  We show that certain homological regularity properties of graded connected
algebras, such as being AS-Gorenstein or AS-Cohen-Macaulay, can be tested by
passing to associated graded rings. In the spirit of noncommutative algebraic
geometry, this can be seen as an analogue of the classical result that, in a
flat family of varieties over the affine line, regularity properties of the
exceptional fiber extend to all fibers. We then show that quantized coordinate
rings of flag varieties and Schubert varieties can be filtered so that the
associated graded rings are twisted semigroup rings. This is a noncommutative
version of the result due to Caldero stating that flag and Schubert varieties
degenerate into toric varieties, and implies that quantized coordinate rings of
flag and Schubert varieties are AS-Cohen-Macaulay.
