Computing noncommutative deformations of presheaves and sheaves of modules

We describe a noncommutative deformation theory for presheaves and sheaves of modules that generalizes the commutative deformation theory of these global algebraic structures, and the noncommutative deformation theory of modules over algebras due to Laudal. In the first part of the paper, we describe a noncommutative deformation functor for presheaves of modules on a small category, and an obstruction theory for this functor in terms of global Hochschild cohomology. An important feature of this obstruction theory is that it can be computed in concrete terms in many interesting cases. In the last part of the paper, we describe noncommutative deformation functors for sheaves and quasi-coherent sheaves of modules on a ringed space $(X, \mathcal{A})$. We show that for any good $\mathcal{A}$-affine open cover $\mathsf{U}$ of $X$, the forgetful functor $\mathsf{QCoh}(\mathcal{A}) \to \mathsf{PreSh}(\mathsf{U}, \mathcal{A})$ induces an isomorphism of noncommutative deformation functors. \emph{Applications.} We consider noncommutative deformations of quasi-coherent $\mathcal{A}$-modules on $X$ when $(X, \mathcal{A}) = (X, \mathcal{O}_X)$ is a scheme or $(X, \mathcal{A}) = (X, \mathcal{D})$ is a D-scheme in the sense of Beilinson and Bernstein. In these cases, we may use any open affine cover of $X$ closed under finite intersections to compute noncommutative deformations in concrete terms using presheaf methods. We compute the noncommutative deformations of the left $\mathcal{D}_X$-module $\mathcal{O}_X$ when $X$ is an elliptic curve as an example.Comment: 22 pages, AMS-LaTeX. Some results from earlier versions have been omitted to focus on the main results in the pape

Noncommutative plane curves

In this paper we study noncommutative plane curves, i.e. non-commutative k-algebras for which the 1-dimensional simple modules form a plane curve. We study extensions of simple modules and we try to enlighten the completion problem, i.e. understanding the connection between simple modules of different dimension.Comment: 31 page

Moduli spaces of reflexive sheaves of rank 2

Let \sF be a coherent rank 2 sheaf on a scheme Y \subset \proj{n} of dimension at least two. In this paper we study the relationship between the functor which deforms a pair (\sF,\sigma), \sigma \in H^0(\sF), and the functor which deforms the corresponding pair (X,\xi) given as in the Serre correspondence. We prove that the scheme structure of e.g. the moduli scheme M_Y(P) of stable sheaves on a threefold Y at (\sF), and the scheme structure at (X) of the Hilbert scheme of curves on Y are closely related. Using this relationship we get criteria for the dimension and smoothness of M_Y(P) at (\sF), without assuming Ext^2(\sF,\sF) = 0. For reflexive sheaves on Y = \proj{3} whose deficiency module M = H_{*}^1(\sF) satisfies Ext^2(M,M) = 0 in degree zero (e.g. of diameter at most 2), we get necessary and sufficient conditions of unobstructedness which coincide in the diameter one case. The conditions are further equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of H_{*}^0(\sF). It follows that every irreducible component of M_{\proj{3}}(P) containing a reflexive sheaf of diameter one is reduced (generically smooth). We also determine a good lower bound for the dimension of any component of M_{\proj{3}}(P) which contains a reflexive stable sheaf with "small" deficiency module M.Comment: 19 page

Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals

We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the more general nongraded setting, via the sequential Cohen-Macaulay property.Comment: 31 pages, 1 figure. Minor changes from previous version. To appear in Advances in Mathematic

Rising externality costs and corporate social responsibility. Case: EU legislation on Electric and Electronic Equipment

Purpose – The purpose of this article is to study how we may identify the link between rising externality costs and Corporate Social Responsibility (CSR) by using a market centric approach to CSR. Design/methodology/approach – We use indicators measuring CSR performances triggered by rising externality costs due to the EU legislation on electric and electronic equipment (EEE). The case study includes three leading companies in the global electric appliances industry. Findings – The EU legislation on EEE has increased the externality costs of the electric appliances industry. Some companies only meet the minimum requirements of the legislation, while others go beyond what is required and engage in CSR. We find that the strongest CSR impact is related to output externalities in our sample in the EEE sector, while the strongest CSR impact in the clothing sector, in an earlier study, is related to input externalities. Practical implications – The findings suggest that governments need to adapt their CSR policies not only to general sector specific features, but in addition to the potential for reducing negative externalities in different parts of the value chain in each sector. Original value – This article contributes to a better understanding of how government policies raise the externality costs of industries which in turn lead these industries to strengthen their CSR performance. The study also demonstrates the usefulness of a market centric approach to CSR

Matrix Factorizations, Minimal Models and Massey Products

We present a method to compute the full non-linear deformations of matrix factorizations for ADE minimal models. This method is based on the calculation of higher products in the cohomology, called Massey products. The algorithm yields a polynomial ring whose vanishing relations encode the obstructions of the deformations of the D-branes characterized by these matrix factorizations. This coincides with the critical locus of the effective superpotential which can be computed by integrating these relations. Our results for the effective superpotential are in agreement with those obtained from solving the A-infinity relations. We point out a relation to the superpotentials of Kazama-Suzuki models. We will illustrate our findings by various examples, putting emphasis on the E_6 minimal model.Comment: 32 pages, v2: typos corrected, v3: additional comments concerning the bulk-boundary crossing constraint, some small clarifications, typo

Evaluation Of Rare Earth Element Extraction From North Dakota Coal-Related Feed Stocks

The rare earth elements consist of the lanthanide series of elements with atomic numbers from 57-71 and also include yttrium and scandium. Due to their unique properties, rare earth elements are crucial materials in an incredible array of consumer goods, energy system components and military defense applications. However, the global production and entire value chain for rare earth elements is dominated by China, with the U.S. currently 100% import reliant for these critical materials. Traditional mineral ores including previously mined deposits in the U.S., however, have several challenges. Chief among these is that the content of the most critical and valuable of the rare earths are deficient, making mining uneconomical. Further, the supply of these most critical rare earths is nearly 100% produced in China from a single resource that is only projected to last another 10 to 20 years. The U.S. currently considers the rare earths market an issue of national security. It is imperative that alternative domestic sources of rare earths be identified and methods developed to produce them. Recently, coal and coal byproducts have been identified as one of these promising alternative resources. This dissertation details a study on evaluation of the technical and economic feasibility of rare earth element recovery from North Dakota lignite coal and lignite-related feedstocks. There were four major goals of this study: i) identify lignite or lignite-related feedstocks with total rare earth element content above 300 parts per million, a threshold dictated by the agency who funded this research as the minimum for economic viability, ii) determine the geochemistry of the feedstocks and understand the forms and modes of occurrence of the rare earth elements, information necessary to inform the development of extraction and concentration methods, iii) identify processing methods to concentrate the rare earth elements from the feedstocks to a target of two weight percent, a value that would be sufficient to leverage existing separation and refining methods developed for the traditional mineral ore industry, and iv) develop a process that is economically viable and environmentally benign. To achieve these overall goals, and to prove or disprove the research hypotheses, the research scope was broken down into three main efforts: i) sampling and characterization of potential feedstocks, ii) laboratory-scale development and testing of rare earth element extraction and concentration methods, and iii) process design and technical and economic feasibility evaluation. In total, 174 unique samples were collected, and several locations were identified that exceeded the 300 ppm total rare earth elements target. The results showed that on a whole sample basis, the rare earths are most concentrated in the clay-rich sediments associated with the coal seams, but on an ash basis in certain locations within certain coal seams the content is significantly higher, an unexpected finding given prior research. At Falkirk Mine near Underwood, North Dakota three coal seams were found to have elevated levels of rare earths, ranging from about 300 to 600 ppm on an ash basis. Additionally, exceptionally high rare earths content was found in samples collected from an outcropping of the Harmon-Hansen coal zone in southwestern North Dakota that contained 2300 ppm on an ash basis. The results dictated that extraction and concentration methods be developed for these rare earth element-rich coals, instead of the mineral-rich sediments. This effort also found that that at a commercial-scale, due to non-uniformity of the rare earths content stratigraphically in the coal seams, selective mining practices will be needed to target specific locations within the seams. The bulk mining and blending practices as Falkirk Mine result in a relatively low total rare earths content in the feed coal entering the Coal Creek Power Station adjacent to the mine. Characterization of the coal samples identified that the predominant modes of rare earths occurrence in the lignite coals are associations with the organic matter, primarily as coordination complexes and a lesser amount as ion-exchangeable cations on oxygen functional groups. Overall it appears that about 80-95% of rare earths content in North Dakota lignite is organically associated, and not present in mineral forms, which due to the weak organic bonding, presented a unique opportunity for extraction. The process developed for extraction of rare earths was applied to the raw lignite coals instead of fly ash or other byproducts being investigated extensively in the literature. Rather, the process uses a dilute acid leaching process to strip the organically associated rare earths from the lignite with very high efficiency of about 70-90% at equilibrium contact times. Although the extraction kinetics are quite fast given commercial leaching operations, there is some tradeoff between extraction efficiency and contact time. However, at shorter contact time there is improved rare earths selectivity that results in a more concentrated product due to limiting extraction of unwanted impurities. There is also a significant difference in the extraction kinetics for the more valuable heavier molecular weight rare earths, which are much faster than the light rare earths. The testing showed that in a one-step process consisting of leaching for two hours with 0.5M sulfuric acid at 40°C, a rare earth concentrate of about 1.4 weight percent rare earths could be achieved with about 70% total rare earths extraction, while also producing a residual coal byproduct that has superior qualities to the feed coal, such as reduced ash content. This represents a concentration factor of 24 over the feed coal. The target of two weight percent rare earths could be achieved by a number of secondary processing methods, such as pH modification or forced air oxidation to selectively precipitate impurities from the rare earths-containing solution. The process developed in this study is simple, highly effective, low cost and novel, with several differentiating benefits compared to methods being developed in the literature. These are made possible by the unique properties of North Dakota lignite coals and the weakly-bonded organic association of the rare earth elements. Key differentiators include the use of the raw coal as the feedstock, the ability to use a mild leaching process, and not needing extensive physical beneficiation processes prior to rare earths extraction. The process is environmentally benign and was demonstrated to be economically viable at the current market conditions. Due to the use of the raw coal as the feedstock, the process can be advantageously integrated with any number of coal utilization processes to augment economics, lower costs and maximize efficiency and synergies. This study evaluated a configuration of rare earths extraction combined with activated carbon production co-located at a combined heat and power facility, and was shown to have highly attractive economics even at small scales representing a first-of-a-kind demonstration system

Deformations of modules of maximal grade and the Hilbert scheme at determinantal schemes

Let R be a polynomial ring and M a finitely generated graded R-module of maximal grade (which means that the ideal I_t(\cA) generated by the maximal minors of a homogeneous presentation matrix, \cA, of M has maximal codimension in R). Suppose X:=Proj(R/I_t(\cA)) is smooth in a sufficiently large open subset and dim X > 0. Then we prove that the local graded deformation functor of M is isomorphic to the local Hilbert (scheme) functor at X \subset Proj(R) under a week assumption which holds if dim X > 1. Under this assumptions we get that the Hilbert scheme is smooth at (X), and we give an explicit formula for the dimension of its local ring. As a corollary we prove a conjecture of R. M. Mir\'o-Roig and the author that the closure of the locus of standard determinantal schemes with fixed degrees of the entries in a presentation matrix is a generically smooth component V of the Hilbert scheme. Also their conjecture on the dimension of V is proved for dim X > 0. The cohomology H^i_{*}({\cN}_X) of the normal sheaf of X in Proj(R) is shown to vanish for 0 < i < dim X-1. Finally the mentioned results, slightly adapted, remain true replacing R by any Cohen-Macaulay quotient of a polynomial ring.Comment: 24 page

Rheological Study of a Direct Liquefaction Feed Slurry

A rheological study was made of a liquefaction startup- feed slurry. The slurry consisted of North Dakota lignite sized to 70% less then 200 mesh and a carrier solvent of an thracene oil. The rheological behavior of the slurries, o with coal concentrations of 0 to 40% at 40 C, was shown to be Newtonian, allowing Bernoulli\u27s equation to be used for direct calculation of pressure drop. Calculated values pre dicted the measured data within 10% for the test conditions. The investigation used an experimental slurry loop, con sisting of a bank of pipes with nominal sizes ranging from 1 inch (2.54 cm) to 3/8 inch (0.95 cm), to determine the actu al pressure drops for the slurries. The pressure drops were found to range from 0.064 psi (4.43xlOJ dynes/cm ) for an thracene oil pumped through the 2.54 cm pipe to 11.71 psi (8.07xl05 dynes/cm2) for the 40% coal slurry pumped through a 0.95 cm pipe. A Brookfield Rheolog viscometer was used to determine viscous behavior of samples taken from the runs. Using lin ear regression analysis, a straight line through zero was obtained for torque vs. angular velocity for each sample with correlation coefficients greater then 0.98. Viscosities of the samples ranged from 23.3 centipoise for anthra cene oil to 286.6 centipoise for the 40% coal slurry. A statistical comparision indicated that calculated and exper imentally measured pressure drops were not significantly different at the 95% confidence level