42,617 research outputs found

    Exercises on derived categories, resolutions, and Brown representability

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    These exercises complement my notes "Derived categories, resolutions, and Brown representability".Comment: 5 pages. To appear in the proceedings of the summer school "Interactions between homotopy theory and algebra" (Chicago, 2004

    Auslander-Reiten duality for Grothendieck abelian categories

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    Auslander-Reiten duality for module categories is generalised to Grothendieck abelian categories that have a sufficient supply of finitely presented objects. It is shown that Auslander-Reiten duality amounts to the fact that the functor Ext^1(C,-) into modules over the endomorphism ring of C admits a partially defined right adjoint when C is a finitely presented object. This result seems to be new even for module categories. For appropriate schemes over a field, the connection with Serre duality is discussed.Comment: 16 page

    Cohomological quotients and smashing localizations

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    The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier's construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S, a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic K-theory and demonstrates the relevance of the telescope conjecture for derived categories. Another application leads to a derived analogue of an almost module category in the sense of Gabber-Ramero. It is shown that the derived category of an almost ring is of this form.Comment: 46 pages; revised versio

    Approximations and adjoints in homotopy categories

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    We provide a criterion for the existence of right approximations in cocomplete additive categories; it is a straightforward generalisation of a result due to El Bashir. This criterion is used to construct adjoint functors in homotopy categories. Applications include the study of (pure) derived categories. For instance, it is shown that the pure derived category of any module category is compactly generated.Comment: 15 pages, added some remarks, another description of the pure derived category of a module category, and an exampl

    Magnetic fields and star formation in spiral galaxies

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    The main observational results from radio continuum and polarization observations about the magnetic field strength and large-scale pattern for face-on and edge-on spiral galaxies are summarized and compared within our sample of galaxies of different morphological types, inclinations, and star formation rates (SFR). We found that galaxies with low SFR have higher thermal fractions/smaller synchrotron fractions than those with normal or high SFR. Adopting an equipartition model, we conclude that the nonthermal radio emission and the \emph{total magnetic field} strength grow nonlinearly with SFR, while the regular magnetic field strength does not seem to depend on SFR. We also studied the magnetic field structure and disk thicknesses in highly inclined (edge-on) galaxies. We found in four galaxies that - despite their different radio appearance - the vertical scale heights for both, the thin and thick disk/halo, are about equal (0.3/1.8 kpc at 4.75 GHz), independently of their different SFR. This implies that all these galaxies host a galactic wind, in which the bulk velocity of the cosmic rays (CR) is determined by the total field strength within the galactic disk. The galaxies in our sample also show a similar large-scale magnetic field configuration, parallel to the midplane and X-shaped further away from the disk plane, independent of Hubble type and SFR in the disk. Hence we conclude that also the large-scale magnetic field pattern does not depend on the amount of SFR.Comment: 5 pages, 3 figures. To be published in "Magnetic Fields in the Universe II (2008)", eds. A. Esquivel et al., Rev. Mex. Astron. Astrof. (SC); typos corrected and references updated 07/05/200

    Cohomological length functions

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    We study certain integer valued length functions on triangulated categories and establish a correspondence between such functions and cohomological functors taking values in the category of finite length modules over some ring. The irreducible cohomological functions form a topological space. We discuss its basic properties and include explicit calculations for the category of perfect complexes over some specific rings.Comment: 17 pages. Version 2: Thorougly revised and extended version. A reference for Theorem 1.1 is included and the space of cohomological functions is now discussed in great detail. Version 3: Slightly revised version, providing few more details. Conjecture 4.8 included. One reference adde

    Polynomial representations of GL(n) and Schur-Weyl duality

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    Polynomial representations of general linear groups and modules over Schur algebras are compared. We work over an arbitrary commutative ring and show that Schur-Weyl duality is the key for an equivalence between both categories.Comment: 4 page

    Large Gravitational Waves and Lyth Bound in Multi Brane Inflation

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    It is shown that multi M5-brane inflation in heterotic M-theory gives rise to a detectable gravitational wave power spectrum with tensor fraction rr typically larger than the projected experimental sensitivity, rexp=0.01r_{exp} = 0.01. A measurable gravitational wave power spectrum entails a large inflationary energy scale and a super-Planckian inflaton variation. They present serious problems for particle theory model building resp. a reliable effective field theory description. These problems are eased or even absent in multi-brane inflation models and multi M5-brane inflation, in particular.Comment: 15 pages, 1 figure; v2: references added; v3: final version published in JCA

    A Non-Linear Roth Theorem for Fractals of Sufficiently Large Dimension

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    Suppose that dβ‰₯2d \geq 2, and that AβŠ‚[0,1]A \subset [0,1] has sufficiently large dimension, 1βˆ’Ο΅d<dim⁑H(A)<11 - \epsilon_d < \dim_H(A) < 1. Then for any polynomial PP of degree dd with no constant term, there exists a point configuration {x,xβˆ’t,xβˆ’P(t)}βŠ‚A\{ x, x-t,x-P(t) \} \subset A with tβ‰ˆP1t \approx_P 1

    Dimension-free Maximal Inequalities for Spherical Means in the Hypercube

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    We extend the main result of Harrow, Kolla, and Schulman -- the existence of dimension-free L2L^2-bounds for the spherical maximal function in the hypercube -- to all Lp,p>1L^p, p > 1. Our approach is motivated by the spectral technique developed by Stein and Nevo and Stein in the context of pointwise ergodic theorems on general groups. We provide an example which demonstrates that no dimension-free weak-type (1-1) bound exists at the endpoint.Comment: 17 page
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