hom.hom (f.f a)) x))
(hy' : (ModuleCat.Hom.hom (L.d a' (a' + 1))) y = 0)
:
(ModuleCat.Hom.hom ((HomologicalComplex.homologyFunctor (ModuleCat R) (ComplexShape.up ℤ) 0).shiftMap f a a' h))
((ModuleCat.Hom.hom (HomologicalComplex.homologyπ K a)) (HomologicalComplex.moduleCatCyclesMk K x (a + 1) ⋯ hx)) = (ModuleCat.Hom.hom (HomologicalComplex.homologyπ L a')) (HomologicalComplex.moduleCatCyclesMk L y (a' + 1) ⋯ hy')
The map in cohomology of a shifted morphism, on classes. For f : K ⟶ L⟦n⟧ and
n + a = a', the induced map K.homology a ⟶ L.homology a' sends the class of a cycle x to the
class of f x, read in L in degree a'.