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In mathematics, the '''Rost invariant''' is an invariant of an [[Absolutely simple group|absolutely simple]] simply connected [[algebraic group]] ''G'' over a field ''k'', which associates an element of the [[Galois cohomology]] group H<sup>3</sup>(''k'', '''Q'''/'''Z'''(2)) to a principal homogeneous space for ''G''. Here the coefficient group '''Q'''/'''Z'''(2) is the [[tensor product]] of the group of [[Root of unity|roots of unity]] of an algebraic closure of ''k'' with itself. {{harvs|txt|first=Markus |last=Rost|authorlink=Markus Rost|year=1991}} first introduced the invariant for groups of type F<sub>4</sub> and later extended it to more general groups in unpublished work that was summarized by {{harvs|txt|last=Serre|year=1995}}.

The Rost invariant is a generalization of the [[Arason invariant]].

==Definition==

Suppose that ''G'' is an absolutely almost simple simply connected algebraic group over a field ''k''. The Rost invariant associates an element ''a''(''P'') of the Galois cohomology group H<sup>3</sup>(''k'','''Q'''/'''Z'''(2)) to a ''G''-torsor ''P''. 

The element ''a''(''P'') is constructed as follows. For any extension ''K'' of ''k'' there is an exact sequence
:<math>0\rightarrow H^3(K,\mathbf{Q}/\mathbf{Z}(2)) \rightarrow H^3_{et}(P_K, \mathbf{Q}/\mathbf{Z}(2)) \rightarrow \mathbf{Q}/\mathbf{Z}</math>
where the middle group is the [[étale cohomology]] group and '''Q'''/'''Z''' is the geometric part of the cohomology. 
Choose a finite extension ''K'' of ''k'' such that ''G'' splits over ''K'' and ''P'' has a rational point over ''K''. Then the exact sequence splits canonically as a direct sum so the étale cohomology group contains '''Q'''/'''Z''' canonically. The invariant ''a''(''P'') is the image of the element 1/[''K'':''k''] of '''Q'''/'''Z''' under the trace map from H{{su|b=et|p=3}}(''P''<sub>''K''</sub>,'''Q'''/'''Z'''(2)) to H{{su|b=et|p=3}}(''P'','''Q'''/'''Z'''(2)), which lies in the subgroup H<sup>3</sup>(''k'','''Q'''/'''Z'''(2)).

These invariants ''a''(''P'') are functorial in field extensions ''K'' of ''k''; in other words the fit together to form an element of the cyclic group Inv<sup>3</sup>(''G'','''Q'''/'''Z'''(2)) of cohomological invariants of the group ''G'', which consists of morphisms of the functor ''K''→H<sup>1</sup>(''K'',''G'') to the functor ''K''→H<sup>3</sup>(''K'','''Q'''/'''Z'''(2)). This element of Inv<sup>3</sup>(''G'','''Q'''/'''Z'''(2)) is a generator of the group and is  called the Rost invariant of ''G''.

==References==

*{{citation|mr=1881703|last=Garibaldi|first= Ryan Skip|title=The Rost invariant has trivial kernel for quasi-split groups of low rank|journal=Comment. Math. Helv. |volume=76 |year=2001|issue=4|pages= 684–711|doi=10.1007/s00014-001-8325-8}}
*{{citation | chapter=Rost invariants of simply connected algebraic groups| last1=Garibaldi | first1=Skip | author1-link=Skip Garibaldi | last2=Merkurjev | first2=Alexander | author2-link=Alexander Merkurjev | last3=Serre | first3=Jean-Pierre | author3-link=Jean-Pierre Serre | title=Cohomological invariants in Galois cohomology | series=University Lecture Series | volume=28 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2003 | isbn=0-8218-3287-5 | zbl=1159.12311 | mr=1999383 }}
*{{citation|mr=1138557|last=Rost|first= Markus|title=A (mod 3) invariant for exceptional Jordan algebras|journal=C. R. Acad. Sci. Paris Sér. I Math. |volume=313 |year=1991|issue= 12|pages= 823–827}} 
*{{citation|mr=1321649 |last=Serre|first= Jean-Pierre|title=Cohomologie galoisienne: progrès et problèmes|series=
Séminaire Bourbaki Exp. No. 783| |journal=Astérisque |volume= 227 |year=1995|issue= 4|pages= 229–257|url= http://www.numdam.org/item?id=SB_1993-1994__36__229_0}}

[[Category:Algebraic groups]]