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Completions of fields September 1, 2009

Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.
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So again, we’re back to completions, though we’re going to go through it quickly. Except this time we have a field {F} with an absolute value {\left \lvert . \right \rvert} like the rationals with the usual absolute value.

 Completions   

Definition 1 The completion {\hat{F}} of {F} is defined as the set of equivalence classes of Cauchy sequences:  (more…)

Lifting idempotents à la Grothendieck August 29, 2009

Posted by Akhil Mathew in algebra, algebraic geometry, commutative algebra.
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I am going to get back shortly to discussing algebraic number theory and discrete valuation rings. But this tidbit from EGA 1 that I just learned today was too much fun to resist. Besides, it puts the material on completions in more context, so I think the digression is justified. 

Lifting Idempotents

The theorem says we can lift “approximate idempotents” in complete rings to actual ones. In detail: 

Theorem 1 Let {A} be a ring complete with respect to the {I}-adic filtration. Then if {\bar{e} \in A/I} is idempotent (i.e. {\bar{e}^2=\bar{e}}) then there is an idempotent { e \in A} such that {e} reduces to {\bar{e}}.   (more…)

The finite presentation trick and completions August 27, 2009

Posted by Akhil Mathew in algebra, category theory, commutative algebra.
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The previous post got somewhat detailed and long, so today’s will be somewhat lighter. I’ll use completions to illustrate a well-known categorical trick using finite presentations.

The finite presentation trick

 Our goal here is:

Theorem 1  Let {A} be a Noetherian ring, and {I} an ideal. If we take all completions with respect to the {I}-adic topology,      

 

\displaystyle \hat{M} = \hat{A} \otimes_A M

for any f.g. {A}-module {M}.   (more…)

Completions of rings and modules August 25, 2009

Posted by Akhil Mathew in algebra, commutative algebra.
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So, we saw in the previous post that completion can be defined generally for abelian groups. Now, to specialize to rings and modules.

 Rings

 The case in which we are primarily interested comes from a ring {A} with a descending filtration (satisfying {A_0 =A}), which implies the {A_i} are ideals; as we saw, the completion will also be a ring. Most often, there will be an ideal {I} such that {A_i = I^i}, i.e. the filtration is {I}-adic. We have a completion functor from filtered rings to rings, sending {A \rightarrow \hat{A}}. Given a filtered {A}-module {M}, there is a completion {\hat{M}}, which is also a {\hat{A}}-module; this gives a functor from filtered {A}-modules to {\hat{A}}-modules. (more…)

Generalities on completions August 23, 2009

Posted by Akhil Mathew in algebra, commutative algebra, topology.
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Today I’ll discuss completions in their algebraic context. All this is really a version of Cauchy’s construction of the real numbers, but it’s also useful in algebra, since one can study a ring through its completions (e.g. in algebraic number theory, as I hope to get to soon).

 Generalities on Completions

 Suppose we have a filtered abelian group {G} with a descending filtration of subgroups {\{G_i\}}. Because of this, we can consider “Cauchy sequences” and “convergence:” 

Definition 1

The sequence {\{x_i\} \subset G}, {i \in \mathbb{N}} is Cauchy if for each {A}, there exists {N} large enough that

 

\displaystyle i,j > N \quad \mathrm{implies} \quad x_i - x_j \in G_A.

The sequence {\{y_i\} \subset G} converges to {y} if for each {A}, there exists {N} large enough that

\displaystyle i>A \quad \mathrm{implies} \quad x_i -y \in G_A. (more…)