[Merged by Bors] - feat(Algebra/Homology): right shifting cochains #8937
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| /-- The additive equivalence `Cochain K L n ≃+ Cochain K L⟦a⟧ n'` when `n' + a = n`. -/ | ||
| @[simps] | ||
| def rightShiftAddEquiv (n a n' : ℤ) (hn' : n' + a = n) : | ||
| Cochain K L n ≃+ Cochain K (L⟦a⟧) n' where | ||
| toFun γ := γ.rightShift a n' hn' | ||
| invFun γ := γ.rightUnshift n hn' | ||
| left_inv γ := by simp | ||
| right_inv γ := by simp | ||
| map_add' γ γ' := by simp |
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Please consider moving this up a bit higher. And then you can deduce all the lemmas about preserving 0 and - etc, by just restating the goal in terms of this AddEquiv, and applying some generic map_zero or map_neg lemma.
Also... should this even be upgraded to a linear equiv?
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Thanks for the suggestion! I have deduced the lemmas that can be deduced from the additive equivalence, and I have also made a linear equivalence version.
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In this PR, we study the behaviour of cochains (of the complex of homomorphisms) with respect to shifts (on the target). In particular, we obtain an additive equivalence `rightShiftAddEquiv K L n a n' h : Cochain K L n ≃+ Cochain K L⟦a⟧ n'` when `n' + a = n`. Co-authored-by: Joël Riou <37772949+joelriou@users.noreply.github.com>
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Pull request successfully merged into master. Build succeeded! And happy new year! 🎉 |
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In this PR, we study the behaviour of cochains (of the complex of homomorphisms) with respect to shifts (on the target). In particular, we obtain an additive equivalence
rightShiftAddEquiv K L n a n' h : Cochain K L n ≃+ Cochain K L⟦a⟧ n'whenn' + a = n.A similar PR shall be necessary for the study of the shift on the source, but that will be more intricate as there will be signs in the definitions...