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1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeJan 24th 2012
   • (edited Jan 24th 2012)
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   Author: Tim_Porter
   Format: MarkdownItexAs we were discussing profinite completions the other day in another thread I thought I would add in some points about completed group algebras at [[profinite group]] and add some mention of pseudo compact algebras to the pre-existing entry on [[pseudocompact ring]]s. It is not clear to me what the connection between these algebras and [[profinite algebras]] should be. These pseudocompact and related linear compact algebras use finite dimensionality instead of finiteness to get a sort of algebraic compactness condition.

   As we were discussing profinite completions the other day in another thread I thought I would add in some points about completed group algebras at profinite group and add some mention of pseudo compact algebras to the pre-existing entry on pseudocompact rings.

   It is not clear to me what the connection between these algebras and profinite algebras should be. These pseudocompact and related linear compact algebras use finite dimensionality instead of finiteness to get a sort of algebraic compactness condition.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJan 24th 2012
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   Author: zskoda
   Format: MarkdownItexI do not know how pseudocompactness fits with profiniteness, but the Gorthendieck's motivation for pseudocompactness for rings was to have good theory of [[formal scheme]]s over such rings.

   I do not know how pseudocompactness fits with profiniteness, but the Gorthendieck’s motivation for pseudocompactness for rings was to have good theory of formal schemes over such rings.

3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeJan 24th 2012
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   Author: Tim_Porter
   Format: MarkdownItexWhat date was Grothendieck's use of them? I did a search on Google and was amazed to find http://arxiv.org/pdf/1008.0599.pdf which uses them with Calabi-Yau algebras! and searching further found a lot more on deformation theory. i was quite amazed.

   What date was Grothendieck’s use of them? I did a search on Google and was amazed to find http://arxiv.org/pdf/1008.0599.pdf which uses them with Calabi-Yau algebras! and searching further found a lot more on deformation theory. i was quite amazed.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeJan 24th 2012
   • (edited Jan 24th 2012)
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   Author: zskoda
   Format: MarkdownItexSurely, deformation theory needs formal schemes. I do not know the dates but it must be around 1960. The generality in van den Bergh's paper, which you [quote](http://arxiv.org/abs/1008.0599) surprises me however.

   Surely, deformation theory needs formal schemes. I do not know the dates but it must be around 1960. The generality in van den Bergh’s paper, which you quote surprises me however.

5. • CommentRowNumber5.
   • CommentAuthorTim_Porter
   • CommentTimeJan 24th 2012
   • (edited Jan 24th 2012)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI have been meaning to include stuff on coGrothendieck categories but have mislaid the classical paper that I wanted to quote from. (Oberst I think?) Should references to some of the deformation theoretic stuff be added to the entry?

   I have been meaning to include stuff on coGrothendieck categories but have mislaid the classical paper that I wanted to quote from. (Oberst I think?)

   Should references to some of the deformation theoretic stuff be added to the entry?

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeAug 31st 2013
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   Author: zskoda
   Format: MarkdownItexPseudocompact algebras are studied already in [[Pierre Gabriel]]'s thesis [[Des Categories Abeliennes]], hence at least in 1960; later they are developed further by Gabriel and Grothendieck in SGA3. For deformation theory we know the importance of Artinian rings. Pseudocompact rings are precisely cofiltered limits of Artinian rings. The book by Dascalescu and Nastasescu on Hopf algebras treats finite topologies and finite duals in the context of gebras (more precisely in duality of coalgebras vs. algebras) in quite much detail. In theorem 1.5.33 they give a theorem that the algebraic dual of a coalgebra is always a pseudocompact topological algebra. These are just remarks for the record, I should think more of this.

   Pseudocompact algebras are studied already in Pierre Gabriel’s thesis Des Categories Abeliennes, hence at least in 1960; later they are developed further by Gabriel and Grothendieck in SGA3. For deformation theory we know the importance of Artinian rings. Pseudocompact rings are precisely cofiltered limits of Artinian rings.

   The book by Dascalescu and Nastasescu on Hopf algebras treats finite topologies and finite duals in the context of gebras (more precisely in duality of coalgebras vs. algebras) in quite much detail. In theorem 1.5.33 they give a theorem that the algebraic dual of a coalgebra is always a pseudocompact topological algebra.

   These are just remarks for the record, I should think more of this.