2 added 448 characters in body

BACKGROUND: Over an algebraically closed field of arbitrary characteristic, most of the basic structure theory of affine (= linear) algebraic groups can be developed concretely without quoting difficult theorems from algebraic geometry, though the study of quotients $G/H$ gets more subtle and can benefit from the scheme setting. (This is even more so when one works over arbitrary fields.)

Similarly, basic finite dimensional (rational) representations of a connected semisimple group can be developed up to a point in down-to-earth ways, including the classification of simple modules by highest weights (though complete reducibility breaks down badly in prime characteristic). But already in characteristic 0 it is useful to work geometrically with the flag variety $G/B$ for a fixed Borel subgroup $B$. Here the classical theorems of Borel-Weil and Bott realize the sheaf cohomology groups $H^i(G/B, \mathcal{L}(\lambda))$ of a line bundle as simple modules or zero. The index $i$ ranges from 0 to $\dim G/B =$ number of positive roots.

More precisely, let $B$ correspond to the negative roots of $G$ relative to a maximal torus $T \subset B$. The characters $\lambda$ of $T$ determine homogeneous line bundles on the projective variety $G/B$. For $\lambda$ not in a root hyperplane after a shift by the half-sum of positive roots $\rho$, exactly one sheaf cohomology group $H^i$ is nonzero; it affords the simple module of dominant highest weight $w(\lambda + \rho)-\rho$ with $w$ of length i in the Weyl group $W=N_G(T)/T$.

In prime characteristic Kempf (1976) proved that higher cohomology all vanishes when $\lambda$ is dominant. But otherwise there can be multiple nonvanishing cohomology groups. (There are partial results by H.H. Andersen and others, though no definitive treatment; the vanishing patterns are conjectured to depend on Kazhdan-Lusztig theory for the dual affine Weyl group.) In any case, the nonzero cohomology groups seldom give simple modules; instead there are parallels with infinite dimensional Verma modules.

Even in this complicated setting, one can invoke the principle from algebraic geometry quoted in the header: Euler characteristic is invariant under base change. This requires suitable realization of the groups, weights, flag variety over $\mathbb{Z}$, using ideas which originated with Chevalley but have only recently been completed by Lusztig (J. Amer. Math. Soc., 2008). The invariance principle should be applied to the individual weight spaces for each fixed weight, so that the alternating sum of dimensions of the weight spaces in the various cohomology groups recovers (up to sign) the classical weight multiplicity. As a result, $$\sum_i (-1)^i \dim H^i(G/B, \mathcal{L}(\lambda))$$ and the associated formal character of the (virtual) $G$-module are given (up to sign) by Weyl's formulas. Approaching this intrinsically in characteristic $p$ seems impossible.

Jantzen's book Representations of Algebraic Groups explains Kempf's theorem and its uses. But further treatment of higher cohomology groups is found mainly in research papers, where the invariance principle is quoted as "standard". It apparently originated in the work of Grothendieck and his school, only a small part of which should be needed as a prerequisite for the algebraic group application.

What source for the invariance of Euler characteristic under base change involves the fewest prerequisites for the limited application in modular representation theory?

ADDED: Concerning sources, it's just been pointed out to me that the last section of Chapter III in Hartshorne's Algebraic Geometry incorporates Mumford's proof of the Grothendieck theorem in question. So this is probably the canonical reference. I confess I haven't become familiar enough with this book, but in any case I still want to understand better how to minimize prerequisites for understanding the special application outlined above.

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# Invariance of Euler characteristic under base change for sheaf cohomology of flag varieties

BACKGROUND: Over an algebraically closed field of arbitrary characteristic, most of the basic structure theory of affine (= linear) algebraic groups can be developed concretely without quoting difficult theorems from algebraic geometry, though the study of quotients $G/H$ gets more subtle and can benefit from the scheme setting. (This is even more so when one works over arbitrary fields.)

Similarly, basic finite dimensional (rational) representations of a connected semisimple group can be developed up to a point in down-to-earth ways, including the classification of simple modules by highest weights (though complete reducibility breaks down badly in prime characteristic). But already in characteristic 0 it is useful to work geometrically with the flag variety $G/B$ for a fixed Borel subgroup $B$. Here the classical theorems of Borel-Weil and Bott realize the sheaf cohomology groups $H^i(G/B, \mathcal{L}(\lambda))$ of a line bundle as simple modules or zero. The index $i$ ranges from 0 to $\dim G/B =$ number of positive roots.

More precisely, let $B$ correspond to the negative roots of $G$ relative to a maximal torus $T \subset B$. The characters $\lambda$ of $T$ determine homogeneous line bundles on the projective variety $G/B$. For $\lambda$ not in a root hyperplane after a shift by the half-sum of positive roots $\rho$, exactly one sheaf cohomology group $H^i$ is nonzero; it affords the simple module of dominant highest weight $w(\lambda + \rho)-\rho$ with $w$ of length i in the Weyl group $W=N_G(T)/T$.

In prime characteristic Kempf (1976) proved that higher cohomology all vanishes when $\lambda$ is dominant. But otherwise there can be multiple nonvanishing cohomology groups. (There are partial results by H.H. Andersen and others, though no definitive treatment; the vanishing patterns are conjectured to depend on Kazhdan-Lusztig theory for the dual affine Weyl group.) In any case, the nonzero cohomology groups seldom give simple modules; instead there are parallels with infinite dimensional Verma modules.

Even in this complicated setting, one can invoke the principle from algebraic geometry quoted in the header: Euler characteristic is invariant under base change. This requires suitable realization of the groups, weights, flag variety over $\mathbb{Z}$, using ideas which originated with Chevalley but have only recently been completed by Lusztig (J. Amer. Math. Soc., 2008). The invariance principle should be applied to the individual weight spaces for each fixed weight, so that the alternating sum of dimensions of the weight spaces in the various cohomology groups recovers (up to sign) the classical weight multiplicity. As a result, $$\sum_i (-1)^i \dim H^i(G/B, \mathcal{L}(\lambda))$$ and the associated formal character of the (virtual) $G$-module are given (up to sign) by Weyl's formulas. Approaching this intrinsically in characteristic $p$ seems impossible.

Jantzen's book Representations of Algebraic Groups explains Kempf's theorem and its uses. But further treatment of higher cohomology groups is found mainly in research papers, where the invariance principle is quoted as "standard". It apparently originated in the work of Grothendieck and his school, only a small part of which should be needed as a prerequisite for the algebraic group application.

What source for the invariance of Euler characteristic under base change involves the fewest prerequisites for the limited application in modular representation theory?