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

By Ulrich Gortz, Torsten Wedhorn

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Iii) We have a = d≥0 a ∩ K[X0 , . . , Xn ]d . An ideal satisfying these equivalent conditions is called homogeneous. (c) Show that intersections, sums, products, and radicals of homogeneous ideals are again homogeneous. (d) Show that a homogeneous ideal p ⊆ K[X0 , . . , Xn ] is a prime ideal if and only if f g ∈ p implies f ∈ p or g ∈ p for all homogeneous elements f and g. (e) Show that every homogeneous ideal a K[X0 , . . , Xn ] is contained in the homogeneous ideal (X0 , . . , Xn ). 21.

1) in P1 (k): The quadric of rank 2 consists of two points; in particular it is not irreducible. The quadric of rank 1 consists of a single point. , the solutions of the corresponding equations over R). As a variety it is isomorphic to P1 (k): We can assume it is given as Q = V+ (X0 X2 − X12 ), and then an isomorphism P1 (k) → Q is given by (x0 : x1 ) → (x20 : x0 x1 : x21 ), cf. 30. The quadric of rank 2 is the union of two diﬀerent lines, and the quadric of rank 1 is a line. 3. , if the rank of a matrix of q is maximal.

G. for local rings). Moreover, if ϕ : A → B is a ring homomorphism, the inverse image of a maximal ideal in B is not necessarily a maximal ideal of A while the analogous statement for prime ideals is true. Thus working with prime ideals we obtain a functorial construction. This is the content of the ﬁrst part of this chapter. 10). Therefore we again endow Spec A with ”functions“. More precisely, we deﬁne a sheaf OSpec A of rings on Spec A. Therefore the second part 41 of this chapter will be a short excursion in which we present the necessary notions from the theory of sheaves, which are generalizations of systems of functions.