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Stream: deprecated: algebraic geometry

Topic: quasicoherent sheaves

John Baez (May 12 2020 at 19:37):

Hi, @Simon Burton! Do you know how Jim is doing right now?

John Baez (May 12 2020 at 19:37):

Todd said he was okay a few weeks ago.

Simon Burton (May 12 2020 at 19:39):

Hi John, yes he is doing well. We are going to talk again tomorrow.

John Baez (May 12 2020 at 19:44):

I need to talk to someone about Dyson's "three-fold way" - a specific question about it - and he's about the only person I can imagine with the knowledge and also the interest in simple fundamental issues that might be willing and able to help me figure it out.

John Baez (May 12 2020 at 19:45):

But he tends to completely dominate the conversation and push it toward whatever he's interested in now, so I'm not sure that'll work. Also he may be sort of digusted with how I stopped talking to him about math.

John Baez (May 12 2020 at 19:45):

So I might try Todd Trimble instead.

John Baez (May 12 2020 at 19:45):

I'm starting to do a lot of "pure math" again, actually secretly mathematical physics, and I need the right people to talk to.

Simon Burton (May 12 2020 at 19:49):

Yes... Jim does not do so well with random questions that don't relate to what he is currently thinking about. But I think you should still try. I don't think he holds any grudge against you.

John Baez (May 12 2020 at 19:50):

Okay. I'll think about it. When he and I were talking a lot, I found that he was in principle interested in talking about what I was doing. It took a long long time for me to communicate whatever it was to him... but then, at some point, his answer would be amazingly helpful, often turning the question into something much deeper.

John Baez (May 12 2020 at 19:50):

But you know all about that.

John Baez (May 12 2020 at 19:51):

So, I'm debating whether I should just figure this damned thing out myself, or talk to him, or someone else.

Simon Burton (May 12 2020 at 19:56):

"But you know all about that." Yes.. and he is always so generous and patient in explaining things to me.

John Baez (May 12 2020 at 20:08):

Yes, that's certainly true too!

John Baez (May 12 2020 at 20:09):

I've considered a career where I just get Jim to explain things to me and then I write them up.

John Baez (May 12 2020 at 20:09):

It could be more useful than what I'm actually doing... but I have this need to do my own stuff.

Simon Burton (May 12 2020 at 20:43):

I'm lucky that many of the things Jim has explained to me, you have already written about it. But there's much more that could be done. So far I haven't made much progress actually finishing writing up something, but I did write an essay for the latest fqxi contest, where I talk about several ideas I learnt from Jim: https://fqxi.org/community/forum/topic/3538 . I remember when he first explained the idea behind Kleinian geometry to me, we spent hours just talking about "the triangle". I had no idea that was even possible.

John Baez (May 12 2020 at 21:04):

I'd be curious how far Jim has gotten into the Langlands program, or whatever really "big" idea he's interested in now.

John Baez (May 12 2020 at 21:04):

But if he started explaining that to me it would take years!

John Baez (May 12 2020 at 21:21):

Do you know what his current top interest is?

Simon Burton (May 12 2020 at 21:22):

Most recently we have been talking about the category of commutative quantales, as a baby step towards understanding the category of quasi-coherent sheaves of an affine or projective variety. These are supposed to be a categorification of quantales. (I'm not sure if I have this exactly right though.) The current question is: is this category of commutative quantales cartesian closed?

Simon Burton (May 12 2020 at 21:28):

Before that was a lot of (infty,1) category theory: spectra, stable homotopy theory, and all that.

John Baez (May 12 2020 at 21:33):

That sounds nice. My student @Jade Master just wrote a paper heavily using commutative quantales... but for an utterly different reason!

John Baez (May 12 2020 at 21:34):

All the stuff you list sounds good. I've finally been getting off my butt and learning more (infinity,1)-category theory. Just a tiny bit.

sarahzrf (May 12 2020 at 21:37):

hmm, i wouldve said that the categorification of a quantale is either a benabou cosmos or a quantaloid :o

John Baez (May 12 2020 at 21:39):

Yeah, actually I'm confused about the analogy Simon mentioned, now that I think about it.

John Baez (May 12 2020 at 21:41):

A commutative quantale is a commutative monoidally cocomplete poset, where "monoidally cocomplete" means the tensor product distributes over colimits. So one traightforward categorification is a symmetric monoidally cocomplete category, which is more or less what I call a symmetric 2-rig.

John Baez (May 12 2020 at 21:42):

But in the poset case the cocompleteness gives you completeness and closedness for free.

John Baez (May 12 2020 at 21:42):

So, in the categorical case you might want to add them on.

John Baez (May 12 2020 at 21:42):

Then you get approximately - or maybe exactly? - a Benabou cosmos.

John Baez (May 12 2020 at 21:43):

Yeah: "Jean Bénabou‘s original definition (see Street 74, p. 1) was that a cosmos is a complete and cocomplete closed symmetric monoidal category. "

sarahzrf (May 12 2020 at 21:44):

yeah lol to be entirely honest i usually remember that by remembering that it's basically a categorification of quantales

John Baez (May 12 2020 at 21:44):

I assume the tensor product distributes over colimits in a Benabou cosmos, though what I just said doesn't make it 100% clear.

sarahzrf (May 12 2020 at 21:44):

it certainly does

sarahzrf (May 12 2020 at 21:44):

it's a left adjoint

sarahzrf (May 12 2020 at 21:45):

er, well, holding either argument fixed gives a left adjoint, to be precise

John Baez (May 12 2020 at 21:45):

Now "the category of quasi-coherent sheaves of an affine or projective variety" certainly is a Benabou cosmos... so I guess that's the idea here.

sarahzrf (May 12 2020 at 21:46):

ah

John Baez (May 12 2020 at 21:46):

We're talking sheaves of modules over the structure sheaf, so they have a "tensor product".

sarahzrf (May 12 2020 at 21:46):

hmmmmm

John Baez (May 12 2020 at 21:46):

So yeah, I think that's the idea.

sarahzrf (May 12 2020 at 21:47):

i learned recently that you can put "sheaf of local rings" equivalently as "local ring object in the topos of sheaves" and i think i like that better

John Baez (May 12 2020 at 21:47):

Yeah, so did Grothendieck.

sarahzrf (May 12 2020 at 21:47):

so would this be equivalent to "module over the ring object in the topos of sheaves"

John Baez (May 12 2020 at 21:48):

Yes, maybe with some slight extra condition.

sarahzrf (May 12 2020 at 21:48):

cool

John Baez (May 12 2020 at 21:48):

The "quasi-coherent" thing always confuses me, but I bet it's in there precisely to make what you said come out true!

sarahzrf (May 12 2020 at 21:49):

:D

John Baez (May 12 2020 at 21:49):

The usual definition of "quasi-coherent sheaf" always confuses me because it looks like the definition of what it means for a module to have a presentation by free modules, and I think "but all modules have a presentation by free modules!"

sarahzrf (May 12 2020 at 21:50):

hmm, i wouldn't know

sarahzrf (May 12 2020 at 21:50):

i don't, like, know algebra :sob:

John Baez (May 12 2020 at 21:50):

Oh.

John Baez (May 12 2020 at 21:50):

Anyway, I think it's a condition thrown in just to rule out certain horrible examples.

John Baez (May 12 2020 at 21:51):

If so, the phrase "quasicoherent" is really poorly chosen.

John Baez (May 12 2020 at 21:51):

It sounds scary.

John Baez (May 12 2020 at 21:51):

I suspect it means "non-idiotic".

John Baez (May 12 2020 at 21:51):

I guess if you're quasicoherent you're non-idiotic, so maybe it makes sense...

John Baez (May 12 2020 at 21:53):

I'd better look up to see if my guess is right: a quasicoherent sheaf on a scheme is just a module of the ring object in the ringed topos that you get from that scheme.

John Baez (May 12 2020 at 22:02):

Hmmph, I can't find out from the nLab whether this is true or false. There a lot of other characterizations of quasicoherent sheaves.

John Baez (May 12 2020 at 22:04):

Someone around here should know!

sarahzrf (May 12 2020 at 22:04):

you can access the nlab?

sarahzrf (May 12 2020 at 22:04):

it's down for me :(

sarahzrf (May 12 2020 at 22:04):

ive used google cache a couple times, is that all

Oscar Cunningham (May 12 2020 at 22:05):

It's up for me

Reid Barton (May 12 2020 at 22:06):

John Baez said:

a module of the ring object in the ringed topos

I'm not entirely certain what this phrase means, but I am pretty sure it would just work out to be "a sheaf of modules", and not necessarily (and probably not) quasicoherent.

sarahzrf (May 12 2020 at 22:06):

oh, what on earth? i can load a page with curl but not with firefox

sarahzrf (May 12 2020 at 22:06):

wait, WAIT, is this the new dns firefox switched to? for fuck's sake

Oscar Cunningham (May 12 2020 at 22:09):

The nLab page on quasicoherent sheaves has a section called 'Synthetic definition using the internal language', but it's all 'big Zariski' whereas we want 'small Zariski'.

sarahzrf (May 12 2020 at 22:10):

omg i turned off secure dns and the nlab loads now :face_palm: are you serious

John Baez (May 12 2020 at 22:11):

Reid Barton said:

John Baez said:

a module of the ring object in the ringed topos

I'm not entirely certain what this phrase means, but I am pretty sure it would just work out to be "a sheaf of modules", and not necessarily (and probably not) quasicoherent.

John Baez (May 12 2020 at 22:12):

Okay. A ringed topos is just a topos with a ring object in it. The category of sheaves (of sets) on a scheme is a topos, and it's a ringed topos, where the structure sheaf gives the ring object.

John Baez (May 12 2020 at 22:12):

Does that help?

John Baez (May 12 2020 at 22:13):

So yes, maybe a module of that ring object is just any old sheaf of modules.

sarahzrf (May 12 2020 at 22:13):

on a scheme, do you mean?

John Baez (May 12 2020 at 22:13):

Yes. I fixed that and another slip.

sarahzrf (May 12 2020 at 22:14):

wouldn't a typical sheaf of modules have to be over a single fixed ring?

John Baez (May 12 2020 at 22:14):

We want them over a sheaf of rings!

sarahzrf (May 12 2020 at 22:14):

exactly

Reid Barton (May 12 2020 at 22:14):

I'm not sure whether (or how) to interpret "module" over a ring object internally, or externally, or whether it matters

John Baez (May 12 2020 at 22:14):

Internally. Sarah was slipping into an external viewpoint.

sarahzrf (May 12 2020 at 22:14):

yeah—i was going external because reid was :)

John Baez (May 12 2020 at 22:15):

Blame each other.

sarahzrf (May 12 2020 at 22:15):

but I am pretty sure it would just work out to be "a sheaf of modules",

—over what ring?

John Baez (May 12 2020 at 22:15):

I thought he meant what makes sense - namely, over the the sheaf of rings called the "structure sheaf", that our scheme is born with.

John Baez (May 12 2020 at 22:16):

Intuitively this is the sheaf of "well-behaved functions" on our scheme.

John Baez (May 12 2020 at 22:16):

For any open set you get a ring of functions.

sarahzrf (May 12 2020 at 22:16):

yeah, but my point was that an ordinary "sheaf of modules" is over some particular ring, but we don't have a ring, we have a sheaf of rings

sarahzrf (May 12 2020 at 22:16):

so it's unclear what a plain old "sheaf of modules" even means

John Baez (May 12 2020 at 22:17):

Okay: in algebraic geometry the "ordinary" concept of "sheaf of modules" is over a sheaf of rings.

sarahzrf (May 12 2020 at 22:17):

oh.

sarahzrf (May 12 2020 at 22:17):

okay :)

John Baez (May 12 2020 at 22:17):

I mean if you're not using sheaves then get outta here! Algebraic geometry is not for you! :upside_down:

Reid Barton (May 12 2020 at 22:17):

Yes, sorry. I mean the concept from algebraic geometry.

Reid Barton (May 12 2020 at 22:18):

i.e. M is a sheaf, and for every U, M(U) is a module over O_X(U), and they all fit together in some reasonable way.

Reid Barton (May 12 2020 at 22:18):

But not quite as reasonable a way as "quasicoherent".

sarahzrf (May 12 2020 at 22:18):

oops!

John Baez (May 12 2020 at 22:20):

Right. So Reid is starting to persuade me that "module of the ring object in the ringed topos associated to a scheme" is just the same as "sheaf of modules of the structure sheaf of our scheme".

Tobias Fritz (May 12 2020 at 22:20):

John Baez said:

So yes, maybe a module of that ring object is just any old sheaf of modules.

That's right, at least according to these slides by Ingo Blechschmidt (p.11). There's also some stuff further down on quasicoherence. Generally, Ingo's work is the place to look for information on the external meaning of internal topos things in the context of algebraic geometry!

John Baez (May 12 2020 at 22:21):

Okay, great.

John Baez (May 12 2020 at 22:21):

The nLab's attempt to be intuitive says:

John Baez (May 12 2020 at 22:21):

A quasicoherent sheaf of modules (often just “quasicoherent sheaf”, for short) is a sheaf of modules over the structure sheaf of a ringed space that is locally presentable in that it is locally the cokernel of a morphism of free modules.

John Baez (May 12 2020 at 22:21):

So it's a sheaf of modules that when you look locally enough has a presentation.

John Baez (May 12 2020 at 22:22):

I guess I have to see a non-quasicoherent sheaf of modules to understand what horrors are being ruled out here.

sarahzrf (May 12 2020 at 22:22):

is it possible that like... the proof of the fact you claimed is normally true is non-constructive

sarahzrf (May 12 2020 at 22:22):

so it doesnt hold in sheaf topoi broadly

John Baez (May 12 2020 at 22:23):

Yeah, maybe something like that!

John Baez (May 12 2020 at 22:26):

It's scary because I think of all algebraic gadgets as having presentations: they're "quotients" (really coequalizer) of free things.

John Baez (May 12 2020 at 22:27):

Like, whenever your gadgets are algebras of a monad this is true.

Reid Barton (May 12 2020 at 22:27):

On an affine scheme X = Spec R, the category of quasicoherent sheaves is equivalent to the category of R-modules, via the global sections functor. So if you have a nonzero sheaf of modules with no nonzero global sections, that's a non-quasicoherent sheaf.

John Baez (May 12 2020 at 22:28):

That last sentence is for an affine scheme, you mean....

John Baez (May 12 2020 at 22:28):

I'm sure you meant that.

sarahzrf (May 12 2020 at 22:28):

nlab appears to say that 'internal choice' holds in a sheaf topos iff it is boolean, and most sheaf topoi are non-boolean i think

Reid Barton (May 12 2020 at 22:28):

Yes, "on an affine scheme" was meant for both sentences.

John Baez (May 12 2020 at 22:28):

:+1:

sarahzrf (May 12 2020 at 22:29):

it's an odd space that has a boolean algebra for a frame

sarahzrf (May 12 2020 at 22:29):

(dunno whether that's sufficient but it's certainly necessary)

Reid Barton (May 12 2020 at 22:29):

I think the issue is with the notion of "presentation" or "free": locally it has to be given by a cokernel between free modules on external sets of generators, not like free modules on some funny sheaf that might only be supported at one point or something.

John Baez (May 12 2020 at 22:30):

I think I'm gonna start by following Reid's suggestion: find a nonzero sheaf of modules on an affine scheme, like the affine line say, that has no nonzero global sections.

John Baez (May 12 2020 at 22:32):

But yes, there's also something funny going on with this concept of presentation, in that it's using a set of generators instead of a sheaf of generators.

John Baez (May 12 2020 at 22:32):

That makes it sounds like a quasicoherent sheaf is like a sheaf that doesn't explode when you bring it to the surface and try to reason about it externally.

John Baez (May 12 2020 at 22:33):

(I'm thinking of certain deep-sea fish here...)

Matteo Capucci (he/him) (May 13 2020 at 07:20):

Screenshot_20200513-092002_Drive.jpg

Matteo Capucci (he/him) (May 13 2020 at 07:21):

I guess this part of Ingo's thesis is spot on

Matteo Capucci (he/him) (May 13 2020 at 07:22):

Especially Remark 8.2 says exactly what you are saying @John Baez : every module is presented by a free module! And this statement holds intuitionistically, hence holds in the sheaf topos as well

Matteo Capucci (he/him) (May 13 2020 at 07:24):

Theorem 8.3 is also nice

sarahzrf (May 13 2020 at 07:24):

so much for my guess :)

Matteo Capucci (he/him) (May 13 2020 at 07:26):

(deleted)

sarahzrf (May 13 2020 at 07:27):

o.O

sarahzrf (May 13 2020 at 07:27):

locally external

Matteo Capucci (he/him) (May 13 2020 at 07:48):

Haha OK maybe that didn't make a lot of sense, I'm deleting it

sarahzrf (May 13 2020 at 07:51):

im just reacting to the phrase itself don't worry :sweat_smile:

Matteo Capucci (he/him) (May 13 2020 at 09:08):

Yeah but still, it didn't make sense at all upon a second reading. Ingo explains it better.

Tim Hosgood (May 13 2020 at 10:50):

as a vague (but formalisable) way of thinking about the quasi-coherent condition: locally free sheaves are just vector bundles, and quasi-coherent sheaves are just vector bundles but where the rank can jump between different open sets, and everything becomes really nice (and somehow “finitely describes”) when you take homology (ie look at things up to quasi-isomorphism)

Tim Hosgood (May 13 2020 at 10:52):

it sounds very vague but i think it can be a helpful first intuition: quasi-coherent sheaves are locally free sheaves glued together by some tower of homotopy data (and if you “truncate” everything and only look at the first two degrees of your homotopy data then you get exactly coherent sheaves)

Tim Hosgood (May 13 2020 at 10:57):

(the one caveat here being that algebraic geometers usually care more about complexes of coh/q-coh/loc free sheaves that just one single sheaf itself)

Jens Hemelaer (May 13 2020 at 11:14):

The free objects in the category of (arbitrary) $O_X$-modules seem to be the ones that can be written as
$O_X \otimes_{\underline{\mathbb{Z}}} (\underline{\mathbb{Z}} \times \mathcal{F})$
for $\mathcal{F}$ an arbitrary sheaf and $\underline{\mathbb{Z}}$ the constant sheaf on the integers.

John Baez (May 13 2020 at 15:50):

Matteo Capucci said:

I guess this part of Ingo's thesis is spot on

Thanks, Matteo, that's great! I bet this thesis could help me understand algebraic geometry better by explaining some of the ideas using more topos theory. Sometimes I find that helps... sometimes it just confuses me... but overall it's much better to have this extra way of thinking about things.

John Baez (May 13 2020 at 15:56):

Jens Hemelaer said:

The free objects in the category of (arbitrary) $O_X$-modules seem to be the ones that can be written as
$O_X \otimes_{\underline{\mathbb{Z}}} (\underline{\mathbb{Z}} \times \mathcal{F})$
for $\mathcal{F}$ an arbitrary sheaf and $\underline{\mathbb{Z}}$ the constant sheaf on the integers.

Nice! If so that makes it clear how much more special quasicoherent sheaves of $O_X$-modules are than sheaves sheaves of $O_X$-modules that merely admit a presentation in terms of free $O_X$-modules. (According to Ingo's thesis the latter are all sheaves of $O_X$-modules.)

Joe Moeller (May 13 2020 at 16:07):

I don't know if this is too late, but this is how I digested what a sheaf of modules over a sheaf of rings is. It's a sheaf values in the category of all modules that matches the given sheaf of rings when you forget down to rings.
image.png

Jens Hemelaer (May 13 2020 at 17:48):

John Baez said:

Nice! If so that makes it clear how much more special quasicoherent sheaves of $O_X$-modules are than sheaves sheaves of $O_X$-modules that merely admit a presentation in terms of free $O_X$-modules. (According to Ingo's thesis the latter are all sheaves of $O_X$-modules.)

For people like me that have to translate Ingo's statements to "external language": if $\mathcal{M}$ is an arbitrary $O_X$-module, there is a surjection
$O_X \otimes_{\underline{\mathbb{Z}}} (\underline{\mathbb{Z}} \times \mathcal{M}) \longrightarrow \mathcal{M}$
and doing the same for the kernel of this map gives a presentation.

Simon Pepin Lehalleur (May 15 2020 at 20:15):

Concrete examples of sheaves of O_X modules which are not quasi-coherent are extensions by zero from open subschemes. Quoting a comment of Brian Conrad on Mathoverflow for a longer list of such examples:

"Canonical flasque resolutions, infinite direct products, extension by zero from a locally closed set (see the discussion of excision early in SGA2), sheaf-Hom (and sheaf-Ext) between quasi-coherent sheaves, topological pullbacks of sheaves (even q-coh. ones) along scheme morphisms,..."

https://mathoverflow.net/q/44563/7878

John Baez (May 15 2020 at 20:24):

What does an "extension by zero from an open subscheme" look like? Could it be something like this? I take $X$ to be the affine line, I choose a closed point $x$, and I make up a sheaf $S$ of $O_X$-modules like this: for any open set $U$ that doesn't contain $x$ I set $S(U) = O_X(U)$, while if $U$ contains $X$ I set $S(U) = \{0\}$. Is that an example?

Simon Pepin Lehalleur (May 15 2020 at 20:27):

Yes. This cannot be quasicoherent since it is a non zero sheaf on an affine scheme with zero global sections.

John Baez (May 15 2020 at 20:33):

Okay, great! Now I'm finally getting a sense of what quasicoherence is like.

Notification Bot (May 15 2020 at 20:34):

This topic was moved here from #general > quasicoherent sheaves by John Baez

Tim Hosgood (May 16 2020 at 03:02):

so you’re describing what’s called a skyscraper sheaf john (if i’m not mistaken). as simon points out, it’s a counterexample because you take it on the affine line. it’s important to note that the same construction on e.g. $\mathbb{P}^1(\mathbb{C})$ would be quasi-coherent (and in fact actually coherent)

John Baez (May 16 2020 at 05:02):

I don't think I was describing a skyscraper sheaf. I thought a skyscraper sheaf $S$ would have something like $S(U) = k$ for all $U$ containing a given point and $S(U) = \{0\}$ for all $U$ not containing it.

John Baez (May 16 2020 at 05:05):

I instead described a sheaf $S$ such that $S(U) = O_X(U)$ for any open set $U$ that doesn't contain $x$ and $S(U) = \{0\}$ if $U$ does contain it.

John Baez (May 16 2020 at 05:06):

I was trying to follow Simon's advice and define an "extension by zero from an open subscheme".

Jens Hemelaer (May 16 2020 at 08:08):

Yes, I think skyscraper sheaves are always quasi-coherent, even on affine schemes.

Simon Pepin Lehalleur (May 16 2020 at 09:44):

I think from a categorical point of view it is perhaps best to define quasicoherent sheaves in algebraic geometry as the right Kan extension of the pseudofunctor from commutative rings to Cat which associates $Mod(A)$ to $A$. This defines $QCoh(X)$ for any presheaf on affine schemes, and so in particular for the functor of points of a scheme. This points to the "right" way of extending $QCoh$ to other geometric objects (stacks, higher stacks,...). The nlab page

https://ncatlab.org/nlab/show/quasicoherent+sheaf

is pretty good on this.

Then one can construct separately an embedding into $O_X$-modules on the associated locally ringed space when $X$ is representable by a scheme and characterize the image.

One could think that having the embedding into $O_X$-modules is necessary to do homological algebra with quasicoherent sheaves, but it is not really the case. If $X$ is quasi-compact and quasi-separated (a very mild assumption), the category $QCoh(X)$ is a Grothendieck abelian category, and in particular has enough injectives.

The fact that quasicoherent sheaves were defined for general locally ringed spaces is, I think, somewhat of an historical accident. When Grothendieck developped scheme theory, one of his inspiration was the theory of coherent sheaves on complex analytic varieties of Cartan, Oka, Serre... so it was natural to define it this way and to specialize to schemes, and it helps when comparing complex algebraic and analytic geometry (as in Serre's GAGA paper). But in complex analytic geometry only coherent sheaves behave well, and in fact the category of quasicoherent sheaves on a general locally ringed space is, IIRC, very badly behaved: it is not always abelian for instance.

The lack of extensions by zero in the quasicoherent world is quite fundamental. It means that the pullback functor $j^*$ has no left adjoint when $j$ is an open immersion. This make the functoriality of quasicoherent sheaves really different from the functoriality shared by many other sheaf theories, like sheaves on complex analytic spaces, l-adic sheaves, or D-modules (the so-called "six operation formalism"). This has played a role in the story of Grothendieck-Verdier duality (the relative, non-smooth version of Serre duality). Recently Scholze has apparently managed to fix this and to have a left adjoint $j_!$ for $j^*$ using his "condensed mathematics" framework: see Lecture XI in

https://www.math.uni-bonn.de/people/scholze/Condensed.pdf

Scholze claims that this should give a much simpler proof of Grothendieck-Verdier duality.

Tim Hosgood (May 16 2020 at 13:20):

(oh woops, yes, i definitely misread your example john)

Tim Hosgood (May 16 2020 at 13:22):

my only other input: quasi-coherent sheaves in the analytic world are pretty badly behaved, but there are a bunch of subtler things that exist to sort of take their place (fréchet quasi-coherent sheaves, for example), so all is not lost, you just have to be a lot more sensitive to analytic data

John Baez (May 16 2020 at 23:09):

Neat! Right now I'm still struggling to get a feeling for quasi-coherent sheaves in plain old algebraic geometry.

Chloe (May 23 2020 at 03:03):

you probably wanna start by trying to understand what quasi-coherent sheaves on a affine schemes look like

Chloe (May 23 2020 at 03:03):

over Spec(A) they are the same thing as A-modules

Chloe (May 23 2020 at 03:05):

the general definition on a locally ringed space in terms of "locally a quotient of free blah blah blah" is a bit annoying to work with, or at least I think so

Chloe (May 23 2020 at 03:06):

but if you think of them as an appropriately globalized version of what happens with affines you will have the right intuition

Chloe (May 23 2020 at 03:08):

this probably is what that definition Simon mentioned using a Kan extension is doing: it says exactly what you want for affines, and then the abstract nonsense takes care of the gluing for you

Chloe (May 23 2020 at 03:09):

of course you can do the same thing in a more hands-on way, and this is important for calculations

Chloe (May 23 2020 at 03:09):

anyways, before I go distracted too much

Chloe (May 23 2020 at 03:10):

I'd suggest thinking about coherent sheaves over spec k[x]

Chloe (May 23 2020 at 03:10):

they are the same thing as finitely generated k[x]-modules, and you can write down exactly what all of those are

Chloe (May 23 2020 at 03:13):

if you want to think of a coherent sheaf as something kind of like a vector bundle, then you should want to understand what the fibers of that bundle look like

Chloe (May 23 2020 at 03:16):

these are very fun things to calculate: you take a point x, which corresponds to some homomorphism from A to a field, and you take the tensor product of your module with that field over A

Chloe (May 23 2020 at 03:22):

if you want to get a sense of the difference between (quasi)coherent sheaves and arbitrary sheaves of modules, I think the best way is to figure out how pullbacks work for both of them and play around with that