Abstract Nonsense

Crushing one theorem at a time

Direct Limit of Rings (Pt. II)


Point of Post: This is a continuation of this post.

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Examples of Direct Limits

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Now that we have defined the direct limit of a directed system of rings, let’s see if we can find explicit direct limits for the examples of directed systems of rings that we gave as examples.

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Our first example has been done before. Namely, the direct limit over this directed system is isomorphic to R^{\oplus\mathbb{N}} since, as is clearly by inspection, all the maps in the proof are really ring homomorphisms.

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We claim that, in this third example case, \displaystyle \bigoplus_{\alpha\in\mathcal{A}}R_\alpha is a direct limit of this system. Indeed, we have the obvious inclusions \displaystyle \varphi_\alpha:R_\alpha\hookrightarrow\bigoplus_{\alpha\in\mathcal{A}}R_\alpha. These certain satisfy \varphi_\alpha\circ f_{\alpha,\alpha}=\varphi_\alpha. Moreover, the uniqueness and existence of morphisms fitting the universal characterization is just a consequence of the way the coproduct is defined.

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Our third example is a fascinating one which (eventually, I hope to talk about more on this blog)–it is one of the first objects of studies of several complex variables. Namely, for our point z\in\mathbb{C} define \mathcal{O}_z to be the germ of holomorphic functions at z to be the set of all germs (or function elements) at z–these are no more than ordered pairs (f,U) where f:U\to\mathbb{C} is a holomorphic function and U\in\mathcal{U}_z. We say that two germs (f,U) and (g,V) at z are equal if f_{\mid U\cap V}=g_{U\cap V}. Thus, formally \mathcal{O}_z consists of equivalence classes of function elements. One can intuitively think about \mathcal{O}_z of consisting of functions which are fundamentally different z, in the sense that they are distinguishable, even locally at z.  This has an obvious ring structure defined by [(f,U)][(g,V)]=[(fg,U\cap V)] and [(f,U)]+[(g,V)]=[(f+g,U\cap V)]. It’s easy to see that this does, in fact, define a ring structure on \mathcal{O}_z. What we claim is that \mathcal{O}_z is the (up to isomorphism) direct limit of the \mathcal{H}(U) in our directed system. Indeed, for each U\in\mathcal{U}_z define \varphi_U:\mathcal{H}(U)\to \mathcal{O}_z given by f\mapsto [(f,U)]. This is evidently a ring homomorphism. Moreover, for any U\leqslant V\in\mathcal{U}_z one has that

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\varphi_V(\text{res}_{U,V}(f))=\varphi_V(f_{\mid V})=[(f_{\mid V},V)]=[(f,U)]=\varphi_{\mid U}(f)

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where this last fact holds since, by definition (recalling that V\subseteq U), f_{\mid U\cap V}=f_{\mid V}=(f_{\mid V})_{\mid U\cap V}. What we now claim is that given any set of maps \omega_U:\mathcal{H}(U)\to R (where R is some ring) such that \omega_V\circ\text{res}_{U,V}=\omega_U, for every U\leqslant V\in\mathcal{U}_z, there exists a unique map j:\mathcal{O}_z\to R such that j\circ\varphi_U=\omega_U. Indeed, define j by the rule \mathcal{O}_z\ni [(f,U)]\mapsto \omega_U(f)\in R. Now, note that this is, indeed, well-defined. For, if [(f,U)]=[(g,V)] then f_{\mid U\cap V}=g_{U\cap V}, but by assumption

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\begin{aligned}\omega_U(f) &=\omega_{U\cap V}\text{res}_{U,U\cap V}(f)\\ &=\omega_{U\cap V}(f_{\mid U\cap V})\\ &=\omega_{U\cap V}(g_{\mid U\cap V})\\ &=\omega_V(\text{res}_{V,U\cap V}(g))\\ &=\omega_V(g)\end{aligned}

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so that j is really well-defined. Moreover, it’s clear that j\circ\varphi_U=\omega_U for all U\in\mathcal{U}_z. Lastly, note that this was the only way we could have hoped to define this j. Indeed, suppose that k:\mathcal{O}_z\to R is another such function and let [(f,U)]\in\mathcal{O}_z be arbitrary. Note then that k([f,U])=k(\varphi_U(f))=g_U(f)=j([(f,U)]), and so j=k. From this we may conclude that the ring \mathcal{O}_z of analytic germs at z is a direct limit of the \mathcal{H}(U) in the directed system as desired.

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This all said, we can say something infinitely more cool. We did a similar construction for the direct limit of modules where we constructed the germs of continuous functions at a point x in some topological space X, as the direct limit of the continuous function spaces C^0(U) for each point U\in\mathcal{U}_x. That said, nothing more could be said about this ring of germs, it had no nice description–it didn’t “look nice”. Intuitively the problem was that, even locally, continuous functions (in general) don’t admit uniformizable, describable behavior (except in extreme, indicative, cases such as x is a discrete point of X, in which case \mathcal{G}_x\cong\mathbb{R}, obviously). But, holomorphic functions are indescribably nicer. How exactly does one define “nicer” in the current context though? Well, it seems clear that how well functions should behave locally at general points tells us how simple the functions really are. Thus, we should expect simpler functions to have simpler (more amenable to description) ring of germs. This is exemplified by the antithesis of \mathcal{G}_z and \mathcal{O}_z for some z\in\mathbb{C}. Really \mathcal{G}_z is such a large, untenable beast that (even after consulting multiple professors) I was unable to find a “nice” description (even in “nice cases”) for this ring. That said, \mathcal{O}_z admits an extremely nice description. Indeed, define \mathbb{C}\{\zeta-z\} to be the ring of convergent power series at z. This is an easy to understand, easy to work with object, and (as the buildup would suggest) \mathcal{O}_z\cong\mathbb{C}\{\zeta-z\}. The beauty of this isomorphism is how simple it is! Namely, we define \nu:\mathcal{O}_z\to\mathbb{C}\{\zeta-z\} by \displaystyle [(f,U)]\mapsto \sum_{n=0}^{\infty}\frac{f^{(n)}(z)}{n!} (\zeta-z)^n (we know this makes sense since f is holomorphic at z). Evidently \nu is a ring homomorphism, it’s clearly an epimorphism since \left[\left(\displaystyle \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}(\zeta-z)^n,D\right)\right]\in\mathcal{O}_z (where D is the maximal disc of convergence of the series, which we know is non-degenerate and so open) is an element of \mathcal{O}_z which maps to \displaystyle \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}(\zeta-z)^n. Moreover, we know that \nu is injective since any two functions mapping to the same power series must be locally equal, and so by definition produce the same equivalence class of function elements. Thus, \nu is an isomorphism! This is a truly beautiful piece of mathematics, making rigorous the notion that locally at a point  (i.e. as we take the limit over shrinking neighborhoods of that point) elements of individual \mathcal{H}(U)‘s are only distinguishable by their power series, that all other considerations are moot.

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References:

[1] Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. Module Theory. Clarendon, 1990. Print.

[4] Lang, Serge. Algebra. Reading, MA: Addison-Wesley Pub., 1965. Print.

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December 4, 2011 - Posted by | Algebra, Ring Theory | , , , , , , , , ,

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