User jacob bell - MathOverflow

Answer by Jacob Bell for Semicontinuity for complexes

2013-05-10T20:42:28Z

Since no one really bothered to write down an answer, I think it might be worth publicising the paper by Heinrich Hartmann "Cusps of Kähler moduli space and stability conditions on K3 surfaces." Proposition 6.4 might be what you want (and before that he's got some useful non tor-independent base change theorems).

A link to the paper would have been useful.

Applications of topological chiral homology and factorization algebras (aka higher Hochschild cohomology)

2013-03-15T18:03:10Z

I recently heard a talk about these topics and found them very interesting. The talk was centered on the formal structure and didn't really focus on examples.

So my question is: what is your favorite application of topological chiral homology? (or its other variants and specialisations)

what is Deligne's cohomological descent (and what are some examples)

2013-03-09T21:46:34Z

As far as I understand Deligne's far reaching generalisation of Čech cohomology is called cohomological descent and is used to endow any variety with a (mixed) Hodge structure. Again, AFAIU, the idea is to resolve the variety, then take intersections of the exceptional pieces, resolve those and so on.

As you can see I don't understand it well, so can someone please help? Also, what is the most ridiculously easy example one can have? (I guess the thing I'm interested in most is very simple examples illustrating the possible behaviours)

The only places I know that discuss this are an SGA, Brian Conrad's notes and Peters-Steenbrink.

Answer by Jacob Bell for Are loop spaces of homotopically equivalent spaces homotopically equivalent?

2013-03-10T18:11:46Z

Is the problem that you want to allow "pathological" topological spaces? I would have thought that for nice spaces all of this were sort of true by definition. (after all, $\Omega(X)=\bullet \times^h_X \bullet$ and homotopy limits are invariant under weak equivalences, no?)

Answer by Jacob Bell for On the local structure of stacks

2013-02-03T20:37:14Z

the one I know without having to look in the literature is 1)

Lemma 2.2.3 of http://arxiv.org/pdf/math/9908167v2.pdf

I think 2) is true as well (maybe you need to add the adjective tameness appropriately?) and for 3) there should be a result of Kresch saying that your stack can be stratified by quotient stacks. But I'd have to look this stuff up.

EDIT:

for 1) I should say etale topology.

for 2) I was thinking about this result (Theorem 4.4 and Proposition 5.1) by Kresch http://www.math.uzh.ch/fileadmin/user/kresch/publikation/geodm.pdf

for 3), the result I was misremembering was Proposition 3.5 of http://arxiv.org/pdf/1002.4372.pdf, and the first paragraph of the proof. (it's for stacks with affine stabilisers)

SECOND EDIT:

There is a paper by Edidin-Hassett-Kresch-Vistoli where the investigate when an Artin stack is a quotient stack. It turns out that this is closely related to the pushforward of the structure sheaf of a smooth atlas to admit a surjection from a vector bundle. Here is the review by Vezzosi. http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=611835&vfpref=html&r=34&mx-pid=1844577

Answer by Jacob Bell for When is a substack closed?

2013-02-27T23:31:48Z

No need to guess, just look it up. http://ens.math.univ-montp2.fr/~toen/cours8.pdf Definition 1.1.

The locus where a sheaf is supported in a certain dimension

2013-02-24T19:53:46Z

I am trying to understand a particular case of this question.

Let $T$ be an affine scheme of finite type over the ground field C. Let $X \to T$ be a morphism. I'll assume this to be the base change of a smooth and projective variety over C. In particular $X \to T$ is flat and proper.

Let $E$ be a coherent sheaf on $X$, which I do not assume to be flat over $T$ (and this is where the problems come in). Fix a number $r$. If $t \in T$ is a (geometric) point of $T$, let $E_t$ be the pullback to the fibre $X_t$. I would like to understand what the locus of points in $T$ such that $E_t$ is supported in dimension at most $r$.

In other words, I want to characterise the subfunctor of $T$, taking an affine $U$ to the set of morphisms $U \stackrel{g}{\to} T$, such that the pullback sheaf $g^*E$ is fiberwise supported in dimension at most $r$.

The condition can be phrased with chern characters (or hilbert polynomials I guess), as if $r$ corresponds to codimension $k$ in $X_t$ then I'm looking at the locus where $E_t$ has chern character with zeros in degrees lower than $k$. If $E$ were flat over $T$ then this locus would be open in $T$.

I suspect this condition for general $E$ should be closed or locally closed, but I can't seem to work it out. One thought would be to use the flattening stratification for $E$ and then impose the condition on chern characters, but that isn't really what we want. The other thought would be to take some locus where $E$ itself has support of dimension at most $r + \dim T$. While it is true that, for any $d$, one can split $E$ in $0 \to E_{\leq d} \to E \to F \to 0$, with $E$ support in dimension at most $d$, I'm not sure taking, say, the schematic image of the (schematic) support of $E_{\leq r + \dim T}$ is the right thing.

What is the structure of the stack of complexes supported in dimension less than r?

2013-02-22T19:36:15Z

Let $X$ be something. (smooth and projective variety over C are my assumptions) The stack $M$ parameterising coherent sheaves on $X$ splits as a disjoint union of open and closed substacks $M_\alpha$, where $\alpha$ ranges over the possible chern characters of the sheaves. (one could equivalently take hilbert polynomials)

What is nice about $M$ is that the stack parameterising coherent sheaves on $X$ with support of dimension $\leq r$ can be written as a union of $M_\alpha$. This implies that $M_{\leq r}$ is an open substack of $M$.

If now we take $D$ to be the stack parameterising complexes on $X$, that is objects of the bounded derived category of coherent sheaves, then the situation becomes more complicated. (see the paper by Lieblich for details) While I still believe that $D$ splits as a disjoint union according to the chern character of the complex*, the deal with supports is a whole other story. Usually, one defines the support of a complex to be the union of the supports of its cohomology sheaves. For example, if one takes $X = P^1$, and considers $E = O \oplus O(-1)[1]$, then the support of $E$ is certainly one-dimensional, but its chern character does not see this.

So my question is: is the subprestack $D_{\leq r}$ of $D$ parameterising complexes with support in dimension at most $r$ a substack of $D$? (in other words, does it satisfy descent?) Is it a locally closed substack?

I'm happy to make further assumptions. For example we could take $A$ to be a tilt via a torsion pair in $Coh(X)$, and we consider the stack parameterising objects in $A$.

*Is it written down anywhere that the chern character of a (relatively perfect) family of complexes is locally constant over the base?

The Grothendieck Ring of Higher Stacks

2013-02-16T13:10:36Z

The Grothendieck ring of varieties is defined to be the free abelian group spanned by isomorphism classes of varieties modulo the cut & paste (or scissor) relations, which say that $[X] = [U] + [Y]$, for any closed $Y \subset X$ and where $U$ is the open complement.

Now, we can equally speak of a Grothendieck ring of schemes, or algebraic spaces, and they all turn out to be isomorphic via the obvious inclusions. (for the scheme case one sees this by noticing that $[X] = [X_\text{red}] - 0$) There is also a Grothendieck ring of Artin stacks (everything here is of finite type over a field of characteristic zero, possibly algebraically closed, and with affine diagonal). This ring turns out to be a localisation of the previous one.

My question is: what happens if we include higher stacks?

I think there is work by Toën on the Grothendieck ring for derived stacks, but here I'm only asking about higher (underived) stacks. Do we have a similar phenomenon to the case of varieties VS schemes? In the sense that $[X] = [\pi_0(X)] - 0$, where $\pi_0$ of a derived stack is its underived truncation?

Answer by Jacob Bell for Geometric fibers of schemes.

2013-02-15T19:04:57Z

If $U \subset Y$ then the fibre product $U \times_Y X$ should be thought of as the preimage $f^{-1}(U)$. Similarly, if you are happy about thinking of points of $Y$ as being morphisms from Specs of fields then the fibre at a point $x: Spec k \to Y$ is the fibre product $Spec k \times_Y X$, often denoted $X_x$ or $X \otimes_Y k$.

Now, if you are working with varieties (or schemes of locally of finite type) over the complex numbers (or an algebraically closed field) then that's that. However in the general setting it is important to understand how your variety behaves under field extensions, and often a property is true for every finite extension if and only if it's true over the algebraic closure, that's a good reason to define the geometric fibres to be $X \otimes_Y \overline{k}$.

I guess the reason they are called geometric is that, first of all there is no Galois group acting, and second there are no phantom points appearing in the following sense. If you take $x^2 + y^2 = -1$ over the reals, then it won't have any real solutions but will have complex solutions.

If $X = Spec R = k[x_1,\ldots,x_n]/(f_1,\ldots,f_r)$ is an affine variety then points of X should correspond to simultaneous solutions to the $f_i$. If you want the solutions to be in some field extension $L$, then these correspond to morphisms of k-algebras $R \to L$. Reversing arrows, these correspond to scheme morphisms $Spec L \to X$.

Answer by Jacob Bell for Recovering an abelian category out of its derived category

2013-02-13T22:37:18Z

The derived category of an abelian category has a t-structure, so obviously that's the first thing you want. To a t-structure corresponds a heart, which is an abelian category, whose derived category might be different than the one you started with. To further complicate matters, as you noted, you could have a heart whose derived category is actually equivalent to the one you started with, but the heart itself is not the original abelian category.

It's not clear what you can recover from a triangulated category alone, or even from a triangulated category + t-structure. If you like algebraic geometry and are willing to consider additional structures then you can recover the abelian category. The derived category of a scheme, considered as a monoidal category (coming from tensor products) recovers the original scheme (and thus the abelian category). The same is true if you start with a variety with an ample canonical bundle, then the category plus the bundle do recover the variety.

Somehow this flexibility of derived categories is a nice feature, as it gives rise to interesting (and hidden?) "symmetries" and "relationships" between spaces.

As per the second question I can only think of what Sasha said, that is stability conditions. Given a stability condition one automatically gets a heart of a t-structure (which again may not have anything to do with the original abelian category) and the slices of the stability condition may be seen as a continuous family of t-structures. It would indeed be really nice to have such a thing as a moduli space of t-structures!

Answer by Jacob Bell for Mathematician trying to learn string theory

2012-12-13T10:32:34Z

the most basic book I know of is Enumerative Geometry and String Theory by Sheldon Katz.

http://books.google.co.uk/books?id=cvNMmma0pQMC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

but of course it doesn't even begin to scratch the surface of the topics you (and not only you) want to understand.

is there some condition I can impose on families of curves on a surface such that the second Ext between the ideal sheaf and the structure sheaf is zero?

2012-12-01T04:50:52Z

(the title got out of hand)

Say I have a surface $X$, then I also have M, the Hilbert scheme of curves and points on X. This can be seen as a moduli space of quotients $O_X \to O_Z$.

If $I_Z$ is the kernel of that map, I would like to impose the condition that $Ext^2(I_Z,O_Z) = 0$. I imagine this is badly behaved, but let me ask anyway:

Is this condition open, or closed, or something on M?

I wouldn't expect to be, but I lack the knowledge to cook up some convincing example.

fpqc sheafification and localisation

2012-10-24T13:31:51Z

I am slightly confused about sheafification at the moment.

I first learned sheaves defined as a subcategory of presheaves, then I was told that sheaves are also a localisation of presheaves, then I was told this was a common feature of localisations (i.e. they are often reflective), but then I was told that not every presheaf can be fpqc-sheafified.

So, what's the deal? What is the correct notion of an fpqc sheaf (localisation vs subcategory)? Or is the problem that the localisation doesn't exist (as can sometimes happen)? But then again, can't one always put a model structure on presheaves so that the homotopy category is sheaves?

Answer by Jacob Bell for "Mathematics talk" for five year olds

2012-10-03T22:41:53Z

perhaps they could play with a bunch of these

http://www.youtube.com/watch?feature=player_embedded&v=VIVIegSt81k#!

got any tricks to build up t-structures on derived categories?

2012-08-02T08:31:39Z

Are there any good tricks to construct a heart of a t-structure? (I'm thinking on the derived category of coherent sheaves of some variety)

I'll start with the only one I know. If $(T,F)$ is a torsion pair on an abelian category $A$ then you can form the tilt inside $D(A)$.

By definition, the tilt is complexes in $D(A)$ such that the minus one cohomology lies in $T$ and the zeroth cohomology lies in $F$ and all other cohomology vanishes. Unfortunately this method is only good for hearts concentrated in two degrees.

An example of such a torsion pair is if you take $Ab$ the category of abelian groups, $T$ torsion groups and $F$ free groups. In any abelian group you can find the largest torsion subgroup and the quotient will be torsion free. This can be generalised to arbitrary integral domains and from that to arbitrary integral schemes, although the geometric meaning of the tilt (if there is one) escapes me.

Another example is Bridgeland's category of perverse coherent sheaves, which can be defined as a tilt. I would be very interested in generalisations of Bridgeland's category. There is also a notion of perverse coherent sheaf by Bezrukavnikov, but from what I understand it is only useful with Artin stacks, as the jump in the dimension of the coarse space given by stabiliser groups is what allows interesting perversities. Comments on these latter perverse sheaves would also be very much appreciated, as I don't really understand the construction.

Answer by Jacob Bell for What is the upper shriek in Grothendieck duality in the non-proper case?

2012-08-02T22:49:46Z

I don't have an answer, but maybe these notes by Lipman help.

From what I understand the upper pling functor $f^!$ is a sort of Frankenstein, definitely for etale maps it's given by ordinary pullback. More generally for smooth maps you have to tensor with the relative canonical bundle.

The only description I know of is via dualising complexes, but that's perhaps not categorical enough for what you want.

If $X$ is a smooth and proper stack, does it admit a smooth and proper atlas?

2012-07-31T11:50:30Z

Fix a ground scheme $S$ (a field say). By atlas for an algebraic stack I mean a smooth and surjective morphism $Y \to X$ from a scheme (or algebraic space or affine scheme) $Y$. If the stack $X$ is smooth then all atlases are smooth schemes over $S$.

What about when $X$ is smooth and proper? Can I find a proper atlas?

If in general the answer is no, what about for Deligne-Mumford stacks?

Comment by Jacob Bell

2013-05-15T17:29:38Z

won't in general the rigidification morphism have higher pushforwards? (so that $\phi_*$ fails to be exact) I guess in characteristic zero this won't happen if you're killing off a finite group.

Comment by Jacob Bell

2013-04-25T18:09:01Z

side remark: when U is affine in X, $(F \otimes G)(U) = F(U) \otimes G(U)$ (at least when $F$ and $G$ are quasi-coherent). I guess abstractly this is just stating the existence of a basis for the topology of $X$ which plays well with the given colimit construction. (I repeat: side remark)

Comment by Jacob Bell

2013-04-23T22:01:16Z

are there examples of contractible but non-affine algebraic varieties? (or contractible but non-Stein analytic spaces?)

Comment by Jacob Bell

2013-03-21T04:16:26Z

cool, thanks for that.

Comment by Jacob Bell

2013-03-21T04:00:46Z

can I ask what the relations are for $K_1$?

Comment by Jacob Bell

2013-03-20T15:14:43Z

can HKR be globalised to schemes?

Comment by Jacob Bell

2013-03-16T20:47:39Z

I often heard the fact that topological chiral homology comes from chiral homology, but also that it's really a completely different thing. Also, I don't understand all details but I find the bit in Lurie's survey where he defines TCH quite readable. So what's the purpose of this comment? It's the following question: is there a readable introduction to chiral homology (which possibly makes clear why TCH is analogous to it)?

Comment by Jacob Bell

2013-03-16T10:08:09Z

thank dylan for the links, although they truly fly way above my head. but they aren't too big on examples, now are they? :P

Comment by Jacob Bell

2013-03-15T00:55:55Z

the idea of basing geometry on shapes sounds very much like the "homotopy type theory" program of Voevodsky and others.

Comment by Jacob Bell

2013-03-15T00:08:10Z

I presume this is all doable, by someone who can actually do it, which obviously isn't me. But the real question here is: why do you care about the colimit?

Comment by Jacob Bell

2013-03-14T17:18:00Z

since you are dealing with quivers with no relations, wouldn't you get the direct sum over all the vertices, quotiented by the images of all the morphisms? For example, for a claw with three arrows you'd get the fibred sum on three terms. Maybe this is wrong, or maybe you've already figured it out but want a more concrete description of a basis.

Comment by Jacob Bell

2013-03-11T13:31:59Z

thanks for the comment

Comment by Jacob Bell

2013-03-11T13:27:55Z

the same could be asked about any derived equivalence and it seems a very interesting (but also very hard!) question. As far as I'm aware, Tarig Abdel Gadir, a former student of Alastair Craw, has done some work on this, although a pdf has yet to appear on the arxiv.

Comment by Jacob Bell

2013-03-11T11:07:17Z

@ricardo: I agree, I put it out there as, to me, this perspective sweeps under the rug the actual maths and makes things look formal and easy. I did say in my second comment that I was cheating. :)

Comment by Jacob Bell

2013-03-10T20:42:05Z

thanks for the explanation