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This article is cited in 20 scientific papers (total in 20 papers)
On deformation of sheaves
I. V. Artamkin
Abstract:
Let $X$ be an algebraic variety over an algebraically closed field $k$, $\mathscr F$ a sheaf on $X$, $A$ and $\widetilde A$ commutative Artinian $k$-algebras, $A=\widetilde A/I$, where $I$ is a one-dimensional ideal, $\mathscr E$ a deformation of $\mathscr F$ with base $\operatorname{Spec}A$, and $\operatorname{Ob}(\mathscr E,A,\widetilde A)\in\operatorname{Ext}^2(\mathscr F,\mathscr F)$ the obstruction to the extension of the deformation to $\operatorname{Spec}\widetilde A$. The author constructs natural trace maps $\operatorname{tr}^i\colon\operatorname{Ext}^i(\mathscr F,\mathscr F)\to H^i(\mathscr O_X)$ and proves that if $\operatorname{Pic}X$ is nonsingular then $\operatorname{tr}^2(\operatorname{Ob}(\mathscr E,A,\widetilde A))=0$. As a consequence, a universal deformation of a simple sheaf $\mathscr F$ on $X$ with nonsingular $\operatorname{Pic}X$ exists if the map $\operatorname{tr}^2$ is injective or, in the case $\operatorname{rk}\mathscr F\ne0$, and $\operatorname{char}k\nmid\operatorname{rk}\mathscr F$, $\operatorname{Ext}^2(\mathscr F,\mathscr F)=H^2(\mathscr O_X)$.
Bibliography: 3 titles.
Received: 23.10.1986
Citation:
I. V. Artamkin, “On deformation of sheaves”, Izv. Akad. Nauk SSSR Ser. Mat., 52:3 (1988), 660–665; Math. USSR-Izv., 32:3 (1989), 663–668
Linking options:
https://www.mathnet.ru/eng/im1199https://doi.org/10.1070/IM1989v032n03ABEH000805 https://www.mathnet.ru/eng/im/v52/i3/p660
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Abstract page: | 600 | Russian version PDF: | 260 | English version PDF: | 17 | References: | 44 | First page: | 1 |
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