J. T. Stafford

Noncommutative curves and noncommutative surfaces

In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing up and down, has been developed.

Noncommutative graded domains with quadratic growth

Letk be an algebraically closed field, and letR be a finitely generated, connected gradedk-algebra, which is a domain of Gelfand-Kirillov dimension two. Write the graded quotient ringQ(R) ofR asD[z,z−1; δ], for some automorphism δ of the division ringD. We prove thatD is a finitely generated field extension ofk of transcendence degree one. Moreover, we describeR in terms of geometric data. IfR is generated in degree one then up to a finite dimensional vector space,R is isomorphic to the twisted homogeneous coordinate ring of an invertible sheaf ℒ over a projective curveY. This implies, in particular, thatR is Noetherian, thatR is primitive when |δ|=∞ and thatR is a finite module over its centre when |δ|<∞. IfR is not generated in degree one, thenR will still be Noetherian and primitive if δ has infinite order, butR need not be Noetherian when δ has finite order.

The quantum coordinate ring of the special linear group

Abstract We prove that, even under the multiparameter definition of Artin, Schelter and Tate, the quantum coordinate ring O q (SL n ( k )) of the special linear group SL n ( k ) satisfies most of the standard ring-theoretic properties of the classical coordinate ring O (SL n ( k )).

Rational Cherednik algebras and Hilbert schemes. II: representations and sheaves

Let H_c be the rational Cherednik algebra of type A_{n-1} with spherical subalgebra U_c = e H_c e. Then U_c is filtered by order of differential operators with associated graded ring gr U_c = \mathbb{C} [ \mathfrac{h} ⊕ \mathfrac{h}^* ]^W, where is the n-th symmetric group. Using the Z-algebra construction from [GS], it is also possible to associate to a filtered H_c - or U_c - module \hat{Φ}(M) on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of U_c and H_c, and we relate it to Hilb(n) and to the resolution of singularities τ : Hilb(n) → \mathfrac{h} ⊕ \mathfrac{h}^* / W. For example, we prove the following. • If c=1/n so that L_c(triv) is the unique one-dimensional simple H_c-module, then \hat{Φ}(e L_c(triv)) ≅ \mathcal{O}_{Z_n}, where Z_n = τ^{-1}(0) is the punctual Hilbert scheme. • If c = 1/n+k for k \in \mathbb{N}, then under a canonical filtration on the finite-dimensional module L_c(triv), gr e L_c(triv) has a natural bigraded structure that coincides with that on H^0( Z_n, \mathscr{L}^k), where \mathscr{L} ≅ \mathcal{O}_{Hilb(n)}(1); this confirms conjectures of Berest, Etingof, and Ginzburg [BEG2, Conjectures 7.2, 7.3]. • Under mild restrictions on c, the characteristic cycle of \hat{Φ}(e Δ_c(μ)) equals \sum_λ K_{μλ}[Z_λ], where K_{μλ} are Kostka numbers and the Z_λ are (known) irreducible components of τ^{-1}(\mathfrak{h}/W)

Non-holonomic modules over Weyl algebras and enveloping algebras

A number of counterexamples in the theory of enveloping algebras and Noetherian rings are presented. In particular, we construct non-holonomic, simple modules over the Weyl algebra A, and the enveloping algebra U(S1 z xS12). Further, we show that U(S1 z x S12) is not weakly ideal invariant and that there exist simple S1 z x S12-modules M and E, with E finite dimensional, such that E | is not Artinian. If M is a module over a ring R, then the Gelfand-Kirillov dimension of M, written GK dim M, is defined in [7]. Suppose that M is a simple module over a Noetherian ring R and that G K d i m R < oc. Then M is called holonomic if G K d i m M = 8 9 The reason for the definition is that in the case when R is a Weyl algebra or the enveloping algebra of a finite dimensional, algebraic Lie algebra, holonomic modules have particularly nice properties.

Examples in non-commutative projective geometry

Let A = k ⊕ ⊕ n ≥ 1 A n connected graded, Noetherian algebra over a fixed, central field k (formal definitions will be given in Section 1 but, for the most part, are standard). If A were commutative, then the natural way to study A and its representations would be to pass to the associated projective variety and use the power of projective algebraic geometry. It has become clear over the last few years that the same basic idea is powerful for non-commutative algebras; see, for example, [ATV1, 2], [AV], [Sm], [SS] or [TV] for some of the more significant applications. This suggests that it would be profitable to develop a general theory of ‘non-commutative projective geometry’ and the foundations for such a theory have been laid down in the companion paper [AZ]. The results proved there raise a number of questions and the aim of this paper is to provide negative answers to several of these.

Endomorphisms of right ideals of the Weyl algebra

Let A = A(k) be the first Weyl algebra over an infinite field k, let P be any noncyclic, projective right ideal of A and set S = End(P). We prove that, as k-algebras, S f A. In contrast, there exists a noncyclic, projective right ideal Q of S such that S End(Q). Thus, despite the fact that they are Morita equivalent, S and A have surprisingly different properties. For example, under the canonical maps, Autk(A) Pick(A) Pick(S). In contrast, Autk(S) has infinite index in Pick(S). Introduction. Given a comnmutative domain R, the (first) Weyl algebra A(R) is defined to be the associative R-algebra (thus R is central subring) generated by elements x and y subject to the relation xy yx = 1. When no ambiguity is possible, we write A for A(R). Let F be a field of characteristic zero. Then A = A(F) is a simple ring and, indeed, may be thought of as one of the nicest and most important examples of simple Noetherian rings. The initial motivation for this paper was the following result of Smith. If P is the noncyclic right ideal P = x2A + (xy + 1)A of A then End(P) t A (as F-algebras) [11]. Note that, as char(F) = 0, A is a simple hereditary ring and so End(P) is automatically Morita equivalent to A. Even worse, as PEDP -ADA [17], the full matrix rings M2(A) and M2(End(P)) are isomorphic. Thus any proof of Smiths result must be fairly subtle and may therefore provide useful invariants for A. The first main aim of this paper is to generalize Smiths result to an arbitrary projective right ideal of A. While the proof of this is harder than that of Smiths result it does provide a more informative proof in the sense that, unlike Smith, we do not (and cannot) require an explicit description of End(P). For the rest of this introduction k will denote a field of arbitrary characteristic and all isomorphisms of rings will be k-algebra isomorphisms. THEOREM A. Let P be a projective right ideal of A = A(k). Then End(P) A if and only if P is a cyclic right ideal of A. (See Theorem 3.1.) This has some easy corollaries: COROLLARY B. Let P and Q be projective right ideals of A = A(k). Then End(P) End(Q) if and only if P = to(Q) for some t E D(k), the division ring of fractions of A, and a E Autk (A), the group of k-automorphisms of A. Received by the editors February 13, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A65, 16A72, 16A19. Supported in part by an NSF grant. (?)1987 American Mathematical Society 0002-9947/87

Invariant differential operators and an homomorphism of Harish-Chandra

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Generating modules efficiently: Algebraic K-theory for noncommutative Noetherian rings

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Naïve noncommutative blowing up

Let g be a reductive complex Lie algebra, with adjoint group G, Cartan subalgebra h and Weyl group W . Then G acts naturally on the algebra of polynomial functions O(g) and hence on the ring of differential operators with polynomial coefficients, D(g). Similarly, W acts on h and hence on D(h). In [HC2], Harish-Chandra defined an algebra homomorphism δ : D(g) → D(h) . Recently, Wallach proved that, if g has no factors of type E6, E7 or E8, then this map δ is surjective [Wa, Theorem 3.1]. The significance of Wallach’s result is that it enables him to give an easy proof of an important theorem of Harish-Chandra about invariant distributions and to give an elegant new approach to the Springer correspondence. The main aim of this paper is to give an elementary proof of [Wa, Theorem 3.1] that also works for all reductive Lie algebras. Set