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26 results

Page 1 of 2

  • A Readily-automated Scheme for Estimating the Critical Dose of Beam-Sensitive Materials

  • Robert Colby
  • Journal:
    / Volume 28 / Issue S1 / August 2022
    Published online by Cambridge University Press:
    22 July 2022, pp. 2236-2237
    Print publication:
    August 2022
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  • Fluence-dependent electron energy loss spectroscopy mapping for beam-sensitive polymers

  • Robert Colby, Donald Carpenter
  • Journal:
    / Volume 27 / Issue S1 / August 2021
    Published online by Cambridge University Press:
    30 July 2021, pp. 1918-1920
    Print publication:
    August 2021
    • Article
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  • Revisiting EELS Fine Structure Analysis for Polymers

  • Robert Colby, Robert Williams, Donald Carpenter, Nuria Bagues, David McComb, Srinivasan Rajagopalan
  • Journal:
    / Volume 26 / Issue S2 / August 2020
    Published online by Cambridge University Press:
    30 July 2020, pp. 474-476
    Print publication:
    August 2020
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  • Level Set Method for Tip Shape Evolution Simulation for Atom Probe Tomography

  • Jie Bao, Zhijie Xu, Robert Colby, Suntharampillai Thevuthasan, Arun Devaraj
  • Journal:
    / Volume 21 / Issue S3 / August 2015
    Published online by Cambridge University Press:
    23 September 2015, pp. 841-842
    Print publication:
    August 2015
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Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings
  • Robert R. Colby, Kent R. Fuller
  • Published online:
    12 August 2009
    Print publication:
    22 March 2004
  • This book provides a unified approach to much of the theories of equivalence and duality between categories of modules that has transpired over the last 45 years. In particular, during the past dozen or so years many authors (including the authors of this book) have investigated relationships between categories of modules over a pair of rings that are induced by both covariant and contravariant representable functors, in particular by tilting and cotilting theories. By here collecting and unifying the basic results of these investigations with innovative and easily understandable proofs, the authors' aim is to provide an aid to further research in this central topic in abstract algebra, and a reference for all whose research lies in this field.

  • 2 - Representable Equivalences

  • Robert R. Colby, University of Iowa, Kent R. Fuller, University of Iowa
  • Book:
    Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings
    Published online:
    12 August 2009
    Print publication:
    22 March 2004, pp 12-27

  • Summary

    We are concerned with equivalences and dualities between subcategories of the categories of modules over rings. Henceforth, by “subcategory” we shall mean full subcategory that is closed under isomorphic images, and all functors between categories of modules are assumed to be additive.

    Suppose C and D are subcategories of Mod-R and Mod-S, respectively. A functor H : C → D is an equivalence if there is a functor T : D → C such that T ∘ H and H ∘ T are naturally isomorphic to the identity functors 1C and 1D, respectively. When this is the case we write C ≈ D. By Theorem A.3.4 these natural isomorphisms can be taken to be of the form μ : T H → 1C and θ : 1D → H T where H μ ∘ θ H = 1H and μT ∘ T θ = 1T. That is, μ and θ, an arrow of adjunction and its quasi-inverse, establish T as a left adjoint of H (see Appendix A). If SVR is a bimodule, then we have functors

    HomR(V, -) : Mod-R ⇄ Mod-S : (- ⊗S V),

    and we say that the equivalence H : C ⇄ D : T is representable by SVR if H and T are naturally isomorphic to the restrictions of these functors, that is,

    H ≅ HomR(V, -)|C and T ≅ (- ⊗S V)|D.

    In this case we shall make the identifications

    and then by Theorem A.2.2 the canonical natural transformations ν and η defined beloware natural isomorphisms when restricted to C and D, respectively.

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