Local Rings
Sorry for the break last Friday.
As long as we’re in the neighborhood — so to speak — we may as well define the concept of a “local ring”. This is a commutative ring which contains a unique maximal ideal. Equivalently, it’s one in which the sum of any two noninvertible elements is again noninvertible.
Why are these conditions equivalent? Well, if we have noninvertible elements 
and 
with 
invertible, then these elements generate principal ideals 
and 
. If we add these two ideals, we must get the whole ring, for the sum contains , and so must contain 
, and thus the whole ring. Thus and cannot both be contained within the same maximal ideal, and thus we would have to have two distinct maximal ideals.
Conversely, if the sum of any two noninvertible elements is itself noninvertible, then the noninvertible elements form an ideal. And this ideal must be maximal, for if we throw in any other (invertible) element, it would suddenly contain the entire ring.
Why do we care? Well, it turns out that for any manifold 
and point 
the algebra 
of germs of functions at 
is a local ring. And in fact this is pretty much the reason for the name “local” ring: it is a ring of functions that’s completely localized to a single point.
To see that this is true, let’s consider which germs are invertible. I say that a germ represented by a function 
is invertible if and only if 
. Indeed, if 
, then 
is certainly not invertible. On the other hand, if , then continuity tells us that there is some neighborhood 
of where . Restricting to this neighborhood if necessary, we have a representative of the germ which never takes the value zero. And thus we can define a function 
for 
, which represents the multiplicative inverse to the germ of .
With this characterization of the invertible germs in hand, it should be clear that any two noninvertible germs represented by 
and 
must have 
. Thus 
, and the germ of 
is again noninvertible. Since the sum of any two noninvertible germs is itself noninvertible, the algebra of germs is local, and its unique maximal ideal 
consists of those functions which vanish at .
Incidentally, we once characterized maximal ideals as those for which the quotient 
is a field. So which field is it in this case? It’s not hard to see that 
— any germ is sent to its value at , which is just a real number.
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March 28, 2011 -
Posted by John Armstrong |
Differential Topology, Ring theory, Structure of Rings, Topology
[...] this would mean evaluating a function at a point, yes, but here we interpret it in terms of the local ring structure of . Given a germ there is a projection , which we write as [...]
Pingback by Tangent Vectors at a Point « The Unapologetic Mathematician | March 29, 2011 |
very useful. might be good to include brief word on the local subrings of Q.
The unapologetic mathematician apologises!