A Representation Theoretic Study of Non-Commutative Symmetric Algebras

Abstract

We study Van den Bergh's noncommutative symmetric algebra S nc (M ) (over division rings) via Minamoto's theory of Fano algebras. In particular, we show S nc (M ) is coherent, and its proj category P nc (M ) is derived equivalent to the corresponding bimodule species. This generalizes the main theorem of [8], which in turn is a generalization of Beilinson's derived equivalence. As corollaries, we show that P nc (M ) is hereditary and there is a structure theorem for sheaves on P nc (M ) analogous to that for P 1 . 2010 Mathematics Subject Classification. Primary 14A22, 16S38; Secondary 16E35. Proof. To prove the first result (which appeared in [5]), one shows that there are D 1 − D 0 -bimodule isomorphisms Hom D1 (M D1 , D 1 ) → Hom k (M, k) and Hom D0 ( D0 M, D 0 ) → Hom k (M, k).The first one takes ψ : M D1 → D 1 to tr D1/k •ψ, and the second is similar.The proofs of the second and third results follow the proof of [4, Lemma 3.2].

An abstract characterization of noncommutative projective lines

“…Even if M does not have left-right dimension (2,2), the space P nc (M ) can still be defined but is usually not noetherian. However, the results of this paper and [6] establish that P nc (M ) is hereditary, satisfies Serre duality, and has the property that each of its objects is a direct sum of its torsion part and a sum of line bundles. Motivated by these results, we refer to spaces of the form P nc (M ) as noncommutative projective lines.…”

“…which is an isomorphism in degrees zero and one. (2) If L satisfies (1)- (6), then the homomorphism (4-1) is an epimorphism.…”

“…Lemma 5.6. Suppose L is linear and satisfies property (6) from Section 4. Then, for each i ∈ Z and j > i + 1, the map…”

“…In light of Proposition 5.7, Theorem 4.2(3) and Theorem 4.2(4) will follow from the next Proposition 5.8. If L is linear and satisfies (6) and (7), then the homomorphism S nc (H 01 ) → H from Corollary 5.5 is an isomorphism.…”

“…The fact that noncommutative symmetric algebras are graded right-coherent has been proven under very general hypotheses [6].…”