User lmn - MathOverflow

Commutativity of Tor

2013-04-09T06:16:38Z

Let $A$ be a commutative ring with $1$ and $M,N$ be $A$-modules. Can you give a quick proof that $\textrm{Tor}_i(M,N) \cong \textrm{Tor}_i(N, M)$ using derived categories?

In his Homological algebra book, Weibel proves this with an argument via a double complexes: the so-called "acyclic assembly lemma", and from what I understand this argument can be essentially reworded into the language of spectral sequences. Hartshorne's discussion (in "Residues and Duality") of derivatives of functors in two variables is quite short, but it's not clear to me if this result (commutativity of Tor) immediately follows from the relevant derived category formalism.

Prorepresentable functors repres. by alg. spaces? Covering spaces by alg. spaces.

2013-03-27T15:47:58Z

Let $X$ be a (reasonable) scheme. I'm curious about constructing the constructing the covering space of a scheme algebraically. The covering space functor $F$ (below) can be represented by a projective family of schemes, and I want to know if there are situations where it can be represented by an algebraic spaces.

Specifically, for a geometric point $z \rightarrow X$, the functor $F$ (on FEt/X) given by $Y \mapsto \textrm{Hom}_X(z, Y) = F(Y)$ is, what Milne calls "strictly pro-representable". That is, there is

(1.) a projective system (= inverse limit system) of schemes in FEt/X $X_i$ and epimorphisms $\phi_{ij}: X_j \rightarrow X_i$

(2.) Elements $f_i \in F(X_i)$ such that $f_i = \phi_{ji} f_j$

(3.) The natural map $\lim_{\rightarrow} \textrm{Hom}(X_i, Z) \rightarrow F(Z)$ induced by the $f_i$ is an isomorphism of sets.

$\textbf{Question: }$ Are there general situations where $F$ is representable by an algebraic space which is not a priori a scheme?

Side note: I suspect that the functor $F$ is not ever representable by a scheme locally of finite type over $X$ (but not finite over $X$), but I don't have a proof. Is this true?

Representability of sheaves of groups

2013-03-18T14:49:46Z

There are lots of natural functors (that define sheaves in the fppf topology) that are not representable by schemes. For example, hilbert schemes of proper non-projective schemes in general need algebraic spaces. However, I know of no examples of such subtleties with group schemes. Every sheaf of groups that I know of is already representable.

Is this a consequence of general theorems? Is it considered easier to show that a sheaf of groups (in the fppf topology) is representable by a scheme than a sheaf of sets? For example, are necessary and sufficient criteria known for a sheaf of groups in the fppf topology to be representable. How about for sheaves of abelian groups?

Betti numbers of Proper nonprojective varieties

2013-02-23T04:05:00Z

This is a question about pathologies.

Let $X/\mathbb{C}$ be an irreducible projective variety smooth over $\mathbb{C}$. Then, the singular cohomology groups $H^i(X, \mathbb{C})$ have a hodge decompositon, and hodge theory tells us that the betti numbers (and hodge numbers) are not completely random. Eg, the odd betti numbers $b_{2i+1}$ are even integers, Hard Lefschetz theorem, even betti numbers $b_{2i}$ are nonzero, ...

We can also see that the even Betti numbers of a variety $X$ as above are nonzero in "another" way: $X$ has a finite surjective map to $\mathbb{P}^n$ where $n= \dim X$. Then, because $X$ is Kahler, such a map induces an injective map on singular cohomology. Hence, the result follows from the corresponding result for projective space.

$\textbf{Question:}$

1.) Are there any nontrivial restrictions on the Betti numbers of smooth, irreducible proper (but non-projective) varieties?

For example, can I have such a 4-fold with Betti numbers $b_0 = b_8 = 1$, but all other $b_i = 0$?

I find the situation a little disconcerting: for example, suppose I have a connected compact topological (or complex) manifold. If I know it's betti numbers are as directly above, then I would like to immediately say that it doesn't have the structure of an algebraic variety. However, the only thing I can say now is that a smooth complex variety with these Betti numbers can't be projective. If such a variety existed, then (by Ehresmann's thm) we can't put it in a proper smooth family over a curve with any projective variety. I'm no expert but that sounds pretty bad.

Understanding Adjointness of Sheaves in Algebraic Geometry

2013-02-22T05:03:04Z

Pushforward and pullback are very basic operations in algebraic geometry, as is the adjointness between them. I worked out a very careful of adjointness of sheaves (below) when I was working out of Hartshorne - however, I still find this theorem somewhat mysterious.

$\textbf{Question:}$ While I am comfortable with using this fairly abstract yet basic theorem, I feel like I should understand it a little better. How do you understand adjointness of sheaves? Is it clearly true if we make some (weak?) additional conditions? Is there a way to think about it to make it more transparent, more believable or even obvious? Please feel very free to work in the case of complex algebraic geometry, etc.

I tried to give a shorter, heuristic proof of adjointness using the etale space of a sheaf - but I got lost checking details. I would be very grateful if someone more knowledgeable could tell me if such a proof exists.

$\textbf{Thm}$ Let $(X, \mathcal{O}_X) \xrightarrow{f} (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces and $\mathcal{F}, \mathcal{G}$ be sheaves of $\mathcal{O}_X, \mathcal{O}_Y$ modules respectively. Then, we have a canonical bijection of sets $$ \textrm{Hom}_{\mathcal{O}_X} (f^*\mathcal{G}, F) = \textrm{Hom}_{\mathcal{O}_Y} (\mathcal{G}, f_*\mathcal{F})$$

Your comments and answers will be very appreciated!

Topologically embedding curves in Jacobian

2013-02-21T04:21:54Z

Let $X$ be an irreducible smooth projective algebraic curve over $\mathbb{C}$. Then, the Abel-Jacobi map gives embedding $X \hookrightarrow Jac(C)$ of $C$ into it's Jacobian. This map induces an isomorphism on $H_1$.

Question: Can it be made obvious, using purely topological reasoning, that a connected, oriented compact 2-dimensional topological manifold of genus $g$ embeds into a torus $\phi: C \rightarrow (S^1)^{2g}$, so that $\phi$ induces an isomorphism on $H_1$?

I don't know how to see this without using the Abel-Jacobi theorem.

Absorbing ramification and factoring finite flat maps

2013-01-25T19:57:37Z

In his Algebraic surfaces book, Beauville gives a result allowing one to "absorb ramification" for certain maps (see below). There are also something similar one can do with number fields. I would really like to know how I should think about these results, (that is, are they specific to these two situations or are they vastly more general.) I'm really hoping for a reference (to EGA?) where such a version is stated, assuming it is true.

This question naturally lends itself to questions 2,3; which are hopefully covered in the same place as #1 (assuming again that they are true).

$\textbf{1.}\textrm{ Absorbing ramification}$

If $K|k$ is a finite extension of number fields, then there are infinitely many finite extensions $E|k$, so that $K\cap E = k$ and $EK|E$ is unramified. (You can get such extensions from the approximation lemma.) (Does this fact have a name?)

$\textbf{Setup:}$ Let $X \xrightarrow{\phi} Y$ be a finite, faithfully flat map of noetherian schemes. Feel free to assume conditions on $X, Y, \phi$, as you need.

Can we have something similar for faithfully flat maps $\phi$? More specifically, let $\phi, X,Y$ be as in the setup. Does there exist $Z \xrightarrow{\tau} Y$ a finite, faithfully flat map so that the base change along $\tau$ is etale. Is there a lemma (like the approximation lemma) showing that there are "many" such maps in good conditions, and allowing us to control the ramification of $\tau$.

-In certain cases, we can absorb ramification for a fibration of a smooth surface (Beauville, p73 Lemma VI.7):

http://books.google.com/books?id=KV1WiV7WmPIC&pg=PA73&dq=Complex+algebraic+surfaces+lemma+VI.7&hl=en&sa=X&ei=e_sCUd2MM4ac2QX8qoCAAg&ved=0CDAQ6AEwAA#v=onepage&q=Complex%20algebraic%20surfaces%20lemma%20VI.7&f=false

$\textbf{2.}\textrm{ Maximal etale subextensions}$

Let $\phi, X,Y$ be as in the setup.

Does there exist a maximal etale subextension? That is, can we factor $\phi$ as $X \rightarrow Y_{et} \xrightarrow{\psi} Y$ with $\psi$ etale, and so that $\psi$ is maximal amongst all such factorizations. That is, if $\phi$ also factors as $$X \rightarrow Y' \xrightarrow{\psi'} Y$$ with $\psi'$ etale, then the map $Y_{et} \xrightarrow{\psi} Y$ factors as $Y_{et} \rightarrow Y' \xrightarrow{\psi'} Y$.

$\textbf{3.}\textrm{ Galois closure}$

The natural question is take $X, Y, \phi$ be as our initial setup, (assume that $\phi$ is separable, $X,Y$ are projective over a field, integral) and ask if there a galois closure $Z \rightarrow X$, so that $Z/Aut(Z/Y) \cong Y$ naturally. However, it's not even clear to me that the automorphism group would be finite (or more generally that the quotient exists as a scheme) - or there exists such $Z$ which is "minimal".

These questions bear a strong relationship to theorems true for number fields. One issue of course is that we use composition of number fields in number theory and in algebraic geometry we have base change along maps which are a little different operations.

Answer by LMN for Complement to an open affine subvariety in an irreducible projective one

2013-01-24T15:14:34Z

aglearner, In his article on abelian varieties Bryden Cais proves the result you mention. You can find the statement/proof on the top of page 4. (math.arizona.edu/~cais/Papers/Expos/AbVar.pdf)

Specifically, he proves that if $X$ is separated, normal, noetherian and $U \subset X$ is a nonempty affine open subset the complement has pure codimension 1. Thus, with it's reduced-induced structure it is a Weil divisor.

Universal property of blowing down

2013-01-23T03:30:53Z

Let $X$ be a smooth algebraic surface over $\mathbb{C}$, and $Y \xrightarrow{\phi} X$ the blowup at a (reduced) point with exceptional divisor $E$. Then, the we have the following universal property: Every morphism from $Y$ to an algebraic variety $Z$ that contracts $E$ to a point factors through $X$ (Beauville, Algebraic Surfaces p.17) I'm not sure to think of this (univ. prop. of blow. down) as a property of smooth surfaces over $\mathbb{C}$ or more generally. Here are some natural questions come to mind:

Is there a more general universal property of blowing down along these lines. I don't expect there to be an answer for arbitrary blowups (say of noeth. schemes), however I would like to replace the field $\mathbb{C}$ by other algebraically closed fields like $\overline{\mathbb{F}_p}$, and if the formalism allows it, even $\mathbb{Q}$. There are a couple of cases:

1.) I can't even figure out a universal property if we stay in the context schemes smooth over $\mathbb{C}$, and blow-up at smooth, irreducible subvarieties $Z$.

2.) Let $X, Y, \phi, E$ as above (in particular, surfaces). Assume that $X$ is reduced, but possibly singular, and allow $\phi$ to be a blow-up at an arbitrary (possibly non-reduced) point. Is there a universal property that blowing-down along the exceptional divisor satisfies in this case? (The way I see it, one issue is to replace "collapsing $E$ to a point" by something else.)

3.) I'm very much interested in the most general version of the universal property one can formulate. If there is a reference where this is covered (EGA?) I would love to see it. I also don't have "good" reasons to believe that a universal property doesn't hold in complete generality (say in the context of noetherian schemes). If you're convinced that this is the case, I'd love to hear your reasoning.

Connectedness principle in algebraic geometry

2013-01-08T19:37:36Z

What is the most general version of the connectedness principle in algebraic geometry? In particular, I'm interested in cases where there is no field available (eg, $Y$ below is the spectrum of something like $\mathbb{Z}_p$), or if it is then it isn't algebraically closed.

In his algebraic geometry book, Hartshorne gives the following version (ex III.11.4):

Let $k = \bar{k}$ and ${X_t }$ be a flat family of closed subschemes of $P^n_k$ parametrized by an irreducible curve $T$ of finite type over $k$. Suppose there is a nonempty open set $U \subset T$ such that for all closed points $t \in U$, $X_t$ is connected. Then $X_t$ is connected for all $t\in T$.

You can also restate this in terms DVR's

Let $X\rightarrow Y$ be a proper faithfully flat map with $Y$ the spectrum of a DVR. Let $y$ be the generic point of $Y$ and assume that $\dim_{k(y)} H^0(X_y, \mathcal{O}_{X_y}) = 1$. In particular, this condition implies that the generic fiber is connected. Then the special fiber is connected.

Answer by LMN for On the blow-up along the diagonal in a product

2013-01-13T19:12:50Z

Chuck, I think you can find references for both of the statements 1,2 that Allen gives in Fulton's Intersection theory, Appendix B, Section 5, and maybe also section 7. There, I think (1) is proved more generally for a regularly embedded subscheme. I asked a question about (2) some time ago. You might find what you are looking for (including how one proves 2, which is easy, and "intuition") here:

http://mathoverflow.net/questions/111430/tangent-bundle-and-normal-bundle-in-self-product

Fulton doesn't include proofs for everything but gives references to EGA. (Since I don't have the book with me, the section numbers might be a little off, maybe someone can correct me.)

Primitive Cohomology Useful?

2013-01-12T05:21:03Z

In her book, after proving the hodge decomposition, Voisin spends time discussing primitive cohomology $H^r(X, \mathbb{C})_{prim} = \ker L^{n-r+1} \subset H^r(X, \mathbb{C})$ (where $L$ is the Lefschetz operator). She proves several general theorems regarding/using primitive cohomology (Hodge index, Lefschetz decomposition, a bilinear form on $H^k(X,\mathbb{C})$ behaving in a controlled way on primitive cohomology) and establishes some technical results (if $\omega$ is a primitive form then there is a formula for $*\omega$ in terms of the Lefschetz operator and $\omega$).

$\textbf{Question: }$ I'm having a hard time understanding why one should care about primitive cohomology. Can you deduce lots of interesting facts about nonsingular complex projective varieties with say the Lefschetz decomposition as was the case with the Hodge decomposition? What are some typical applications? I'd really like some examples to illustrate if/how primitive cohomology is useful.

Specifically, I am interested in how primitive cohomology could be useful on a "daily basis". For example, let $X, Y$ be smooth complex projective varieties. Sometimes one can deduce that there are no surjective maps $X \xrightarrow{\phi} Y$ because such maps induce injective maps on cohomology (which preserve Hodge structure). Can primitive cohomology give a more refined obstruction to the existence of $\phi$ in certain cases?

$\textbf{Computing}$

1.) How about primitive cohomology? This depends on the choice of a Kahler form. Do the dimensions of the primitive cohomology groups not depend on the choice of Kahler form? It's not clear to me if primitive cohomology of abelian varieties depends only on the dimension. Does one know the dimensions of primitive cohomology groups of an abelian variety? How about other classes of varieties? For a K3 surface, it seems like one can give the dimensions of primitive cohomology groups independent of kahler form, the main point is $h^{1,1}$, where primitive cohomology has dimension 19.

$\textbf{Functoriality}$

2.) A surjective map of smooth complex projective varieties is injective on cohomology and a map of hodge structures. A finite surjective map pulls back ample divisiors to ample divisors, so if we choose kahler classes appropriately, then such a map induces a map on primitive cohomology. Does a more general class of maps induce maps on primitive cohomology (if we choose kahler classes appropriately)?

Stein factorization and flatness

2013-01-08T02:43:34Z

$\textbf{Question: }$ Suppose $X \xrightarrow{\phi} Y$ is a proper faithfully flat map of noetherian schemes and let $X \xrightarrow{f} Y' \xrightarrow{g} Y$ be the Stein factorization. Is either of $g$ or $f$ necessarily flat? I’d really like to see examples if these are false. In general, for a composite as above with $f$ faithfully flat, we have that $g$ is faithfully flat or flat iff $\phi$ is.

More generally, it seems like if $\phi = g \circ f$ is a factorization of a proper flat map with $g$ finite then $g$ is also flat. But I can't prove this.

This came up when I was thinking about how I would prove the connectedness principle in algebraic geometry (Let $X\rightarrow Y$ be a proper faithfully flat map with $Y$ the spectrum of a DVR. Then if the generic fiber is connected, so is the special fiber), and you can prove this with ZMT. The above result would give you a different proof. If $f$ in the Stein factorization above is faithfully flat then so is $g$ (by general properties) but because the generic fiber is connected $g$ is also an isomorphism on an open set, hence an isomorphism.

Googling around shows that it also came up in the comments here http://mathoverflow.net/questions/65267/global-sections-of-flat-scheme-also-flat but without a resolution.

Points in sites (etale, fppf, ... )

2012-12-30T07:39:45Z

I asked a part of this in an earlier question, but that part of my question didn't receive precedence.

http://mathoverflow.net/questions/117229/etale-site-is-useful-examples-of-using-the-small-fppf-site

Let $X$ be a scheme (assume it is as nice as you like). There is a description of "points" in the (small) etale site $X_{et}$, and these are the geometric points of $X$. More generally, I've heard that the notion of "points" makes sense in any site (maybe "any" is a little too strong?).

1.) Can you give me a reference defining "points" in other sites. Specifically, I am interested in the small fppf site over a scheme and the big etale site. Is the notion of "points" a useless notion in sites other Zariski and small etale?

2.) What are "points" in other sites "supposed" to do? Is there an analogy that we keep in mind (as to why they are called points)? In the case of the Zariski site, the "points" have a natural structure of a locally ringed space - (the local rings being the stalks in the Zariski site) and this gives a canonically associated locally ringed space to a given site. An analogy similar to this doesn't seem to hold in the small etale site over a scheme.

3.) To whatever a "point" is, I expect one would have a naturally associated local ring. Is this the case in the small fppf site over a (nice?) scheme? This is of course the case in the etale and Zariski site. The small fppf site over a scheme seems a little strange, since limits tend not to be directed.

Recovering torsion in singular homology from cplx of singular chains

2013-01-02T18:35:07Z

For a simply connected simplicial complex, a theorem of Whitehead (Derived categories for the working mathematician, bottom of page 2) explains that the associated chain complexes with coefficients in $\mathbb{Z}$ $$K \textrm{ : } \rightarrow C_n(X) \rightarrow C_{n-1}(X) \cdots $$ contains more information than the singular homology/cohomology groups (two such simplicial complexes are homotopic iff there is a certain relation between their associated chain complexes involving the chain complex of another simplicial complex).

Question: Let $X$ be a compact simplicial complex. Can one recover the torsion in $H^i(X, \mathbb{Z})$ from knowing the complex of simplicial chains (with $\mathbb{Q}$-coefficents)? Is there a procedure to do this?

Finite extension of projective space

2012-12-30T21:25:27Z

Let $k$ be an arbitrary field, we work with schemes $X$ of finite type over $k$. Does every irreducible projective scheme have a finite surjective morphism to a projective space $\mathbb{P}^n_k$?. What if I just assume that $X$ is equidimensional. Does the same argument work?

We know that a proper $k$-scheme with this property must be also be projective by formal properties of ample line bundles.

I would do this by projecting from sufficiently general points, and this probably works (maybe not in complete generality) but I can't help but think there is a cleaner argument (that also doesn't require possibly assuming that the field is infinite, algebraically closed or of characteristic $0$).

Feel free to assume our schemes have basic niceness properties.

Etale site is useful - examples of using the small fppf site?

2012-12-26T04:11:49Z

Edit: After the answers and comments, I'm hoping for a little bit of elaboration (in the comment to the answer below.) Also, question 2 was discussed here:

http://mathoverflow.net/questions/117595/points-in-sites-etale-fppf

There, Davidac897 gave a nice description of points in more general sites. I would be interested if there is a different description of points in the small fppf site and big etale site over an (as nice as you like) scheme (similar to the very concrete description of points that we have for the small etale site).

Thanks to everyone for all the helpful comments and answers!

The title very much sums it up. The etale site is extremely useful and the basic applications are well known. Milne also devotes time to the Flat site in his Etale Cohomology book. I am hoping that someone can give me example applications.

1.) I'm most interested in the (small) Flat site. What do you typically use this for? Let $X$ a scheme over $\mathbb{F}_P$ and $\alpha\alpha_P$ be the sheaf on the small fppf site over $X$ defined by group scheme $\mathbb{F}_P[t]/(t^p)$. The sequence of sheaves

$$ 0 \rightarrow \alpha\alpha_P \rightarrow \mathbb{G}_a \xrightarrow{F} \mathbb{G}_a \rightarrow 0$$ ($F$ is the map $z \mapsto z^p$) makes sense in the (small) Etale site, but is typically not exact there. However, it becomes exact in the (small) fppf site. This is useful, because a ses of sheaves yields a long exact sequence, and hence relations that one (at least I) cannot so easily express without the Flat site. I don't even know if this example is typical, or if there are many other examples on these lines (or many examples not along these lines).

$\textrm{Principal Homogenous Spaces}$: Cohomology in the flat site calculates the set of principal homogenous spaces over a scheme (wrt a group scheme $G/X$).

2.) What are points in the (small) flat site? I am not able to dream up a good description of these (or where to look).

Oriented finite CW complex with prescribed homology?

2012-12-26T15:32:50Z

Moore spaces are finite CW complexes with prescribed homology, but they may be non-orientable and even not topological manifolds. Are there oriented, connected CW complexes with prescribed homology (with $\mathbb{Z}$-coefficents?) If such a complex is also a topological manifold then there are additional restrictions (Poincare duality) so we can't expect such spaces to be homotopic to topological manifolds.

As a side note: Should I consider CW complexes as the "most general & nice" class of topological space that people commonly work with upto homotopy. (This is really a Yes/No question intended to be interpreted as basic. For example, the class of CW complexes (upto homotopy) includes all other classes of topological spaces I know: topological manifolds (with or without boundary), simplicial complexes, delta complexes. In fact, except the class of topological manifolds (with or without boundary), all of these describe the same class of topological spaces upto homotopy) For me, topological manifolds should have a countable basis.

Infinitesimal deformations and moving cycles

2012-12-20T17:30:26Z

The wonderful responses to an earlier question http://mathoverflow.net/questions/111464/self-intersection-and-the-normal-bundle motivated me to ask the following question:

Let $Y \subset X$ be a subvariety of a variety $X$. Infinitesimal deformations of $Y$ in $X$ are subschemes of $X \times \textrm{Spec } k[\epsilon]/(\epsilon^2)$ flat over $\textrm{Spec } k[\epsilon]/(\epsilon^2)$ and with closed fiber $Y$. Such subschemes correspond bijectively to sections of the normal bundle $\mathcal{N}_{Y/X}$. (Hartshorne, III.9)

$\textbf{Question:}$ Do infinitesmal deformations of a regularly embedded subvariety $Y \subset X$ of codimension $d$ naturally determine cycles in $X$ (rationally equivalent to $Y$)? This seems like a bit of a long shot, but comments of Charles and Donu in the linked question seem to suggest that something like this is true.

If this were true, it would be important for both the linked question, and in it's own right. References where I can learn the relevant material be greatly appreciated.

Self-intersection and the normal bundle

2012-11-04T15:03:16Z

Let $X/k$ be a surface nonsingular and proper over an algebraically closed field $k$. Let $C \subset X$ be a nonsingular curve. Then it is clear that the self-intersection $(C \cdot C)_X$ is $\textrm{deg}_C ( \mathcal{N}_{X/C} )$ , basically a matter of definition in intersection theory. More generally, if $X/k$ is a proper variety of dimension $k$, and $Y \subset X$ is a cartier divisor, the the class $[Y\cdot Y] \in A_{k-2}(Y)$ is the class of the line bundle $\mathcal{O}_X(Y) \vert_{Y} = \mathcal{N}_{Y/X}$ . Both of these results are fairly easy to prove. I'm asking for something a little different:

$\textbf{Question:}$ I imagine these results are "intuitively clear" at some level to geometers. Let's stick to complex algebraic varieties. In the setting of surfaces, can one explain why the normal bundle controls the number of points that divisors linearly equivalent to $C$ meet $C$? I want to say that this "follows" because we can consider the normal bundle as a "tubular neighborhood", but I don't know how to do this precisely, or how to finish the argument. How about in the higher dimensional case?

Hodge numbers of reduction mod $p$

2012-12-17T19:52:42Z

Let $X$ be a projective variety defined over a number field $K$, and $p \in \textrm{Spec }\mathcal{O}_K$ a maximal ideal, so that reduction mod $p$ makes sense, and the resulting scheme (mod $p$) $\bar{X}$ is smooth over the relevant finite field. Assume that $X$ smooth over $K$.

1.) In the case that $X$ is a curve, is there a short argument to show that the geometric genus of $X$ and of $\bar{X}$ are the same? Certainly if $X$ is a plane curve this is clear.

2.) The hodge numbers $h^{p,q}_X = \dim H^p(X, \Omega^q)$ make sense in all characteristics. Are the hodge numbers preserved under reduction mod $p$, that is, $h^{p,q}_X = h^{p,q}_\bar{X}$ ?

3.) The Weil conjectures tell us that we can recover the Betti numbers of $X$ (considered as a complex manifold) from the zeta function of $\bar{X}$. There are many smooth projective varieties that have reduction $\bar{X}$ mod $p$ and the Weil conjectures tell us that all of them have the same Betti numbers. Can one prove this without using the Weil conjectures, perhaps with Etale cohomology?

4.) More generally, if $\mathcal{L}$ is a locally free sheaf on $X$, and $\bar{\mathcal{L}}$ denotes the reduction mod $p$, I would guess that the numbers $\dim H^p(X, L)$ and $\dim H^p(\bar{X}, \bar{L})$ don't match up - but I don't have a good example.

I am interested in proofs (not using the Weil conjectures if possible).

Torsion in cohomology of smooth manifolds

2012-12-19T19:54:49Z

I've been interested in the possible (singular) cohomology groups of complex projective algebraic varieties, and there are lots of theorems that give various restrictions on these (Hodge decomposition, Lefshetz theorem, ... ). I realized that I would like to know more about the what is true for smooth manifolds hence my questions:

1.) One can construct CW complexes that have prescribed (reduced) homology groups (coeffs in $\mathbb{Z}$), these are the Moore spaces. However, they aren't even topological manifolds in general. Can one construct compact oriented smooth manifolds that have prescribed singular cohomology groups $H^i(X, \mathbb{Z})$, provided that after we remove torsion our sequence of groups satisfy Poincare duality? Should one expect that this is "generally possible" but it may be hard to actually construct examples?

2.) If $X$ is a compact oriented smooth manifold, is there any regularity in the torsion subgroups of it's cohomology: $H^i_{sing} (X, \mathbb{Z})$? (Eg, poincare duality gives regularity between the various torsion free parts.) How about if $X$ is a nonsingular complex projective variety?

3.) For $X$ a smooth oriented manifold, it seems like compactly supported cohomology contains more information than ordinary cohomology. Can one recover ordinary cohomology $H^i_{sing}(X, \mathbb{Z})$ from compactly supported cohomology $H^i_c(X, \mathbb{Z})$? How about if we take coefficents in $\mathbb{Q}$?

I'd love to see "typical", or common examples where various phenomena appears.

Schemes with no nonconstant maps to lower dimensional schemes

2012-12-14T20:35:17Z

Fix an algebraically closed field $k$ (arbitrary characteristic), all schemes will be of finite type over $k$.

(Property *): I'm interested in (classes of) examples of schemes $X$ (irreducible, of dimension $n$) so that any morphism of schemes $\phi: X \rightarrow Y$ with $\dim Y < n$ is constant.

There are two examples I know of. Projective spaces $\mathbb{P}^n_k$ have this property and simple abelian varieties do too. (One may also put arbitrary non-reduced structures on these, see below).

$\textbf{Claim}$: More generally, if $X$ is a proper irreducible scheme so that every effective divisor is ample (so proper=projective), then $X$ has property (*). (Projective spaces have this property by definition, and simple abelian varieties do too by a general result from Mumford's book.)

Proof: (Eisenbud-Harris give a similar argument for the case of projective space). Let $X$ be as in the theorem and $\phi: X \rightarrow Y$ be a morphism with $Y$ of smaller dimension than $n = \dim X$. Without loss of generality, we may assume $\phi$ is surjective (hence we can pullback cartier divisors). Choose an effective Cartier divisor $D$ and a point $p \not \in |D|$ the support of $D$, but in the image of $\phi$ (since $\phi$ surjective). The pullback of an effective Cartier divisor is also effective, Cartier - hence ample. The pullback of the point will contain a complete curve (hence these two subschemes of $X$ meet by the Nakai-Moishezon criteria - contradicting that $p \not \in |D|$. $\square$

Easy observations

(1) Having property $*$ is not stable under blowing-up (Blow up of $\mathbb{P}^2$ is $\mathbb{P}^1 \times \mathbb{P}^1$.)

(2) If a scheme $X$ satisfies the claim, then by definition, so does $X_{red}$. Further, any thickening of $X$ has property $*$.

$\textbf{Proof: }$ To check that an $X$ satisfying the claim satisfies $*$, we did calculations in the intersection ring. This is invariant under changing the non-reduced structure. $\square$

Questions

Main question: Are there other (families of?) examples of schemes satisfying (*)?

(1) Does every scheme (no finiteness conditions!) have a dense affine open subset? This came up when I was thinking about this, and I realized I can't prove it offhand. Certainly it is true for irreducible schemes, and suffices to show it for connected schemes.

(2) Do you suspect that the only examples also satisfy the claim above? That is, have every effective divisor ample?

(3) Certainly all examples of schemes satisfying $*$ must be connected. Are there connected, but not irreducible examples?

I thought this was a little interesting (and admittidenly, I have no applications in mind.)

Cohomology of vector bundles via Intersection Theory

2012-12-12T05:37:48Z

Let $X$ be a smooth projective variety over a fixed field $k$ (take $k = \mathbb{C}$ if necessary). For a vector bundle $E$ on $X$, $ch(E)$ will be in the Chow ring.

$\textbf{Question 1: }$ If $\mathcal{E}$ is a locally free sheaf of rank $n$ on $X$, (with associated vector bundle $E$) can one recover the dimensions of the sheaf cohomology groups $\dim_k H^i(X, \mathcal{E})$ from the total chern class $\textrm{ch}(E)$? How about just $\dim_k H^0(X, \mathcal{E})$? If not that, how about if $E$ is just a line bundle? Can we at least determine if $\mathcal{E}$ has global sections?

$\textbf{Question 2: }$ In the case $k = \mathbb{C}$, can one recover the dimensions of the singular cohomology groups $\dim_k H^i_{sing} (X, k)$ from total chern classes of various bundles? We can recover the Euler characteristic of $X$ as $\int_X c_n(T_X)$. In the case of curves, we can even recover the geometric genus (since this is a degenerate case: the Euler characteristic and geometric genus encode the same information). Can we recover the geometric genus of $X$ if $\dim X > 1$ from chern classes of various bundles?

$\textbf{Question 3: }$ Is there a good example to indicate the kind of information that $\textrm{ch}(T_X)$ carries about $X$ beyond it's Euler characteristic?

$\textbf{Question 4: }$ Colloquially, people refer to the Chow ring as giving a "homology theory". In the case $k = \mathbb{C}$, can one recover the usual (singular) homology groups $H_i(X,\mathbb{Z})$ from the Chow groups? If not, what about $H_i(X, \mathbb{Q})$?

2-dimensional galois representations

2012-12-09T00:29:13Z

Class field theory gives us a framework in which we can understand one-dimensional galois representations. I'd like to learn about the galois representations attached to modular forms, but I'm having trouble navigating the immense amount of material available on the web. Can someone give me a reference for the construction? Is this covered in Shimura's "Introduction to the arithmmetic theory of automorphic functions" book?

Basis for hodge decomposition of Elliptic K3 Surfaces

2012-12-06T18:52:08Z

We know the hodge numbers of K3 Surfaces. To work out some ideas, I'd like to know an explicit basis for the hodge decomposition $H^{p,q}$ of a smooth Elliptic K3 Surface over $\mathbb{C}$ (for all $p,q$, but no other restriction on the surface). Are there some such surfaces where particularly "nice" bases are known?

Scheme defined over $\mathbb{Z}$

2012-11-29T04:37:24Z

I'd like to check a definition:

If $X$ is a scheme, what does it mean to say that $X$ is "defined over $\textrm{Spec }\mathbb{Z}$"? Is this a precise statement? Certainly this statement requires that $X$ is finite type over $\textrm{Spec }\mathbb{Z}$.

If $X$ is a projective of affine variety over $\textrm{Spec }\mathbb{C}$ (with choice of embedding) we can ask if the coefficients of the equations defining it are integers, and maybe call such a scheme to defined over $\textrm{Spec }\mathbb{Z}$. Other than not being "coordinate free", it is also easy to get schemes that are better said to be "defined over $\textrm{Spec }\mathbb{Z}[1/N]$" (in particular, there are lots of examples when a construction is defined over the latter scheme, and a goal is to make it defined over $\textrm{Spec }\mathbb{Z}$).

$\textbf{Question:}$ If $X$ is a scheme, does the phrase "defined over $\textrm{Spec }\mathbb{Z}$" mean that the structure map to $\textrm{Spec }\mathbb{Z}$ is fppf? (or is the base change to say $\textrm{Spec }\mathbb{Q}$ of a scheme with such a structure map).

Depth zero, high dimension

2012-11-28T04:05:08Z

$\textbf{Question: }$We know that the depth of a noetherian local ring is at most the dimension. Do there exist noetherian local rings with high dimension but zero depth? If not, what's the smallest possible depth for a noetherian local ring of dimension $n$?

Genus of non-complete intersections

2012-11-26T03:58:10Z

Suppose $X\subset \mathbb{P}_k^N$ a nonsingular curve is a complete intersection of hypersurfaces $F_1, \cdots, F_{N-1}$ (of degrees $d_1, \cdots, d_{N-1}$ resp). Then, we know that the canonical divisor on $X$ is $\mathcal{O}_X(d_1 + \cdots + d_{N-1} - n - 1)$. Hence, intersection theory on projective space gives a formula for the genus of $X$ entirely in terms of the various $d_i$. Specifically, $$2g - 2 = d_1\cdots d_{N-1} (d_1 + \cdots + d_{N-1} - N - 1)$$

$\textbf{Question:}$ Suppose $X$ is a nonsingular curve in $\mathbb{P}^N$, and $X = V(F_1, \cdots, F_m)$ is not a complete intersection. Can one get a similarly simple formula for the genus of $X$, perhaps entirely in terms of the degrees of the $F_i$? Is this too much to ask? I can't even simply describe the canonical divisor.

Non-cyclotomic abelian extensions

2012-11-19T04:48:27Z

Suppose $L|\mathbb{Q}$ is an abelian extension of number fields. Then, all the roots of unity are certainly contained in the maximal abelian extension $L^{ab}$ of $L$. Why is it obvious that if $L \ne \mathbb{Q}$ then $L^{ab} \ne \mathbb{Q}^{ab}$.

Comment by LMN

2013-04-11T06:04:00Z

@ulrich, Thanks!

Comment by LMN

2013-04-09T06:54:59Z

Thanks for you comments Sasha!

Comment by LMN

2013-04-09T06:23:13Z

Sasha, so it seems like #1 isn't mainstream. Is that right?

Comment by LMN

2013-03-27T22:46:05Z

Thanks Dan! This is interesting.

Comment by LMN

2013-03-01T01:35:16Z

@Qing: Thanks !

Comment by LMN

2013-02-23T18:19:18Z

anon, sorry to be silly - but since this isn't my area I just want to make sure. When you speaks of betti numbers in characteristic $p$, you referring to the algebraic de Rham complex?

Comment by LMN

2013-02-23T16:05:38Z

Thanks Dmitri, Donu. This is very helpful!

Comment by LMN

2013-02-23T07:59:27Z

No problem :)

Comment by LMN

2013-02-22T19:26:49Z

@ayanta, no problem, and Thanks for your comments!

Comment by LMN

2013-02-22T14:34:14Z

Thanks Donu!

Comment by LMN

2013-02-22T05:47:42Z

Sasha, Thanks !

Comment by LMN

2013-02-22T05:47:08Z

ayanta, I'm asking for something a little different, I already worked out a proof for myself. You're of course right, the proof splits up as you say :)

Comment by LMN

2013-02-21T21:24:04Z

Thanks Davidc897, and thanks everyone for your great answers!

Comment by LMN

2013-02-21T04:46:39Z

Ah, yes. Thanks Eric.

Comment by LMN

2013-02-03T00:47:59Z

John, I'm just trying to understand the question. When you say "Hodge map" are you explicitly referring to the hodge star operator? I'm a little confused, since you say the plural "Hodge maps" later (which makes sense in this context). I'm just being a little careful. Is this standard terminology?