QUANTUM TORIC DEGENERATION OF QUANTUM FLAG AND SCHUBERT VARIETIES
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Springer Science+Business Media New York (2020)
QUANTUM TORIC DEGENERATION OF QUANTUM FLAG AND SCHUBERT VARIETIES L. RIGAL
P. ZADUNAISKY∗
Universit´e Sorbonne Paris Nord LAGA, CNRS, UMR 7539 93430 Villetaneuse, France
Departamento de Matem´aticas Universidad CAECE Buenos Aires, Argentina
[email protected]
[email protected]
Abstract. 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.
1. Introduction Let k be a field, and let A be a noetherian commutative algebra over k. If we put an ascending filtration on A then we can build the Rees ring of the filtration, which is a free k[t]-algebra R such that A ∼ = R/(t−λ)R for all λ ∈ k× , while R/tR is isomorphic to the associated graded ring. In geometric terms, if k is algebraically closed then the variety associated to R is a flat family over the affine line, whose generic fiber is isomorphic to Spec A and whose fiber over 0 is isomorphic to Spec grA; in this context the fiber over 0 is called a degeneration of Spec A. A standard result from algebraic geometry states that if the fiber over 0 is regular (resp. Gorenstein, Cohen–Macaulay, or any other of a long list of properties) then all fibers are regular (resp. Gorenstein, Cohen–Macaulay, etc.) Of course, the idea of studying a ring by imposing a filtration and passing to the associated graded ring is a basic tool in an algebraist’s toolbox, and can be applied outside of a geometric context. In particular the hypothesis of commutativity is not necessary for filtered-to-graded methods to work. However, in the spirit of noncommutative algebraic geometry, we should look at the case where A is noetherian, N-graded and connected (i.e., A0 = k) with an eye on the geometric case. Although in this case there are no varieties associated to our algebras as DOI: 10.1007/S00031-020-09615-y Supported by a CONICET Postdoctoral fellowship Received February 21, 2019. Accepted June 9, 2020. Corresponding Author: P. Zadunaisky, e-mail: [email protected] ∗
L. RIGAL, P. ZADUNAISKY
in the commutative setting, we have suitable analogues of the notions of being regular, or Gorenstein, or Cohen–Macaulay, defined in purely homological terms (see subsubsection 2.2.4 ). Hence it makes sense to ask whether
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