# Algebra. A graduate course by Isaacs I.M.

By Isaacs I.M.

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**Example text**

So a is a natural isomorphism. We also note that the hom functor as a bifunctor is faithful. Taking for example, Jr4 : X 1-4 d(A, X), we have for a : X ~ Y, h" : A I~ Aa, thus }{' is right multiplication by a, and choosing A = Ix we find that Aa = 0 for all A implies a = 0; similarly for htl . 7. II. Then the left adjoint S is right exact alld the right adjoint T is left exact. II hm'e arbitrary products and coprodllcts, then T preserves products alld 5 preserves coproducts. Proof.

O. e. ker aT #- AT and so aT #- o. 2). If this is exact. then (A/l)T = AT /l T = 0, hence A/l = O. Now let ker /l = (B', i) and consider the composition B' ~ B ~ C. This is zero, hence so is the result of applying T and it gives rise to a map (ker /l) T ~ ker /l T. 1) is exact at B. BT ~ (coker A)T is zero, and it • There is a useful test for exactness in the case of adjoint functors. 6) where in the case of additive categories ~ is an isomorphism of abelian groups which is natural in X and Y. 6) is merely a bijection of sets (still natural in X and Y).

Then av = ipav = i(pa - qf3)v = ill v = 0, and hence v = 0; this means that 11 is epic. 6, a' is also epic. 7. 6 in an abelian category, if a is epic, the1l so is a'. Dually, if in a pushout diagram a is monic, then so is a'. • Exercises 1. Show that Ens has an initial and a final object, but no zero object. 2. Show that in Rg the inclusion Z ~ Q is monic and epic but not an isomorphism. Is the inclusion Z ~ R an epimorphism? 3. Show that in a concrete category every monomorphism is injective. 2 Functors on abelian categories 41 4.