By Ulrich Gortz, Torsten Wedhorn

ISBN-10: 3834806765

ISBN-13: 9783834806765

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**Additional info for Algebraic geometry 1: Schemes**

**Sample text**

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. Equipped with this machinery we can construct OSpec A in the third part of this chapter. Topological spaces endowed with sheaves of rings are called ringed spaces. In fact, (Spec A, OSpec A ) will be always in the subcategory of so-called locally ringed spaces. Locally ringed spaces isomorphic to (Spec A, OSpec A ) will be called aﬃne schemes, and we will show that A → (Spec A, OSpec A ) deﬁnes an anti-equivalence from the category of rings with the category of aﬃne schemes.

This leads us to the following deﬁnition. 2. Let A be a ring. The set Spec A of all prime ideals of A with the topology whose closed sets are the sets V (a), where a runs through the set of ideals of A, is called the prime spectrum of A or simply the spectrum of A. The topology thus deﬁned is called the Zariski topology on Spec A. 42 2 Spectrum of a Ring If x is a point in Spec A, we will often write px instead of x when we think of x as a prime ideal of A. 1) where we considered the case A = k[T1 , .

2) D(f ) ∩ D(g) = D(f g). 4. Let (fi ) be a family of elements in A and let g ∈ A. Then D(g) ⊆ i D(fi ) if and only if there exists an integer n > 0 such that g n is contained in the ideal a generated by the fi . Proof. 3. Applying this to g = 1 it follows that (D(fi ))i is a covering of Spec A if and only if the ideal generated by the fi is equal to A. 5. Let A be a ring. The principal open subsets D(f ) for f ∈ A form a basis of the topology of Spec A. 22). In particular, the space Spec A is quasi-compact.

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

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