Module homomorphism

Sometimes, in order to emphasize the underlying ring, one says D-module homomorphism or D-homomorphism instead of homomorphism. 

To verify the second property note that the composition of / with continuous maps corresponding to singular simplices yields group homomorphisms / Cj [X TZ) —> Ci Y TZ), for alH > 0. It is easy to see that these homomorphisms commute with the boundary operators, and therefore induce the 7 .-module homomorphisms mentioned above. 

Let us now return to the simplicial context. For any trisp A, the algebraic structure introduced in Subsection 3.2.3, and more generally in Subsection 3.4.2, can be summarized as a sequence of 72 -modules and 72 -module homomorphisms... 

Suppose we have a pair (Af,P) where M. is a finite locally free sheaf of Os-modules and F — M a homomorphism of Os-modules. Put... 

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. 

If moreover 8 consists of flat R-modules then for every R-module M the homomorphism - induces isomorphisms in... 

We will call "minimal" a surjective homomorphism u F - + N of finitely generated R-modules such that F is free and kertu) c m F. It is elementary that such a minimal homomorphism exists for every finitely generated N We will denote B"(X ) lm(f ) and Z"(K ) ker(fn) for every cohomological complex (K, f )... 

It easy to verify that this is a homomorphism of 5-modules. 

A2.5) Lemma. For every homomorphism of k-algebras f 5 —> R and for every R-module 1 there is an exact sequence of 5-modules ... 

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. 

The D-modules M and M are called isomorphic, if there exists a bijective homomorphism from M to M. Sometimes, we shall write M = M in order to indicate that M and M are isomorphic. 

The following theorem is called the Homomorphism Theorem for modules over associative rings with 1. 

Let G be a group functor, X a set functor. An action of G on X is a natural map G x X - X such that the individual maps G(R) x X(R)- X(R) are group actions. These will come up later for general X, but the only case of interest now is X(R)= V R, where V is a fixed k-module. If the action of G(R) here is also R-linear, we say we have a linear representation of G on V. The functor GLV(R) = Aut (F R) is a group functor a linear representation of G on V clearly assigns an automorphism to each g and is thus the same thing as a homomorphism G - GLK. If V is a finitely generated free module, then in any fixed basis automorphisms correspond to invertible matrices, and linear representations are maps to GL . 

This is primarily a technical chapter introducing another algebraic tool. We will use it at once to complete the proof of the smoothness theorem (11.6) and then draw on it throughout the rest of the book. To begin, we call a ring homomorphism A - B flat if, whenever M - JV is an injection of /1-modules, then MaB- N<8>aB is also an injection. For example, any localization A->S 1A is flat. Indeed, an element m a/s in M S = S lM is zero... 

Induction now shows that we can take any polynomial in the f with coefficient in (p0(A) and reduce it to have all exponents less than q. Hence A is a finitely generated module over B = cp0 A). This implies first of all that under A - B the dimension cannot go down. But since G is connected, A modulo its nilradical is a domain (6.6), and from (12.4) we see then that the kernel of k. Let M be the kernel, a maximal ideal of B. As B injects into A, we know BM injects into AM, and thus Am is a nontrivial finitely generated BM-module. By Nakayama s lemma then MAm Am, and so MA A. Any homomorphism x A- A/MA - fc then satisfies q>(x) = y. ... 

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