Post #839262

2026-03-23 00:34 UTC

New blog post up on my website: How Algebraic is Algebraic Geometry? https://cbaberle.com/Blog/How+Algebraic+is+Algebraic+Geometry%3F This post was the result of me trying to wrap my head around some of the central concepts of modern algebraic geometry and its generalizations from the perspective of someone more at home in type theory, category theory, programming languages, etc. In particular, I&#39;ve been watching much of the recent work on &quot;synthetic algebraic geometry&quot; with some interest, and also have had a passing interest in the programme of trying to develop geometry over &quot;the field with one element,&quot; due to my longstanding interest in the Riemann Hypothesis and the various research programmes it has spawned. If any of that sounds interesting to you, then I hope that this blog post can be as illuminating for you to read as it was for me to write :) Tagging a few people who I think might find this interesting and/or whose writings on these and related topics have been stimulating to my thinking on this subject: @jcreed @johncarlosbaez @thosgood @dwarn Comments, feedback, and criticism are much welcomed! In particular, please do let me know if I got anything wrong! Like I said, I&#39;m very much an outsider to these subjects trying to understand them better for myself.

Replies (7)

@johncarlosbaez@mathstodon.xyz 2026-03-23 00:50

@cbaberle - Hi! For some reason this line doesn't compile when I'm reading your blog: \require{AMScd} \begin{CD} \mathfrak{F}[X] @>X \mapsto n>> N \\ @VX \mapsto X^{-1}VV @vvv \\ \mathfrak{F}[X, X^{-1}] @>>> N[n^{-1}] \end{CD} This stuff is interesting. I don't know if anyone has gone much further than Durov in doing actual hard-core algebraic geometry with finitary commutative monads. The "frameworky" stuff is a lot of fun, but then I'd like to see it applied. (Not that I'm very good at algebraic geometry, mind you!)

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@highergeometer@mathstodon.xyz 2026-03-23 02:49

@cbaberle It makes me think of the work of Sasha Rosenberg https://newprairiepress.org/ebooks/1/ on noncommutative algebraic geometry, whereby you embed schemes in this framework by taking their symmetric monoidal category of quasicoherent sheaves. Others (Toën and Vaquié) have done "schemes over an general symmetric monoidal category" is a similar way, see eg https://mathoverflow.net/questions/89475/connections-between-various-generalized-algebraic-geometries-toen-vaqui%C3%A9-durov @jcreed @johncarlosbaez @thosgood @dwarn

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@julesh@mathstodon.xyz 2026-03-23 13:20

@cbaberle After reading just the introduction this sounds like something I wished I'd been able to read years ago. Algebraic geometry is a subject I always looked at and thought this looks like it must be useful for *something*, but I don't have enough motivation to get through the difficulty because I really don't care about the geometry of polynomials

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@thosgood@mathstodon.xyz 2026-03-24 00:07

@cbaberle @jcreed @johncarlosbaez @dwarn i put this on my reading list for coffee break tomorrow, thanks for the tag!

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@oantolin@mathstodon.xyz 2026-03-24 11:59

@cbaberle Excellent post summarizing a ton of work! (I have to admit I was a little nervous there would be no credit assigned —except to Ingo Blechschmidt, Felix Cherubini, Thierry Coquand, Matthias Hutzler, David Wärn, Hugo Moeneclaey— until I got to the end where all the references are! 😅) One thing I noticed was wrong is this sentence: "For commutative rings A, B the homomorphisms A → B are in natural bijection with continuous maps Spec(B) → Spec(A)." This is very far from true! Take A = B = Z, for example. There is a unique ring homomorphism A → B but there are uncountably many continuous functions Spec(B) → Spec(A): fix 0 and permute the primes however you want amongst themselves. One nearby correct statement is, of course, that you can equip the topological space Spec(A) with the structure of a scheme and then ring homomorphisms A → B are in natural bijection with morphisms of schemes Spec(B) → Spec(A).

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@hallasurvivor@sunny.garden 2026-03-25 17:33

@cbaberle I'm glad I finally found the time to read this! I really enjoyed it ^_^. Thanks for sharing!

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@2something@transfem.social 2026-03-23 14:44

@cbaberle@mathstodon.xyz @johncarlosbaez@mathstodon.xyz @jcreed@mastodon.social @thosgood@mathstodon.xyz @dwarn@mathstodon.xyz Thank you for writing this!

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