Friday, 27 June 2008

dg.differential geometry - Is there an easy way to describe the sheaf of smooth functions on a product manifold?

Let Mfd be the category of smooth manifolds (over mathbbR,mathbbC) and LRS be the category of locally ringed spaces (over mathbbR,mathbbC). Then the functor MfdtoLRS is full and faithful and indeed, you may define Mfd to be the full subcategory of LRS, which consists of those locally ringed spaces, which are locally isomorphic to mathbbRn together with the sheaf of smooth functions. Unfortunately, the functor MfdtoLRS does not preserve products; see below.



It should be remarked that products in LRS do exist (even infinite ones); take the obvious product in RS and "make it local" by introducing new points, namely prime ideals in the stalks, and take the localizations of the stalks as the new stalks. Now if we take the product of two manifolds M,N in LRS, we get as a topological space the usual product MtimesN; however, the structure sheaf consists only of those functions, which are locally of the form (u,v)mapstof(u)g(v) for smooth functions f,g defined locally on M,N, or sums and also quotients of these functions. These functions are also smooth with respect to the usual smooth structure on MtimesN, but the reverse is not true: For example it seems to be true that mathbbR2tomathbbR,(u,v)mapstoexp(uv) is not such a functions (however, I don't know how to prove this).



However, I think that Stone-Weierstraß implies that these simple functions are dense within all smooth functions. Thus, we may regard MtimesN in Mfd as the "completion" of MtimesN in LRS.

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