see also at Galois group
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Classical Galois theory classifies field extensions. It is a special case of a classification of locally constant sheaves in a topos by permutation representations of the fundamental groupoid/fundamental group.
Even more generally one can define a Galois group associated to a presentable symmetric monoidal stable (infinity,1)-category. There is an analogue of the Galois correspondence in this setting, see Mathew 14.
We discuss the classical/traditional case of Galois theory, which concerns the classification of field extensions. Below in Galois theory for schemes and then in Galois theory in a topos we discuss how this is a special case of a more general concept of Galois theory in a topos.
We call a field extension a Galois extension if is algebraic and there exists a subgroup of the automorphism group such that is the field of elements that are invariant under .
If is a Galois extension, we define the Galois group to be
This means that we have
Let be a fixed algebraic closure of . If is any collection of non-zero polynomials, the splitting field of over is the subfield of generated by and the zeros of the polynomials in .
We call separable if it has no multiple zero in .
We call separable over if the irreducible polynomial of over is separable.
A subextension is called separable over if each is separable over .
We call normal over if for each the polynomial splits completely in linear factors in .
Let be a field and a subfield of
Denote by the set of subfields of for which is a finite Galois extension of . Then , when partially ordered by inclusion is a directed poset.
The following assertions are equivalent:
is a Galois extension of .
There is a set of separable polynomials such that is the splitting field of over .
If these conditions are satisfied, then there is a group isomorphism
where on the right we have the limit over the poset of subfield of the contravariant functor .
Since each group is finite, the above isomorphism can be used to equip with a profinite topology (i.e. take the limit in the category of topological groups, where each has the discrete topology), making it into a profinite group. We henceforth consider as a profinite group in this way.
(main theorem of classical Galois theory)
Let be a Galois extension of fields with Galois group . Then the intermediate fields of correspond bijectively to the closed subgroups of .
More precisely, the maps
defined by
and
are bijective and inverse to each other. This correspondence reverses the inclusion relation: corresponds to and to .
If corresponds to , then we have
is finite precisely if is open (in the profinite topology on )
if is open;
is Galois with (as topological groups);
for every we have that corresponds to ;
is Galois precisely if is a normal subgroup of ;
(as topological groups) if is Galois.
This appears for instance as Lenstra, theorem 2.3.
This suggests that more fundamental than the subgroups of a Galois group are its quotients by subgroups, which can be identified with transitive -sets. This naturally raises the question of what corresponds to non-transitive -sets.
Let be a commutative ring and a module over .
A collection of elements of is called a basis of (over ) if for every there is a unique collection of elements of such that for all but finitely many and .
If has a basis it is called free (over ). If is free with basis a finite set of cardinality , then we say that is free with rank (over ). In this case, is a finitely generated free module.
Let be a finitely generated free -module with basis and let be -linear. Then
for certain , and the trace of is defined by
This is an element of that only depends on , and not on the choice of basis. It is easily checked that the map is -linear.
Let be a ring, an -algebra, and suppose that is free with finite rank as an -module. For every the map defined by is -linear, and the trace or is defined to be . The map is easily seen to be -linear and to satisfy for .
The -module (underlying which is the hom-set in the category of modules) is clearly free over with the same rank as . Define the -linear map by
for .
If for an -algebra the the morphism is an isomorphism we call separable over , or a free separable -algebra if we wish to stress the condition that is finitely generated and free as an -module.
Recall the notion of separable elements
Let be a field and an algebraic closure of . The separable closure of is defined by
We have that is a subfield of and that precisely if is a perfect field, in particular if the characteristic of is 0.
From xyz it follows that the inclusion is Galois.
The Galois group is called the absolute Galois group of .
Let be a field and its absolute Galois group. Then there is an equivalence of categories
The classical Galois theory for fields is a special case of a general geometric/topos theoretic statement about locally constant sheaves and the action of the fundamental group on their fibers.
A morphism of schemes is a finite étale morphism if there exists a covering of by affine open subsets , such that
for each the open subschemes of is affine,
and equal to , where is a free separable -algebra.
In this situation we also say that is a finite étale covering of .
A morphism from a finite étale covering to a finite étale covering is a morphism of schemes such that .
This defines the category of finite étale covers of .
Let be a connected scheme. Then there exists a profinite group – the fundamental group of – uniquely determined up to isomorphism, such that the category of finite étale coverings is equivalent to the category of finite permutation representations of (finite sets, with the discrete topology, on which acts continuously).
This appears for instance as Lenstra, main theorem 1.11. It is fully discussed in SGA1.
The profinite group, , is often called the étale fundamental group of the connected scheme . In SGA1, Grothendieck also considers coverings with profinite fibres, and a profinitely enriched fundamental groupoid. In the above the actual group depends on the choice of a fibre functor given by a geometric point of . Different choices of fibre functor produce isomorphic groups. Taking two such fibre functors yields a -torsor for either version of . This is important in attacks on Grothendieck's section conjecture.
The disjoint union of copies of corresponds, under this theorem, to a finite set of elements on which acts trivially.
The fact that for there are no other finite étale coverings of is thus expressed by the group being trivial .
The same is true for , where is an algebraically closed field.
If is an arbitrary field, then is the absolute Galois group of ; i.e. the Galois group of the separable closure over . In this case theorem is a reformulation of classical Galois theory.
In particular, if is a finite field, then .
Let , where is the ring of integers in an algebraic number field . Let be the maximal algebraic extension of that is unramified at all non-zero prime ideals of . Then is the Galois group of over .
More generally, if , , then is the Galois group, over , of the maximal algebraic extension of that is unramified at all non-zero prime ideals of not dividing .
In this section we explain the connection between the main theorem of Galois theory for schemes, theorem , and classical Galois theory.
We denote by a field. It is our purpose to show that the opposite category of the category of free separable -algebras is equivalent to the category of finite -sets, for a certain profinite group . This is a special case of the main theorem , with . In the general proof we shall use the contents of this section only for algebraically closed . In that case, which is much simpler, the group is trivial, so that the category of finite -sets is just the category of finite sets.
(…)
In SGA1, Grothendieck introduced an abstract formulation of the above theory in terms of Galois categories. A Galois category is a category, , satisfying a small number of properties together with a fibre functor , preserving those properties. The theory is more fully described in the entry on Grothendieck's Galois theory.
One notices that classical Galois theory has an equivalent reformulation in topos theory. That puts it into a wider general abstract context and leads to a topos-theoretic general Galois theory.
The étale morphisms corresponds precisely to the locally constant sheaves on with respect to the étale topology, in that it is equivalently a morphism for which there is an étale cover such that is a constant sheaf on each .
For a field let be its small étale site. And
the sheaf topos over it. This topos is a
Then Galois extensions of correspond precisely to the locally constant objects in . The full subcategory on them is the Galois topos .
The Galois group is the fundamental group of the topos.
Accordingly in topos theory Galois theory is generally about the classification of locally constant sheaves. The Galois group corresponds to the fundamental group of the topos .
(…)
In the context of higher topos theory, there are accordingly higher analogs of Galois theory. According to shape theory, any (∞,1)-topos has an associated fundamental ∞-groupoid , which in general is a pro-space whose -truncation is the ordinary fundamental groupoid of the underlying 1-topos. Classical topos-theoretic Galois theory states that locally constant sheaves (of sets) on a locally connected topos are equivalent to representations of , i.e., functors . This generalizes to higher topoi as follows:
Let be a locally n-connected (n+1,1)-topos, . Then there is an equivalence of categories
where is the subcategory of locally constant objects.
This generalization of Galois theory is discussed in (Grothendieck 75, Hoyois 13, Hoyois 15). For further discussion in the case , see
Between January and June 1981, Grothendieck wrote about 1600 manuscript pages of a work with the above title. The subject is the absolute Galois group, of the rational numbers and its geometric action on moduli spaces of Riemann surfaces. This will (one day be) discussed at Long March. Other entries that relate to this include anabelian geometry, children's drawings (in other words Dessins d’enfants, which is the study of graphs embedded on surfaces, their classification and the link between this and Riemann surfaces) and the Grothendieck-Teichmuller group.
The anabelian question is: how much information about the isomorphism class of an algebraic variety, is contained in the étale fundamental group of ? Grothendieck calls varieties which are completely determined by their étale fundamental group, anabelian varieties. His anabelian dream was to classify the anabelian varieties in all dimensions over all fields. This can be seen to relate to questions of the étale homotopy types of varieties.
(…)
Lascar group (a Galois group of first order theories)
geometric homotopy groups in an (∞,1)-topos, fundamental group of a topos, fundamental ∞-groupoid of an (∞,1)-topos
Original articles:
Michael Barr, Abstract Galois Theory, J. Pure Appl. Algebra 19 (1980) 21–42 [pdf, pdf]
Michael Barr, Abstract Galois Theory II, J. Pure Appl. Algebra 25 3 (1982) 227–247 [doi:10.1016/0022-4049(82)90080-9, pdf, pdf]
Introduction:
Hendrik Lenstra, Galois theory for schemes , Mathematisch Instituut Universiteit van Amsterdam (1985) [pdf]
Some of the material above is taken from this.
Ravi Vakil, Kirsten Wickelgren, Universal covering spaces and fundamental groups in algebraic geometry as schemes, (pdf).
Monographs:
Francis Borceux, George Janelidze, Galois Theories, Camb. Stud. Adv. Math. 72, Cambridge: Cambridge University Press (2001)
Jean-Pierre Tignol, Galois’ Theory of Algebraic Equations, World Scientific (2001) [doi:10.1142/4628]
The locally simply connected case is discussed for instance in
Galois theory in a presentable symmetric monoidal stable (infinity,1)-category is developed in
Galois theory in (higher) topos theory
SGA4, Expose VIII, Proposition 2.1.
Marc Hoyois, Higher Galois theory, J. Pure Appl. Alg. 222:7, (2018) 1859-1877 arXiv:1506.07155 doi
Joseph Rennie, The Unreasonable Efficacy of the Lifting Condition in Higher Categorical Galois Theory I: a Quasi-categorical Galois Theorem [arXiv:2409.03347]
See also
Tamás Szamuely, Galois groups and fundamental groups, Cambridge Studies in Adv. Math. 117 (2009)
Alexander Grothendieck, letter to Larry Breen 17/2/1975 pdf, scan
Alexander Grothendieck, La Longue Marche a Travers la Théorie de Galois pdf
Last revised on September 6, 2024 at 09:04:05. See the history of this page for a list of all contributions to it.