Noncommutative Blowups of Elliptic Algebras

We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element ∈ T 1 , T / g T is a twisted homogeneous coordinate ring of an elliptic curve E at an infinite order automorphism). Given an effective div...

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Bibliographic Details
Published inAlgebras and representation theory Vol. 18; no. 2; pp. 491 - 529
Main Authors Rogalski, D., Sierra, S. J., Stafford, J. T.
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.04.2015
Subjects
Associative Rings and Algebras
Commutative Rings and Algebras
Mathematics
Mathematics and Statistics
Non-associative Rings and Algebras
Noetherian graded rings
Noncommutative blowing up
16W50
Noncommutative surfaces
Sklyanin algebras
16E65
14A22
14H52
16P40
Noncommutative projective geometry
18E15
16S38
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Summary:We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element ∈ T 1 , T / g T is a twisted homogeneous coordinate ring of an elliptic curve E at an infinite order automorphism). Given an effective divisor d on E whose degree is not too big, we construct a blowup T ( d ) of T at d and show that it is also an elliptic algebra. Consequently it has many good properties: for example, it is strongly noetherian, Auslander-Gorenstein, and has a balanced dualizing complex. We also show that the ideal structure of T ( d ) is quite rigid. Our results generalise those of the first author in Rogalski (Advances Math. 226 , 1433–1473, 2011 ). In the companion paper Rogalski et al. ( 2013 ), we apply our results to classify orders in (a Veronese subalgebra of) a generic cubic or quadratic Sklyanin algebra.
ISSN:1386-923X
1572-9079
DOI:10.1007/s10468-014-9506-7