41 research outputs found
Non-commutative crepant resolutions for some toric singularities. II
No full textUsing the theory of dimer models Broomhead proved that every 3-dimensional Gorenstein aftine tonic variety Spec R admits a tonic non-commutative crepant resolution (NCCR). We give an alternative proof of this result by constructing a tilting bundle on a (stacky) crepant resolution of Spec R using standard tonic methods. Our proof does not use dimer models.The first author is a FWO [PEGASUS]2 Marie Sklodowska-Curie fellow at the Free University of Brussels (funded by the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 665501 with the Research Foundation Flanders (FWO)). During part of this work she was also a postdoc with Sue Sierra at the University of Edinburgh.
The second author is a senior researcher at the Research Foundation Flanders (FWO). While working on this project hewas supported by theFWOgrant G0D8616N: "Hochschild cohomology and deformation theory of triangulated categories".Spenko, S (corresponding author), Univ Libre Bruxelles, Dept Math, Campus Plaine CP 213,Blvd Triomphe, B-1050 Brussels, Belgium.
[email protected]; [email protected]
Tilting Bundles on Hypertoric Varieties
No full textRecently McBreen and Webster constructed a tilting bundle on a smooth hypertoric variety (using reduction to finite characteristic) and showed that its endomorphism ring is Koszul. In this short note we present alternative proofs for these results. We simply observe that the tilting bundle constructed by Halpern-Leistner and Sam on a generic open Geometric Invariant Theory substack of the ambient linear space restricts to a tilting bundle on the hypertoric variety. The fact that the hypertoric variety is defined by a quadratic regular sequence then also yields an easy proof of Koszulity.This work was supported by grant G0D8616N: "Hochschild cohomology and deformation theory of triangulated categories'' (to M.V.d.B.).Van den Bergh, M (corresponding author), Univ Hasselt, Dept WNI, Univ Campus, B-3590 Diepenbeek, Belgium.
[email protected]
Semi-orthogonal decompositions of GIT quotient stacks
No full textIf G is a reductive group acting on a linearized smooth scheme X then we show that under suitable standard conditions the derived category D(X-ss/G) of the corresponding GIT quotient stack Xss/G has a semi-orthogonal decomposition consisting of derived categories of coherent sheaves of rings on X-ss//G which are locally of finite global dimension. One of the components of the decomposition is a certain non-commutative resolution of X-ss//G constructed earlier by the authors. As a concrete example we obtain in the case of odd Pfaffians a semi-orthogonal decomposition of the corresponding quotient stack in which all the parts are certain specific non-commutative crepant resolutions of Pfaffians of lower or equal rank which had also been constructed earlier by the authors. In particular this semi-orthogonal decomposition cannot be refined further since its parts are Calabi-Yau. The results in this paper complement results by Halpern-Leistner, Ballard-Favero-Katzarkov and DonovanSegal that assert the existence of a semi-orthogonal decomposition of D(X/G) in which one of the parts is D(X-ss/G).S. Spenko is a FWO [PEGASUS]2 Marie Sklodowska-Curie fellow at the Free University of Brussels (funded by the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 665501 with the Research Foundation Flanders (FWO)). During part of this work she was also a postdoc with Sue Sierra at the University of Edinburgh. Partly she was supported by yL'Oreal-UNESCO scholarship "For women in science". M. Van den Bergh is a senior researcher at the Research Foundation Flanders (FWO). While working on this project he was supported by the FWO Grant G0D8616N: "Hochschild cohomology and deformation theory of triangulated categories". Substantial progress on this project was made during visits of the authors to each other's host institutions. They respectively thank the University of Hasselt and the University of Edinburgh for their hospitality and support.Van den Bergh, M (corresponding author), Univ Hasselt, Dept WNI, Univ Campus, B-3590 Diepenbeek, Belgium.
[email protected]; [email protected]
The Frobenius morphism in invariant theory
No full textLet R be the homogeneous coordinate ring of the Grassmannian G = Gr(2, n) defined over an algebraically closed field of characteristic p > 0. In this paper we give a completely characteristic free description of the decomposition of R, considered as a graded R-p-module, into indecomposables ("Frobenius summands"). As a corollary we obtain a similar decomposition for the Frobenius pushforward of the structure sheaf of G and we obtain in particular that this pushforward is almost never a tilting bundle. On the other hand we show that R provides a "noncommutative resolution" for R-p when p >= n - 2, generalizing a result known to be true for toric varieties. In both the invariant theory and the geometric setting we observe that if the characteristic is not too small the Frobenius summands do not depend on the characteristic in a suitable sense. In the geometric setting this is an explicit version of a general result by Bezrukavnikov and Mirkovid on Frobenius decompositions for partial flag varieties. We are hopeful that it is an instance of a more general "p-uniformity" principle. (C) 2019 Elsevier Inc. All rights reserved.The first author is supported by an EPSRC postdoctoral fellowship EP/R005214/1.
The second author is a FWO [PEGASUS]2 Marie Sklodowska-Curie fellow at the Free University of Brussels (funded by the European Union Horizon 2020 research and innovation programme under the Marie Skiodowska-Curie grant agreement No 665501 with the Research Foundation Flanders (FWO)). During part of this work she was also a postdoc with Sue Sierra at the University of Edinburgh.
The third author is a senior researcher at the Research Foundation Flanders (FWO). While working on this project he was supported by the FWO grant GOD8616N: "Hochschild cohomology and deformation theory of triangulated categories"
Catalan numbers and noncommutative Hilbert schemes
No full textWe find an explicit S-n -equivariant bijection between the integral points in a certain zonotope in R- n , combinatorially equivalent to the permutahedron, and the set of m-parking functions of length n. This bijection restricts to a bijection between the regular S n -orbits and (m, n)-Dyck paths, the number of which is given by the Fuss-Catalan number A( n) (m, 1). Our motivation came from studying tilting bundles on noncommutative Hilbert schemes. As a side result we use these tilting bundles to construct a semi-orthogonal decomposition of the derived category of noncommutative Hilbert schemes.We thank Cesar Ceballos for very helpful discussions, in particular for teaching us about parking functions which lead to the simple proof of Corollary 1.6. We further thank him for making us aware of a plethora of incarnations and
variations of Catalan numbers and for providing references. While this work was in its final stages, the second author attended Tudor P ˘a durariu’s interesting lecture in the workshop “Representation theory and flag or quiver varieties” (Paris, June 2022) during which he reported on joint We thank Cesar Ceballos for very helpful discussions, in particular for teaching us about parking functions which lead to the simple proof of Corollary 1.6. We further thank him for making us aware of a plethora of incarnations and variations of Catalan numbers and for providing references. While this work was in its final stages, the second author attended Tudor P ˘a durariu’s interesting lecture in the workshop “Representation theory and
flag or quiver varieties” (Paris, June 2022) during which he reported on join
Catalan numbers and noncommutative Hilbert schemes
No full textWe find an explicit S-n -equivariant bijection between the integral points in a certain zonotope in R- n , combinatorially equivalent to the permutahedron, and the set of m-parking functions of length n. This bijection restricts to a bijection between the regular S n -orbits and (m, n)-Dyck paths, the number of which is given by the Fuss-Catalan number A( n) (m, 1). Our motivation came from studying tilting bundles on noncommutative Hilbert schemes. As a side result we use these tilting bundles to construct a semi-orthogonal decomposition of the derived category of noncommutative Hilbert schemes.We thank Cesar Ceballos for very helpful discussions, in particular for teaching us about parking functions which lead to the simple proof of Corollary 1.6. We further thank him for making us aware of a plethora of incarnations and
variations of Catalan numbers and for providing references. While this work was in its final stages, the second author attended Tudor P ˘a durariu’s interesting lecture in the workshop “Representation theory and flag or quiver varieties” (Paris, June 2022) during which he reported on joint We thank Cesar Ceballos for very helpful discussions, in particular for teaching us about parking functions which lead to the simple proof of Corollary 1.6. We further thank him for making us aware of a plethora of incarnations and variations of Catalan numbers and for providing references. While this work was in its final stages, the second author attended Tudor P ˘a durariu’s interesting lecture in the workshop “Representation theory and
flag or quiver varieties” (Paris, June 2022) during which he reported on join
Multilingual reading proficiency in an emerging parallel-language environment
No full textRapid changes have taken place in the linguistic environment of higher education in Europe, where many students attend parallel-language courses, leading to a use of English (officially a foreign language) for academic purposes alongside the local language. This study investigated the relationship of Swedish students’ reading habits and abilities in Swedish and English. Their reading abilities were assessed with an interview and a Swedish and English reading test, and their reading habits with an interview, questionnaire, and Author Recognition Test. The study found correlation between English reading ability and some of the reading habits measures which is more reminiscent of situations where English is an official language. This was reflected in the students’ reading habits. Their leisure reading included both Swedish and English material, and their choice between the two depended primarily on factors such as quality and availability, and not language. So for these students there is little difference between reading difficulty in L1 and L2. These results suggest that many students in the parallel-language environments are highly biliterate, implying very different EAP requirements than encountered elsewhere. Implications are discussed.</p
Non-Commutative Crepant Resolutions for Some Toric Singularities I
No full textWe give a criterion for the existence of noncommutative crepant resolutions (NCCRs) for certain toric singularities. In particular, we recover Broomhead's result that a three-dimensional toric Gorenstein singularity has an NCCR. Our result also yields the existence of an NCCR for a four-dimensional toric Gorenstein singularity, which is known to have no toric NCCR.YN This work was supported by EPSRC grant [EP/M008460/1 to S.S.]; M.V.D.B. is a senior researcher at the Research Foundation Flanders (FWO). While working on this project he was supported by the FWO grant G0D8616N: "Hochschild cohomology and deformation theory of triangulated categories."Van den Bergh, M (corresponding author), Univ Edinburgh, Sch Math, James Clerk Maxwell Bldg,Kings Bldg, Edinburgh EH9 3FD, Midlothian, Scotland.
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Two-stage hepatectomy in resection of colorectal liver metastases – a single-institution experience with case-control matching and review of the literature
No full textTwo-stage hepatectomy (TSH) has been proposed for patients with bilateral liver tumours who have a high risk of posthepatectomy liver failure after one-stage hepatectomy (OSH). This study aimed to determine the outcomes of TSH for extensive bilateral colorectal liver metastases
Cystatin F Regulates Proteinase Activity in IL-2-activated Natural Killer Cells
No full textCystatin F is a unique member of the cystatin family of cysteine protease inhibitors, which is synthesized as an inactive dimer and it is activated by N-terminal cleavage in the endolysosomes. It is expressed in the cells of the immune system: myeloid cells and the cells involved in target cell killing: natural killer (NK) cells and cytotoxic T cells (CTLs). Upon activation of the NK cells with interleukin 2 (IL-2), cystatin F was found upregulated and co-localized in cytotoxic granules with cathepsin C (CatC) and CatV. However, cystatin F inhibits the CatC in cells only when its N-terminal part is processed. Although cystatin F could inhibit both CatV and CatC, the IL-2 stimulation of the YT cells resulted in an increased CatV activity, while the CatC activity was unchanged. The incubation of IL-2 activated NK cells with a cysteine proteinase inhibitor E-64d increased the cystatin F dimer formation. Our results suggest that cystatin F not only inhibits CatV, but it is processed by the CatV in order to inhibit the CatC activity in cytotoxic granules. The regulation of the CatC activity in the cytotoxic granules of the NK cells by the cystatin F could be important for the processing and activation of granule-associated serine proteases - granzymes
