A non-commutative homogeneous coordinate ring for the degree six del Pezzo surface

作者：Smith S Paul^*

来源：Journal of Algebra, 2012, 354(1): 95-109.

DOI：10.1016/j.jalgebra.2011.11.030

• Homogeneous coordinate ring
• Noncommutative
• del Pezzo surface
• Graded algebra

Let R be the free C-algebra on x and y modulo the relations x(5) = yxy and y(2) = xyx endowed with the Z-grading deg x = 1 and deg y = 2. The ring R appears, in somewhat hidden guise, in a paper on quiver gauge theories. Let B-3 denote the blow up of CP2 at three non-colinear points. The main result in this paper is that the category of quasi-coherent O-B3-modules is equivalent to the quotient of the category of Z-graded R-modules modulo the full subcategory of modules that are the sum of their finite dimensional submodules. This reduces almost all representation-theoretic questions about R to algebraic geometric questions about the del Pezzo surface B-3. For example, the generic simple R-module has dimension six. Furthermore, the main result combined with results of Artin, Tate, Van den Bergh, and Stephenson implies that R is a noetherian domain of global dimension three.

• 出版日期2012-3-15