Homomorphism Theorem

In the third section, we shall prove a Homomorphism Theorem and two Isomorphism Theorems for schemes of finite valency. All three of these results naturally generalize the finite versions of Emmy Noether s corresponding theorems for groups. 

The following theorem is called the Homomorphism 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. 

Let be a homomorphism from a group G to a group H. Consequences of the so-called homomorphism theorem are ... 

In Proposition 4.8 and in Section 6.3 we will use the Spectral Theorem to simplify calculations in SU (2), In Section 6.6 we will use the homomorphism T between S(7(2) and SO(3) to make some calculations about SO(3) that would be harder to make directly. 

Let us start by stating the definitions and theorems we use from topology. We will use the notion of local homomorphisms. 

Proof. We shall use the description of (C2) in terms of matrices given in Theorem 1.14. Suppose Z is a T-invariant O-dimensional subscheme in (C2), and corresponds to a triple of matrices (Bi, B2, i). Recall that it is given as follows Define a iV-dimensional vector space V as H°(Oz), and a 1-dimensional vector space W. Then the multiplications of coordinate functions z, z2 6 C define endomorphisms Bi, B2. The natural map Oc2 —> Oz defines a linear map i W V. Prom this construction, V is a T-module, and W is the trivial T-module. The pair (Bi,B2) is T-equivariant, if it is considered as an element in Hom(V, Q V), where Q is 2-dimensional representation given by the inclusion T C SU(2). (This follows from that (Zi,z2) is an element in Q.) And i is also a T-equi variant homomorphism W —> V. 

Theorem 43.1 ([DeSt98, GHZ02, Gra03]) Any (r, q)-polycycle P admits a cell-homomorphism into (r, q) and such homomorphism is defined uniquely by a flag and its image. 

Example. Let

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A 1.2) Theorem. Let f (k,n) — (R,m) be a local homomorphism of local noetherlan rings containing a field K isomorphic to their residue fields k/n s R/m = K. Consider the following conditions ... 

Theorem (Yoneda s Lemma). Let E and F be (set-valued) Junctors represented by k-algebras A and B. The natural maps E- F correspond to k-algebra homomorphisms B- A. 

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