First Isomorphism Theorem

The following theorem is called the First Isomorphism Theorem for schemes. 

Note that the Homomorphism Theorem and the First Isomorphism Theorem deal with factorizations over arbitrary closed subsets, whereas the Second Isomorphism Theorem deals with factorizations over normal closed subsets. 

The Homomorphism Theorem and the two Isomorphism Theorems were first proved in [35]. The thin case was already proved in 1929 by Emmy Noether cf. [32 I. 2],... 

The first section of this chapter provides general observations on modules over associative rings with 1. The collection includes the Homomorphism Theorem and the Isomorphism Theorem for modules over associative rings with 1. 

There are only two groups of order 4 that are not isomorphous and so have different multiplication tables. Derive the multiplication tables of these two groups, G4 and G4. [Hints. First derive the multiplication table of the cyclic group of order 4. Call this group G4. How many elements of G4 are equal to their inverse Now try to construct further groups in which a different number of elements are equal to their own inverse. Observe the rearrangement theorem.]... 

Proof. First we assume that / is 6tale near x, and prove that f is an isomorphism. This part can be reduced to Theorem 2 by making a fibre product ... 

To prove Theorem 1 all 3 of the methods mentioned in the introduction can be used. There is a projective method, based on Seven s idea of projecting cones. Or using complete local rings, we can reduce the UFD property for o (x closed) to the UFD property for o. which is isomorphic by Th. 10, 6, to k [[fi,..., tn]]. The most fax-reaching method is the cohomological one, due to Auslander and Buchsbaum, by which it can be proven that all regular local rings are UFD s. We shall present the first two methods. For the last, cf. Zaxiski-Samuel, appendix, vol. 2. 

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