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Chloe (May 22 2020 at 19:21):does anyone have any nice intuition for uhh
what it means, geometrically, when a scheme is affine?
Chloe (May 22 2020 at 19:22):this is kind of a vague question, and im not quite sure what kind of answer im looking for...
Chloe (May 22 2020 at 19:23):so let me just point to some random thoughts and hope someone picks up on it lmao
Chloe (May 22 2020 at 19:24):if I have some scheme (or variety, or whatever...), which happens to be affine, that seems more like a structure on X than a property of X to me
Chloe (May 22 2020 at 19:24):namely it's "the structure of being isomorphic to spec(A) for some ring A"
Chloe (May 22 2020 at 19:25):(of course we know which ring A has to be if this is the case: it's just the ring of global functions on X)
John Baez (May 22 2020 at 19:26):That sounds good to me except I'd say
"the structure of being equipped with an isomorphism to spec(A) for some ring A".
Chloe (May 22 2020 at 19:26):hmm yeah you're right I should be more precise...
John Baez (May 22 2020 at 19:27):Of course one can demote any structure to a property using existential quantifiers, and then you get the property
"the property of being isomorphic to spec(A) for some ring A".
John Baez (May 22 2020 at 19:27):But one of the great lessons of category theory is that such properties are less useful than the structures they came from!
Chloe (May 22 2020 at 19:27):yes!
John Baez (May 22 2020 at 19:28):If a scheme is equipped with an isomorphism to spec(A) for some ring A, then you can act like it "is" spec(A) and start doing things with it.
John Baez (May 22 2020 at 19:28):But mere existence is like knowing you've got money in the bank but not knowing which bank.
Chloe (May 22 2020 at 19:28):okay so i'll be explicit with my motivation
there's this theorem of serre that characterizes affine schemes (if X is Nice, then X is affine iff it has no nontrivial quasicoherent cohomology)
Chloe (May 22 2020 at 19:29):and i am trying to understand a proof that goes through this result
John Baez (May 22 2020 at 19:29):This sounds like he's treating it as a property.
John Baez (May 22 2020 at 19:29):"having no nontrivial cohomology" is a mere property...
Chloe (May 22 2020 at 19:30):i've heard people use this theorem to justify saying that affineness is somehow like being contractible... but that seems WAY too coarse to me
John Baez (May 22 2020 at 19:30):It's possible his theorem could be enhanced to be a theorem that gave you structure.
John Baez (May 22 2020 at 19:31):That is, if you read the proof you might see that by choosing A, B, and C for your scheme you get an isomorphism between it and an affine scheme.
John Baez (May 22 2020 at 19:31):But maybe the statement of the theorem is only giving you conditions under which there exists A, B, and C for your scheme.
Chloe (May 22 2020 at 19:32):it can be upgraded to a structure, and it's not so hard to do
the cohomological statement is basically encoding the fact that taking global sections is exact
John Baez (May 22 2020 at 19:32):It's funny because on YouTube there's a lecture by Serre, I think, complaining about theorems where you have to peek into the proof to get the full use of the result.
Chloe (May 22 2020 at 19:32):and the affine scheme can be recovered as spec of the global sections of the structure sheaf
Chloe (May 22 2020 at 19:32):tbh i dont wanna think about serre's theorem too much
Chloe (May 22 2020 at 19:33):also i should probably step back and set this up a bit more categorically
Chloe (May 22 2020 at 19:33):there's an adjunction between affine schemes and arbitrary schemes
John Baez (May 22 2020 at 19:33):I think Serre's theorem is calling to you, begging you to state it more clearly.
Chloe (May 22 2020 at 19:34):I think you're right, but I think that I have to understand some simpler stuff before I can even get there
John Baez (May 22 2020 at 19:34):So yeah, if you can take your scheme and take global sections of the structure sheaf and canonically get an affine scheme from that if your scheme was secretly affine (as a property) to begin with, you are are in great shape.
Chloe (May 22 2020 at 19:34):the inclusion from affines to all schemes has a left adjoint
John Baez (May 22 2020 at 19:34):I think the adjunction approach is good.
Chloe (May 22 2020 at 19:35):which is exactly "take spec of the global functions"
Chloe (May 22 2020 at 19:35):so there is a canonical map from a scheme X to its "affinization"; it's either the unit or the co-unit of this adjunction (i can never get that straight without a bit of effort...)
John Baez (May 22 2020 at 19:36):Affinization sounds a bit like abelianization, which is a left adjoint.
Chloe (May 22 2020 at 19:36):affinization is the left adjoint yes
John Baez (May 22 2020 at 19:37):Any group has a canonical map to the underlying group of its abelianization, and this is a unit id => RL.
Chloe (May 22 2020 at 19:37):ahh yes it's the unit
John Baez (May 22 2020 at 19:37):Yes, so then you can just ask if this is an isomorphism, and that's a property.
Chloe (May 22 2020 at 19:37):yup
John Baez (May 22 2020 at 19:38):It sounds like you're dealing with a "reflective subcategory" here.
Chloe (May 22 2020 at 19:38):so maybe if i was categorically woke enough that would satisfy me
Chloe (May 22 2020 at 19:38):but i would really like to understand this more geometrically
John Baez (May 22 2020 at 19:39):I think you should be happy if affine schemes are a reflective subcategory of schemes.
Chloe (May 22 2020 at 19:39):they are, and I am
Chloe (May 22 2020 at 19:39):well, to an extent
John Baez (May 22 2020 at 19:39):This means that if you take an affine scheme, and look at its underlying scheme, and then affinize that, you get back to where you started up to an isomorphism - the counit of the adjunctions.
John Baez (May 22 2020 at 19:40):And this implies that affine schemes are a full subcategory of schemes (which you probably knew already), or more philosophically that being an affine scheme is a mere property of schemes.
John Baez (May 22 2020 at 19:41):Whenever you have a reflective subcategory you've got a subcategory of "nice" gadgets which are just gadgets with an extra property, and a way of "nicefying" any gadget.
Chloe (May 22 2020 at 19:41):yup
Chloe (May 22 2020 at 19:41):the example i usually jump to is sheafification
Chloe (May 22 2020 at 19:43):anyways i would like to explain why i don't find this satisfying... tho I'm not sure I can quite get at it
Chloe (May 22 2020 at 19:46):maybe what i really want is to have a better understand of what the affinization functor is actually doing in general
John Baez (May 22 2020 at 19:48):That sounds good. I'd like to understand that too! I was having a lot of fun at one point thinking about the process that takes a variety with a line bundle over it, and creates the vector space of sections of that bundle, and the map from original variety to the projectivization of the dual of that vector space.
Chloe (May 22 2020 at 19:52):so one way that u:X->Aff(X) can fail to be an isomorphism is if X doesn't have enough functions on it
Chloe (May 22 2020 at 19:52):"the coordinate ring doesn't separate points"
Chloe (May 22 2020 at 19:53):an extreme example would be if you take projective space, which has no non-constant global functions
Chloe (May 22 2020 at 19:53):so it's affinization is a point
John Baez (May 22 2020 at 19:53):Right. Is there another way?
Chloe (May 22 2020 at 19:53):there is
Chloe (May 22 2020 at 19:54):if you take A^2-{0}, with a point removed
Chloe (May 22 2020 at 19:54):that's not affine
Chloe (May 22 2020 at 19:54):it's affinization is the full plane, and the unit is the inclusion
Chloe (May 22 2020 at 19:55):the reason is something to do with Hartog's theorem: any regular function on the punctured plane can be extended over the puncture, because it's codimension 2
Chloe (May 22 2020 at 19:56):Chloe said:
if you take A^2-{0}, with a point removed
sorry I forgot a word here, I meant "a plane with a point removed"
John Baez (May 22 2020 at 20:02):I read what you meant, not what you wrote!
John Baez (May 22 2020 at 20:03):You can edit stuff here if you want to appear perfect in retrospect.
Chloe (May 22 2020 at 20:03):ahh, that's good to know
John Baez (May 22 2020 at 20:03):Okay, so the map can fail to be one-to-one, or fail to be onto. Makes sense.
Chloe (May 22 2020 at 20:03):yup
John Baez (May 22 2020 at 20:04):There should be some nice name for when it just fails to be onto - when your scheme is a "chunk" of an affine scheme
Chloe (May 22 2020 at 20:04):the failures of one-to-one-ness are kinda intuitive for me, cuz ive dealt with the phrase "separates points" in enough different contexts...
Chloe (May 22 2020 at 20:05):the failures of surjectivity seem pretty mysterious to me tho
Chloe (May 22 2020 at 20:05):I'm not sure if there is a good name for that
Chloe (May 22 2020 at 20:05):"quasi-affine" is close, it means "an open subset of something affine"
Chloe (May 22 2020 at 20:06):but something can be quasi-affine while being affine itself...
John Baez (May 22 2020 at 20:06):I was thinking of "quasi", but I only knew "quasiprojective".
John Baez (May 22 2020 at 20:06):Okay, we don't need words for bad properties, like "quasi-affine but not affine".
Chloe (May 22 2020 at 20:14):okay sure
John Baez (May 22 2020 at 20:16):So is the map from a quasiaffine scheme to its affinization always injective, or whatever is the right concept like that in algebraic geometry? (Monic in some category, I guess.)
Chloe (May 22 2020 at 20:18):yes, it should be
Chloe (May 22 2020 at 20:19):you will always get enough functions to separate points, because you can restrict things from the bigger space...
John Baez (May 22 2020 at 20:19):Right!
John Baez (May 22 2020 at 20:19):So is a scheme quasiaffine iff the map to its affinization is monic?
John Baez (May 22 2020 at 20:20):That would be nice. If it's not true, maybe there's some other concept than "quasiaffine" which is good to think about.
Chloe (May 22 2020 at 20:20):hmm
Chloe (May 22 2020 at 20:20):I don't think that's true
John Baez (May 22 2020 at 20:20):Which side of the "iff" fails - or both?
John Baez (May 22 2020 at 20:21):Well, you said that one side doesn't fail, so it must be the other.
Chloe (May 22 2020 at 20:21):I think any scheme that can be "embedded" inside an affine will have that be monic
John Baez (May 22 2020 at 20:21):Oh, so quasiaffines are just open in affines.
Chloe (May 22 2020 at 20:21):you have to be careful what you mean by "embedded", but open embeddings are certainly too restrictive
Chloe (May 22 2020 at 20:21):yah
Chloe (May 22 2020 at 20:21):closed subschemes are fine too
Chloe (May 22 2020 at 20:22):or locally closed...
Chloe (May 22 2020 at 20:22):but also more exotic things
Oscar Cunningham (May 22 2020 at 20:22):You missed a pun opportunity, 'affineness' could have been 'affinity'.
Chloe (May 22 2020 at 20:22):like "the germ of a point" (aka spec of one of the local rings)
John Baez (May 22 2020 at 20:22):I never learned enough about stuff like "embeddings" in algebraic geometry - much less more exotic things.
Chloe (May 22 2020 at 20:22):or "the formal nhbd of a point" (something like spec of the completion of a local ring; think power series)
John Baez (May 22 2020 at 20:22):But okay, sure, I know about things like "the nth infinitesimal neighborhood of a point"
John Baez (May 22 2020 at 20:23):Okay, "formal neighborhood".
Chloe (May 22 2020 at 20:23):actually I would guess that just having a map into something affine which is monic will be enough
Chloe (May 22 2020 at 20:30):yup that will be enough
Chloe (May 22 2020 at 20:31):hmm this is a bit slippery
Chloe (May 22 2020 at 20:31):the statement i am claiming is that if X admits a monomorphism into any affine scheme, then in fact the affizination map is monic
Chloe (May 22 2020 at 20:32):ahh and when i write it that way it's obvious: a map from X into some affine scheme factors through the affinization, so if that's not monic none of them can be
Chloe (May 22 2020 at 22:00):okay im coming back to this
Chloe (May 22 2020 at 22:00):i think that this situation looks a lot like... convexity
Chloe (May 22 2020 at 22:01):if I have a non-convex subset of R^n or something, and an affine function on it, I can extend that uniquely to the convex hull
Chloe (May 22 2020 at 22:02):well, uniquely in a way that retains convexity
Chloe (May 22 2020 at 22:02):this is... kinda similar if you squint??
Chloe (May 22 2020 at 22:02):if I have a polynomial defined on A^2-{0}, I can extend it uniquely to a polynomial on A^2
Chloe (May 22 2020 at 22:03):(by "polynomial" I should mean "regular function")
Chloe (May 22 2020 at 22:04):in situations where the unit map from X to its affinization is injective, what it's doing is basically adding in every point where the value of a function is constrained by the values on X
John Baez (May 22 2020 at 22:07):That sounds nice to think about.
Chloe (May 22 2020 at 22:12):it's kind of strange tho
Chloe (May 22 2020 at 22:12):like...
Chloe (May 22 2020 at 22:13):I can replace R^n with an "abstract convex space" which is some kind of finitary algebraic theory...
Chloe (May 22 2020 at 22:13):and then the "convexity" is a structure of the object that I'm talking about...
Chloe (May 22 2020 at 22:14):and "take the convex hull" is something like forming something by generators and relations....
Chloe (May 22 2020 at 22:14):now what the heck, if anything, does any of this analogize to in the schemes situation...
Chloe (May 22 2020 at 22:15):also, can you think of any other examples that look like this?
Chloe (May 22 2020 at 22:16):i guess any sort of algebraic object works if i am talking about homomorphisms
Chloe (May 22 2020 at 22:16):but convex sets are weird because they also "feel geometric"...
Chloe (May 22 2020 at 22:16):whereas schemes seem to only live on the geometric side
John Baez (May 22 2020 at 22:17):Is the "affinization" of a scheme also something you can describe using generators and relations?
John Baez (May 22 2020 at 22:17):In some nice way?
Chloe (May 22 2020 at 22:20):I have no idea
Chloe (May 22 2020 at 22:20):that seems like an extremely weird move to even try...
Chloe (May 22 2020 at 22:20):algebra is supposed to be dual to geometry...
Chloe (May 22 2020 at 22:21):but hmmm
Chloe (May 22 2020 at 22:21):im not sure what the "generators" could be, but the "relations" had better come from functions on my original scheme...
Sarah Griffith-Siqueira (May 23 2020 at 21:16):hi chloe. exercise 2.17 on page 81 of hartshorne is, to my interpretation: a scheme is affine iff it permits a finite partition of unity with each bump function supported on an open affine subset
Sarah Griffith-Siqueira (May 23 2020 at 21:23):to some extent this pushes the problem back because if we're trying to understand what makes something affine geometrically, it may not appear promising to say well such and such open subsets are affine. but what it suggests to me is that affineness is defined by ease of patching local information together
Sarah Griffith-Siqueira (May 23 2020 at 21:28):this sits well with my intuition about vanishing of cohomology (to the extent I have any) and also seems reasonable in relation to contractibility. In a topological context the minimal thing you need to have interesting global obstructions is a failure of contractability. in a scheme context the minimal thing you need to have interesting global obstructions is a failure of a good partition of unity, ie a failure of affineness
John Baez (May 23 2020 at 21:41):Sarah Griffith-Siqueira said:
hi chloe. exercise 2.17 on page 81 of hartshorne is, to my interpretation: a scheme is affine iff it permits a finite partition of unity with each bump function supported on an open affine subset
I'll have to look at that. "Partition of unity" is a phrase I associate with smooth manifolds. Is this exercise special to the case of schemes that are complex manifolds, or is "partition of unity" being defined in some more general algebraic way.
I'm afraid my copy of Hartshorne is not at home...
... oh, never mind, I found it. This is not what I'd call a partition of unity, but you could say it's some kind of algebraic generalization of that. Interesting!
Sarah Griffith-Siqueira (May 23 2020 at 21:47):this construction comes up when you prove that an affine scheme is always compact, and almost never afterwards, so in introductory sources you almost always have some aside like "(this is really a 'partition of unity' argument)"
Sarah Griffith-Siqueira (May 23 2020 at 21:48):but nobody makes a definition out of it
Chloe (May 24 2020 at 02:30):i think that's a pretty appealing picture
Chloe (May 24 2020 at 02:31):i have more to say about it but i have to think a bit to formulate it better
Simon Pepin Lehalleur (May 24 2020 at 08:23):Affinization maps for general schemes are not often considered in algebraic geometry because the affinization can be complicated. In particular, if is of finite type over a ring (even a field) then by construction the affinization has a morphism to which is in general not of finite type. There are however important cases where it stays of finite type, like algebraic groups and certain homogeneous spaces. See for instance https://arxiv.org/abs/1509.03059 section 3.2 for a discussion of the case of algebraic groups and a counter example to finite typeness of affinization.
Peter Arndt (May 24 2020 at 13:03):I have an algebraic intuition to offer, about affineness and affinization.
The construction of an affine scheme out of a ring is actually the solution to an algebraic problem: Given a ring R, is there an initial local ring L that R maps to? (categorically: are local rings a reflective subcategory of rings?) The answer is no: Suppose e.g. that there was such an initial local ring L for the integers Z. From Z we can map to the local ring Z_(2). In Z_(2) the element 2 is not invertible, so 2 can not be invertible in L in order to have a factorization Z-->L-->Z_(2). The same argument applies to all other primes, so actually we can't invert anything in the passage Z-->L. But without inverting something it's impossible to get a local ring.
However, there is a _set_ of jointly initial local rings: Any map from Z to a local ring factors through one of the Z_(p). We can use this to solve a relaxed version of our algebraic problem.
[I don't know if you know that a sheaf on a topological space can equivalently be defined as a local homeomorphism to the space. In this picture the stalks of the sheaf are the fibers of that local homemorphism. Anyway, that is true, and good mental image for the following story, but not necessary to read it.]
We can see our given ring as a sheaf of rings on the one point space. Now there _is_ an initial local ring under R, if we allow it to live over a different space than the one point space - that space is Spec(R) and the initial "local ring" is the usual structure sheaf on Spec(R): The stalks are local rings (which makes the structure sheaf a local ring from the internal point of view in the category of sheaves).
Intuitively you give the ring R just enough space to spread out into a collection of local rings, such that these form an initial family -- you have exactly one initial localization for each prime ideal. Looking at maps of ringed spaces in the opposite - i.e. the algebraic - direction, we have a map ({point},R)-->(Spec(R),O_Spec(R)). And you can check that this map is the initial map from R to a local ring, if we see a local ring S as a ringed space ({point},S)
(it is also the initial map to a local ring in the now generalized sense of local rings "living over spaces", and even in the sense of locally ringed toposes)
So let's see a scheme as a local homemorphism whose fibers are local rings.
Then "affine" means: "arises as the initial collection of local rings from some ring".
And affinization is sort of a reversal of our above problem: Above we were given R and asking for the best R-->L, i.e. the closest L to the right of R. Now we are given L and asking for the closest R left of it. More explicitly: Given a collection of local rings, namely the stalks of the structure sheaf of your scheme, find the ring (living over the one point space) that is closest to giving rise to all these local rings by localizations. How do you do this? Well, you take the limit of the diagram consisting of all the stalks and all the partial localizations sitting under them. I.e. you take the limit of the diagram consisting of the values of the structure sheaf on all the opens. But that limit is exactly the ring of global sections, by the sheaf condition. So this is affinization.