Mannheimer Manuskripte 177 gk-mp-9403/3 SOME CONCEPTS OF ...

CONCEPTS OF MODERN ALGEBRAIC GEOMETRY 27has maps between the objects. Let Φ : R → S be a ring homomorphism. If I isany ideal of S, then Φ −1 (I) is an ideal of R. The reader is advised to check that ifP is prime then Φ −1 (P) is again prime. Hence, Φ ∗ : P ↦→ Φ −1 (P) is a well-definedmap Spec(S) → Spec(R). Indeed, it is even continuous because the pre-image of aclosed set is again closed. Let X = (Spec(S), O S ) and Y = (Spec(R), O R ) be twoaffine schemes. The map Φ induces also a map on the level of the structure sheavesΦ ∗ : O R → O S . The pair (Φ ∗ ,Φ ∗ ) of maps fulfills certain compatibility conditions whichmakes them to a homomorphism of schemes.We will not work with schemes in general later on but let me give at least for completenessthe definition here.Definition. (a) A scheme is a pair X = (|X|, O X ) consisting of a topological space|X| and a sheaf O X of rings on X, such that X is locally isomorphic to affine schemes(Spec(R), O R ). This says that for every point x ∈ X there is an open set U containing x,and a ring R (it may depend on the point x) such that the affine scheme (Spec(R), O R )is isomorphic to the scheme (U, O X|U ). In other words there is a homeomorphismΨ : U → Spec(R) such that there is an isomorphism of sheavesΨ # : O R∼ = Ψ∗ (O X |U ) .Here the sheaf Ψ ∗ (O X |U ) is defined to be the sheaf on Spec(R) given by the assignmentΨ ∗ (O X |U )(W) := O X (Ψ −1 (W)),for every open set W ⊆ Spec(R).(b) A scheme is called an affine scheme if it is globally isomorphic to an affine scheme(Spec(R), O R ) associated to a ring R.Fact. The category of affine schemes is equivalent to the category of commutative ringswith unit with the arrows (representing the maps) reversed.There are other important concepts in this theory. First, there is the concept of ascheme over another scheme. This is the right context to describe families of schemes.Only within this framework it is possible to make such useful things precise as degenerations,moduli spaces etc. Note that every affine scheme is in a natural way a schemeover Spec(Z), because for every ring R we have the natural map Z → R, n ↦→ n · 1 .Taking the dual map introduced above we obtain a homomorphism of schemes.If R is a K−algebra with K a field then we have the map K → R, α ↦→ α · 1, which is aring homomorphism. Hence, we always obtain a map: Spec(R) → Spec(K) = ({0},K).By considering the coordinate ring R(V ) of an affine variety V over a fixed algebraicallyclosed field K and assigning to it the affine scheme Spec(R(V )) we obtain a functorfrom the category of varieties over K to the category of schemes over K. The schemes