Ivancevic_Applied-Diff-Geom

402 Applied Differential Geometry: A Modern Introductionfor every n ∈ Z. When A n = 0 for n < 0, one speaks of cochain complexes.The d n are called coboundary operators.The morphisms of the category S • (C) are sequences f • = (f n ) : A • →B • where, for each n ∈ Z, f n : A n → B n is a morphism of C, and in thediagram· · · −→ A n−1 d n−1 ✲ An d n ✲ A n+1 −→ · · ·f n−1 | ↓ f n | ↓ f n+1 | ↓ (3.220)· · · −→ B n−1 d n−1 ✲ Bn d n ✲ B n+1 −→ · · ·all squares are commutative; one says the f n commute with the coboundaryoperators. One has Im d n+1 ⊂ Ker d n ⊂ A n for every n ∈ Z; the quotientH n (A • ) = Ker d n / Im d n+1 is called the nth cohomology object of A • . From(3.220) it follows that there is a morphismdeduced canonically from f • , andH n (f • ) : H n (A • ) → H n (B • )(A • , f • ) ⇒ (H n (A • ), H n (f • ))is a covariant functor from S • (C) to C.The cohomology exact sequence: if three cochain complexes A • , B • , C •are elements of a short exact sequence of morphisms0 −→ A • −→ B • −→ C • −→ 0then there exists an infinite sequence of canonically defined morphismsd n : H n (C • ) → H n−1 (A • ) such that the sequence· · · −→ H n (A • ) −→ H n (B • ) −→ H n (C • ) −→ H n−1 (A • ) −→ · · ·is exact, that is the image of each homomorphism in the sequence is exactlythe kernel of the next one.The dual to the category S • (C) is the category of S • (C) of generalizedchain complexes. Its objects and morphisms are get by formal inversion ofall arrows and lowering all indices.