Algebraic bundles over Pn and problems of linear algebra
Algebraic bundles over Pn and problems of linear algebra
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DOI：
10.1007/bf01681435
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发表时间：
1978-07
影响因子：
0.4
通讯作者：
I. N. Bernshtein;I. Gel'fand;S. I. Gel'fand
中科院分区：
文献类型：
--
作者：
I. N. Bernshtein;I. Gel'fand;S. I. Gel'fand
IN Bernshtein，IM Gel'fand和SI Gel'fand UDC 513.015。7 i.射影空间pn上的代数向量丛的描述引起了代数几何领域许多专家的注意（见[1-3]）。最近，随着Atiyah和Ward [4]以及Belavin和Zakharov [5]的杰出论文的发表，人们对这个问题的兴趣进一步增加，在这些论文中，描述了CP 3上的丛与四维球面上规范场的联系。在本说明中，它是如何在pn上的丛的分类减少到一个线性代数问题，即，关于（n+ i）个变量的外（Grassman）代数的有限维分次表示的分类。在Barth [2]和Drinfel'd和Manin [3]中有这种约化的特殊情况。独立获得的Beilinson [6]与我们的结果接近。我们要向于表示深切的谢意. I.马宁关于[3]的报告引起了我们对这些问题的兴趣。设E是代数闭域k上的（n+1）维线性空间，A是E上的外代数.我们在A上引入一个分次模，通过设置deg~=--i，对于~ o。A-模是指一个分次生成的分次A-模，记法F=~ Fj。设~是自由A-模类，我们称A-模为V，V ′ ~-
IN Bernshtein, IM Gel'fand, and SI Gel'fand UDC 513.015. 7 i. The description of the algebraic vector bundles over projective space pn has attracted the attention of many specialists in algebraic geometry (see [1-3]). Recently, interest in this problem has increased even more in connection with the remarkable papers of Atiyah and Ward [4] and Belavin and Zakharov [5], in which the connection of bundles over CP 3 with gauge fields on the four-dimensional sphere is described. In the present note it is shown how the classification of bundles over pn reduces to a problem of linear algebra, viz., to the classification of finite-dimensional graded representations of the exterior (Grassman) algebra on (n+ i) variables. There are special cases of such a reduction in Barth [2] and Drinfel'd and Manin [3]. Independently obtained, Beilinson [6] is close to our result. We want to express profound gra> itude to Yu. I. Manin, whose report on [3] stimulated our interest in these questions.2. Let E be an (n+ l)-dimensional linear space over an algebraically closed field k, A be the exterior algebra on the space E. We introduce a grading on A, by setting deg~=--i for~~ o By a A-module we shall mean a finitely generated graded A-module; notation F=~ Fj. Let~ be the class of free A-modules; we shall call A-modules V, V'~-