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

2009/06/13 by S. Paul Smith, Smith, S. Paul
Mathematics · #14A22 #16S38 #16W50 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.0906.2481

openalex publication_date 2009/06/13 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

Let R be the free algebra on x and y modulo the relations x⁵=yxy and y²=xyx\nendowed with the grading deg x=1 and deg y=2. Let B₃ denote the blow up of the\nprojective plane at three non-colliear points. The main result in this paper is\nthat the category of quasi-coherent sheaves on B₃ is equivalent to the\nquotient of the category of graded R-modules modulo the full subcategory of\nmodules M such that for each m in M, (x,y)ⁿm=0 for n sufficiently large.\nThis is proved by showing the R is a twisted homogeneous coordinate ring (in\nthe sense of Artin and Van den Bergh) for B₃. This reduces almost all\nrepresentation-theoretic questions about R to algebraic geometric questions\nabout the del Pezzo surface B₃. For example, the generic simple R-module has\ndimension six. Furthermore, the main result combined with results of Artin,\nTate, and Van den Bergh, imply that R is a noetherian domain of global\ndimension three, and has other good homological properties.\n

Related

• The space of Penrose tilings and the non-commutative curve with homogeneous coordinate ring k/(y²)
• The non-commutative scheme having a free algebra as a homogeneous coordinate ring
• The Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings
• On stacky surfaces and noncommutative surfaces
• Homogeneous coordinate rings as direct summands of regular rings