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 in the sense of [RZ12]. This is a noncommutative version of the result due to Caldero [C02] 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.

我们表明，可以通过传递给相关的分级环来检验渐变连接代数的某些同源正则性质，例如AS-Gorenstein或AS-Cohen-Macaulay。本着非交换代数几何的精神，这可以看作是经典结果的类似物，该经典结果是在仿射线上的扁平变体族中，优异纤维的规则性扩展到所有纤维。然后我们表明，可以对标志变种和舒伯特变种的量化坐标环进行滤波，以使相关的渐变环成为[RZ12]意义上的扭曲半群环。这是结果的非可交换版本，这是由于Caldero [C02]指出旗帜和舒伯特品种退化为复曲面品种，并暗示旗帜和舒伯特品种的量化坐标环是AS-Cohen-Macaulay。