# Upper Motives of Outer Algebraic Groups

## Summary

Let *G* be a semisimple affine algebraic group over a field *F*. Assuming that *G* becomes of inner type over some finite field extension of *F* of degree a power of a prime *p*, we investigate the structure of the Chow motives with coefficients in a finite field of characteristic *p* of the projective *G*-homogeneous varieties. The complete motivic decomposition of any such variety contains one specific summand, which is the most understandable among the others and which we call the *upper* indecomposable summand of the variety. We show that every indecomposable motivic summand of any projective *G*-homogeneous variety is isomorphic to a shift of the upper summand of some (other) projective *G*-homogeneous variety. This result is already known (and has applications) in the case of *G* of inner type and is new for *G* of outer type (over *F*).

## Keywords

Algebraic Group Galois Group Linear Algebraic Group Chow Group Indecomposable Summand## Preview

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## References

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