Understanding Direct and Inverse Limits

[lugita15]

I can't seem to make head or tail of the description of direct and inverse limits of abelian groups in problems 8 and 10 of the attached excerpt from Dummitt and Foote. Does anyone have a simpler or more intuitive definition of these two notions, or just an explanation of Dummit and Foote's definitions? Perhaps the definition would simplify considerably if instead of an arbitrary partially ordered indexing set, we had a sequence of abelian groups indexed by the natural numbers?

Any help would be greatly appreciated.

Thank You in Advance.

 

[mathwonk]

the idea is to have a way of building up something complicated from simpler things. one way to do this is as a direct sum or direct product, or a union or intersection. these limits are generalizations of those constructions. a direct limit is a quotient of a sum and an inverse limit is a subobject of a product. In both cases there is some relation between the different pieces being put together that must be respected.

e.g. if you have a collection of objects and you take their direct sum, the objects are sort of separated from each other in that sum. if you also have some maps between various summands, you can use that after forming the sum, by equating elements in different summands that are mapped to each other by the maps.

also in a product you can consider only sequences of elements that are all mapped to each other and you get a sub-object of the product.
e.g. given a bunch of sets you can take their disjoint union, which considers all the sets as totally different. But if those sets are subsets of the same huge set, they may overlap and some may be subsets of the others. if we take all subsets of a given set, and take their disjoint union and identify elements of two subsets if one is mapped to the other by an inclusion relation, and take the equivalence relation this generates, we just get back the usual union, i.e. the originals set. So any set is in as sense the direct limit of its subsets.

given a point of the complex plane, we can also consider for each nbhd of that point, all analytic functions defined and analytic in that neighborhood. If we define two functions as equivalent provided they agree on some smaller neighborhood of that point, then we get as the direct limit of those families of functions, just the family of all convergent power series at that point.


suppose we take an open cover of the plane and in each open set consider all functions analytic in that open set. then choosing one function from each open set, i.e. an element of the product of those families of functions, we can require that any two functions should agree on the overlap of their domains. if we do that we get the subset of elements of the product that patch together into entire functions. Thus the set of entire functions is the inverse limit of that collection of locally analytic functions.

thus inverse and direct limits are natural constructions that mimic these ways of building up local and global families from open covers.


i.e. you will need to work problems 9 and 11 in order to appreciate the definitions in problems 8 and 10.

i attach a small extract from my class notes of a decade ago in algebraic geometry (sheaf theory).

 

[mathwonk]

did that help? or did your head just pop off?