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Stream: learning: questions

Topic: Grothendieck construction in geometry

Matteo Capucci (he/him) (Dec 20 2020 at 18:25):

Hi folks, does anyone know what Grothendieck invented his constructon for? Is it used in algebraic geometry?
My guess would be that you use it to transform a sheaf (resp. stack) into a discrete (resp. indiscrete) fibration. If this is the case, is this the same thing as constructing the étalé space (resp. étalé ??) of the sheaf (resp. stack)?

John Baez (Dec 20 2020 at 18:28):

I think @Joe Moeller started translating the section of Grothendieck's work where he introduced this construction... that would be the way to find out the answer to this.

Jens Hemelaer (Dec 20 2020 at 21:42):

To a presheaf $\mathcal{F}$ in $\mathbf{PSh}(\mathcal{C})$, there is a corresponding opfibration
$\pi : \mathcal{D} \to \mathcal{C}.$
Further, the slice topos $\mathbf{PSh}(\mathcal{C})/\mathcal{F}$ is then given by $\mathbf{PSh}(\mathcal{D})$, and the slice topos projection
$\mathbf{PSh}(\mathcal{D}) \simeq \mathbf{PSh}(\mathcal{C})/\mathcal{F} \to \mathbf{PSh}(\mathcal{C})$
is the étale geometric morphism induced by $\pi$.

Now suppose that $\mathbf{PSh}(\mathcal{C}) \simeq \mathbf{Sh}(X)$ for some topological space $X$. This happens iff $\mathcal{C}$ is a preorder. Then $\mathcal{D}$ is again a preorder, and $\mathbf{PSh}(\mathcal{D}) \simeq \mathbf{Sh}(Y)$ for some topological space $Y$. This topological space $Y$ is then the étale space associated to the presheaf $\mathcal{F}$.

For sheaves it works more or less in the same way, but only some discrete opfibrations correspond to sheaves, and the discrete opfibrations come equipped with nontrivial Grothendieck topologies (lifting the Grothendieck topology on the base site), leading to a more difficult situation.

fosco (Dec 21 2020 at 08:49):

Matteo Capucci said:

Hi folks, does anyone know what Grothendieck invented his constructon for? Is it used in algebraic geometry?
My guess would be that you use it to transform a sheaf (resp. stack) into a discrete (resp. indiscrete) fibration. If this is the case, is this the same thing as constructing the étalé space (resp. étalé ??) of the sheaf (resp. stack)?

In a sense, yes, it is the same thing: if $O(X)$ is the category of opens of a space, there is a tautological functor $\iota : O(X) \to {\sf Top}/X$ sending an open $U\subseteq X$ into the map $U\hookrightarrow X$. Yoneda extension of $\iota$ gives you a functor ${\sf PSh}(X) \to {\sf Top}/X$, that must have a right adjoint, precisely the left Kan extension of the Yoneda embedding $O(X) \to {\sf PSh}(X)$ along $\iota$.

$Lan_y\iota$ sends a presheaf F into its associated bundle space, and this bundle is a local homeomorphism if and only if F was a sheaf; on the other hand, "nerves tend to map into the subcategory of fibrant objects", so the image of every bundle over $X$ is a sheaf, precisely the sheaf of sections of the bundle.

fosco (Dec 21 2020 at 09:02):

Similarly, given a category $\mathcal C$, there is a functor $\iota : \mathcal C \to {\sf Cat}/\mathcal C$ sending an object $C$ to the slice over C, or the coslice under C, according to you wanting to consider fibrations / copresheaves or opfibrations / presheaves.

fosco (Dec 21 2020 at 09:03):

What is the Yoneda extension $Lan_y\iota$ of this second $\iota$, if not the good old category of elements of a presheaf?

$Lan_y\iota(F) = \displaystyle \int^A \mathcal C/A\times FA$

fosco (Dec 21 2020 at 09:04):

furthermore, this last functor also has a right adjoint: I will let you find it. :-)

Reid Barton (Dec 21 2020 at 15:59):

I was under the impression that the original purpose of the Grothendieck construction was primarily to justify not working directly with pseudofunctors at all, but rather the fibered categories which would be their images under the construction.

Reid Barton (Dec 21 2020 at 16:01):

In fact many examples arise more naturally as fibered categories than as pseudofunctors, at least if you want to be picky about the details, because describing them as pseudofunctors involves making choices (like choices of pullback diagrams, or of tensor products).

John Baez (Dec 21 2020 at 17:26):

All this math is wonderful but I'm getting more and more curious about @Matteo Capucci's original question, which was "does anyone know what Grothendieck invented his construction for?"

John Baez (Dec 21 2020 at 17:27):

So I'm gonna have to find out...

John Baez (Dec 21 2020 at 17:28):

According to @Joe Moeller and @Christina Vasilakopoulou he did it here:

• Categories fibrees et descente. Chapter 6 of Etale Covers and the Fundamental Group, Seminaire de Geometrie Algebrique de l’Institut des Hautes Etudes Scientifiques* (SGA 1), 1961.

(sorry I can't easily put the accents in).

John Baez (Dec 21 2020 at 17:36):

Here's how the chapter describing the Grothendieck construction starts:

Contrary to what was said in the introduction to the previous chapter, it proved impossible to do descent in the category of preschemes, even in special cases, without having already carefully developed the language of descent in general categories.

The notion of "descent'' provides the general framework for all processes of "gluing'' objects, and therefore of "gluing'' of categories. The most classic case of gluing relates to the data of a topological space $X$ and a cover of $X$ by open sets $X_i$; if we consider for all $i$ a fibre bundle (say) $E_i$ above $X_i$, and for every pair $(i, j)$ an isomorphism $f_{ji}$ of $E_i| X_{ij}$ on $E_j | X_{ij}$ (where we put $X_{ij} = X_i \cap X_j$), satisfying a well-known condition of transitivity (which we write in a short way $f_{kj} f_{ji} = f_{ki}$ ), we know that there is a fibre bundle $E$ on $X$, defined up to isomorphism [...]

Thus the notion of "gluing'' objects, like the "localisation'' of a property, is related to the study of certain types of "base changes'' $X '\to X$. In algebraic geometry many other types of base change, and in particular faithfully flat morphisms $X '\to X$, should be considered as corresponding to a process of "localisation'' relative to preschemes, or other objects, "over'' $X$. This type of localization is used just as much as the simple topological localization (which is a special case of this more general idea). The same is true (to a lesser extent) in Analytic Geometry.

John Baez (Dec 21 2020 at 17:40):

Unfortunately the whole book is not translated yet, so I can't read (in English) the section where fibred categories are introduced, and it's hard to see where the Grothendieck construction is first actually applied.

Joe Moeller (Dec 21 2020 at 17:42):

Once when I was reading Hartshorne, I realized that there's a slick way to define sheaf of modules by using the category of modules over all rings rather than thinking of just categories of modules over a single ring. I don't know if this was primarily the motivation.

Joe Moeller (Dec 21 2020 at 17:43):

Tim Hosgood has lots of translations. I don't know if he finished this chapter of SGA yet though.

John Baez (Dec 21 2020 at 17:44):

Joe Moeller said:

Once when I was reading Hartshorne, I realized that there's a slick way to define sheaf of modules by using the category of modules over all rings rather than thinking of just categories of modules over a single ring. I don't know if this was primarily the motivation.

I think something like that was the motivation, but Grothendieck wouldn't have invented all this complicated stuff just to handle that one example. So I want to know what his first applications of the Grothendieck construction were. They must be in this book.

Joe Moeller (Dec 21 2020 at 17:47):

When I was working on my translation, I was only interested in seeing how much of the 2-equivalence he proved in there, so I didn't get to see him using it. The answer to how much he proved of the 2-equivalence is "basically none".

John Baez (Dec 21 2020 at 18:53):

Right, you mentioned that to me once. It's interesting. I imagine he only developed the stuff he needed.

Tim Hosgood (Dec 21 2020 at 19:22):

I haven’t started on the mammoth task of the SGA yet, but I recently finished a translation of the first part of FGA 3, which talks about descent along faithfully flat morphisms, but from a much more geometric perspective than the part in SGA mentioned above

Tim Hosgood (Dec 21 2020 at 19:23):

but this part of FGA introduces fibred categories and all of this language too

Tim Hosgood (Dec 21 2020 at 19:25):

might answer the question, but might not! it’s an interesting thing that I’ll look into later when I get a chance :)

John Baez (Dec 21 2020 at 19:42):

The portion of SGA1 where Grothendieck introduces his famous construction has a lot to say about "descent along faithfully flat morphisms", so I imagine that has a lot to do with why he introduced it!

What's a faithfully flat morphism? :rolling_eyes: I don't really want to know the definition, I want to know why it matters.

Is it like a covering space, or maybe a branched cover? That would count as an explanation of why it matters: after all, in SGA1 Grothendieck is busy talking about the etale fundamental group, which consists of deck transformations of something like branched covers.

Tim Hosgood (Dec 21 2020 at 19:55):

faithfully flat morphisms ensure that the inverse image of quasi-coherent sheaves is an exact functor

Tim Hosgood (Dec 21 2020 at 19:56):

it’s basically, when combined with quasi-compactness, “exactly the same as a descent morphisms”

Tim Hosgood (Dec 21 2020 at 19:57):

a nice example is that any locally free sheaf of algebras of finite type that is everywhere non-zero

Tim Hosgood (Dec 21 2020 at 19:57):

such things always give you a faithfully flat (and quasi compact) morphism

Tim Hosgood (Dec 21 2020 at 19:57):

not sure how helpful that is as a motivation...

Jens Hemelaer (Dec 21 2020 at 19:58):

I agree with @Tim Hosgood.

I'd say that étale is really the notion corresponding to the idea of "local homeomorphism" or n-sheeted cover without branchpoints.
Faithfully flat morphisms can be much weirder: for example the projection of an affine line to a point is faithfully flat.
Branched covers are also often faithfully flat (but not étale).

For example, $\mathrm{Spec}(\mathbb{Z}[\sqrt{D}]) \to \mathrm{Spec}(\mathbb{Z})$ is faithfully flat.
Here $\mathbb{Z}[\sqrt{D}]$ is free as $\mathbb{Z}$-module so in particular flat.

John Baez (Dec 21 2020 at 21:05):

Tim Hosgood's explanation of "faithfully flat" helped me only a tiny amount. Wanting the "inverse image of quasi-coherent sheaves to be an exact functor" sounds like some technical condition, and I can actually understand the individual words separately, but it doesn't give me any intuition for why Grothendieck would care enough about faithfullly flat descent to write hundreds of pages about it. I'm not an algebraic geometer and have no intention of becoming one. But it's helpful to know that faithfully flat morphisms can be much weirder than étale morphisms.... so maybe covering space ideas are not what motivated his interest in "faithfully flat descent".

Jens Hemelaer (Dec 22 2020 at 08:21):

Tim Hosgood said:

it’s basically, when combined with quasi-compactness, “exactly the same as a descent morphisms”

I think descent is the reason that faithfully flat morphisms "still work". If you restrict a sheaf along a (qc) faithfully flat morphism then you don't lose any information, and you can recover the original sheaf from it. Similar to how you can restrict a sheaf to an open covering and not lose any information (in both cases, you have to remember the gluing maps).

However, faithfully flat morphisms are not the most general descent morphisms, "pure" morphisms are a bit more general. Here are some examples of pure but not faithfully flat morphisms. So, I don't know why the notion of faithfully flat is more popular... maybe because it is useful in proofs that pulling back sheaves is exact.

Tim Hosgood (Dec 22 2020 at 14:24):

let me try to motivate this "inverse image" property a tiny bit (but this really isn't something that I'm an expert in). quite often in algebraic geometry, we're interested in base changes: we have a bunch of objects (schemes, varieties, or other "space-looking gadgets") that live over some base object (let's call it $S$); we then want to know, given some morphism $\alpha\colon S'\to S$, if we can transfer all of this stuff to live over $S'$ instead of $S$. If the morphism $\alpha$ is faithfully flat, then we know that complexes of quasi-coherent sheaves living over $S$ can be pulled back "exactly" to complexes of quasi-coherent sheaves over $S'$.

Tim Hosgood (Dec 22 2020 at 14:27):

but really Wikipedia gives a great summary of why we care about flatness (note that faithfully flat just means "surjective and flat"): to quote/paraphrase from https://en.wikipedia.org/wiki/Flat_morphism ...

1. flatness is a _generic_ property, i.e. an arbitrary (not too pathological) morphism can always be restricted to a _flat_ morphism on some Zariski open subset (and the open subsets in the Zariski topology are "big"), and so, vaguely, "most morphisms are flat in most places"
2. flatness (or, more precisely, failing to be flat) tells us where things jump in dimension, which is something that happens a lot in algebraic geometry (e.g. coherent sheaves are basically vector bundles but where the rank can "jump" when passing from one open subset to another, and quasi-coherent sheaves are basically coherent sheaves but where the rank can jump to infinity in some places)

Tim Hosgood (Dec 22 2020 at 14:28):

again, I'm not really an algebraic geometer, so maybe a proper one would say something different, but these two points are a helpful way to understand why we often care about flatness

John Baez (Dec 22 2020 at 17:27):

Thanks, Tim, that's very helpful. I find 2. quite helpful, since I feel I have some intuition for how, say, kernels of vector bundle maps are not always vector bundles because the dimension can 'jump', and how they're still coherent sheaves.

John Baez (Dec 22 2020 at 17:30):

But I don't quite understand what you mean "failing to be flat tells us where things jump in dimension". Can we say something like this: if a morphism $\alpha: S' \to S$ is flat, and we have a vector bundle over $S$, and we pull it back along $\alpha$ then it's still a vector bundle, no "jumps" are introduced?

Hmm, that doesn't sound right, I'd imagine you can pull back a vector bundle along any map and still get a vector bundle. Can you tell me a true sentence containing the words "flat" and "jump in dimension" that helps explain your remark?

Tim Hosgood (Dec 23 2020 at 03:50):

sure! here’s a nice fact:
if $f\colon X\to Y$ is a faithfully flat morphism of (locally Noetherian) schemes, then, for any closed $Z\subset Y$, the codimension of $Z$ in $Y$ is equal to the codimension of $f^{-1}(Z)$ in $X$

Tim Hosgood (Dec 23 2020 at 03:51):

similarly, if we pull back a quasi-coherent sheaf by a flat morphism then the projective dimension does not get bigger

Tim Hosgood (Dec 23 2020 at 03:52):

but it’s very important (for descent things) that faithfully flat morphisms also tell us properties about the image in terms of properties of the domain

Tim Hosgood (Dec 23 2020 at 03:53):

eg faithfully flat morphisms tell us that the image is normal (or reduced) if the domain is normal (or reduced)

Tim Hosgood (Dec 23 2020 at 03:54):

similarly for locally Noetherian, if you ask that your morphism also be quasi-compact

Tim Hosgood (Dec 23 2020 at 03:54):

there are hundreds of these sorts of lemmas, and i always have to look them up because it’s very easy (for me) to get confused between them all!

Tim Hosgood (Dec 23 2020 at 03:55):

a takeaway point though is just that “faithfully flat morphisms are general enough that they cover lots of examples that we come across, but refined enough that they ensure lots of nice properties”

John Baez (Dec 23 2020 at 04:59):

Thanks. None of those sentences have "faithfully flat" and "jump in dimension" in them, so I'm still confused about what this means:

flatness (or, more precisely, failing to be flat) tells us where things jump in dimension, which is something that happens a lot in algebraic geometry (e.g. coherent sheaves are basically vector bundles but where the rank can "jump" when passing from one open subset to another...

Tim Hosgood (Dec 23 2020 at 12:13):

you’re right, I sort of dodged around you question! let me try doing that one more time, just in case the following satisfies you:

Under some finiteness/Noetherian conditions, given a morphism $f\colon X\to Y$ of varieties, the map $Y\to\mathbb{N}$ given by $y\mapsto\dim f^{-1}(y)$ is upper semicontinuous. If $f$ is flat then this map is constant.

Tim Hosgood (Dec 23 2020 at 12:16):

and to maybe answer your (fair) request: if there is a jump in dimension in the fibres of a morphism, then it cannot be flat

Tim Hosgood (Dec 23 2020 at 12:18):

if you throw in lots of conditions (locally Noetherian, finite type, etc etc) then it turns out that the open set on which a morphism is flat (recalling the first property that I mentioned, about being generic) is exactly the open set defined by saying that the fibres don’t jump in dimension

Tim Hosgood (Dec 23 2020 at 12:19):

I admit, I have no idea exactly which conditions you need to impose for this to be true, but I think it’s a sort of “standard” amount of conditions

John Baez (Dec 23 2020 at 16:36):

Okay, thanks! So this stuff is not about "pulling back vector bundles" along a morphism, which was in my original guess about what you were talking about. It's about jumps in dimensions of fibers of a morphism. That's what I was not getting before. So okay: I'll just remember "if the fibers have a jump in dimension, the morphism can't be flat.... and if it is flat and nice in other ways, the dimensions of the fibers don't jump".

John Baez (Dec 23 2020 at 16:37):

That's about the most I can hope to remember from this encounter. :upside_down:

Tim Hosgood (Dec 23 2020 at 16:49):

sorry, my original comment about locally free sheaves and jumping dimensions etc was confusing!

Tim Hosgood (Dec 23 2020 at 16:50):

honestly that’s about all i can remember about flat morphisms without looking things up too 🙃

John Baez (Dec 23 2020 at 16:52):

The stuff that's easy to remember is actually the most important stuff.

Dan Doel (Dec 23 2020 at 16:53):

Are dimensions of fibers essentially measuring how many dimensions a map is 'projecting out,' or the like?

Dan Doel (Dec 23 2020 at 17:01):

So $f$ is exhibiting $X$ as a fixed number of extra dimensions added to $Y$?

John Baez (Dec 23 2020 at 17:03):

Look at these twisted ribbons:

twisted ribbons

John Baez (Dec 23 2020 at 17:04):

Just look at one... at most points it looks 2-dimensional, but there are some points where it "pinches down to a point".

John Baez (Dec 23 2020 at 17:05):

What we're doing here is mapping the ribbon down to the plane (your field of view, or the plane of the page).

John Baez (Dec 23 2020 at 17:06):

Most places the dimension of the fiber is zero. But at those "pinch points" the dimension of the fiber is one. I.e., there's a whole line of points on the ribbon there, all mapped to a single point on the plane.

John Baez (Dec 23 2020 at 17:10):

If these ribbons were infinitely fat, they would be planes, so mathematically what we have here are certain maps $f : \mathbb{R}^2 \to \mathbb{R}^2$. And most of the fibers of $f$ are points, but some are lines.

Dan Doel (Dec 23 2020 at 17:12):

Oh, I guess it doesn't say anything about $Y$, but about the image of $f$.

John Baez (Dec 23 2020 at 17:12):

Right.

Dan Doel (Dec 23 2020 at 17:13):

$Y$ could have extra dimensions that don't matter for the image.

Dan Doel (Dec 23 2020 at 17:14):

I guess what I said might be the 'faithfully flat' thing.

John Baez (Dec 23 2020 at 17:14):

In my example of $f: Y \to X$, $X$ and $Y$ have the same dimension at each point, but some $x \in X$ have a higher-dimensional subset of $Y$ mapping to them than others.

John Baez (Dec 23 2020 at 17:15):

In general, the dimension of the fiber can jump up - suddenly get bigger - but not jump down, I think.

Dan Doel (Dec 23 2020 at 17:17):

I'm not really sure what that means. Like, in your picture, the 1-dimensional fibers are right next to 0-dimensional fibers on both sides top to bottom.

Dan Doel (Dec 23 2020 at 17:18):

Or in any direction, really.

John Baez (Dec 23 2020 at 17:20):

What I mean is that as you approach one of those funny points on the ribbon, the dimension of the fiber is 0, 0, 0, 0, ... and all of a sudden it discontinuously jumps up to 1.

Dan Doel (Dec 23 2020 at 17:22):

Oh, okay. So, like, the fiber of a point can be above its neighborhood, but not below.

John Baez (Dec 23 2020 at 17:23):

Right. The dimension of the fiber is an upper semicontinuous function of position:

upper semicontinuous function

John Baez (Dec 23 2020 at 17:24):

(though it doesn't look like this example: it jumps up at isolated points).