Jakob

I’m a mathematician working as a computer programmer, based in Regensburg, Germany. Sometimes I'm found thinking about constructive algebraic geometry.

Also interested in literature, history, sociology, economics, philosophy and FLOSS software.

Books: Jakob (npub1g76…g80h)
Pronouns: All

Last Notes

2026-10-13 20:34:10 GMT

Can anyone recommend a privacy friendly alternative to google forms?

2026-09-10 10:55:16 GMT

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I don't know if they are more active elsewhere, but at least @npub1tuz…3hxk @npub13hc…qw66 and [@gro_tsen_test](https://mastodon.sdf.org/@gro_tsen_test ) all have registered accounts on mastodon

2026-05-27 20:48:12 GMT

I'm realizing more and more how important decentralization really is. Thinking of Wikipedia of course, but also of Signal for example. Every community FLOSS project and perhaps every progressive project whatsoever should think about decentralization from the very beginning, especially with the growing authoritarianism we are facing right now. It's just so much easier to kill a centralized project than a decentralized one.
I worked years on bringing everyone to Signal and now telling everyone to move on to e.g. @npub17gp…nrxn won't be easy. But I guess it's better to start now than when it's too late…

2026-04-14 04:53:35 GMT

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You don't have to link to y**tube, he's on @npub17gp…7p4g:
@npub152l…zxw2

2026-02-05 16:30:45 GMT

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I think then you need local rings, not just fields. Ingo Blechschmidt's thesis is the canonical reference for this, unfortunately the link to the pdf seems to be down right now https://www.ingo-blechschmidt.eu/

2026-02-05 16:16:29 GMT

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I mean this is not true. Hom(R, -) restricted to the category of fields is not enough to recover the ring (for example Z/4 and Z/2 have the same homomorphisms to fields), much less if you take the colimit of this functor.
If you want to make do with fields, you need the additional info of the topology and the structure sheaf. The nice thing of the functorial view is that you don't need to worry about these things if you allow yourself to plug arbitrary rings in your functor.

2026-02-05 16:06:56 GMT

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The "character" point of view, i.e. looking at equivalence classes of homomorphisms to fields ("characters" on R) is relatively common in Berkovich geometry.
There you don't consider rings R but Banach rings (rings with a complete norm) and you don't look at homomorphisms to abstract discrete fields, but to fields with a complete absolute value. If you take the analogous colimit you arrive at the Berkovich spectrum https://en.wikipedia.org/wiki/Berkovich_space of the Banach ring.
Similarly as in algebraic geometry, every character has a unique smallest representative (the completed residue field) which in turn is completely determined by the pullback of its absolute value which gives you a multiplicative seminorm on R (rather than the pullback of the zero element in the algebraic geometry case, which gives you a prime ideal).

2026-02-05 16:01:01 GMT

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Note given f ∈ R and p ∈ Spec(R) (an equivalence class of homomorphisms to a field), while f(p) is not well-defined, the question whether f(p) = 0 (i.e. whether f is mapped to the zero element of a field) *is* well-defined (the map from the colimit to the classical spectrum which maps a homomorphism to a field to its kernel is well-defined because it is compatible with the morphisms of the index category).
In fact, this map to the classical spectrum is bijective, the map in the inverse direction maps a prime ideal to κ(p). Another way look at this is, that the equivalence class in the colimit of some homomorphism χ : R → k has a canonical "smallest" representative - the residue field of its kernel. (The universal property of the quotient R / Ker(χ) and then of its Fraction field is exactly what shows that χ factors over the residue field.
So this shows that we don't need to consider homomorphisms to arbitrary fields, it is enough to consider homomorphisms to residue fields of prime ideals, or equivalently just prime ideals.
But I often find it more natural to allow arbitrary fields here.
Often times it is even more natural to not take the colimit and instead look at the functor Hom(R, -) : Field -> Set, or even more natural to look at the functor Hom(R, -) : Ring -> Set on the full category of rings - this is the "functorial view" of algebraic geometry

2026-02-05 15:51:46 GMT

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Sorry for hijacking your thread again, but I think you might like the following way of coming up with the spectrum (and the residue fields). Say you know the theory of manifolds. There, if you have a function f ∈ R = C^∞(X) and a point p ∈ X, you can evaluate f(p) and this is a morphism R → ℝ .
Now let's say someone told us of this idea of Grothendieck that one can regard elements f ∈ R of an arbitrary ring as functions on some space, and we are supposed to make this idea work. We don't know yet what the points of our space should be, but given a point p, there should be a way to evaluate f(p) and this should be a homomorphism to some field (but we don't have a canonical candidate field like ℝ available).
So let's just *define* a point to be a pair (k, χ) where k is a field and χ is a ring homomorphism R → k. Then we can certainly evaluate a function f at our point and get and element of some field - just plug f into χ . But we must be a little bit careful - if we have two homomorphisms χ : 𝑅→𝑘 and χ' : 𝑅→𝑘', they should be considered to describe the same point if there exists a common field extension k'' of both k and k' such that both ways of getting from R to k'' are the same (whether we interpret f(p) to be χ(f) or χ'(f), they are the same field element in some bigger field).
So let's quotient by this equivalence relation. Formally, we take the colimit Spec(R) := colim_(k ∈ Field) Hom(R, k). (Note that the index category of fields is directed except for the fact that it falls into disjoint parts corresponding to the characteristic of the fields. If we work in fixed characteristic (for example over some fixed ground field), then we have a directed index category.)

2025-12-25 15:48:49 GMT

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Yes, the English -> German translations are:
B -> H
B flat -> B
B sharp -> His (on a well-tempered piano this is the same as C)
It's also interesting that while (except for H, the flat version of which is B) the flat version of a note is indicated with the -es ending (C flat -> Ces), the flat sign ♭ itself is called b in German 😅 I guess the sign ♭ is also a stylized b. So for example you would say that the B-major scale has two b, namely (B (B flat in English) and Es (E flat in English)). https://de.wikipedia.org/wiki/B_%28Notenschrift%29?wprov=sfla1
So the name B for the English H-flat makes a little bit of sense, because it is somehow the most canonical flat note (the first one in the circle of quints, appearing already in the F-major scale).
I hope this wasn't too confusing 😅 I don't know about the history unfortunately.

2025-12-25 15:33:11 GMT

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And a minor (pun intended) nitpick: It's not H-moll but h-Moll 😅 In H-Dur the H is capitalzed.

2025-12-25 15:21:21 GMT

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And then we Germans say Cis/Ces for C-sharp/C-flat etc. And if we put two sharp-signs in front of C we say Cis-is (I'm not sure how that's called in English).

2025-12-24 19:04:20 GMT

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And then in German we refer to the note called *B* in English by the letter *H* 🥲

2025-09-14 17:29:25 GMT

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I’ve known about this plan for a while, but I never really understood what the expected benefits are. Is this written up somewhere?