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By Bergman G.M.

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Extra resources for A companion to S.Lang's Algebra 4ed.

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This 3-tuple is universal for making a commuting diagram:’’ Incidentally, why the term ‘‘fibered’’? , with a map into) a set Z, the inverse image in X of each element z ∈Z is called a ‘‘fiber’’ of the map X → Z. ) The ‘‘fibered product’’ P of two sets X and Y over Z is so called because for each z ∈Z, the fiber of P at z is the direct product of the fibers of X and Y at z. The term ‘‘fibered’’ is sometimes extended, as Lang will do on the next page, to the dual case of coproducts, though there is no intrinsic motivation for it there.

For an example of a covariant representable functor, let us fix any integer n, and consider the construction associating to every group G the set of its elements of exponent n, which we might write En (G) = {x ∈G x n = e}. It is easy to see that a group homomorphism G1 → G 2 will carry elements B G. M. 40 of exponent n in G1 to elements of exponent n in G 2 , making En a functor Group → Set. I claim that this functor is in fact isomorphic to hZ . Intuitively, the idea is that to determine a n homomorphism from Zn to a group G, one simply has to specify where the generator [1] ∈Zn is to be sent, and the possible choices are precisely the elements of exponent n, so the set of homomorphisms Zn → G is in natural bijective correspondence with the set of such elements.

Inverse limit and completion. This is an interesting subject, but one which we generally don’t have time to cover in Math 250A. Some time I hope to go through this section and set down full commentary. I give below a few errata supplied by Bjorn Poonen who covered this section in Fall 2003, and some comments of my own resulting from reading the points to which those errata applied. 50, beginning of last line of first paragraph [=]: lim should be lim_ (as on the preceding line); in ← earlier printings, the first two displays are also missing the arrow beneath the symbol.

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