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Izvestiya: Mathematics, 2001, Volume 65, Issue 5, Pages 941–975
DOI: https://doi.org/10.1070/IM2001v065n05ABEH000358
(Mi im358)
 

This article is cited in 24 scientific papers (total in 24 papers)

Krichever correspondence for algebraic varieties

D. V. Osipov

Steklov Mathematical Institute, Russian Academy of Sciences
References:
Abstract: We construct new acyclic resolutions of quasicoherent sheaves. These resolutions are connected with multidimensional local fields. The resolutions obtained are applied to construct a generalization of the Krichever map to algebraic varieties of any dimension.
This map canonically produces two $k$-subspaces $B\subset k((z_1))\dots((z_n))$ and $W\subset k((z_1))\dots((z_n))^{\oplus r}$ from the following data: an arbitrary algebraic $n$-dimensional Cohen–Macaulay projective integral scheme $X$ over a field $k$, a flag of closed integral subschemes $X=Y_0 \supset Y_1 \supset\dots\supset Y_n$ such that $Y_i$ is an ample Cartier divisor on $Y_{i-1}$ and $Y_n$ is a smooth point on all $Y_i$, formal local parameters of this flag at the point $Y_n$, a rank $r$ vector bundle $\mathscr F$ on $X$, and a trivialization of $\mathscr F$ in the formal neighbourhood of the point $Y_n$ where the $n$-dimensional local field $B\subset k((z_1))\dots((z_n))$ is associated with the flag $Y_0\supset Y_1\supset\dots\supset Y_n$. In addition, the map constructed is injective, that is, one can uniquely reconstruct all the original geometric data. Moreover, given the subspace $B$, we can explicitly write down a complex which calculates the cohomology of the sheaf $\mathscr O_X$ on $X$ and, given the subspace $W$, we can explicitly write down a complex which calculates the cohomology of $\mathscr F$ on $X$.
Received: 21.03.2000
Bibliographic databases:
Document Type: Article
MSC: 14F05
Language: English
Original paper language: Russian
Citation: D. V. Osipov, “Krichever correspondence for algebraic varieties”, Izv. Math., 65:5 (2001), 941–975
Citation in format AMSBIB
\Bibitem{Osi01}
\by D.~V.~Osipov
\paper Krichever correspondence for algebraic varieties
\jour Izv. Math.
\yr 2001
\vol 65
\issue 5
\pages 941--975
\mathnet{http://mi.mathnet.ru//eng/im358}
\crossref{https://doi.org/10.1070/IM2001v065n05ABEH000358}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1874355}
\zmath{https://zbmath.org/?q=an:1068.14053}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0042220544}
Linking options:
  • https://www.mathnet.ru/eng/im358
  • https://doi.org/10.1070/IM2001v065n05ABEH000358
  • https://www.mathnet.ru/eng/im/v65/i5/p91
  • This publication is cited in the following 24 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Abstract page:670
    Russian version PDF:221
    English version PDF:25
    References:93
    First page:1
     
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