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Harry Gindi
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What information does completing the the completion of a stalk at a point x (with respect to its maximal ideal) give us in a locally ringed space accomplishthe holomorphic, analytic, or algebraic cases?

Let $(X,\mathcal{F})$ be a locally ringed space. InIn a bonus exercise last year, we were asked to compute the completion in general of such a stalk on a smooth manifold of dimension $n$ (it is isomorphic to the ring of formal power series over $\mathbb{R}$ in $n$ unknowns). It's clear that this is a bad case to work with, since smooth manifolds admit bump functions (which allows us to prove that there exists a nonzero element in the intersection of all (finite) powers of the maximal ideal), and therefore the completion contains very little data.

However, what kind of "stuff" does this technique allow one to do in the analytic/holomorphic cases? Similarly, in the algebraic case, we can often look at the henselization for the same information, but I still am not really sure why one would want to do so in the first place. That is, what geometric idea corresponds to the idea of "completion" (including henselization and strict henselization depending on the context) in the same way that localization at a prime corresponds to taking stalks geometrically?

What does completing the the stalk at a point x (with respect to its maximal ideal) in a locally ringed space accomplish?

Let $(X,\mathcal{F})$ be a locally ringed space. In a bonus exercise last year, we were asked to compute the completion in general of such a stalk on a smooth manifold of dimension $n$ (it is isomorphic to the ring of formal power series over $\mathbb{R}$ in $n$ unknowns). It's clear that this is a bad case to work with, since smooth manifolds admit bump functions (which allows us to prove that there exists a nonzero element in the intersection of all (finite) powers of the maximal ideal), and therefore the completion contains very little data.

However, what kind of "stuff" does this technique allow one to do in the analytic/holomorphic cases? Similarly, in the algebraic case, we can often look at the henselization for the same information, but I still am not really sure why one would want to do so in the first place. That is, what geometric idea corresponds to the idea of "completion" (including henselization and strict henselization depending on the context) in the same way that localization at a prime corresponds to taking stalks geometrically?

What information does the completion of a stalk at its maximal ideal give us in the holomorphic, analytic, or algebraic cases?

In a bonus exercise last year, we were asked to compute the completion in general of such a stalk on a smooth manifold of dimension $n$ (it is isomorphic to the ring of formal power series over $\mathbb{R}$ in $n$ unknowns). It's clear that this is a bad case to work with, since smooth manifolds admit bump functions (which allows us to prove that there exists a nonzero element in the intersection of all (finite) powers of the maximal ideal), and therefore the completion contains very little data.

However, what kind of "stuff" does this technique allow one to do in the analytic/holomorphic cases? Similarly, in the algebraic case, we can often look at the henselization for the same information, but I still am not really sure why one would want to do so in the first place. That is, what geometric idea corresponds to the idea of "completion" (including henselization and strict henselization depending on the context) in the same way that localization at a prime corresponds to taking stalks geometrically?

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Harry Gindi
  • 19.6k
  • 16
  • 123
  • 215

Let $(X,\mathcal{F})$ be a locally ringed space. In a bonus exercise last year, we were asked to compute the completion in general of such a stalk on a smooth manifold of dimension $n$ (it is isomorphic to the ring of formal power series over $\mathbb{R}$ in $n$ unknowns). It's clear that this is a bad case to work with, since smooth manifolds admit bump functions (which allows us to prove that there exists a nonzero element in the intersection of all (finite) powers of the maximal ideal), and therefore the completion contains very little data.

However, what kind of "stuff" does this technique allow one to do in the analytic/holomorphic cases? Similarly, in the algebraic case, we shouldcan often look at the henselization or strict henselization for the same information, but I still am not really sure why one would want to do so in the first place. That is, what geometric idea corresponds to the idea of "completion" (including henselization and strict henselization depending on the context) in the same way that localization at a prime corresponds to taking stalks geometrically?

Let $(X,\mathcal{F})$ be a locally ringed space. In a bonus exercise last year, we were asked to compute the completion in general of such a stalk on a smooth manifold of dimension $n$ (it is isomorphic to the ring of formal power series over $\mathbb{R}$ in $n$ unknowns). It's clear that this is a bad case to work with, since smooth manifolds admit bump functions (which allows us to prove that there exists a nonzero element in the intersection of all (finite) powers of the maximal ideal), and therefore the completion contains very little data.

However, what kind of "stuff" does this technique allow one to do in the analytic/holomorphic cases? Similarly, in the algebraic case, we should look at the henselization or strict henselization, but I still am not really sure why one would want to do so in the first place. That is, what geometric idea corresponds to the idea of "completion" (including henselization and strict henselization depending on the context) in the same way that localization at a prime corresponds to taking stalks geometrically?

Let $(X,\mathcal{F})$ be a locally ringed space. In a bonus exercise last year, we were asked to compute the completion in general of such a stalk on a smooth manifold of dimension $n$ (it is isomorphic to the ring of formal power series over $\mathbb{R}$ in $n$ unknowns). It's clear that this is a bad case to work with, since smooth manifolds admit bump functions (which allows us to prove that there exists a nonzero element in the intersection of all (finite) powers of the maximal ideal), and therefore the completion contains very little data.

However, what kind of "stuff" does this technique allow one to do in the analytic/holomorphic cases? Similarly, in the algebraic case, we can often look at the henselization for the same information, but I still am not really sure why one would want to do so in the first place. That is, what geometric idea corresponds to the idea of "completion" (including henselization and strict henselization depending on the context) in the same way that localization at a prime corresponds to taking stalks geometrically?

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Harry Gindi
  • 19.6k
  • 16
  • 123
  • 215

What does completing the the stalk at a point x (with respect to its maximal ideal) in a locally ringed space accomplish?

Source Link
Harry Gindi
  • 19.6k
  • 16
  • 123
  • 215
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