Infinite-dimensional vector bundles in algebraic geometry (an introduction)
Infinite-dimensional vector bundles in algebraic geometry (an introduction)
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发表时间：
2003-09
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通讯作者：
V. Drinfeld
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作者：
V. Drinfeld
雷诺（Raynaud）和格鲁森（Gruson）证明了离散（无限维）向量空间族的代数几何概念是合理的。作者引入了泰特空间族的概念（“泰特”意为“局部线性紧”），并声称它是局部的。这个定义考虑到环的K_{-1}不一定是零。然而，我们证明了K_{-1}在Nisnevich凝固后总是消失的。作为Tate空间族的离散对应物，我们引入了几乎射影模的概念。讨论了一类Tate空间的维量变形和行列式变形的概念。上述技术有两种不同的应用。首先，我们澄清了光滑仿射流形Y的形式环的ind-格式的结构，这允许在Y上定义一个无零的微分形式的“精炼”动机积分，它是一个三角化范畴的对象，而不是它的K_0群的元素。其次，我们证明了在穿孔光滑流形上有限维向量束族的上同调研究中，Tate空间的几乎射影模和族是自然出现的。由这个上同调产生的正则中心扩展允许将射影平面上向量束堆栈的“Uhlenbeck紧化”解释为某一类广义向量束的精细模空间。
Raynaud and Gruson showed that there is a reasonable algebro-geometric notion of family of discrete (infinite-dimensional) vector spaces. The author introduces a notion of family of Tate spaces ("Tate"means"locally linearly compact") and claims that it is local. The definition takes in account that the K_{-1} of a ring is not necessarily zero. However, we prove that K_{-1} always vanishes after Nisnevich sheafification. As a discrete counterpart of families of Tate spaces, we introduce the notion of almost projective module. We discuss the notions of dimension torsor and determinant gerbe of a family of Tate spaces. The above technique has two different applications. First, we clarify the structure of the ind-scheme of formal loops of a smooth affine manifold Y. This allows to define a"refined"motivic integral of a differential form on Y with no zeros, which is an object of a triangulated category rather than an element of its K_0 group. Second, we show that almost projective modules and families of Tate spaces appear naturally in the study of the cohomology of a family of finite-dimensional vector bundles on a punctured smooth manifold. The canonical central extension that comes from this cohomology allows to interpret the"Uhlenbeck compactification"of the stack of vector bundles on the projective plane as the fine moduli space of a certain type of generalized vector bundles.