Naive Noncommutative Blowing Up

Let B(X,L,s) be the twisted homogeneous coordinate ring of an irreducible variety X over an algebraically closed field k with dim X > 1. Assume that c in X and s in Aut(X) are in sufficiently general position. We show that if one follows the commutative prescription for blowing up X at c, but in this noncommutative setting, one obtains a noncommutative ring R=R(X,c,L,s) with surprising properties. In particular: (1) R is always noetherian but never strongly noetherian. (2) If R is generated in degree one then the images of the R-point modules in qgr(R) are naturally in (1-1) correspondence with the closed points of X. However, both in qgr(R) and in gr(R), the R-point modules are not parametrized by a projective scheme. (3) qgr R has finite cohomological dimension yet H^1(R) is infinite dimensional. This gives a more geometric approach to results of the second author who proved similar results for X=P^n by algebraic methods.Comment: Latex, 42 page

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 g of degree 1, T/gT 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 the companion paper "Classifying Orders in the Sklyanin Algebra", we apply our results to classify orders in (a Veronese subalgebra of) a generic cubic or quadratic Sklyanin algebra.Comment: 39 pages. Minor changes from previous version. The final publication is available from Springer via http://dx.doi.org/10.1007/s10468-014-9506-

Algebras in which every subalgebra is noetherian

We show that the twisted homogeneous coordinate rings of elliptic curves by infinite order automorphisms have the curious property that every subalgebra is both finitely generated and noetherian. As a consequence, we show that a localisation of a generic Skylanin algebra has the same property.Comment: 5 pages; comments welcome; v2 only minor changes, most suggested by refere

PA6 nanofibre production: A comparison between rotary jet spinning and electrospinning

© 2018 by the authors. Polymer nanofibres are created from many different techniques, with varying rates of production. Rotary jet spinning is a relatively new technique for making nanofibres from both polymer solutions and melt. With electrospinning being by far the most widespread processing method for polymer nanofibres, we performed a direct comparison of polyamide 6 (PA6) nanofibre production between these two methods. It was found that electrospinning produced slightly smaller-diameter fibres, which scaled with a decrease in solution viscosity. In comparison, rotary jet spun fibres could be produced from a reduced range of polymer concentrations and exhibited therefore slightly larger diameters with greater variation. Crystallinity of the fibres was also compared between the two techniques and the bulk polymer, which showed a decrease in crystallinity compared to bulk PA6

Some Noncommutative Minimal Surfaces

In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is to determine the minimal models within any birational class. In this paper we show that the generic noncommutative projective plane (corresponding to the three dimensional Sklyanin algebra R) as well as noncommutative analogues of P^1 x P^1 and of the Hirzebruch surface F_2 (arising from Van den Bergh's quadrics R) satisfy very strong minimality conditions. Translated into an algebraic question, where one is interested in a maximality condition, we prove the following theorem. Let R be a Sklyanin algebra or a Van den Bergh quadric that is infinite dimensional over its centre and let A be any connected graded noetherian maximal order containing R, with the same graded quotient ring as R. Then, up to taking Veronese rings, A is isomorphic to R. Secondly, let T be an elliptic algebra (that is, the coordinate ring of a noncommutative surface containing an elliptic curve). Then, under an appropriate homological condition, we prove that every connected graded noetherian overring of T is obtained by blowing down finitely many lines (line modules).Comment: 42 pages; Third and final version to appear in Advances in Mathematics: minor revisions from second versio

Formation and action of lignin-modifying enzymes in cultures of Phlebia radiata supplemented with veratric acid

Transformation of veratric (3,4-dimethoxybenzoic) acid by the white rot fungus Phlebia radiata was studied to elucidate the role of ligninolytic, reductive, and demeth(ox)ylating enzymes. Under both air and a 100% O2 atmosphere, with nitrogen limitation and glucose as a carbon source, reducing activity resulted in the accumulation of veratryl alcohol in the medium. When the fungus was cultivated under air, veratric acid caused a rapid increase in laccase (benzenediol:oxygen oxidoreductase; EC 1.10.3.2) production, which indicated that veratric acid was first demethylated, thus providing phenolic compounds for laccase. After a rapid decline in laccase activity, elevated lignin peroxidase (ligninase) activity and manganese-dependent peroxidase production were detected simultaneously with extracellular release of methanol. This indicated apparent demethoxylation. When the fungus was cultivated under a continuous 100% O2 flow and in the presence of veratric acid, laccase production was markedly repressed, whereas production of lignin peroxidase and degradation of veratryl compounds were clearly enhanced. In all cultures, the increases in lignin peroxidase titers were directly related to veratryl alcohol accumulation. Evolution of 14CO2 from 3-O14CH3-and 4-O14CH3-labeled veratric acids showed that the position of the methoxyl substituent in the aromatic ring only slightly affected demeth(ox)ylation activity. In both cases, more than 60% of the total 14C was converted to 14CO2 under air in 4 weeks, and oxygen flux increased the degradation rate of the 14C-labeled veratric acids just as it did with unlabeled cultures

Ring-theoretic blowing down: I

One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand-Kirillov dimension 3). Earlier work of the authors classified the connected graded noetherian subalgebras of Sklyanin algebras using a noncommutative analogue of blowing up. In order to understand other algebras birational to a Sklyanin algebra, one also needs a notion of blowing down. This is achieved in this paper, where we give a noncommutative analogue of Castelnuovo's classic theorem that (-1)-lines on a smooth surface can be contracted. The resulting noncommutative blown-down algebra has pleasant properties; in particular it is always noetherian and is smooth if the original noncommutative surface is smooth. In a companion paper we will use this technique to construct explicit birational transformations between various noncommutative surfaces which contain an elliptic curve.Comment: 40 page