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nLab > Latest Changes: Novikov field

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- CommentRowNumber1.
- CommentAuthorMike Shulman
- CommentTimeJul 6th 2019
- (edited Jul 8th 2019)
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Author: Mike Shulman
Format: MarkdownItexCreated page about Novikov fields, another kind of generalized power series field. I'm looking for a good notation for these along the lines of $k[[x]]$ for power series rings, $k((x))$ for Laurent series fields, $k((x^G))$ for Hahn series fields; the only notation in the literature seems to be $\Lambda(k)$, which doesn't suggest its meaning to me and clashes with exterior algebras. <a href="https://ncatlab.org/nlab/revision/Novikov+field/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Novikov+field">current</a>

Created page about Novikov fields, another kind of generalized power series field. I’m looking for a good notation for these along the lines of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[[x]]</annotation></semantics></math>$ for power series rings, $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k((x))</annotation></semantics></math>$ for Laurent series fields, $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>G</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k((x^G))</annotation></semantics></math>$ for Hahn series fields; the only notation in the literature seems to be $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Λ</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Lambda(k)</annotation></semantics></math>$ , which doesn’t suggest its meaning to me and clashes with exterior algebras.

v1, current
- CommentRowNumber2.
- CommentAuthorUlrik
- CommentTimeJul 8th 2019
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Author: Ulrik
Format: MarkdownItexHow about $k((x^{(G)}))$, using parentheses to indicate a finiteness condition, in analogy to $k^G$ and $k^{(G)}$? Or, simply $k_{\text{Nov}}((x^G))$ in analogy to $k_{\text{Puis}}((x^G))$ for the notation for (generalized) Puiseux series? (E.g. in Efrat's book.)

How about $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k((x^{(G)}))</annotation></semantics></math>$ , using parentheses to indicate a finiteness condition, in analogy to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>k</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">k^G</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>k</mi> <mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">k^{(G)}</annotation></semantics></math>$ ?

Or, simply $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi> <mtext>Nov</mtext></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>G</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k_{\text{Nov}}((x^G))</annotation></semantics></math>$ in analogy to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi> <mtext>Puis</mtext></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>G</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k_{\text{Puis}}((x^G))</annotation></semantics></math>$ for the notation for (generalized) Puiseux series? (E.g. in Efrat’s book.)
- CommentRowNumber3.
- CommentAuthorMike Shulman
- CommentTimeJul 8th 2019
- PermaLink
Author: Mike Shulman
Format: MarkdownItexWhat does $k^{(G)}$ mean?

What does $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>k</mi> <mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">k^{(G)}</annotation></semantics></math>$ mean?
- CommentRowNumber4.
- CommentAuthorUlrik
- CommentTimeJul 8th 2019
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Author: Ulrik
Format: MarkdownItexFinitely supported functions. I think it's quite common; here's an [example reference](https://books.google.de/books?id=PUNwDwAAQBAJ&pg=PA649&lpg=PA649#v=onepage&q&f=false).

Finitely supported functions. I think it’s quite common; here’s an example reference.
- CommentRowNumber5.
- CommentAuthorMike Shulman
- CommentTimeJul 8th 2019
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Author: Mike Shulman
Format: MarkdownItexThat link doesn't work for me, but I'll take your word for it. However, finitely supported functions is *not* what we mean here!

That link doesn’t work for me, but I’ll take your word for it. However, finitely supported functions is not what we mean here!
- CommentRowNumber6.
- CommentAuthorUlrik
- CommentTimeJul 8th 2019
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Author: Ulrik
Format: MarkdownItexYes, I understood that, but I figured that the context was sufficiently different that the notational device could be reused to denote another finiteness condition. Besides, there is already the notation $k^{(G)}$ for the finitely supported functions, so then $k((x^{(G)}))$ must mean something different, but related to $k((x^G))$. Anyway, this perhaps illustrates that $k_{\text{Nov}}((x^G))$ is the safer choice.

Yes, I understood that, but I figured that the context was sufficiently different that the notational device could be reused to denote another finiteness condition. Besides, there is already the notation $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>k</mi> <mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">k^{(G)}</annotation></semantics></math>$ for the finitely supported functions, so then $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mrow><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k((x^{(G)}))</annotation></semantics></math>$ must mean something different, but related to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>G</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k((x^G))</annotation></semantics></math>$ .

Anyway, this perhaps illustrates that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi> <mtext>Nov</mtext></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>G</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k_{\text{Nov}}((x^G))</annotation></semantics></math>$ is the safer choice.
- CommentRowNumber7.
- CommentAuthornLab edit announcer
- CommentTimeJul 22nd 2019
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Author: nLab edit announcer
Format: MarkdownItexFukaya Category Ammar Husain <a href="https://ncatlab.org/nlab/revision/diff/Novikov+field/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Novikov+field/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Novikov+field">current</a>

Fukaya Category

Ammar Husain

diff, v3, current

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nLab > Latest Changes: Novikov field