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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Published in:Algebras and representation theory 2015-04, Vol.18 (2), p.491-529
Main Authors: Rogalski, D., Sierra, S. J., Stafford, J. T.
Format: Article
Language:English
Subjects:
Associative Rings and Algebras
Commutative Rings and Algebras
Mathematics
Mathematics and Statistics
Non-associative Rings and Algebras
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description 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.
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subjects Associative Rings and Algebras
Commutative Rings and Algebras
Mathematics
Mathematics and Statistics
Non-associative Rings and Algebras
title Noncommutative Blowups of Elliptic Algebras
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