# 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?

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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I assume you mean that if $X$ is a proper (or projective) scheme over a dvr $R$ & $F$ is an $R$-flat coherent sheaf on $X$ (e.g., line bundle if $X$ is $R$-flat, such as $R$-smooth $X$) then Euler characteristic of $F$ on generic & special fibers coincide. Mumford gives a proof 5.5 pages into sec. 5 of Ch. II of "Abelian Varieties", allowing $R$ to be any noetherian local ring. I can see two minor simplifications offhand: by using an affine base one can replace the higher direct images sheaves with cohomology modules, and by taking $R$ to be dvr the algebra may be "easier". Maybe that helps? –  BCnrd Oct 14 '10 at 17:35
@BCnrd: I'll have to track down this reference, which looks promising if the proof is relatively self-contained. The base change situation for flag varieties should avoid any delicate issues, since one is starting out over the integers with good geometric properties. –  Jim Humphreys Oct 14 '10 at 20:11
Dear Jim: The "delicate" issue which makes Grothendieck's theorem so fantastic is that even in geometrically nice cases the dimensions of the individual cohomologies can jump up under specialization, as you know. So it is the cancellation of such jumping in the alternating sum which requires a real insight into the nature of cohomology of coherent sheaves on projective (or proper) varieties. Mumford explains it pretty well, and I think is essentially self-contained (granting basic finiteness theorems for cohomology of coherent sheaves on projective schemes over a dvr or field, say). –  BCnrd Oct 14 '10 at 20:55
@BCnrd: Mumford's section II.5 does seem appropriate, though for me it's nontrivial to translate the general language there into the specifics I'm interested in. The Corollary at the top of p.50 is the heart of the matter, including the essential upper semicontinuity I didn't state explicitly: this should apply to each weight space, while implying overall that each $H^i$ is at least as big as the classical one. If there is no literature covering the special case of the flag variety, your comments should be converted to an answer. –  Jim Humphreys Oct 17 '10 at 12:54
Dear Jim: I can't imagine how a proof for flag varieties could be given in a manner any easier than the general case, so I'd be surprised if there's literature specifically on that case. But have you succeeded to do the translation that you wish to have (perhaps with guidance from a colleague)? Doesn't seem to merit being an answer if still unclear to you what's going on with the argument. –  BCnrd Oct 18 '10 at 2:37