Abstract
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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The first author is partially supported by NSF grants DMS-0900981 and DMS-1201572.
The second author was partially supported by an NSF Postdoctoral Research Fellowship, grant DMS-0802935.
The third author is a Royal Society Wolfson Research Merit Award Holder.
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Rogalski, D., Sierra, S.J. & Stafford, J.T. Noncommutative Blowups of Elliptic Algebras. Algebr Represent Theor 18, 491–529 (2015). https://doi.org/10.1007/s10468-014-9506-7
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DOI: https://doi.org/10.1007/s10468-014-9506-7
Keywords
- Noncommutative projective geometry
- Noncommutative surfaces
- Sklyanin algebras
- Noetherian graded rings
- Noncommutative blowing up