User simon pepin lehalleur - MathOverflow

Weil restriction of abelian schemes along finite étale (resp. finite locally free) morphisms

2013-04-10T16:49:02Z

Q: Is there a simple proof of the fact that the Weil restriction of an abelian scheme along a finite étale morphism is an abelian scheme ?

Details: Let $S$ be a scheme and $f:S'\rightarrow S$ a finite étale morphism. Let $A/S'$ be an abelian scheme. Then the following argument shows that the Weil restriction $\mathfrak{R}_{S'/S}$(A) (see section 7.6 of the book Néron Models by Bosch, Lütkebohmert and Raynaud) which a priori is just an fppf sheaf of abelian groups on $Sch/S$ is representable by an abelian scheme over $S$:

1) Theorem 1.5 of M. Olsson's paper Hom stacks and restriction of scalars (Duke Math. J. 134 (2006), 139-164.) gives a general criterion for a Weil restriction to be representable by an algebraic space, which applies here to show that $\mathfrak{R}_{S'/S}(A)$ is representable by an algebraic space over $S$.

By passing to a Galois cover, $\mathfrak{R}_{S'/S}(A)$ decomposes as a product of abelian schemes, so is representable by an algebraic space over $S$.

2) The arguments used to prove Proposition 7.6.5 (f) and (h) of Néron Models show that $\mathfrak{R}_{S'/S}(A)$ is smooth and proper as an algebraic space. Moreover the formation of the Weil restriction commutes with base change and its fibers are connected by Proposition A.5.9 of the book Pseudo-reductive Groups by Conrad,Gabber and Prasad.

3) By 1) and 2), $\mathfrak{R}_{S'/S}(A)$ is an abelian algebraic space in the sense of Section I.1 of the book Degeneration of abelian varieties by Chai and Faltings. By Theorem 1.9 of loc. cit. (due to Raynaud) it is actually an abelian scheme.

The "problem" with this proof is that 3) is a relatively delicate result. Of course, if $A$ is projective over $S$, the Weil restriction is automatically a scheme and we can do without 1) and 3). However, the question of when an abelian scheme is projective over the base is subtle in general: it is known to hold over a noetherian normal base but the argument is a key part of the proof of 3) anyway.

My motivation is to understand the push-forward functoriality of Deligne 1-motives over a base, and I would like to be able to consider general abelian schemes over general base schemes.

1) Does anyone know a simpler proof ?

2) If one assumes only that $S'/S$ is finite locally free, is $\mathfrak{R}_{S'/S}(A)$ (which might not be proper anymore) still a scheme ?

Answer by Simon Pepin Lehalleur for Picard groups of non-projective varieties

2010-08-05T16:30:27Z

The first thing to consider is the case of affine curves : let $k$ be an algebraically closed field, $C/k$ a smooth affine curve, $\bar{C}/k$ its smooth projective compactification, $\bar{C}=C\cup{p_0,p_1,...,p_n}$, $J=J(\bar{C})$ the jacobian, $\theta:C\rightarrow J$ the map induced by the choice of the base point $p_0$. Then $Pic^0(C)$ is identified with the quotient $J/\langle\theta(p_1),\ldots,\theta(p_n))\rangle$. This is always divisible but depends somehow on what this subgroup of the groups of rational points of an abelian variety look like (does it land in the torsion, etc.).

Let's think about it over $\mathbb{C}$ : there you have the quotient of a complex torus by a finitely generated subgroup : when this subgroup is not discrete the quotient does look like it is not representable as the $\mathbb{C}-$points of a scheme.

*Edit : * As Emerton pointed out in the comments, in this case the correct "geometric" object is the 1-motive associated to C. But there is a general construction of Picard 1-motives associated to varieties over a field of characteristic 0 due to Barbieri-Viale and Srinivas, which encode the $Pic^0$ geometrically :

Albanese and Picard 1-motives Luca Barbieri-Viale - Vasudevan Srinivas Mémoires de la SMF 87 (2001), vi+104 pages

http://arxiv.org/abs/math/9906165

Answer by Simon Pepin Lehalleur for Algebraic geometry examples

2010-08-11T23:10:04Z

A small but illuminating exemple : isolated singularities consisting of affine $k$ lines meeting at the origin in $\mathbb{A}^n$. One can show easily that the analytic type of the singularity depends on whether lines are coplanar by computing Zariski tangent cone, that there is a 1-dimensional moduli of analytical types for 4 lines in the plane (cross-ratio) or that seminormalization of such a singularity always gives the maximally non-coplanar case. When you introduce flat families, you can also look at flat limits of families of lines in this way, and explain where the "missing" tangent vector goes.

Answer by Simon Pepin Lehalleur for Uniqueness/motivation for the Suslin-Voevodsky theory of relative cycles.

2010-08-10T14:27:13Z

I will just sum up the situation as I see it (too big for the comment box).

One important goal is to set up a good intersection theory for cycles without quotienting by rational equivalence, and using it to get a composition product for finite correspondences, which are by definition elements of groups of the form $c_{equi}(X\times_S Y/X,0)$

It is true that the variety of definitions of cycle groups in the paper is somewhat confusing. There are 16 possible groups because starting from the "bare" notion of relative cycles (def. 3.1.3) there are 4 binary conditions : being effective, being equidimensional, having compact support (c, PropCycl), and being "special", i.e satisfying the equivalent conditions of lemma 3.3.9 (everything except Cycl and PropCycl). So you have

1)$z_{equi}(X/S,r)\subset z(X/S,r)\subset Cycl(X/S,r) \supset Cycl_{equi}(X/S,r)$

and their effective counter-parts.

2)$c_{equi}(X/S,r)\subset c(X/S,r)\subset PropCycl(X/S,r) \supset PropCycl_{equi}(X/S,r)$

and their effective counter-parts.

(1) is then a "subline" of 2))

In a sense, the most satisfying definition would be to use only cycles which are flat over $S$ (the $\mathbb{Z}Hilb$-groups, or the closely related $z_{equi}$) but pullbacks along arbitrary morphisms are not defined there in general.

With the groups Cycl, thanks to the relative cycle condition built in Cycl, you have pullbacks along arbitrary morphism, but only with rational coefficients (thm 3.3.1, the denominators of the multiplicities are divisible by residue characteristics)

The main interest of the "special" relative cycles $z(-,-)$ is in their definition : they admit integral pullbacks ! Then you have the small miracle that this condition is stable by those pullbacks and you get a subpresheaf. This means that using them you can set up intersection theory with integral coefficients even on singular car p schemes.

All this zoology simplifies when $S$ is nice : there are some results when $S$ is geometrically unibranch, but the nicest case is $S$ regular, in which the chains of inclusions I wrote down collapse, you are left with two distinctions which are reasonable from the point of view of classical intersection theory : effective/non-effective, general/with compact support. Furthermore, the intersection multiplicities are computed by the Tor multiplicity formula, so the Suslin-Voevodsky theory is really an extension of local intersection theory of regular rings as in Serre's book.

Answer by Simon Pepin Lehalleur for Relation between motivic homotopy category and the derived category of motives

2010-08-08T12:35:24Z

The short answer is that they are very different, but become quite similar if you 1) stabilize, i.e invert smash product by $\mathbb{P}^1_k$ on the homotopy side and invert tensor product by the Tate motive of the motivic side and 2) pass to rational coefficients. This is the analogue of the similar result in topology : the rationalized stable homotopy category is equivalent to the derived category of $\mathbb{Q}$-vector spaces.

The precise comparison result if you do those two operations was announced by Morel in http://www.mathematik.uni-muenchen.de/~morel/Splittinggrassman.pdf and a proof was written down recently by Deglise and Cisinski in the preprint http://www.math.univ-paris13.fr/~deglise/docs/2009/DM.pdf paragraph 15.2

Even the stable, integral versions are quite different : one way to quantify this is to say that spectra in $SH_{\mathbb{A}^1}(k)$ represent generalized cohomology theories - oriented ones like motivic cohomology, algebraic K-theory, algebraic cobordism but also non-oriented like Balmer-Witt groups, Hermitian K-theory - while objects in $DM_{k}$ represent only "oriented cohomology theories with additive group law" (in the sense of Quillen) : see e.g the Déglise-Cisinski preprint above, paragraph 10.3

For the different between unstable versions, a good simple example is the case of curves of genus greater that 1 : their effective motives are non-trivial (weight one effective motivic cohomology detects Pic ) while their unstable homotopy type is in a sense completely disconnected. This is essentially the reason why unstable $\mathbb{A}^1$-homotopy seems most interesting for "nearly rational" varieties, see the papers of Asok and Morel.

Answer by Simon Pepin Lehalleur for Work of ICM 2010 plenary speakers (and other humans)

2010-08-06T12:38:14Z

Raman Parimala is an algebraist and algebraic geometer. She studies problems related to the existence of rational points on algebraic varieties over various fields (both "dimension one" : local and global, and "higher dimensional" fields, like function fields of curves over local fields, etc.), in particular varieties associated to algebraic groups : quadrics, Severi-Brauer varieties, varieties linked to algebras with involutions... The methods include Galois cohomology, K-theory, unramified cohomology on the one hand and the classical algebraic theory of quadratic forms on the other.

As a personal note, I would say this is the area of algebraic geometry which is most satisfying from the point of view of sophisticated cohomological/K-theoretic tools (including the whole machinery developped in the wake of Morel-Voevodesky's $\mathbb{A}^1$-homotopy theory) because one can make a lot of computations of otherwise intractable invariants.

A few collaborators : Bayer-Fluckiger, Colliot-Thélène, Gille, Quequiner, Srinivas, Suresh, Tignol...

A few of her important results (some with said collaborators):

The first proof for classical groups of Serre's conjecture II on Galois cohomology of algebraic groups over fields of cohomological dimension 2
Examples of zero cycles of degree 1 without rational points on projective homogeneous varieties
Results on the u-invariant (i.e whether a quadratic forms in enough variables is automatically isotropic, like in Meyer's theorem for number fields) of function fields over p-adic fields

...

She gave a lecture in the Suslin birthday conference in Saint Petersburg in july : see the end of the following webpage, which hosts videos of all the talks :

http://www.pdmi.ras.ru/EIMI/2010/ag/program.html

Answer by Simon Pepin Lehalleur for Intuitive crutches for higher dimensional thinking

2010-08-06T07:17:57Z

This is not so much a crutch as a way to explore the upper bound of purely visual exploration of space : Jeff Weeks has made a nice computer program which allows one to fly around some compact 3-manifolds. I find it a nice way to get some intuition of global topological feature in higher dimensions.

http://www.geometrygames.org/CurvedSpaces/index.html.en

Answer by Simon Pepin Lehalleur for Can a mathematical definition be wrong?

2010-08-06T06:51:22Z

An example from algebraic geometry :

At some point during the redaction of the EGA by Grothendieck and Dieudonne, Grothendieck discovered how to make parts of the theory work without finiteness (noetherian) hypotheses on schemes, by strenghtening finiteness for morphisms (finite presentation instead of finite type). The study of morphisms of finite presentation was carried out in EGAIV.

Unfortunately, some definitions of properties of morphisms were made before this discovery. In particular, the definition of a proper morphism in EGA only includes finite type and not finite presentation, mostly because it was first used for noetherian schemes. This can lead (and lead some fine french mathematicians) to spend entire classes repeating the words "morphismes propres de présentation finie"...

Answer by Simon Pepin Lehalleur for What should be learned in a first serious schemes course?

2010-07-27T09:58:07Z

A small suggestion : the deformation to the normal cone is a nice construction that I would have liked to see in a first course. It illustrate the use of blow-ups, the degeneration of a family with constant fibers (an highly non-obvious concept the first times you see it) and how important intuitions from differential geometry - tubular neighbourhoods - have a non-trivial translation to algebraic geometry.

Answer by Simon Pepin Lehalleur for Why the Killing form?

2010-07-27T15:00:05Z

A less algebraic answer, but one that really helped me to understand the role of the Killing form, is that it induces the unique G-invariant riemannian metric on symmetric spaces $G(\mathbb{R})/K$ (K maximal compact subgroup), another fact which was very dear to Cartan as well...

Answer by Simon Pepin Lehalleur for Why do so many textbooks have so much technical detail and so little enlightenment?

2010-07-27T14:02:57Z

I hope no one will object my raising this question from the dead...

One point which has been alluded to by Tracer Tong but which is worth emphasizing is that it is sometimes very difficult to justify the usefulness of a fundamental concept without starting a whole new book. Just saying "This gets very important later on" may satisfy the lecturer/writer who knows what he is talking about but will leave the student with an aftertaste of argument by authority.

This happens most often with exercises : it is very tempting for the author to take an example or a theorem from a more advanced corner of his subject and strip it down of its fancy apparel.

I'll list a few examples of mathematical concepts I encountered in this way "before their times" and came out with the first impression that those were silly and unmotivated - and changed my mind when I learned about them in a more thorough manner :

Hyperbolic geometry (!!)
p-adic numbers (!!!)
Dirichlet series
Milnor K-theory

I don't know the best option here... It is nice to see glimpses of more exciting subjects, but sometimes it is more a way to satisfy the (quite natural) inclination of the teacher for what lays further down the road.

Answer by Simon Pepin Lehalleur for Grothendieck's Galois Theory today

2010-07-27T10:48:08Z

I don't know much about this topic, but I was recently recommended the paper An extension of the Grothendieck Galois theory of Grothendieck by Joyal and Tierney as an enlightening abstract generalisation in the language of toposes. It seems that it predates some of the other references given above, but might be worth reading.

Answer by Simon Pepin Lehalleur for Geometric flavored textbook on algebra

2010-07-27T10:09:44Z

One fundamental aspect of the connection between geometry/topology and algebra is the analogy between galois theory of fields and covering theory in algebraic topology and algebraic geometry. A nice book on the subject has appeared recently : Tamas Szamuely's Galois Groups and Fundamental Groups.

Comment by Simon Pepin Lehalleur

2013-04-15T18:21:33Z

@Jason Starr: which paper of Murre are you referring to ? Is it the Bourbaki seminar on unramified functors ? @nosr: because of Raynaud's theorem, most of the algebraic spaces (e.g. semi-abelian algebraic spaces of constant rank) I need are schemes. On the other hand, I can work with algebraic spaces, since I am ultimately interested in the complex of sheaves and the associated objects in triangulated categories of mixed motives. It is more a matter of not being familiar with the technology.

Comment by Simon Pepin Lehalleur

2013-04-10T20:05:03Z

@Emerton: this shows that the restriction is an algebraic space, but how do you prove that it is a scheme ? Indeed, I see that 1) is overkill, even in the case where $S'/S$ is only finite locally free. I will edit the question accordingly.

Comment by Simon Pepin Lehalleur

2013-02-08T19:15:21Z

$M_{g,n}$ is very far from being affine. For every $g\geq 3$, there is a complete curve passing through any point. This follows from the existence of a compactification with boundary of codimension >1 (the Satake compactification). The problem is only the complement of a very ample divisor is guaranteed to be affine.

Comment by Simon Pepin Lehalleur

2013-02-04T06:36:28Z

Also, there is a potential confusion with the notion of "Hodge-Tate representation" in p-adic Hodge theory. According to Faltings' theorem, the cohomology of any smooth proper variety over a p-adic field is Hodge-Tate (see definition 2.3.4 and theorem 2.2.3 in the Brinon-Conrad lecture notes, math.stanford.edu/~conrad/papers/notes.pdf) so this does not quite match the notions 1-4) (which are closer to "the motive of X is a mixed Tate motive", I guess)

Comment by Simon Pepin Lehalleur

2013-02-04T06:28:31Z

The paper "Eigenvalues of Froebenius and Hodge numbers" from Kisin and Lehrer discusses the relations between 1), 2) and 3), using p-adic Hodge theory.

Comment by Simon Pepin Lehalleur

2012-12-01T20:45:58Z

The book of Farb and Margalit, "A Primer on mapping class groups", seems a very accessible introduction to (some aspects of) the subject, and the lengthy overview at the beginning makes some historical comments: press.princeton.edu/titles/9495.html

Comment by Simon Pepin Lehalleur

2012-10-04T12:41:10Z

I do not feel competent enough to give a proper answer, but it seems that the key-word you are looking for is "p-Lie algebra". For an algebraic group over a field of char. p>0, you have (at least) three infinitesimal invariants, of increasing strength : the Lie algebra, the p-Lie algebra, and the formal group. This hierarchy collapses in char. 0 where the Lie algebra determines the formal group. The p-Lie algebra can be exponentiated "up to height 1". For p-Lie algebras (also called restricted Lie algebras), see Borel, Linear Algebraic Groups, I.3.1, and Pseudo-reductive groups, App. A7

Comment by Simon Pepin Lehalleur

2010-09-23T18:42:15Z

Yes, thank you.

Comment by Simon Pepin Lehalleur

2010-08-11T21:27:29Z

Sorry, should have been $SomeLieRings\rightarrow Sets$ and $Somep-Groups\rightarrow Sets$.

Comment by Simon Pepin Lehalleur

2010-08-11T21:26:27Z

Why not just think about it as a equivalence between the two forgetful functors $SomeLieRings\rightarrow\Sets$ and $Somep-groups\rightarrow\Sets$ ? I guess the fact that the exponential map is a bijection is common in "unipotent" contexts.

Comment by Simon Pepin Lehalleur

2010-08-11T07:58:13Z

Yet another approach to the finiteness of the Betti numbers (which proves much more). Complex algebraic varieties can be compactified in the following way : let $X$ be such a variety, then there exists $\bar{X}$ smooth projective in which $X$ is a dense Zariski open and $D=\bar{X}-X$ is a strict normal crossing divisor in $\bar{X}$. You can then relate the cohomology groups of $X$ to those of $\bar{X}$ and $D$ (which are finite dimensional) in De Rham theory by using logarithmic differential forms : see e.g. the treatment of this in Claire Voisin's book on Hodge theory.

Comment by Simon Pepin Lehalleur

2010-08-11T00:14:29Z

@TOM : Sorry, I was completely wrong... I had forgotten that the group $G_{\mathbb{R}}$ can have compact factors, not defined over $\mathbb{Q}$ as is the case with $SU(2)\times SU(2)$ in Mumford's construction.

Comment by Simon Pepin Lehalleur

2010-08-10T21:52:36Z

It is a well-established tradition in France to keep the (scheme-theoretic, what else ?) definition of "algebraic variety" in limbo, just to keep the students from getting bogged down into concreteness :-)

Comment by Simon Pepin Lehalleur

2010-08-10T20:16:36Z

Another small addition : the kähler differential approach described in the last paragraph is described in the following paper : mathjournals.org/mrl/2008-015-004/… , along with extensions to other holonomy groups (Calabi-Yau, Hyperkähler,...)

Comment by Simon Pepin Lehalleur

2010-08-10T17:59:34Z

Evaluating the strength and reach of a proof "to be verified" of an incredibly hard question in a subtle and error-prone field... What could go wrong ;-?