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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 4th 2013
   • (edited Dec 4th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added the following reference to _[[Berkovich space]]_. Judging from the abstract this sounds like I nice unifying perspective. But I haven't studied it yet * Oren Ben-Bassat, Kobi Kremnizer, _Non-Archimedean analytic geometry as relative algebraic geometry_ ([arXiv:1312.0338](http://arxiv.org/abs/1312.0338)) > We show that Berkovich analytic geometry can be viewed as algebraic geometry in the sense of Toën-Vaquié-Vezzosi over various categories. The objects in these categories are vector spaces over complete valued fields which are equipped with additional structure. The categories themselves will be quasi-abelian and this is needed to define certain topologies on the categories of affine schemes. We give new definitions of categories of Berkovich analytic spaces and in this way we also define (higher) analytic stacks. We characterize in a categorical way the G-topology or the topology of admissible subsets used in analytic geometry. We demonstrate that the category of Berkovich analytic spaces embeds fully faithfully into the categories which we introduce. We also include a treatment of quasi-coherent sheaf theory in analytic geometry proving Tate's acyclicity theorem for quasi-coherent sheaves. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

   I have added the following reference to Berkovich space. Judging from the abstract this sounds like I nice unifying perspective. But I haven’t studied it yet

   • Oren Ben-Bassat, Kobi Kremnizer, Non-Archimedean analytic geometry as relative algebraic geometry (arXiv:1312.0338)

   We show that Berkovich analytic geometry can be viewed as algebraic geometry in the sense of Toën-Vaquié-Vezzosi over various categories. The objects in these categories are vector spaces over complete valued fields which are equipped with additional structure. The categories themselves will be quasi-abelian and this is needed to define certain topologies on the categories of affine schemes. We give new definitions of categories of Berkovich analytic spaces and in this way we also define (higher) analytic stacks. We characterize in a categorical way the G-topology or the topology of admissible subsets used in analytic geometry. We demonstrate that the category of Berkovich analytic spaces embeds fully faithfully into the categories which we introduce. We also include a treatment of quasi-coherent sheaf theory in analytic geometry proving Tate’s acyclicity theorem for quasi-coherent sheaves. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 4th 2013
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   Author: David_Corfield
   Format: MarkdownItexDoes this help us towards the elusive Berkovichian cohesion?

   Does this help us towards the elusive Berkovichian cohesion?