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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 22nd 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWe don't have a page for local cohomology. Is it there under some other name? Someone has made Grothendieck's book on the subject [available](http://matematicas.unex.es/~navarro/res/localcohomology.pdf). Akhil Mathew has a number of [posts](https://amathew.wordpress.com/tag/local-cohomology/) on the topic.

   We don’t have a page for local cohomology. Is it there under some other name?

   Someone has made Grothendieck’s book on the subject available.

   Akhil Mathew has a number of posts on the topic.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 22nd 2016
   • (edited Sep 22nd 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn as far as local cohomology is the operation of torsion approximation (as recalled in Akhil's first article in the series [here](https://amathew.wordpress.com/2012/08/09/local-cohomology-i/)) then this would be one of the operations of "arithmetic cohesion" recalled [here](https://ncatlab.org/nlab/show/differential%20cohesion%20and%20idelic%20structure#ArithmeticCohesion), I suppose. This is amplified in [Barthel-Heard 15](https://ncatlab.org/nlab/show/fracture+theorem#BarthelHeard15). Which reminds me: I had given up on "arithmetic cohesion" because it only worked for non-unital rings. But yesterday Thomas Nikolaus tells me about him generalizing some key facts to non-unital ring theory. For instance the basic fact of "Kontsevich-style non-commutative algebraic geometry" or "non-commutative motives" that compactly generated stable $\infty$-categories are $\infty$-categories of modules over $A_\infty$-ring(oid)s generalizes to the non-unital case.

   In as far as local cohomology is the operation of torsion approximation (as recalled in Akhil’s first article in the series here) then this would be one of the operations of “arithmetic cohesion” recalled here, I suppose. This is amplified in Barthel-Heard 15.

   Which reminds me: I had given up on “arithmetic cohesion” because it only worked for non-unital rings. But yesterday Thomas Nikolaus tells me about him generalizing some key facts to non-unital ring theory. For instance the basic fact of “Kontsevich-style non-commutative algebraic geometry” or “non-commutative motives” that compactly generated stable $\infty$-categories are $\infty$-categories of modules over $A_\infty$-ring(oid)s generalizes to the non-unital case.

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 22nd 2016
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIt seems Barthel-Heard-Valenzuela aim for full generality: "a simple abstract categorical framework which allows the construction of local cohomology and homology in general categorical contexts", just given a collection of compact objects of a closed symmetric monoidal, presentable stable category. >Which reminds me: I had given up on “arithmetic cohesion” because it only worked for non-unital rings. Was that after [this](https://nforum.ncatlab.org/discussion/6178/sphere-augmentation-and-cohesive-arithmetic-geometry/)?: >I am beginning to wonder if the appearance of nonunital $E_\infty$-geometry in "arithmetic cohesion" is more of a virtue than a bug.

   It seems Barthel-Heard-Valenzuela aim for full generality: “a simple abstract categorical framework which allows the construction of local cohomology and homology in general categorical contexts”, just given a collection of compact objects of a closed symmetric monoidal, presentable stable category.

   Which reminds me: I had given up on “arithmetic cohesion” because it only worked for non-unital rings.

   Was that after this?:

   I am beginning to wonder if the appearance of nonunital $E_\infty$-geometry in “arithmetic cohesion” is more of a virtue than a bug.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeSep 22nd 2016
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, I had been beginning to wonder that way, but then I didn't really sort out what to make of it. Something to look into again.

   Yes, I had been beginning to wonder that way, but then I didn’t really sort out what to make of it. Something to look into again.