Faithfully Embedded Closed Subsets

Let T and U be closed subsets of S such that T C U. Recall that T is said to be faithfully embedded in U if, for any two elements y in X and s in yT, each faithful map x from y, z to yU extends faithfully to a bijective map from yT to yxT. 

Lemma 6.2.1 Thin dosed subsets of S are faithfully embedded in S. 

For each element x in yT, we define xx to be the uniquely defined element in yxf, where t stands for the uniquely defined element in T with x yt. Then X is a bijective map from yT to yxT satisfying yx = yx an(i = -W We claim that x is faithful. 

In order to show this we pick two elements v and w in yT, and we denote by t the uniquely determined element in T satisfying w e vt. We have to show that wx vxt. 

The following two propositions show that the property of being faithfully embedded is inherited to closed subsets and to quotient schemes. 

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The second section of this chapter deals with faithfully embedded closed subsets of S. We mainly discuss the question to which extent the property of being faithfully embedded is inherited from given quotient schemes of closed subsets of S to other quotient schemes of closed subsets of S. 

In our last result, we do not need the full strength of the notion of a faithfully embedded closed subset. In order to weaken this concept we fix closed subsets... 

Morphisms are related to faithful maps, which lead naturally to the notion of a faithfully embedded closed subset. Such subsets provide an appropriate language for an attempt to establish so-called recognition theorems. These theorems deal with the question of which schemes are quotient schemes of thin schemes. We shall come back to recognition theorems and their role in scheme theory later in this preface. 

In the sixth chapter, we introduce faithful maps and faithfully embedded closed subsets. In particular, we define a closed subset to be schurian if it is faithfully embedded in itself. This chapter is also the place where we prove the first recognition theorems. 

A closed subset of S will be called schurian if it is faithfully embedded in itself. 

In Section 6.5, we assume S to have finite valency. We shall prove that closed subsets of S which are generated by a single symmetric element of valency 2 are faithfully embedded in S. We also establish the corresponding recognition theorem. After that we shall look at closed subsets of S in which each nonidentity element has valency 2. 

In the last of the seven sections of this chapter, we investigate closed subsets T of S which have finite valency and satisfy O 1 (T) C 0 (T). We shall give a sufficient criterion for T to be faithfully embedded in S. 

Proposition 6.2.2 says that the property of being faithfully embedded is inherited to closed subsets. Given closed subsets T and U in S with T C U, we shall now ask ourselves under which hypotheses the property of being faithfully embedded in S can be lifted from T to U. 

Theorem 6.2.8 LetT and U be closed subsets of S such thatT C U. Assume that U contains a closed subset V of S which covers T. Assume further that h G U 1 ThT C V and that V is faithfully T-embedded in S. Then, if U//T is faithfully embedded in S//T, U is faithfully embedded in S. 

Theorem 6.3.1, together with Proposition 6.2.2, also shows that, if S is schurian, each closed subset of S is faithfully embedded in S. Lemma 6.2.1 says that thin closed subsets are faithfully embedded in S. The set of all closed... 

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