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2 results

  • ON ENDOMORPHISMS OF EXTENSIONS IN TANNAKIAN CATEGORIES

  • Part of
  • PAYMAN ESKANDARI
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 110 / Issue 1 / August 2024
    Published online by Cambridge University Press:
    07 November 2023, pp. 103-114
    Print publication:
    August 2024
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  • We prove analogues of Schur’s lemma for endomorphisms of extensions in Tannakian categories. More precisely, let $\mathbf {T}$ be a neutral Tannakian category over a field of characteristic zero. Let E be an extension of A by B in $\mathbf {T}$. We consider conditions under which every endomorphism of E that stabilises B induces a scalar map on $A\oplus B$. We give a result in this direction in the general setting of arbitrary $\mathbf {T}$ and E, and then a stronger result when $\mathbf {T}$ is filtered and the associated graded objects to A and B satisfy some conditions. We also discuss the sharpness of the results.

  • Local solutions to positive characteristic non-Archimedean differential equations

  • Part of
  • João Pedro Pinto dos Santos
  • Journal:
    Compositio Mathematica / Volume 143 / Issue 6 / November 2007
    Published online by Cambridge University Press:
    01 November 2007, pp. 1465-1477
    Print publication:
    November 2007
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  • In the complex domain, one can integrate (solve) holomorphic ordinary differential equations (ODEs) near a non-singular point. We study the existence of solutions in the case of a positive characteristic base field k which is complete with respect to a non-Archimedean absolute value. ODEs are substituted by modules over a ring of analytic functions endowed with an action of all differential operators. The monodromy groups associated to the corresponding category are computed.