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Stream: learning: questions

Topic: Exponentiable affine schemes

John Baez (Jan 20 2024 at 18:55):

The category of affine schemes over $k$, denoted $\mathsf{AffSch}_k$, is the opposite of the category of commutative algebras over the field $k$. We say an affine scheme $X$ is exponentiable if the functor

$X \times - : \mathsf{AffSch}_k \to \mathsf{AffSch}_k$

has a right adjoint. In other words, maps $X \times Y \to Z$ are in natural bijection to maps $Y \to Z^X$.

The most famous example of an exponentiable affine scheme is the walking tangent vector $T$, whose corresponding commutative algebra is $k[x]/\langle x^2 \rangle$. Lawvere pointed out that this means the tangent bundle of any affine scheme $X$ is simply $X^T$: you can think of a tangent vector in $X$ either as a map $T \to X$ or as a map $1 \to X^T$.

So, what are all the exponentiable affine schemes?

@Todd Trimble reported a wonderful fact, namely that an affine scheme is exponentiable iff its corresponding commutative algebra is finite-dimensional!

He says this was proved by Susan Niefield.

Where's the reference to this - and even better, how do you prove it? Is it hard to prove?

Todd Trimble (Jan 20 2024 at 19:00):

I forget where, but no, I don't think it's hard to prove. The corresponding fact about the coordinate algebra $A$ is that $A \otimes_k -$ preserves limits. I'll think about the proof if no one else comes along (and I hope I remembered correctly!).

Todd Trimble (Jan 20 2024 at 19:05):

See Theorem 4.3 here.

David Michael Roberts (Jan 20 2024 at 23:05):

What happens when k is generalised from being a field? Say to a local ring, a Dedekind domain, or all the way to a ring?

Todd Trimble (Jan 20 2024 at 23:07):

I should have said, but see Theorem 4.3. ;-) Just kidding. For $k$ a commutative ring, the underlying module of the coordinate algebra needs to be finitely generated and projective: that's necessary and sufficient.

John Baez (Jan 20 2024 at 23:31):

Let me give it a try. An endofunctor on a locally presentable category has a left adjoint iff it preserves colimits, and $\mathsf{CommAlg}_k$ is locally presentable. So $\mathsf{AffSch}_k$ is "co-locally presentable", and for some affine scheme $X$ we have

$X \times - : \mathsf{AffSch}_k \to \mathsf{AffSch}_k$

has a right adjoint iff it preserves limits.

I might as well do the case where $k$ is any commutative ring.

John Baez (Jan 20 2024 at 23:31):

So we just need to show $X \times -$ preserves limits iff the corresponding commutative algebra has an underlying $k$-module that's finitely generated and projective.

John Baez (Jan 20 2024 at 23:34):

But let's work purely with commutative algebras and let $A \in \mathsf{CommAlg}_k$.

John Baez (Jan 20 2024 at 23:36):

The product in $\mathsf{AffSch}_k$ is the coproduct in $\mathsf{CommAlg}_k$ which is none other than $\otimes_k$, "tensoring over $k$".

John Baez (Jan 20 2024 at 23:37):

So we just need to show $A \otimes_k - \mathsf{CommAlg}_k \to \mathsf{CommAlg}_k$ preserves limits iff the underlying $k$-module of $A$ is finitely generated projective.

John Baez (Jan 20 2024 at 23:40):

This sounds almost like some standard facts about $k$-modules, but we have to be a little careful since our condition here is that $A \otimes_k -$ preserves limits in $\mathsf{CommAlg}_k$, not in $k\mathsf{Mod}$.

John Baez (Jan 20 2024 at 23:43):

Hmm, but the forgetful functor $\mathsf{CommAlg}_k \to k\mathsf{Mod}$ is a right adjoint, so it preserves limits. And this functor is also monadic, right? - so it also reflects limits - right?

John Baez (Jan 20 2024 at 23:44):

If I'm not screwing up, this means $A \otimes_k -$ preserves limits in $\mathsf{CommAlg}_k$ iff it preserves limits in $k\mathsf{Mod}$. Is this correct? I'm not very good at this stuff....

John Baez (Jan 20 2024 at 23:45):

Anyway, if I haven't screwed up it's enough to show $A \otimes_k - : k\mathsf{Mod} \to k\mathsf{Mod}$ preserves limits iff $A$ is a finitely generated projective module.

Todd Trimble (Jan 21 2024 at 01:08):

Oh shoot, yes. I should have said $A \otimes_k -$ preserves limits earlier. My bad. I'll go back and edit in a bit.

Todd Trimble (Jan 21 2024 at 01:08):

Anyway, if $A$ is f.g. projective, then so is the linear dual $A^\ast = k\mathsf{Mod}(A, k)$ (and in fact they're dual to each other). But if $B$ is f.g. projective, what that really means is that $k\mathsf{Mod}(B, -): k\mathsf{Mod} \to k\mathsf{Mod}$ preserves colimits (projective gives you preservation of finite colimits, and finitely generated gives you preservation of filtered colimits, and the two together give you preservation of all colimits).

Todd Trimble (Jan 21 2024 at 01:08):

Okay, so what? Well, a functor $F: k\mathsf{Mod} \to k\mathsf{Mod}$ that preserves colimits must be of the form $C \otimes_k -$ for some $C$, and that $C$ is $F(k)$. So for $B$ f.g. projective, we get $F = k\mathsf{Mod}(B, -) \cong k\mathsf{Mod}(B, k) \otimes_k -$. The right side is more simply $B^\ast \otimes_k -$. But for $B = A^\ast$, that is even more simply still, $A \otimes_k -$.

Todd Trimble (Jan 21 2024 at 01:09):

So we've managed to show that for $A$ f.g. projective, that $A \otimes_k - \cong k\mathsf{Mod}(A^\ast, -)$. But now lookie here: the right side, being a covariant hom-functor, preserves limits!! Which is what you wanted to show for $A \otimes_k -$.

Todd Trimble (Jan 21 2024 at 01:09):

It's kind of a sneaky twisted little argument. Ultimately it may be easier to think about all this from the standpoint of monoidal duals, but anyway it's a pleasant argument.

John Baez (Jan 21 2024 at 01:28):

Wow, this is all very nice. To really get the desired "iff" - an affine scheme is exponentiable iff the corresponding commutative algebra is f.g. projective - we need the converse of the work you did. I think a key bit here is that the internal hom

$[B,-] : k\mathsf{Mod} \to k\mathsf{Mod}$

preserves colimits iff $B$ is f.g. projective.

Todd Trimble (Jan 21 2024 at 01:32):

Yes indeed. All this is part and parcel of Morita theory, where objects of the Cauchy completion are tantamount to adjoint pairs of bimodules, harking back to Lawvere's metric spaces where he first introduced categorical Cauchy completion: he defined this in terms of adjoint bimodules. (You and I and maybe Joe was there too -- we were talking about this once.)

John Baez (Jan 21 2024 at 01:33):

The nLab reminds me that the external hom

$k\mathsf{Mod}(B, -) : k\mathsf{Mod} \to \mathsf{Set}$

preserved filtered colimits iff $B$ is finitely generated, and this is really a general fact about varieties in universal algebra, nothing special about modules of rings.

Todd Trimble (Jan 21 2024 at 01:33):

Insofar as the Cauchy completion of a ring is its category of f.g. projective modules, you see how this fits together.

John Baez (Jan 21 2024 at 01:33):

Umm, no I don't.

Todd Trimble (Jan 21 2024 at 01:34):

Yeah, I think there the result should be about finitely presentable algebras, if you're speaking about varieties generally.

Todd Trimble (Jan 21 2024 at 01:35):

Sorry. Let me slow down. (I often feel hurried up in these Zulip discussions, unfortunately. Whereas in real life, I'm a slow-talking Southern boy.)

John Baez (Jan 21 2024 at 01:36):

I was busy thinking about the more humdrum issue of the last step: why

$k\mathsf{Mod}(B, -) : k\mathsf{Mod} \to \mathsf{Set}$

preserved finite colimits iff $B$ projective. I guess this is also easy.

John Baez (Jan 21 2024 at 01:37):

Todd Trimble said:

Sorry. Let me slow down. (I often feel hurried up in these Zulip discussions, unfortunately. Whereas in real life, I'm a slow-talking Southern boy.)

Yeah, I like it when in "real life" (meaning Zoom) you patiently demonstrate something step by step on the whiteboard.

Todd Trimble (Jan 21 2024 at 01:37):

Yes. Finite biproducts (so finite coproducts) are automatic. The projective part is about preserving coequalizers.

Todd Trimble (Jan 21 2024 at 01:38):

I don't know. Should we say more about that?

Todd Trimble (Jan 21 2024 at 01:42):

I'll take that as a "sure".

John Baez (Jan 21 2024 at 01:42):

Sure! I was trying to formulate a confusion I'm having....

John Baez (Jan 21 2024 at 01:42):

.... but I was confused about what the confusion was.

John Baez (Jan 21 2024 at 01:43):

It's about "homming out of projectives preserves epis" versus "homming out projectives preserves coequalizers".

John Baez (Jan 21 2024 at 01:44):

I guess we're in a situation (namely a module category) where every epi is a regular epi, so we can use that to show the "homming out of B preserves epis" iff "homming out of B preserves coequalizers".

Todd Trimble (Jan 21 2024 at 01:45):

Right. Well, I think it's a bit tricky when you put it in that generality, so I personally would do this: first show that for any module $B$, that $\mathsf{Mod}(B, -)$ preserves short exact sequences $0 \to X \to Y \to Z$.

Todd Trimble (Jan 21 2024 at 01:46):

(I don't think that preserving regular epis is the same as preserving coequalizers, even if you're in a regular category!)

John Baez (Jan 21 2024 at 01:47):

(Okay, thanks.)

Todd Trimble (Jan 21 2024 at 01:48):

So then, the only missing piece would be to show preservation of exactness at $Z$ in a short exact sequence

$0 \to X \to Y \to Z \to 0$.

That's where you need only preservation of epis.

Todd Trimble (Jan 21 2024 at 01:49):

So this mode of argumentation is pretty tied to abelian categories.

Todd Trimble (Jan 21 2024 at 01:50):

Maybe someone listening in can explain it better! But this is what I have at the moment.

John Baez (Jan 21 2024 at 01:57):

Okay, I think I know this classic sort of stuff; i.e. having played with homological algebra enough I know I can prove the facts you're mentioning, at least if I think a bit. I was just starting to wonder how much of this motto

"homming out of B preserves filtered colimits iff B is finitely generated and finite colimits iff B is projective."

generalizes to an arbitrary variety. This is part of trying to upgrade my understanding of how homological algebra fits into more general ideas.

The first part of the motto works in any variety. But not the second?

Todd Trimble (Jan 21 2024 at 01:59):

For the first part of the motto, replace "finitely generated" by "finitely presentable".

John Baez (Jan 21 2024 at 01:59):

Whoops!

James Deikun (Jan 21 2024 at 02:00):

So modules of rings are special in that finitely generated modules are automatically finitely presentable?

Todd Trimble (Jan 21 2024 at 02:01):

Yes, that's true.

Todd Trimble (Jan 21 2024 at 02:01):

Oh wait. Is that true? I'm suddenly worried about Noetherian....

John Baez (Jan 21 2024 at 02:02):

Ugh, I'm confused enough that I decided to delete my comment just now.

John Baez (Jan 21 2024 at 02:05):

I now believe that finitely generated R-modules are not necessarily finitely presented, but finitely generated projective ones are. So maybe this is why people run around acting like "finitely generated" and "projective" are the key ideas when playing with modules, when more generally, for other varieties, maybe we should be saying "finitely presented" and... something.

Todd Trimble (Jan 21 2024 at 02:05):

Okay, the whole point is that f.g. projective means having a retract of a f.g. free module -- and those last behave very well.

John Baez (Jan 21 2024 at 02:06):

I'm looking for a slogan, valid for varieties, of the form

"homming out of B preserves filtered colimits iff B is finitely presented and finite colimits iff B is ????."

John Baez (Jan 21 2024 at 02:07):

So then homming out of an object preserves all colimits iff it's finitely presented and ????.

Todd Trimble (Jan 21 2024 at 02:07):

Yeah, hm. Not thinking especially clearly at the moment...

John Baez (Jan 21 2024 at 02:09):

Anyway, you have satisfied my original desire to understand exponentiable affine schemes. You've done it so elegantly that I'm itching to generalize this result to categories that are opposites of varieties... but it's not really important.

Todd Trimble (Jan 21 2024 at 02:09):

You deserve an answer to that question though.

Todd Trimble (Jan 21 2024 at 02:10):

(And feel I should know this. I think I need to step away from the computer for a moment!)

James Deikun (Jan 21 2024 at 02:13):

If tensoring with something preserves finite limits it's called "flat"... if Homming out of something preserves finite colimits it is called "????".

David Kern (Jan 21 2024 at 02:40):

The nlab calls [[tiny+object]] one that preserves all small colimits, and does not seem to have anything in-between these and a [[small+object]] (preserving directed colimits). They do, however, mention that Kelly called them small-projective, so maybe finite-projective would be an appropriate name.

David Kern (Jan 21 2024 at 02:44):

John Baez said:

I was busy thinking about the more humdrum issue of the last step: why

$k\mathsf{Mod}(B, -) : k\mathsf{Mod} \to \mathsf{Set}$

preserved finite colimits iff $B$ projective. I guess this is also easy.

Speaking of this nlab page, they example they give for categories of modules (or any with a zero object) seems to show that this will not be true if you consider $k\mathsf{Mod}$ as a $\mathsf{Set}$-enriched category, that is if you put $\mathsf{Set}$ as the codomain of the corepresentable functor, so you do need to do as Todd Trimble above and work "abelianly".

Todd Trimble (Jan 21 2024 at 02:44):

Yes. The only thing I'll say at the moment is that if the ambient category is a variety of algebras, then this $E(e, -): E \to \mathsf{Set}$ preserving colimits is quite rare, since typically coproducts in that category behave tend to behave very differently from coproducts of sets. This might help explain why we're not coughing up names that readily in this generality.

Todd Trimble (Jan 21 2024 at 02:45):

Oh, you were making the same point. (And I was a principal author of that nLab article!)

Graham Manuell (Jan 21 2024 at 23:49):

I haven't read all the above discussion, but this was indeed proved by Niefield. I didn't find it completely trivial to use her argument to construct the exponential explicitly, so I gave an explicit description in terms of generators and relations in Theorem 4.2.1 of by PhD thesis. (Technically I do it for commutative quantales, but change the joins to sums and the situation for commutative rings is identical.)

Kevin Arlin (Jan 22 2024 at 04:02):

Yeah, I think that following on David’s and Todd’s recent points, there is no completion of the slogan for when maps out of an object in an arbitrary variety commute with finite colimits because that simply never happens. The misleading thing about module categories is that they’re presheaf categories in the enriched sense, and it’s reasonable to hope for objects maps out of which commute with all colimits in presheaf categories, but basically never otherwise. Just consider when maps out of even a free algebra on one generator commute with coproducts: only if coproducts are disjoint, which is highly uncharacteristic of algebraic categories. In fact I wouldn’t be shocked if this property will only happen in a variety that can be presented via a theory with only unary operations, ie one of presheaf type.

Regular projective objects, maps out of which commute with regular epis, are certainly very important in algebraic categories since specifically monadic categories are characterized as being exact with a regular projective generator. These things are retracts of frees and so maps out will commute with whatever colimits maps out of frees commute with, which seems like all one can reasonably ask for to me.

Kevin Arlin (Jan 22 2024 at 04:05):

I think we also can’t usually ask for maps out of the regular projectives to commute with coequalizers, see maps out of $\mathbb Z$ for the variety of abelian groups seen as an ordinary category—this would be a lot like asking the forgetful functor to commute with coequalizers, when really it probably only commutes with quotients of equivalence relations.