3 years ago

On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

We already said in

We already said in Section 4.1 that C is equivalent to the category of topologicalcovers of X. We identify C with C {1} where {1} denotes the trivial group.We recall that an object Y of a category in which coproducts exist is calledconnected if it is not isomorphic to a coproduct Y 1∐Y2 where Y 1 and Y 2 areobjects of this category not isomorphic to the initial object.For an arbitrary discrete group π 1 we define the following category:C(π 1 ) π 1 -setsThe objects are discrete sets with a left or right action of the group π 1 .The morphisms are π 1 -equivariant maps between these sets.We warn the reader that when discussing the fundamental group π 1 in algebraicgeometry one often considers sets with left π 1 -actions whereas in topology oneusually considers sets with right π 1 -actions. Therefore the author decided to berather pedantic about this point, especially after he was trapped when he was notpaying attention to it.We choose a universal cover ˜p : ˜X → X of X in the sense of topology. Wedenote by Aut(˜X) the group of deck transformations of ˜X over X. Then we defineπ top1 (X, ˜X) to be the opposite group to Aut( ˜X). Given a (connected) cover p :Y → X the group Aut(˜X) acts from the right on the set of cover morphismsHom(˜X, Y). Hence there is a left π top1 (X, ˜X)-action on this set. This defines afibre functor from the category C to the category of sets with a left π top1 (X, ˜X)-action and makes C into a Galois category.Here we have to relax Grothendieck’s terminology a little bit: We also allowquotients by discrete groups rather than only finite ones. Also we assume that thefibre functor maps to the category of discrete sets with a group action of a discretegroup rather than only to the category of finite sets together with a continuousaction of a profinite group.Conversely, given a fibre functor F there is always a group π top1 (X, F ) calledthe automorphism group of the functor F . A map between two covers Y 1 and Y 2over X is uniquely determined by the π top1 (X, F )-equivariant map from F (Y 1 )to F (Y 2 ). The main content of Galois theory and the theory of the fundamentalgroup in this setup is that a fibre functor induces an equivalence of categoriesbetween C and C(π 1 ).The connection with the fundamental group defined via loops is as follows:We let F x0 (Y) := p −1 (x 0 ) be the fibre of p : Y → X above a point x 0 ofX. Lifting loops based at x 0 to paths in Y defines a right action of the “loop“-fundamental group π top1 (X, x 0 ) on the set F x0 (Y). Now we fix a point ˜x 0 on thefibre F x0 (˜X) of a universal cover. Then we compare the left Aut( ˜X)-action withthe right π top1 (X, x 0 )-action in this point: For every automorphism ϕ there is aunique element γ in the fundamental group such that ϕ · ˜x 0 = ˜x 0 · γ. This defines32

an isomorphism between these two groups that depends on the choice of ˜x 0 . Infact, given another point ˜x ′ 0 of F x 0(˜X) there is a unique element α ∈ π top1 (X, x 0 )such that ˜x ′ 0 = ˜x 0 · α. We then computeϕ · ˜x ′ 0 = ϕ · (˜x 0 · α) = (ϕ · ˜x 0 ) · α = (˜x 0 · γ) · α = ˜x ′ 0 · (α−1 γα).That is, with respect to ˜x ′ 0 we obtain an isomorphism of π top1 (X, x 0 ) with Aut(˜X)that differs from the isomorphism with respect to ˜x 0 by conjugation with α.If we fix a point ˜x 0 of ˜X in the fibre ˜p −1 (x 0 ) we can identify Hom(˜X, Y) withthe set F x0 (Y) by associating to a morphism ϕ : ˜X → Y the point ϕ(˜x 0 ). Underthis identification the right action of π top1 (X, x 0 ) on F x0 (Y) becomes a left actionon Hom(˜X, Y) and it is this point where the group actions change their side whenpassing from topology to algebraic geometry and vice versa.For a cover p : Y → X we recover the group of its automorphisms as follows:The group Aut(˜X) acts on Hom(˜X, Y). We choose a point on this latter set,i.e. we choose a map from ˜X to Y, and denote by H the subgroup of Aut( ˜X)stabilising this point. This identifies Y with the quotient H\ ˜X. An element ofAut(˜X) induces an automorphism of Y if and only if it normalises H. Since theelements acting trivially on Y are precisely those of H we get an isomorphismbetween the group of cover automorphisms of Y over X and NH/H where NHdenotes the normaliser of H in Aut( ˜X).The same can be done for covers with a G-action. So we assume a finite groupG of automorphisms acts from the left on X. The following constructions werealready sketched in [SGA1, Remarque IX.5.8] and we will fill out some of thedetails:For a connected C-cover p : Y → X we define the following C G -cover:and a left G-action on Y × G viaY × G → X(y, h) ↦→ h · p(y)G × (Y × G) → Y × Gg , (y, h) ↦→ (y, gh)This clearly is a connected object of C G . The object so associated to Y is the sameas the fibre product of Y with (X × G) with G-action as described above over X.Every connected G-cover of X × G is dominated by a G-cover of the formY × X (X × G) where Y → X is a connected topological cover. Indeed, forgettingthe G-action, a connected G-cover of X × G becomes a cover of X consisting ofexactly |G| components. If we choose Y to dominate each of these componentsit is not complicated to obtain a G-morphism from Y × G onto the G-cover ofX × G we started with.33

The Path Ahead: Fundamentals of Generative ... - Right 4 Your Type
Groups of integral representation type - MSP
Generic process improvement approach applied to IT projects - PMI ...
Project Insight - Power Generation Construction Projects in Asia-Pacific
Carbon reductions generate positive ROI - Carbon Disclosure Project
Projected Costs of Generating Electricity - OECD Nuclear Energy ...
Renewable Projects Portal for CLP Group - OSIsoft
Building Next Generation Design Support Programmes - See Project
GALOIS THEORY - Tata Institute of Fundamental Research
On The Structure of Certain Galois Cohomology Groups
The Fundamental Theorem of Galois Theory and Normal Subgroups
Generic Galois groups for q-difference equations
periods of eisenstein series: the galois case - Project Euclid
fields and galois theory - Neil Strickland - University of Sheffield
Introduction to Differential Galois Theory
Galois module structure of local unit groups
On Galois Algebras with an Inner Galois Group - Bradley Bradley
Supersingular Galois representations and a generalization of a ...
Galois groups with restricted ramification Romyar Sharifi Harvard ...
On Galois Algebras with a Unique Galois Group - Bradley Bradley
Automorphy lifting for residually reducible l-adic Galois ...
Connected Linear Groups as Differential Galois Groups (with C ...
1 The Galois Group of a Quadratic 2 The Galois Group of a Cubic
Model Theory of Absolute Galois Groups - FreiDok - Albert-Ludwigs ...
A family of polynomials with Galois group $ PSL_5 (2) $ over ...
Wildly ramified Galois representations and a generalization of a ...