By Eduard Casas-Alvero, Gerald E. Welters, Sebastian Xambo-Descamps

ISBN-10: 3540152326

ISBN-13: 9783540152323

**Read Online or Download Algebraic Geometry. Proc. conf. Sitges (Barcelona), 1983 PDF**

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**Extra resources for Algebraic Geometry. Proc. conf. Sitges (Barcelona), 1983**

**Sample text**

9 1. Every ideal in a finitely generated monoid is finitely generated (as an ideal). 2. Every exact submonoid of a fine (resp. saturated) monoid is fine (resp. saturated). 3. A face of an integral monoid is an exact submonoid. Every face of a fine monoid is finitely generated (as a monoid), and monogenic (as a face). 4. 3) of a fine monoid (resp. saturated) is fine (resp. saturated). 5. The equalizer of two maps of integral monoids P → M is an exact submonoid of P × P . The equalizer of two maps from a fine (resp.

Our assumption implies that F + Ggp is a facet of P + Ggp , and hence (P + Ggp )/(F + Ggp ) is a one-dimensional sharp monoid. Since P /(F +G) is also one-dimensional, the map P /(F +G) → (P +Ggp )/(F +Ggp ) is almost surjective. This means that for every p ∈ P , there is a positive m such that mp belongs to P + F gp + Ggp . But this implies that φ(p) ≥ 0 for every p, and hence that φ ∈ C(P )∨ . We conclude that C(P )∨ = C(P )∨ and hence that C(P ) = C(P ). Hence for every p ∈ P , there exists a positive integer m such that mp ∈ F ⊕ Q.

Let P be a fine monoid, let S := Spec P with its Zariski topology, and let p be a point of S. 9) says that there exists an f ∈ P such that f = F . Then {p : p ∈ {p }− } = SF := {p : F ∩ p = ∅} = Sf := {p : f ∈ p } 2. CONVEXITY, FINITENESS, AND DUALITY 47 is open in S. Thus the set of generizations of each point is open, and hence a subset of S is open if and only if it is stable under generization. This shows that the topology of S is entirely determined by the order relation among the primes of P .