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Stream: learning: questions

Topic: D-modules

GhaS Shee (Sep 03 2020 at 04:16):

Hi, I am a free math learner. And I would like to ask here;

I understood that

• Differential operators D can be seen as variables over function spaces.
• the solution space of D can be seen as its zero .
• Thus it establishes sheaves on sheaves .

And now, I have a natural question.
The solving problem of zero points of D-modules can be seen as the Galois group by which algebraic equations can be solved ? Are there any differences ?

GhaS Shee (Sep 03 2020 at 04:46):

And another question is that
"Does the action of taking exponential on function f equal to giving its projective object in that D-module category?"

Matteo Capucci (he/him) (Sep 03 2020 at 10:02):

Hi Ghas, I'm not really sure I understand your questions entirely but surely there's a theory of solvability of DE similar to that of algebraic equations, à là Galois so to say: indeed, it's aptly called 'differential Galois theory'.
Observe this theory is orthogonal to the usual solvability theory, which is usually non-effective, meaning it proves existence of solutions without telling you how to write them down.
Differential Galois theory, instead, is focused on the 'algebraic solvability', which means finding a solution expressible in terms of elementary functions. Compare this with algebraic Galois theory and its concept of 'solvability by radicals' vs fundamental theorem of algebra.

GhaS Shee (Sep 03 2020 at 11:42):

@Matteo Capucci Thank you for the reply!
I definitely did not know what "Differential Galois Theory" denotes for !
( And this might be the very term what I was seeking for. )
Now I might be able to bridge nicely this term and Kashiwara's work which is in my curiousity :-)

GhaS Shee (Sep 04 2020 at 08:46):

And I got another natural question.
In presheaves, with yoneda embedding and yoneda Lemma, we can illustrate how mathematical objects can be composed.
Have we found already this kind of the category A such that A^op -> Set is the category of D-modules ?

Matteo Capucci (he/him) (Sep 04 2020 at 08:49):

I don't know enough microanalysis to answer you, though I doubt D-modules form a presheaf category. The motivation is an educated guess given the fact they probably form an abelian category and topoi (presheaf category are always topoi) do not mix well with those (AFAIK)

Morgan Rogers (he/him) (Sep 04 2020 at 08:52):

GhaS Shee said:

I understood that

• Differential operators D can be seen as variables over function spaces.
• the solution space of D can be seen as its zero .
• Thus it establishes sheaves on sheaves .

I would love if you could expand on this line of reasoning a bit more. I don't understand how the third point follows from the preceding two.

Morgan Rogers (he/him) (Sep 04 2020 at 09:04):

GhaS Shee said:

Have we found already this kind of the category A such that A^op -> Set is the category of D-modules ?

If you give me a definition of what D-modules are, I should be able to give you some concrete answers!

GhaS Shee (Sep 04 2020 at 09:08):

@Matteo Capucci
Thank you for the clarification !
Indeed differential operators like grad and div are not commutative.

@[Mod] Morgan Rogers
As @Matteo Capucci pointing out, maybe I was thinking too abstract.
I now just thought the D-modules of ordinal polynomials whose variables are 0-tensor can be seen as an abelian category ?

Morgan Rogers (he/him) (Sep 04 2020 at 09:09):

I have no idea whether that's accurate without someone telling me what a D-module of an ordinal polynomial is :wink:

Morgan Rogers (he/him) (Sep 04 2020 at 09:11):

I'm not being facetious; if you have the time, please explain what these things are (as far as you understand them :blush: ) and I will do my best to help you understand the category of them.

GhaS Shee (Sep 04 2020 at 09:18):

@[Mod] Morgan Rogers
Thank you for that ! I would like to try that :-)

GhaS Shee (Sep 04 2020 at 09:38):

let me show the example of two variables first.
over the polynomials like $p(x,y) = Σ_i x^{a_i} y^{b_i}$
then differential operators $d/dx$ , $d/dy$ are commutative .
This should be true when we replace polynomials with formal series.
So, we can handle elementary functions on this setting.

This function space is abelian and satisfying sheaf condition.
Now I would like to construct sheaves over it.

let us call this sheaf F.
differential operators are defined as function
D : F -> F
D : f |-> f'
D is generated by $d/dx$ and $d/dy$.
addition is obvious and the multiplication is their composition.

Maybe now I think we can define the distance of two functions with wasserstein metrics.
so, it seems to me that this kind of Differential Operator category satisfies sheaf condition now.
@[Mod] Morgan Rogers what do you think ?

GhaS Shee (Sep 04 2020 at 09:52):

Matteo Capucci said:

I don't know enough microanalysis to answer you, though I doubt D-modules form a presheaf category. The motivation is an educated guess given the fact they probably form an abelian category and topoi (presheaf category are always topoi) do not mix well with those (AFAIK)

I have just remembered that abelian category need not to be abelian about its multiplication.
So, I do not know why D-modules are not abelian category now.

Morgan Rogers (he/him) (Sep 04 2020 at 09:54):

So... the function space is the space of all formal power series in two variables? And you put a topology on this with some metric (I don't see how a Wassestein metric will work, but I'm not intimately familiar with its definition). And you're saying a differential operator acts on (a particular category of) sheaves over this space?

Morgan Rogers (he/him) (Sep 04 2020 at 10:02):

GhaS Shee said:

This function space is abelian and satisfying sheaf condition.

Mostly I'm confused about this statement. What are we taking sheaves over here? I presume there is some space (maybe some scheme?) and we are considering the sheaf over it that sends each open to a ring of polynomial functions over that open, subject to some constraints? Or are we taking an arbitrary space and just taking the constant sheaf at this polynomial ring?

GhaS Shee (Sep 04 2020 at 10:03):

In Hartshorne's book, there might be a concept "Hom-sheaf" .
Do you think this does not match it ?

Morgan Rogers (he/him) (Sep 04 2020 at 10:04):

GhaS Shee said:

I have just remembered that abelian category need not to be abelian about its multiplication.
So, I do not know why D-modules are not abelian category now.

Beware: the 'abelian' in "abelian category" refers to a category having certain structural properties, and is distinct from the sense in group theory referring to commutativity.

Morgan Rogers (he/him) (Sep 04 2020 at 10:05):

GhaS Shee said:

In Hartshorne's book, there might be a concept "Hom-sheaf" .
Do you think this does not match it ?

Any sheaf is defined on some space (or more loosely as an object of some topos, but I think that's a distraction here). I'm asking what space, or what kind of space, you're working over here.

GhaS Shee (Sep 04 2020 at 10:08):

I might have already built the space.
Yes, the function space of formal series.

Morgan Rogers (he/him) (Sep 04 2020 at 10:10):

Okay! It doesn't make sense to say a space satisfies the sheaf condition: the space itself determines a sheaf condition. So, what topology are you putting on this space?

GhaS Shee (Sep 04 2020 at 10:10):

the topology generated by the distance I mentioned.

Morgan Rogers (he/him) (Sep 04 2020 at 10:12):

To quote wikipedia,

In mathematics, the Wasserstein or Kantorovich–Rubinstein metric or distance is a distance function defined between probability distributions on a given metric space M.

I don't understand how this can give a metric on a space of formal power series, would you mind clarifying?

Morgan Rogers (he/him) (Sep 04 2020 at 10:16):

My own instinct would have been to go for a sup quasi-metric (where the distance between two power series is the supremum of the differences between their coefficients, which may be infinite), although that results in a space with a lot of connected components.

GhaS Shee (Sep 04 2020 at 10:16):

Function space can be seen as graphs.
So if we choose a two-dimenstional function f(x,y) then
we have a surface.

So, the Distance of two funcitons can be seen as the distance of two surfaces.

Wasserstein distance is defined with a cost function which is needed when we carry all points in a surface into another surface.
And thus it requires integral over the cost function.

Morgan Rogers (he/him) (Sep 04 2020 at 10:17):

But these power series are formal; there's no reason for them to converge to real values anywhere.

Morgan Rogers (he/him) (Sep 04 2020 at 10:20):

So do you want to restrict to a class of power series which converge everywhere on $\mathbb{R}^2$? Or perhaps almost everywhere, so the integral of the difference between two functions is defined (although possibly infinite)?

GhaS Shee (Sep 04 2020 at 10:22):

[Mod] Morgan Rogers said:

But these power series are formal; there's no reason for them to converge to real values anywhere.

Yes I was missing that point .
So, because function defined on local space is equivalent to that on Global space,
we might be able to restrict to some good region.

GhaS Shee (Sep 04 2020 at 10:23):

[Mod] Morgan Rogers said:

So do you want to restrict to a class of power series which converge everywhere on $\mathbb{R}^2$? Or perhaps almost everywhere, so the integral of the difference between two functions is defined (although possibly infinite)?

Yes, this might be smarter .. I do not know .

Morgan Rogers (he/him) (Sep 04 2020 at 10:24):

GhaS Shee said:

So, because function defined on local space is equivalent to that on Global space, we might be able to restrict to some good region.

There are power series, like $\sum_n n! x^n$, which converge only at a single point, so that's still not ideal. I know it seems like I'm being pedantic, but in order to define a sheaf (on a space), or check that something is a sheaf (on a space), you need to know what that space is.

Morgan Rogers (he/him) (Sep 04 2020 at 10:26):

It might be easiest to just reduce back to polynomials for the time being, where the (quasi-)metric you're proposing works. Let's see what happens on the space of polynomials in two variables, and then hash out generalisations later.

GhaS Shee (Sep 04 2020 at 10:28):

Indeed :big_smile: ! Thank you for that organization!

Morgan Rogers (he/him) (Sep 04 2020 at 10:29):

So, suppose I have a sheaf on my space of polynomials. That's a functor: it sends each open collection of polynomials to some set, and each restriction of opens to a mapping between the corresponding sets (contravariantly) subject to some technical conditions.

Morgan Rogers (he/him) (Sep 04 2020 at 10:33):

From your description of differential operators, they look like natural endo-transformations of sheaves...

Morgan Rogers (he/him) (Sep 04 2020 at 10:33):

(sorry for the jargon, just muttering to myself)

Morgan Rogers (he/him) (Sep 04 2020 at 10:40):

Okay, from the latter part of the explanation, it seems like we've ended up going in the wrong direction, so let's back up a bit. Here's what I suspect you wanted, roughly.
I have some space, say a two-dimensional manifold, for argument's sake $\mathbb{R}^2$. On this space, I have a sheaf which sends each open set to the collection of smooth functions which converge in every point of that open set. The restriction maps are just the usual function restriction maps, so the sheaf conditions are easily checked (a smooth function on an open set is uniquely determined by its restrictions to a covering collection of open sets, subject to agreement on intersections of these covering opens). Now this special sheaf $F$ comes equipped with a collection $\mathcal{D}$ of differential operators, each of which is a natural transformation $D:F \to F$ which sends each smooth function to its derivative with respect to that operator, as you originally described. This $\mathcal{D}$ isn't a "sheaf on a sheaf" - that doesn't mean anything, as far as I know.

Morgan Rogers (he/him) (Sep 04 2020 at 10:41):

So, now I want to know (if this context seems like a reasonable one for what you were thinking of): what is a $D$-module?

GhaS Shee (Sep 04 2020 at 10:44):

[Mod] Morgan Rogers said:

Now this special sheaf $F$ comes equipped with a differential operator $D:F \to F$ which sends each smooth function to its derivative, as you originally described. This isn't a "sheaf on a sheaf" - that doesn't mean anything :wink: .

I do not understand this part now. what was wrong ?

GhaS Shee (Sep 04 2020 at 10:48):

I think you have said it once before.
And now you said the same thing without pointing out what was wrong too.

Morgan Rogers (he/him) (Sep 04 2020 at 10:51):

A sheaf is a functor defined on the lattice of open sets of some space. These differential operators act on one specific sheaf; there's no mapping on open sets involved here. A sheaf is not the same thing as a space.

GhaS Shee (Sep 04 2020 at 10:54):

@[Mod] Morgan Rogers
This your reply is also the same thing what you said above.
I have already showed the way to see the sheaf as a space .

Morgan Rogers (he/him) (Sep 04 2020 at 10:56):

Even if we topologize the sheaf $F$, making $F(U)$ into a space for each open set $U$ in the way you described so that we can define sheaves on each of these spaces, I don't yet see how $D$ will be a sheaf on these spaces.

GhaS Shee (Sep 04 2020 at 10:59):

No, we should see F as a point on a space now. D, in turn, should be sheaf.
So, we regard F as a point in the set of sheaves.
Then we can define topology on the set.
Then we can see open sets on it .
And so on ..

Morgan Rogers (he/him) (Sep 04 2020 at 11:03):

Well, there are ways to turn sheaves into spaces, such as taking the étale space corresponding to $F$ over $\mathbb{R}^2$, but the topology won't quite be the one you described earlier. Alternatively, you can take the function space you describe, which has a related mapping down to the underlying space, but does not directly correspond to the sheaf $F$.

Morgan Rogers (he/him) (Sep 04 2020 at 11:04):

I take that back; I'm just overly accustomed to working with sheaves of sets.

Morgan Rogers (he/him) (Sep 04 2020 at 11:05):

Okay, we have a sheaf of spaces, and performing the Grothendieck construction we get a space over our base space, and you're claiming a differential operator should be a sheaf on that?

Morgan Rogers (he/him) (Sep 04 2020 at 11:06):

[I would still argue against calling $D$ or $\mathcal{D}$ or anything a "sheaf on a sheaf". A sheaf and its corresponding space are different things, even if the correspondence gives an equivalence of categories between them.]

Morgan Rogers (he/him) (Sep 04 2020 at 11:12):

So, if you are saying a differential operator should be a sheaf on this function space... I suppose an operator $D$ will send an open set in the function space to the (space of) functions obtained by applying that operator to functions in the open set? Is that how you're thinking of this as a sheaf?

GhaS Shee (Sep 04 2020 at 11:14):

[Mod] Morgan Rogers said:

Well, there are ways to turn sheaves into spaces, such as taking the étale space corresponding to $F$ over $\mathbb{R}^2$, but the topology won't quite be the one you described earlier. Alternatively, you can take the function space you describe, which has a related mapping down to the underlying space, but does not directly correspond to the sheaf $F$.

I might see the etale part, I might expect you are taking zariski topology on ideals.
And I do not understand the latter part.
I just use a forgetful functor in order to make a space from a sheaf.

GhaS Shee (Sep 04 2020 at 11:16):

[Mod] Morgan Rogers said:

So, if you are saying a differential operator should be a sheaf on this function space... I suppose an operator $D$ will send an open set in the function space to the (space of) functions obtained by applying that operator to functions in the open set? Is that how you're thinking of this as a sheaf?

Yes! Exactly!

Morgan Rogers (he/him) (Sep 04 2020 at 11:49):

So what do the restriction mappings look like? If $U \subseteq V$ are open sets of functions, we need a restriction map $D(V) \to D(U)$. What will that be?

Morgan Rogers (he/him) (Sep 04 2020 at 11:52):

GhaS Shee said:

I just use a forgetful functor in order to make a space from a sheaf.

I want to ask "what forgetful functor?" in order to get to the bottom of what you think a sheaf is, but let's stay on track with differential operators for the time being.

GhaS Shee (Sep 04 2020 at 11:59):

I do not understand why you are trolling the same question.
We have seen that once.

Oh, is the restriction NOT meaningful?

Morgan Rogers (he/him) (Sep 04 2020 at 12:18):

Yes, there is a problem with the contravariance here.

GhaS Shee (Sep 04 2020 at 12:20):

Do you mean it might be possible with covariance ?

Morgan Rogers (he/him) (Sep 04 2020 at 12:21):

It would, since we can just take the subset $D(U) \subseteq D(V)$ in that direction. But then we're not dealing with a sheaf any more :thinking:

GhaS Shee (Sep 04 2020 at 12:21):

That might make sense !
Actually I am mixing presheaves and copresheaves.
This is because I have less experience with cohomology and homology.

I have finally understood that I was handling them in a bad manner ... orz

Morgan Rogers (he/him) (Sep 04 2020 at 12:27):

We can carry on discussing, but I feel I should point you to something related: the cotangent sheaf. In the context of schemes, it is the (universal) sheaf of derivations over a sceme. Derivations are 'things that behave like differential operators' in the sense of satisfying the Leibniz rule and so forth. I've seen it presented in the context of smooth manifolds somewhere, but typically geometers prefer to turn it in a space and get the cotangent bundle (which I suspect you would also have a preference for).

Morgan Rogers (he/him) (Sep 04 2020 at 12:28):

I'm not an expert on these, but I hope you should be able to find a reference on the subject written in language that is familiar to you.

Morgan Rogers (he/him) (Sep 04 2020 at 12:30):

Maybe you can teach me what a $D$-module is anyway, so we can think about the category of them while you work out these foundational details?

GhaS Shee (Sep 04 2020 at 12:41):

[Mod] Morgan Rogers said:

I'm not an expert on these, but I hope you should be able to find a reference on the subject written in language that is familiar to you.

Exactly !
I am just trying to start the learning however I did not know where and when to start.

[Mod] Morgan Rogers said:

Maybe you can teach me what a $D$-module is anyway, so we can think about the category of them while you work out these foundational details?

Thank you very much, I really appreciate for your kindness :-)

Morgan Rogers (he/him) (Sep 04 2020 at 12:43):

It's what this community is for! Also feel free to take your time and reply tomorrow or whenever you've had time to do some thinking. The topic structure here means there is no time pressure. :innocent:

John Baez (Sep 04 2020 at 21:53):

Matteo Capucci said:

I don't know enough microanalysis to answer you, though I doubt D-modules form a presheaf category. The motivation is an educated guess given the fact they probably form an abelian category and topoi (presheaf category are always topoi) do not mix well with those (AFAIK).

Good point! Note that in an abelian category the projection $\pi_2 : 0 \times x \to x$ is an isomorphism for all $x$, while in an elementary topos the projection $\pi_1 : 0 \times x \to 0$ is an isomorphism for all $x$. Thus, a category that's abelian and a topos has all objects isomorphic to each other... by a unique isomorphism!

So, up to equivalence there's just one category that's abelian and a topos: the category with one object.

I'm not 100% sure that categories of D-modules are abelian, but it sure looks like it. If so, they aren't topoi.

GhaS Shee (Sep 05 2020 at 07:31):

Yesterday @[Mod] Morgan Rogers takes covariant Functor whose codomain is a set of functions .
Rather, I thought we should take a set of differential operators as its codomain, thus it might be a sheaf.
Look at the introduction of this paper.
http://www.ihes.fr/~celliott/D_modules.pdf

GhaS Shee (Sep 05 2020 at 07:41):

@[Mod] Morgan Rogers Ah, you have already mentioned it ! ( Oh, now I have found that you played with me! )

GhaS Shee (Sep 05 2020 at 08:06):

Let me organize. ( I do not understand what I do not understand now. )
First, I was using terms often mixing "a presheaf on X" and "the category of presheaves on X ".
Second, I was introduced to some wrong direction in constructing a D-module.
Third, a single presheaf $X : \hat{C}$ need not to be a category.
( however we can think it as a category by grothendieck construction. That is $\hat{C} / X$. )

GhaS Shee (Sep 05 2020 at 08:23):

We can think of the category of presheaves $Psh( X )$ .
If this category was both a topos and an abelian category, then it might be the category of single object, as @Matteo Capucci pointed out and @John Baez explained.
( They, all the objects in the category, are isomorphic to 0-object, hence it is the category $\mathscr{1}$ . )
A sheaf $D$ which consists of a bunch of D-modules can be defined as a Hom-sheaf in the above paper.

The category of D-modules is a subcategory of the category of Hom-sheaves, which satisfies the law of derivation.
(Is this named Leibniz Rule?)

GhaS Shee (Sep 05 2020 at 08:46):

Then this is a category whose morphisms are differential operators and whose objects are bunches of differential operators.
A differential operator is an ideal.

This category can have

• 0-operator
• kernel

And how about these properties of Abelian category ?

• ( How about coker ? )
• ( Are coproducts and products isomorphic ? )
• ( 0 = 1 ? )

GhaS Shee (Sep 05 2020 at 08:56):

Now let me remind my first question.

GhaS Shee said:

And I got another natural question.
In presheaves, with yoneda embedding and yoneda Lemma, we can illustrate how mathematical objects can be composed.
Have we found already this kind of the category A such that A^op -> Set is the category of D-modules ?

GhaS Shee (Sep 05 2020 at 09:10):

Now let me cheat the paper I showed the link above.
The presheaf $Der( O_X )$ is a subpresheaf of $End(O_X)$ over X .
So, $Der( O_X ) : \hat{O_X}$ .
So, the answer to my first question seems to be $O_X$ .

( Here I wonder that this kind of construction is "pure" since the $Psh( O_X )$ includes a lot of non-differential homomorphisms . The D-module presheaves can be representable as the yoneda embedding of $O_X$ and I wonder this decomositions of a presheaf are all differential operators. )

We might be able to do grothendieck construction here.
So, finding some proper D-module sheaf $D$, we might be able to think of $\hat{O_X} / D$.

( And I do not understand why these are not an abelian category and what being a non-abelian category means. )
( My example which I showed yesterday might not be proper. In order to think of D-modules now I think we should also handle the time operator $t$ together with $x(t), y(t)$ . )

Morgan Rogers (he/him) (Sep 05 2020 at 09:25):

GhaS Shee said:

[Mod] Morgan Rogers Ah, you have already mentioned it ! ( Oh, now I have found that you played with me! )

I wasn't playing with you! Just throwing out ideas in the hope of stumbling across what you were getting at :relieved:

GhaS Shee (Sep 05 2020 at 09:27):

@[Mod] Morgan Rogers
Oh, I am sorry for that! :grinning_face_with_smiling_eyes:

Morgan Rogers (he/him) (Sep 05 2020 at 09:40):

GhaS Shee said:

Look at the introduction of this paper.
http://www.ihes.fr/~celliott/D_modules.pdf

Okay, so using the definition from this paper, I have something to go on! First, our space $X$ is an algebraic variety, which comes equipped with the Zariski topology, and with this topology we can talk about the category of sheaves of sets on $X$, which is to say $\mathbf{Sh}(X)$, which is a good place to start. This category is a topos. However, as is the way with algebraic geometry, the first thing we do is to build a local ring object in this category, specifically the structure sheaf $\mathcal{O}_X$. This is a special sheaf of rings which you probably know something about: it sends each open set of $X$ to a ring (which we think of as a ring of functions), such that the stalk of this sheaf at a point $x$ in $X$ is a local ring (thought of as the local ring of functions which are non-singular at $x$).

We can build other sheaves out of this one in a few ways. Since $\mathcal{O}_X$ is a ring object inside $\mathbf{Sh}(X)$, it makes sense to talk about modules over it, and in particular to consider it as a module over itself. Then the endomorphisms of $\mathcal{O}_X$ as an $\mathcal{O}_X$-module form a ring object too, which is the sheaf $\mathcal{End}(\mathcal{O}_X)$. The sheaf of derivations is a subsheaf of this one; it is easy to define it as a sub-presheaf, as is done in the paper, and harder to show that it is a genuine sheaf.

Morgan Rogers (he/him) (Sep 05 2020 at 09:44):

The ring of differential operators which we end up with, $D_X$, is the subsheaf of $\mathcal{End}(\mathcal{O}_X)$ "generated by" the derivation subsheaf and the canonical embedding of $\mathcal{O}_X$ into this endomorphism ring. When we say "generated by", we are again understanding these as subring objects of a ring object, so it makes sense to talk about the subring they generate.

Morgan Rogers (he/him) (Sep 05 2020 at 09:44):

Example 2.3 in that paper should definitely by the basis for your intuition about what we're dealing with here.

Morgan Rogers (he/him) (Sep 05 2020 at 09:49):

Skipping over the next few points in the paper, we arrive at "A $D$-module is a sheaf of $D_X$-modules". That is, one last time, $D_X$ is a sheaf of rings (aka a ring object in $\mathbf{Sh}(X)$) and we consider the category of modules over it as a ring. In particular, the category of all $D$-modules (for a fixed algebraic variety $X$) is an abelian category, since this holds for the category of modules of any ring in any topos. I can go into detail about why if you like, but at least we have an answer to your original question!

GhaS Shee (Sep 05 2020 at 09:59):

[Mod] Morgan Rogers said:

In particular, a category of $D$-modules is an abelian category, since this holds for the category of modules of any ring in any topos.

You mean that the category of D-modules is abelian ( but a single element of this category is not an abelian category. )
Oh, this seems to well organize the whole argument ! Thanks a lot !

Morgan Rogers (he/him) (Sep 05 2020 at 10:02):

Yes, be careful to distinguish the concept of abelian category from the concept of commutative ring or commutative action, they are definitely different!

Morgan Rogers (he/him) (Sep 05 2020 at 10:03):

No problem, I'm glad I could see through things enough to help :grinning_face_with_smiling_eyes:
Unfortunately, extracting how kernels, cokernels, etc are concretely computed in such a category may be a lot more work, even though you know they exist.

GhaS Shee (Sep 05 2020 at 11:26):

GhaS Shee said:

You mean that the category of D-modules is abelian ( but a single element of this category is not an abelian category. )

Wait, this is not done well.
SGL says $Psh(C)$ is a topos for small category C. ( Not a single presheaf is. )
So if the category of D-modules is abelian, it is equivalent to the category $\mathbb{1}$.

[Mod] Morgan Rogers said:

Skipping over the next few points in the paper, we arrive at "A $D$-module is a sheaf of $D_X$-modules". That is, one last time, $D_X$ is a sheaf of rings (aka a ring object in $\mathbf{Sh}(X)$) and we consider the category of modules over it as a ring. In particular, a category of $D$-modules is an abelian category, since this holds for the category of modules of any ring in any topos. I can go into detail about why if you like, but at least we have an answer to your original question!

Is it really an abelian category ?
( I doubt that the category of modules is a ring. )

Morgan Rogers (he/him) (Sep 05 2020 at 13:01):

It sounds like you're conflating some of the things involved here, and that's the source of the confusion. Indeed, $\mathbf{PSh}(\mathcal{C})$ is a topos, which is a special kind of category. The objects of that category are the presheaves of sets on $\mathcal{C}$. That is, each individual presheaf of sets $F$ is an object of the topos. We can construct the slice category $\mathbf{PSh}(\mathcal{C})/F$, and this is also a topos, whose properties depend on the choice of the presheaf $F$.

Returning to our situation, the category of $D$-modules has objects consisting of abelian group objects in $\mathbf{Sh}(X)$ (which are precisely sheaves of abelian groups over $X$), equipped with an action of the ring object $D_X$; the morphisms are group homomorphisms which respect the action. This is just like the ordinary situation where we look at the category of modules of an ordinary ring. This category is not a topos.

Morgan Rogers (he/him) (Sep 05 2020 at 13:02):

I keep talking about ring objects and abelian group objects, so it would probably help if I clarified that.

Morgan Rogers (he/him) (Sep 05 2020 at 13:17):

Groups and rings are models of algebraic theories. We are classically used to seeing the axioms of these theories presented in terms of elements. However, the axioms can be presented in a more convenient categorical form by thinking of the operations as arrows. This is expressed using, for example, Lawvere theories. Generally, a theory consists of a signature (the underlying data of the theory) and axioms (conditions the data must satisfy in order to be a valid model of the theory); for more complicated theories, we need larger fragments of logic, but let's stick to algebraic logic for now.

For the example of groups, the signature consists of a single sort, $G$, and three function symbols, $e: 1 \to G$, $i: G \to G$, $m: G \times G \to G$. The formalism of algebraic logic allows us to make sense of the terms "$1$" and "$G \times G$"; let's just say that they will mean what you expect them to. These function symbols represent the identity element, the inverse map and the group operation, respectively, and so the axioms of the theory are equations expressing the relations between these, such as $m(x,i(x)) = e$ and $m(x,e) = x$.

Now a model of this theory in a category with finite products (in this case in our topos $\mathbf{Sh}(X)$) is a choice of object, $[G]$, and choices of morphisms $[e]:1 \to [G]$ (where $1$ is the terminal object of the category), $[i]:[G] \to [G]$ and $[m]:[G] \times [G] \to [G]$, which satisfy the equations, in that for any morphism - aka generalised element - $x:X \to G$, the composites of $x$ with these structural morphisms satisfy the equations given by the axioms. Such a model is also called a group object in the topos, because it behaves just like a group. Indeed, a group object in Set is exactly a group in the ordinary sense.

As I've hinted, the abelian group objects in a topos are exactly the sheaves of abelian groups. Local ring objects are not so simple, because expressing the axioms of local rings requires a larger fragment of logic. But in any case, looking at this definition it should be clear that a ring object is typically more complicated than a ring in the external sense: it is just some data in this category that behaves (categorically) like a ring should.

GhaS Shee (Sep 05 2020 at 13:43):

For which operation are you arguing it is group ?
And would you give me an example of a sheaf which is not a group object in terms of polynomials ?

GhaS Shee (Sep 05 2020 at 13:49):

Is there some answer after you mentioned that sheaves on X was abelian?

GhaS Shee (Sep 05 2020 at 13:51):

I began to think you are just a logician since your answers always have some hole.

Morgan Rogers (he/him) (Sep 05 2020 at 13:54):

The holes are because it normally takes a lecture course of 20 hours or more to teach sheaf theory, and I'm trying to boil it down to point you in the right direction without spending my whole day teaching you :upside_down:

GhaS Shee (Sep 05 2020 at 13:56):

So what is the answer ?

GhaS Shee (Sep 05 2020 at 14:00):

It might be easy to give an example if you are really seeing it correctly.

GhaS Shee (Sep 05 2020 at 14:07):

And we can always teach something without holes. It is just to show an example.

Morgan Rogers (he/him) (Sep 05 2020 at 14:07):

Anyway... as a concrete example let's take $X = \mathbb{C}^2$ as our base space. The category of set-valued sheaves, $\mathbf{Sh}(X)$, contains many many sheaves. For example, for each ordinary set $A$ there is the "locally constant sheaf" $\Delta_A$ that sends each open set $U$ to $A \times C(U)$, where $C(U)$ is the collection of connected components of $U$. This is not a group object in any intrinsic way. The data of such a sheaf is just some sets.

A ring object in this category consists of a sheaf of sets, plus the extra data of morphisms (which are natural transformations) satisfying extra conditions like I sketched above. A particular example is the sheaf $P$ of polynomials in two variables; the underlying sheaf of sets sends each open set to the set of polynomials defined on that open set. This has a ring structure, because we can define the operations $+,\times:P \times P \to P$ and so on. Another ring object has as its underlying sheaf the sheaf of holomorphic functions on each open set.

There are many ring objects. We pick one of them to be the structure sheaf. This is one particular ring object inside the topos of sheaves which we are interested in studying. Either of the two ring objects I have just mentioned could be chosen as a structure sheaf on $\mathbb{C}^2$. From that structure sheaf we can derive another ring object, the sheaf $D_X$ in the notation of the paper from earlier.

In a similar way, there are many abelian group objects. We can put these together into a category, which we might call $\mathbf{Ab}(X)$. This is not the same category as $\mathbf{Sh}(X)$. Yet another category, the category of $D$-modules, has as its objects abelian group objects equipped with a $D_X$-action. This is a different category again, but has a forgetful functor to $\mathbf{Ab}(X)$.

Morgan Rogers (he/him) (Sep 05 2020 at 14:09):

GhaS Shee said:

I began to think you are just a logician since your answers always have some hole.

PS none of this would work without a healthy dose of logic, so don't take digs at logicians! :stuck_out_tongue_closed_eyes:

Morgan Rogers (he/him) (Sep 05 2020 at 14:10):

Note that some confusion may arise from the name "structure sheaf". It makes it sound like an ordinary sheaf (of sets), but it actually comes equipped with extra data, hence the "structure".

Morgan Rogers (he/him) (Sep 05 2020 at 14:16):

Oh I should give an example of a group object. The simplest example is to take the structure sheaf itself. Just like classically, the data of a ring object contains the data of its underlying abelian group. So we can forget the multiplicative structure to turn any ring object (and in particular the structure sheaf) into an abelian group object. It's a bit lazy, but I don't have any nice examples to hand except to turn the locally constant sheaves I mentioned into abelian groups by taking $A$ to be the underlying set of a group; the group multiplication lifts in a nice way to give an operation on the sheaf $\Delta_A$.

GhaS Shee (Sep 05 2020 at 14:53):

I am afraid I might see what you taught me about "group object" and "ring object" and that the category of D-modules is NOT $Sh(X)$. There seems no problem now !
Oh, my godness... And I am really sorry for the digging words.

GhaS Shee (Sep 05 2020 at 15:00):

Indeed if we handle groups from outside of the group by its action, we must have operator $inv$ and so on .
I did not understand the "internalization" processes well and I might have found that you really explained it in a super nice way just now.

Morgan Rogers (he/him) (Sep 05 2020 at 16:06):

Hooray, I am glad it ended up making some sense! A challenge for you now is to externalise this category.
What do I mean by that? Well, you have a completely formal description of how this category is constructed. To really understand the category, the first step is to extract a concrete description of the objects. I've explained enough for you to reach that point, although by the time you're finished, you'll likely end up with a category of things with quite complicated descriptions, even in the intuitive examples I gave earlier. Nonetheless, this is what you would need in order to make direct arguments involving individual $D$-modules (objects in this category).

In order to make the situation more manageable, you may want to find an equivalent category with a simpler description. In the nicest possible situation, since this is an abelian category, that could look like an ordinary ring whose category of modules is equivalent to the category of $D$-modules on $\mathbf{Sh}(X)$ ((I'm not confident that such an equivalence will exist, but it's an example of how we might make this easier to work with, since the description of a category of modules for an ordinary ring is much simpler)). There are other constructions that give abelian categories that may provide options too. Which descriptions end up being useful really depends what features of $D$-modules you care about!

Morgan Rogers (he/him) (Sep 05 2020 at 16:07):

Anyway, good luck! And feel free to share any insights or epiphanies you have here.

GhaS Shee (Sep 06 2020 at 04:45):

@[Mod] Morgan Rogers
Thank you very much at all.
I will do my best :-)