Torsor (algebraic Geometry)
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In algebraic geometry, a torsor or a principal bundle is an analog of a principal bundle in algebraic topology. Because there are few open sets in Zariski topology, it is more common to consider torsors in
étale topology In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale t ...
or some other flat topologies. The notion also generalizes a
Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base fiel ...
in abstract algebra. The category of torsors over a fixed base forms a
stack Stack may refer to: Places * Stack Island, an island game reserve in Bass Strait, south-eastern Australia, in Tasmania’s Hunter Island Group * Blue Stack Mountains, in Co. Donegal, Ireland People * Stack (surname) (including a list of people ...
. Conversely, a prestack can be stackified by taking the category of torsors (over the prestack).


Definition

Given a smooth
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. ...
''G'', a ''G''-torsor (or a principal ''G''-bundle) ''P'' over a scheme ''X'' is a scheme (or even
algebraic space In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, ...
) with an action of ''G'' that is locally trivial in the given
Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is c ...
in the sense that the base change Y \times_X P along some covering map Y \to X is isomorphic to the trivial torsor Y \times G \to Y (''G'' acts only on the second factor). Equivalently, a ''G''-torsor ''P'' on ''X'' is a
principal homogeneous space In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non ...
for the
group scheme In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups ha ...
G_X = X \times G (i.e., G_X acts simply transitively on P.) The definition may be formulated in the sheaf-theoretic language: a sheaf ''P'' on the category of ''X''-schemes with some Grothendieck topology is a ''G''-torsor if there is a covering \ in the topology, called the local trivialization, such that the restriction of ''P'' to each U_i is a trivial G_-torsor. A line bundle is nothing but a \mathbb_m-bundle, and, like a line bundle, the two points of views of torsors, geometric and sheaf-theoretic, are used interchangeably (by permitting ''P'' to be a stack like an
algebraic space In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, ...
if necessary). It is common to consider a torsor for not just a group scheme but more generally for a
group sheaf In mathematics, a group functor is a group-valued functor on the category of commutative rings. Although it is typically viewed as a generalization of a group scheme, the notion itself involves no scheme theory. Because of this feature, some authors ...
(e.g., fppf group sheaf).


Examples and basic properties

Examples * A \operatorname_n-torsor on ''X'' is a principal \operatorname_n-bundle on ''X''. *If L/K is a
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past partici ...
Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base fiel ...
, then \operatorname L \to \operatorname K is a \operatorname(L/K)-torsor (roughly because the Galois group acts simply transitively on the roots.) This fact is a basis for Galois descent. See
integral extension In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' ...
for a generalization. Remark: A ''G''-torsor ''P'' over ''X'' is isomorphic to a trivial torsor if and only if P(X) = \operatorname(X, P) is nonempty. (Proof: if there is an s: X \to P, then X \times G \to P, (x, g) \mapsto s(x)g is an isomorphism.) Let ''P'' be a ''G''-torsor with a local trivialization \ in étale topology. A trivial torsor admits a section: thus, there are elements s_i \in P(U_i). Fixing such sections s_i, we can write uniquely s_i g_ = s_j on U_ with g_ \in G(U_). Different choices of s_i amount to 1-coboundaries in cohomology; that is, the g_ define a cohomology class in the sheaf cohomology (more precisely Čech cohomology with sheaf coefficient) group H^1(X, G). A trivial torsor corresponds to the identity element. Conversely, it is easy to see any class in H^1(X, G) defines a ''G''-torsor on ''X'', unique up to an isomorphism. If ''G'' is a connected algebraic group over a finite field \mathbf_q, then any ''G''-bundle over \operatorname \mathbf_q is trivial. ( Lang's theorem.)


Reduction of a structure group

Most of constructions and terminology regarding principal bundles in algebraic topology carry over in verbatim to ''G''-bundles. For example, if P \to X is a ''G''-bundle and ''G'' acts from the left on a scheme ''F'', then one can form the associated bundle P \times^ F \to X with fiber ''F''. In particular, if ''H'' is a closed subgroup of ''G'', then for any ''H''-bundle ''P'', P \times^H G is a ''G''-bundle called the
induced bundle In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in ...
. If ''P'' is a ''G''-bundle that is isomorphic to the induced bundle P' \times^H G for some ''H''-bundle ''P''', then ''P'' is said to admit a reduction of structure group from ''G'' to ''H''. Let ''X'' be a smooth projective curve over an algebraically closed field ''k'', ''G'' a semisimple algebraic group and ''P'' a ''G''-bundle on a relative curve X_R = X \times_ \operatornameR, ''R'' a finitely generated ''k''-algebra. Then a
theorem of Drinfeld and Simpson In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
states that, if ''G'' is simply connected and
split Split(s) or The Split may refer to: Places * Split, Croatia, the largest coastal city in Croatia * Split Island, Canada, an island in the Hudson Bay * Split Island, Falkland Islands * Split Island, Fiji, better known as Hạfliua Arts, entertain ...
, there is an
étale morphism In algebraic geometry, an étale morphism () is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy ...
R \to R' such that P \times_ X_ admits a reduction of structure group to a Borel subgroup of ''G''.http://www.math.harvard.edu/~lurie/282ynotes/LectureXIV-Borel.pdf


Invariants

If ''P'' is a parabolic subgroup of a smooth affine group scheme ''G'' with connected fibers, then its degree of instability, denoted by \deg_i(P), is the degree of its Lie algebra \operatorname(P) as a vector bundle on ''X''. The degree of instability of ''G'' is then \deg_i(G) = \max \. If ''G'' is an algebraic group and ''E'' is a ''G''-torsor, then the degree of instability of ''E'' is the degree of the
inner form In mathematics, an inner form of an algebraic group G over a field K is another algebraic group H such that there exists an isomorphism \phi between G and H defined over \overline K (this means that H is a ''K-form'' of G) and in addition, for ever ...
^E G = \operatorname_G(E) of ''G'' induced by ''E'' (which is a group scheme over ''X''); i.e., \deg_i (E) = \deg_i (^E G). ''E'' is said to be ''semi-stable'' if \deg_i (E) \le 0 and is ''stable'' if \deg_i (E) < 0.


Examples of torsors in applied mathematics

According to
John Baez John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...
,
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
,
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge t ...
,
position Position often refers to: * Position (geometry), the spatial location (rather than orientation) of an entity * Position, a job or occupation Position may also refer to: Games and recreation * Position (poker), location relative to the dealer * ...
and the phase of a quantum-mechanical
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ma ...
are all examples of torsors in everyday physics; in each case, only relative comparisons can be measured, but a reference point must be chosen arbitrarily to make absolute values meaningful. However, the comparative values of relative energy, voltage difference, displacements and phase differences are ''not'' torsors, but can be represented by simpler structures such as real numbers, vectors or angles. In basic calculus, he cites indefinite integrals as being examples of torsors.


See also

* Beauville–Laszlo theorem *
Moduli stack of principal bundles In algebraic geometry, given a smooth projective curve ''X'' over a finite field \mathbf_q and a smooth affine group scheme ''G'' over it, the moduli stack of principal bundles over ''X'', denoted by \operatorname_G(X), is an algebraic stack gi ...
*
Fundamental group scheme In mathematics, the fundamental group scheme is a group scheme canonically attached to a scheme over a Dedekind scheme (e.g. the spectrum of a field or the spectrum of a discrete valuation ring). It is a generalisation of the étale fundamental ...


Notes


References

*Behrend, K
The Lefschetz Trace Formula for the Moduli Stack of Principal Bundles.
PhD dissertation. * *{{Citation , last1=Milne , first1=James S. , title=Étale cohomology , url=https://books.google.com/books?isbn=978-0-691-08238-7 , publisher=
Princeton University Press Princeton University Press is an independent Academic publishing, publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, ...
, series=Princeton Mathematical Series , isbn=978-0-691-08238-7 , mr=559531 , year=1980 , volume=33


Further reading

*Brian Conrad
Finiteness theorems for algebraic groups over function �fields
Algebraic geometry