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Minors and resolutions of non-commutative schemes

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Abstract

We develop the theory of minors of non-commutative schemes. This study is motivated by applications in the theory of non-commutative resolutions of singularities of commutative schemes. In particular, we construct a categorical resolution for non-commutative curves and in the rational case show that it can be realized as the derived category of a quasi-hereditary algebra.

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Notes

  1. Note that in the book [31] the notations for the orthogonal subcategories are opposite to ours. The latter seems more usual, especially in the representation theory, see, for instance, [2, 19]. In [17] the objects of the right orthogonal subcategory \(\mathscr {C}^\perp \) are called \(\mathscr {C}\) -closed.

  2. Actually, Neeman uses this term for triangulated categories, but we will use it for abelian categories too.

  3. Note that in this situation \(\phi ^*=\phi ^{-1}\).

  4. In the affine case this notion was introduced in [14]. Actually, the main results of this section are just global analogues of those from [14].

  5. Note that all \(\mathscr {B}\)-modules from \(\mathsf {H}(\mathscr {A}\text{- }\mathsf {Inj})\) are injective.

  6. We do not know whether the last condition implies the Cohen–Macaulay property, as it is in the commutative case.

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Acknowledgements

The results of this paper were mainly obtained during the stay of the second author at the Max-Plank-Institut für Mathematik. Its final version was prepared during the visit of the second and the third author to the Institute of Mathematics of the Köln University.

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Authors and Affiliations

  1. Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931, Cologne, Germany

    Igor Burban

  2. Institute of Mathematics, National Academy of Sciences of Ukraine, Tereschenkivska str. 3, Kiev, 01601, Ukraine

    Yuriy Drozd & Volodymyr Gavran

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  1. Igor Burban
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  2. Yuriy Drozd
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  3. Volodymyr Gavran
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Correspondence to Yuriy Drozd.

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Burban, I., Drozd, Y. & Gavran, V. Minors and resolutions of non-commutative schemes. European Journal of Mathematics 3, 311–341 (2017). https://doi.org/10.1007/s40879-017-0128-6

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