A Course in Homological Algebra by Peter J. Hilton, Urs Stammbach

ll->JI -> 0, 0->A->I2 ->J2 ->0, with 11 ,12 injective. Show that II (f;J2 ~ 12 (f;JI • To what statement is this dual?

We therefore leave the details to the reader. 1. (a) Show that the zero module 0 is characterized by the property: To any module M there exists precisely one homomorphism cp: O-+M. (b) Show that the dual property also characterizes the zero module. 7. 2. Give a universal characterization of kernel and cokernel, and show that kernel and cokernel are dual notions. 3. 5. 4. Let rp : A -+ B. Characterize im rp, rp - \ Bo for Bo ~ B, without using elements. What are their duals? Hence (or otherwise) characterize exactness.

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