Residue Complex for Sklyanin Algebras of Dimension Three

Abstract

We describe the residue complex for three-dimensional Sklyanin algebras, which are the interesting special cases of quantum polynomial rings in three variables. In particular, we show that the multiplicities of the point modules in the residue complex are all one, just as in the classical case of commutative polynomial rings in three variables. We explain why the residue complex in the quantum case has the same multiplicities of point modules as that in the commutative case (even if there are fewer point modules in the quantum case) by pointing out two quantum anomalies. Academic PressContents.

The Injective Spectrum of a Noncommutative Space

“…Thus we have embeddings R/ ≤ M 1 and R/ ≤ M 2 . Since each of M 1 , M 2 is tiny, we have R/ ≤ M n 1 1 and R/ ≤ M n 2 2 for positive integers n 1 , n 2 by Lemma 8.5. Now, the Krull dimension of each of M 1 , M 2 must equal the Krull dimension of R/ .…”

“…Keeping with tradition, 1 The author was partially supported by a postdoctoral fellowship from the Mathematical Sciences Research Institute and a Baylor University summer sabbatical. we use the following quotation from Manin, "To do geometry you don't really need a space; all you need is a category of sheaves on this would-be space" [13, p. 83].…”

Sklyanin algebras and Hilbert schemes of points

“…Suppose that B K is not surjective. Then Lemma 6.15 (1) implies that GKdim Γ * (Coker(B)) = 1 and so, by Proposition 6.17, there is a quotient complex…”

“…Returning to k-algebras, this implies that the second row of (6.5) is also exact. Applying Γ * to that sequence gives the complex (1) .…”

Noncommutative curves and noncommutative surfaces

“…If A is a Z-algebra and m ∈ Z, then we may define the shifted Z-algebra A(m) by A(m) ij = A(m) i+m,j+m . Clearly Gr(A(m)) ∼ = Gr(A) and QGr(A(m)) ∼ = QGr(A) but A comes from a graded ring if and only if A ∼ = A (1). If C is a noncommutative Bondal-Polishchuk P 2 , then it is easy to see that the sequence of objects (O(n)) n is determined up to shift by the categorical properties of C and hence that the associated Z-algebra is also determined up to shifting (see [88] for the details).…”