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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Differential operators, $n$-branch curve singularities and the $n$-subspace problem
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by R. C. Cannings and M. P. Holland PDF
Trans. Amer. Math. Soc. 347 (1995), 1439-1451 Request permission


Let $R$ be the coordinate ring of a smooth affine curve over an algebraically closed field of characteristic zero $k$. For $S$ a subalgebra of $R$ with integral closure $R$ denote by $\mathcal {D}(S)$ the ring of differential operators on $S$ and by $H(S)$ the finite-dimensional factor of $\mathcal {D}(S)$ by its unique minimal ideal. The theory of diagonal $n$-subspace systems is introduced. This is used to show that if $A$ is a finite-dimensional $k$-algebra and $t \geqslant 1$ is any integer there exists such an $S$ with \[ H(S) \cong \left ( {\begin {array}{*{20}{c}} A & {\ast } \\ 0 & {{M_t}(k)} \\ \end {array} } \right ).\] Further, the Morita classes of $H(S)$ are classified for curves with few branches, and it is shown how to lift Morita equivalences from $H(S)$ to $\mathcal {D}(S)$.
    S. Brenner, Endomorphism algebras of vector spaces with distinguished sets of subspaces, J. Algebra 6 (1967), 100-114. K.A. Brown, The Artin algebras associated with differential operators on singular affine curves, Math Z. 206 (1991), 423-442. R.C. Cannings and M.P. Holland, Right ideals of rings of differential operators, J. Algebra 167 (1994), 116-141. โ€”, Differential operators and finite dimensional algebras, J. Algebra (to appear). R.C. Cannings, M.P. Holland, and G. Masson, Gorenstein curve singularities and self-dual diagonal systems of vector spaces, in preparation. I.M. Gelfand and V.A. Ponomarev, Problems of linear algebra and classification of quadruples of subspaces in a finite dimensional vector space, Colloq. Math. Soc. Jรกnos Bolyai, vol. 5, North-Holland, Amsterdam, 1970. H. Kraft and C. Riedtmann, Geometry of representations of quivers, Representations of Algebras (P. Webb, ed.), London Math. Soc. Lecture Note Ser., vol. 116, Cambridge Univ. Press, Cambridge, 1986, pp. 109-146. S.P. Smith and J.T. Stafford, Differential operators on an affine curve, Proc. London Math. Soc. (3) 56 (1988), 229-259.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 1439-1451
  • MSC: Primary 16S32; Secondary 14H20
  • DOI:
  • MathSciNet review: 1273480