### 713 Assignments for Fall 2017

1. p.133 #1 -- "get-acquainted special"

2. Consider the set P of all positive integers. Define an arrow from to if and only if divides . Show that this provides the structure of a (non-concrete) category on the positive integers.

3. Prove that a group is a (non-concrete) category with one object in which all arrows are invertible, and that a morphism of groups is a functor of such categories.

4. p.137 #4

BONUS 1 p.133 #3 and p.137 #1

5. p.140 #1

6. p.146 #4

7. Prove that a lattice is a category in which any two objects have at most one arrow between them, and that has products and coproducts.

8. p.153 #3

9. p.162 #5

10. Prove Thm.3 p.166.

11. p.167 #6

BONUS 2 p.167 #7

12. p.170 #13.(b) and p.173 #4.(b)

13. p.178 #4

14. p.184 #2

15. p.192 #1

16. p.192 #4

17. p.511 #2

18. p.511 #7

19. p.325 #1

BONUS 3 Prove Prop.18 p.324 for f.g. free/projective modules.

20. p.328 #3

BONUS 4 Same for f.g. free/projective modules.

21. p.333 #2

22. p.334 #6

23. Prove that the coproduct in CRng is the tensor product.

24. Determine the monics and epics in the category P of Assignment 1.

25. p.500 #4

26. Show that the isomorphism in (32) p.321 is natural in ModK.

BONUS 5 Find and verify all natural isomorphisms in ModK from Chapters V and IX.8--12.

BONUS 6 Regard a group as in Assignment 2. Prove that an automorphism is inner if and only if it is naturally equivalent to 1G .

27. p.516 #4--5

28. p.520 #4

29. p.520 #5.(b)

30. p.520 #12

From the handout:

31. p.1 #1

32. p.1 #2

33. p.2 #1

The remaining assignments are due Wed 13 Dec by 4:00 pm.

34. p.2 #3

35. p.3 #3, 4, 5

BONUS 7 p.3 #6