## Algebraic geometry 1: Schemes by Ulrich Gortz, Torsten Wedhorn

By Ulrich Gortz, Torsten Wedhorn

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Let (X, OX ) be a prevariety. The topological space X is noetherian (in particular quasi-compact) and irreducible. Proof. 19. 49. (Comparison with diﬀerential/complex manifolds) In diﬀerential geometry (resp. complex geometry) the notion of a diﬀerentiable manifold (resp. a complex manifold) is often deﬁned by charts with diﬀerentiable (resp. holomorphic) transition maps. This is problematic in our situation because we cannot consider open subsets of aﬃne algebraic sets again as aﬃne algebraic sets.

Of course we do not obtain all spaces with functions in this way. We will now deﬁne prevarieties as those connected spaces with functions that can be glued together from ﬁnitely many spaces with functions attached to irreducible aﬃne algebraic sets. 49). 15) Deﬁnition of prevarieties. We call a space with functions (X, OX ) connected , if the underlying topological space X is connected. 46. (1) An aﬃne variety is a space with functions that is isomorphic to a space with functions associated to an irreducible aﬃne algebraic set.

Fm ) we have X = V (I(X)) = C(Z). In particular we see that for every closed subset Z ⊆ Pn (k) there exist homogeneous polynomials f1 , . . , fr ∈ k[X0 , . . , Xn ] such that Z = V+ (f1 , . . , fm ). 22) Change of coordinates in projective space. ,n ∈ GLn+1 (k) be an invertible (n + 1) × (n + 1)-matrix. The map k n+1 → k n+1 described by A maps one-dimensional subspaces to one-dimensional subspaces and induces a map Pn (k) → Pn (k). It is given by n (x0 : · · · : xn ) → ( n a0i xi : · · · : i=0 ani xi ) i=0 31 and we obtain a morphism of prevarieties which we denote by ϕA .