This article contains a survey of a new algebro-geometric approach for working with iterated algebraic loop groups associated with iterated Laurent series over arbitrary commutative rings and its applications to the study of the higher-dimensional Contou-Carrre symbol. In addition to the survey, the article also contains new results related to this symbol.

The higher-dimensional Contou-Carrre symbol arises naturally when one considers deformation of a flag of algebraic subvarieties of an algebraic variety. The non-triviality of the problem is due to the fact that, in the case , for the group of invertible elements of the algebra of -iterated Laurent series over a ring, no representation is known in the form of an ind-flat scheme over this ring. Therefore, essentially new algebro-geometric constructions, notions, and methods are required. As an application of the new methods used, a description of continuous homomorphisms between algebras of iterated Laurent series over a ring is given, and an invertibility criterion for such endomorphisms is found. It is shown that the higher- dimensional Contou-Carrre symbol, restricted to algebras over the field of rational numbers, is given by a natural explicit formula, and this symbol extends uniquely to all rings. An explicit formula is also given for the higher-dimensional Contou-Carrre symbol in the case of all rings. The connection with higher-dimensional class field theory is described.

As a new result, it is shown that the higher-dimensional Contou-Carrre symbol has a universal property. Namely, if one fixes a torsion-free ring and considers a flat group scheme over this ring such that any two points of the scheme are contained in an affine open subset, then after restricting to algebras over the fixed ring, all morphisms from the -iterated algebraic loop group of the Milnor -group of degree to the above group scheme factor through the higher-dimensional Contou-Carrre symbol.

Bibliography: 67 titles.

本文包含对一种新的代数几何方法的调查，该方法用于处理与任意交换环上的迭代 Laurent 级数相关的迭代代数环群及其在高维 Contou-Carrre 符号研究中的应用。除了调查，文章还包含与此符号相关的新结果。

当考虑代数簇的代数子簇的标志的变形时，自然会出现更高维的 Contou-Carrre 符号。问题的重要性在于，在 的情况下，对于 的代数的可逆元素群-在一个环上迭代 Laurent 级数，在这个环上没有以 ind-flat 方案的形式表示。因此，本质上需要新的代数几何结构、概念和方法。作为所用新方法的应用，给出了环上迭代Laurent级数代数之间连续同态的描述，并找到了这种自同态的可逆性判据。结果表明，仅限于有理数域上的代数的高维 Contou-Carrre 符号由一个自然显式公式给出，并且该符号唯一地扩展到所有环。在所有环的情况下，还给出了高维 Contou-Carrre 符号的明确公式。描述了与高维类场论的联系。

一个新的结果表明，高维 Contou-Carrre 符号具有普遍性。也就是说，如果固定一个无扭环并考虑该环上的平群方案，使得该方案的任何两个点都包含在一个仿射开子集中，那么在限制到固定环上的代数之后，来自-通过高维 Contou-Carrre 符号将 Milnor - 群的迭代代数环群迭代到上述群方案因子。

参考书目：67 个标题。