By P. J. Hilton, U. Stammbach (auth.)
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Additional info for A Course in Homological Algebra
We shall show that (j is essential. e. let He H <;; I where the 1. Modules 38 first inclusion is strict. By the maximality of B the intersection H nE is non-trivial, hence HIB na E is non-trivial. It follows that a is essential. By the first part of the proof E admits no proper essential monomorphism, whence it follows that a: E'::::'" liB is an isomorphism. The sequence B>---+I~E now splits by the embedding of E in I. 3. 3. Let E 1 , E2 be two maximal essential extensions of A contained in injective modules I}> 12, Then El ~ E2 and every injective module I containing A also contains a submodule isomorphic to El .
B is an essential extension of A if and only if, for every o=l= b E B, there exists A E A such that AbE A and Ab =l= O. Proof. Let B be an essential extension of A, and let H be the submodule generated by bE B. e. there exists A E A such that 0 =l= Ab E A. Conversely, let H be a non-trivial submodule of B. For 0 =l= h E H there exists A E A such that 0 =l= Ah E A. Therefore H nA =l= 0, and B is an essential extension of A. 0 Let A be a submodule ofa A-module M. Consider the set rp of essential extensions of A, contained in M.
3. Show that the category of groups has a generator. 4. Show that, in the category of groups, there is a one-one correspondence between elements of G and morphisms Z-+G. 5. e. each space has a base-point and morphisms are to preserve base-points, see [21J). 6. 7. Show that if a category has a zero object, then every initial object, and every terminal object, is isomorphic to that zero object. Deduce that the category of sets has no zero object. 2. Functors Within a category (t we have the morphism sets (t(X, Y) which serve to establish connections between different objects of the category.
A Course in Homological Algebra by P. J. Hilton, U. Stammbach (auth.)