Commutative algebra [Lecture notes] by Jason P. Bell

By Jason P. Bell

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Example text

6 1) (2) t here is a unique h : K ----t K such t hat k 0 h = k. T hus h = i' 0 i = 1[(. Similarly, i 0 i' = l K , . Consequent ly, i is an isomorphism. 8 Theorem. Let C be a topological construct. If (X , ~) , (Y,1)) are C- objects and j , 9 : (X ,O ----t (Y , 1)) C-morphism s, th en th e f ollowing is valid: a) Let K = {x E X : f (x ) = g(x )} be endowed with the initi al Cr-siru cture ~ K with respect to th e in clusi on map i : K ----t X . T hen i : (K , ~K) ----t (X,~) is th e equaliz er of j and g .

B) a) (indirect) . Suppose t hat f is not surjective. Then t here is a y' E Y such t hat y' if- I[X] . l y E J[X] 1 otherwi se are C-morphisms (cf. 2. 1)) such that 1 0 f = 8 0 f and 1 =f:. is . f is not an epimorphism . (3) If f is surjective, t hen for every y E Y t here is an z E X such t hat f (x ) = y . e. a = {3. T husf is an epimorp hism. In a topological cons truct C a C- mo rphism f : (X,O ---t 40 CHAPTER 1. TOPOLOGICAL CONSTR UCTS (Y, Tf) is an a) extrem al monomorphism if and only if it is an embedding, i.

All concepts are du alized and the role of bireflector s (dually: bicoreflectors) in t he realm of topological const ruc ts is clarified. Concerning th e historical development of all t hese categorical concepts and resul ts th e inte reste d read er in referred t o an article of H. E. Strecker [73] .

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