Species and non-commutative P1's over non-algebraic bimodules

Abstract

Abstract. We study non-commutative projective lines over not necessarily algebraic bimodules. In particular, we give a complete description of their categories of coherent sheaves and show they are derived equivalent to certain bimodule species. This allows us to classify modules over these species and thus generalize, and give a geometric interpretation for, results of C. Ringel [22].Throughout this paper, K, K 0 , and K 1 denote fields of characteristic = 2.

An abstract characterization of noncommutative projective lines

“…(3) If L satisfies (1)- (7), then the homomorphism (4-1) is an isomorphism, and H ii+1 is 2-periodic and not of type (1, 1), (1,2) [12] have helices. In addition, although the noncommutative curves studied in [7] are noetherian, they are not noncommutative curves of genus zero in the sense of [12] since they are not necessarily Ext-finite. Nevertheless, Theorem 4.2 implies they also have helices.…”

“…Proposition 5.7. If L is linear and satisfies (6) and (7), and j ∈ Z, then (1) H j,j+1 is 2-periodic, and (2) H j,j+1 is not of type (1, 1), (1,2) or (1, 3).…”

“…Although one has to make some restrictions on M so that the category P nc (M ) is homologically well behaved (see the statement of Theorem 4.2 (5)), once these restrictions are made, P nc (M ) behaves in many respects like the commutative projective line. If, for example, M has left-right dimension (2, 2) and we let P nc (M ) denote the category of finitely generated graded right S nc (M )-modules modulo those that are eventually zero, then the category P nc (M ) is noetherian and has many properties in common with the commutative projective line [7]. 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.…”

Witt’s Theorem for Noncommutative Conic Curves

“…(2, 2)-bimodule, which relies on the commutativity of End(L) and End(L). However, it is proven in [3] that the commutativity assumption is not necessary.…”

“…In this section, we explicitly define diagram (1)(2)(3) and prove that it is commutative. First recall that if σ is an auto-equivalence of H then the abelian group j ≥0 Hom H (σ −j L, L) can be made into a Z-graded k-algebra by defining multiplication as follows: for a ∈ Hom H (σ −i L, L) and b ∈ Hom H (σ −j L, L), we let a · b := a • σ −i (b).…”