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Schubert varieties are arithmetically Cohen-Macaulay. (English) Zbl 0541.14039
Let G be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic. Let B be a Borel subgroup and $$Q\supset B$$ a parabolic subgroup of G. The closure of a B-orbit in G/Q is called a Schubert variety. We prove in this paper the following theorems:
Theorem 1. Let X and Y be unions of Schubert varieties in G/Q taken with the reduced subscheme structure. Then their scheme theoretic intersection $$X\cap Y$$ is reduced. - Theorem 2. Let X be a union of Schubert varieties in G/Q taken with the reduced subscheme structure and L a line bundle on G/Q with $$H^ 0(G/Q,L)\neq 0.$$ Then $$H^ i(X,L)=0$$ for $$i>0$$ and $$H^ 0(G/Q,L)\to H^ 0(X,L),$$ the restriction map, is surjective. - Theorem 3. Let X be a Schubert variety in G/Q. Then X is Cohen-Macaulay and in the projective embedding given by any ample line bundle on G/Q it is arithmetically Cohen-Macaulay.
The proofs proceed by reducing to characteristic p$$>0$$. The key point is the fact that the p-th power map $${\mathcal O}_{G/Q}\to F_*{\mathcal O}_{G/Q}$$ where F is the absolute Frobenius morphism, has a splitting which gives compatible splitting of the corresponding map for any Schubert variety in G/Q. This is a further development of the methods of the papers ”Frobenius splitting and cohomology vanishing for Schubert varieties” by V. B. Mehta and the author [preprint (to appear)] and ”Projective normality of flag varieties and Schubert varieties” by S. Ramanan and the author [Invent. Math. (to appear)].

##### MSC:
 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 14M15 Grassmannians, Schubert varieties, flag manifolds
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##### References:
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