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12) To construct h1 we observe that our assumption ϕ = 0 implies that the kernel of I 0 −→ I 1 is contained in the kernel of the vertical arrow I 0 −→ J 0 . Since J 0 is injective we can construct h1 : I 1 −→ J 0 which produces a commutative diagram I0 ... ... ... .. . i ........................................... .... .... . .. .... 1 ... ......... I1 h J0 Now we modify the given vertical arrow I 1 −→ J 1 by subtracting the composition of h1 and the horizontal arrow I 0 −→ J 1 . To this modiﬁed arrow we can apply the previous argument and it becomes clear how to construct these hν by induction.

4 The Functors Ext and Tor 31 (iii) This map is universal: If we have another Ψ : N × M −→ X with an abelian group X which satisﬁes (i) and (ii) then we can ﬁnd a ϕ : N ⊗R M −→ X such that Ψ = ϕ ◦ Ψ. It is easy to construct N ⊗R M , we form the free abelian group which is generated by pairs (n,m) ∈ N × M and divide by the subgroup generated by elements of the form (n1 + n2 ,m) − (n1 ,m) − (n2 ,m) (n,m1 + m2 ) − (n,m1 ) − (n,m2 ) (nr,m) − (n,rm). If our ring R is commutative then we can give N ⊗R M the structure of an R-module: We simply deﬁne r(n ⊗ m) = nr ⊗ m = n ⊗ rm.

E. we get a map ∞ ◦h : CY∞ (V ) −→ CX (U ) (resp. ◦ h : OY (V ) −→ OX (U )). A better formulation is obtained if we introduce the sheaf (see the following sections on f∗ ,f ∗ and the adjointness formula) h∗ (CY∞ ) on X: For any open subset U ⊂ X the space of section h∗ (CY∞ )(U ) consists of functions f : U −→ which have the following property: For any point p ∈ U we can ﬁnd a neighborhood Up of p and an open set Vh(p) ⊂ Y such ∞ that h(Up ) ⊂ Vh(p) and we can ﬁnd a section f ∈ CX (Vh(p) ) so that f = f ◦ h.