Borel Subgroups and Flag Manifolds
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PCMI Summer 2015 Undergraduate Lectures on Flag Varieties 0. What
PCMI Summer 2015 Undergraduate Lectures on Flag Varieties 0. What are Flag Varieties and Why should we study them? Let's start by over-analyzing the binomial theorem: n X n (x + y)n = xmyn−m m m=0 has binomial coefficients n n! = m m!(n − m)! that count the number of m-element subsets T of an n-element set S. Let's count these subsets in two different ways: (a) The number of ways of choosing m different elements from S is: n(n − 1) ··· (n − m + 1) = n!=(n − m)! and each such choice produces a subset T ⊂ S, with an ordering: T = ft1; ::::; tmg of the elements of T This realizes the desired set (of subsets) as the image: f : fT ⊂ S; T = ft1; :::; tmgg ! fT ⊂ Sg of a \forgetful" map. Since each preimage f −1(T ) of a subset has m! elements (the orderings of T ), we get the binomial count. Bonus. If S = [n] = f1; :::; ng is ordered, then each subset T ⊂ [n] has a distinguished ordering t1 < t2 < ::: < tm that produces a \section" of the forgetful map. For example, in the case n = 4 and m = 2, we get: fT ⊂ [4]g = ff1; 2g; f1; 3g; f1; 4g; f2; 3g; f2; 4g; f3; 4gg realized as a subset of the set fT ⊂ [4]; ft1; t2gg. (b) The group Perm(S) of permutations of S \acts" on fT ⊂ Sg via f(T ) = ff(t) jt 2 T g for each permutation f : S ! S This is an transitive action (every subset maps to every other subset), and the \stabilizer" of a particular subset T ⊂ S is the subgroup: Perm(T ) × Perm(S − T ) ⊂ Perm(S) of permutations of T and S − T separately. -
Algebras in Which Every Subalgebra Is Noetherian
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 142, Number 9, September 2014, Pages 2983–2990 S 0002-9939(2014)12052-1 Article electronically published on May 21, 2014 ALGEBRAS IN WHICH EVERY SUBALGEBRA IS NOETHERIAN D. ROGALSKI, S. J. SIERRA, AND J. T. STAFFORD (Communicated by Birge Huisgen-Zimmermann) Abstract. We show that the twisted homogeneous coordinate rings of elliptic curves by infinite order automorphisms have the curious property that every subalgebra is both finitely generated and noetherian. As a consequence, we show that a localisation of a generic Skylanin algebra has the same property. 1. Introduction Throughout this paper k will denote an algebraically closed field and all algebras will be k-algebras. It is not hard to show that if C is a commutative k-algebra with the property that every subalgebra is finitely generated (or noetherian), then C must be of Krull dimension at most one. (See Section 3 for one possible proof.) The aim of this paper is to show that this property holds more generally in the noncommutative universe. Before stating the result we need some notation. Let X be a projective variety ∗ with invertible sheaf M and automorphism σ and write Mn = M⊗σ (M) ···⊗ n−1 ∗ (σ ) (M). Then the twisted homogeneous coordinate ring of X with respect to k M 0 M this data is the -algebra B = B(X, ,σ)= n≥0 H (X, n), under a natu- ral multiplication. This algebra is fundamental to the theory of noncommutative algebraic geometry; see for example [ATV1, St, SV]. In particular, if S is the 3- dimensional Sklyanin algebra, as defined for example in [SV, Example 8.3], then B(E,L,σ)=S/gS for a central element g ∈ S and some invertible sheaf L over an elliptic curve E.WenotethatS is a graded ring that can be regarded as the “co- P2 P2 ordinate ring of a noncommutative ” or as the “coordinate ring of nc”. -