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By Giuliano Sorani

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Extra info for An introduction to real and complex manifolds.

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F Proof (a) ⇒ (b): For A ⇒ B consider the fiber product diagram g K p1  A p2 G (id,f ) G A  (id,g) A×B p2 Then K → A is the diﬀerence kernel of f and g. In fact, consider h : C → A with f h = gh. If one has (id, f )h = (h, f h) = (h, gh) = (id, g)h , 28 then there exists a uniquely determined morphism ˜:C→K h with ˜ = h = p2 h ˜. p1 h Furthermore, if q1 : A×B → A is the first projection, then p1 = q1 (id, f )p1 = q1 (id, g)p2 = p2 . Hence we obtain a uniquely determined morphism K → A with the desired property.

A1 47 G ? BA2 G 0. If we continue like this, we obtain a naive double complex I ∗,∗ with a co-argumentation A∗ → I ∗,∗ which is a resolution of A∗ in the category of complexes, and where the first diﬀerential dI is given by the composition p,∗ dp,∗ ↠ BI p+1,∗ → ZI p+1,∗ → I p+1,∗ , I : I so that BI p+1,∗ is really the image of dpI i and ZI p,∗ is the kernel of dp+1,∗ , and moreover I p,∗ p,∗ p,∗ p ∗,∗ HI = ZI /BI = H (I , dI ). 7. Let F : A → B and G : B → C be left exact functors between abelian categories with enough injectives, where F maps injectives to G-acyclic objects.

P+1,q dII G C p+1,q y G ... G C p+1,q−1 y G ... dp,q−1 II ... e. dp,q+1 dp,q dI = 0 for all p, q ∈ Z. I II + dII 44 (c) The double complex associated to a naive double complex as in (a) is the double complex p p,q where dp,q II is replaced by (−1) dII . e. d|C p,q = dp,q I + dII . The following construction is very important for the treatment and definition of spectral sequences. 9 Let T ot(I ∗,∗ ) be the total complex associated to the double complex, which is associated to the naive double complex I ∗,∗ .