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**Example text**

B the same way we did before. Given a morphism W A ! a/b; a 2 A; b 2 B: Also, given an A-module L, we can construct the B-module L ˝A B. A/, f W Am ! B/, and Gr. f / W B m D Am ˝A B ! 9. For the property (1), namely the fact that Gr is local, we send the reader to [23], Ch. I, §1. For the property (2), we have to give explicitly a cover by open affine subfunctors. i1 ; : : : ; ir /, 1 Ä i1 < < ir Ä m, and the map I W Ar ! y1 ; : : : ; ym /, with yik D xk for k D 1; : : : ; m and yj D 0 otherwise.

It is customary to denote a differentiable manifold and its underlying topological space with the same symbol. In this section, we follow this convention, however starting from the next section, we shall mark the difference between the manifold M and its underlying topological space jM j. We shall also do the same for algebraic varieties. Given two manifolds M and N , and the respective sheaves of smooth functions 1 CM and CN1 , a morphism f from M to N , viewed as ringed spaces, is a morphism jf j W M !

The functor of points is a categorical device to bring back our attention to the points of a scheme; however the notion of point needs to be suitably generalized to go beyond the points of the topological space underlying the scheme. Grothendieck’s idea behind the definition of the functor of points associated to a scheme is the following. If X is a scheme, for each commutative ring A, we can define the set of the A-points of X in analogy to the way the classical geometers used to define the rational or integral points on a variety.