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Steinberg group (K-theory)

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In algebraic K-theory, a field of mathematics, the Steinberg group of a ring is the universal central extension of the commutator subgroup of the stable general linear group of .

It is named after Robert Steinberg, and it is connected with lower -groups, notably and .

Definition

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Abstractly, given a ring , the Steinberg group is the universal central extension of the commutator subgroup of the stable general linear group (the commutator subgroup is perfect and so has a universal central extension).

Presentation using generators and relations

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A concrete presentation using generators and relations is as follows. Elementary matrices — i.e. matrices of the form , where is the identity matrix, is the matrix with in the -entry and zeros elsewhere, and — satisfy the following relations, called the Steinberg relations:

The unstable Steinberg group of order over , denoted by , is defined by the generators , where and , these generators being subject to the Steinberg relations. The stable Steinberg group, denoted by , is the direct limit of the system . It can also be thought of as the Steinberg group of infinite order.

Mapping yields a group homomorphism . As the elementary matrices generate the commutator subgroup, this mapping is surjective onto the commutator subgroup.

Interpretation as a fundamental group

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The Steinberg group is the fundamental group of the Volodin space, which is the union of classifying spaces of the unipotent subgroups of .

Relation to K-theory

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K1

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is the cokernel of the map , as is the abelianization of and the mapping is surjective onto the commutator subgroup.

K2

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is the center of the Steinberg group. This was Milnor's definition, and it also follows from more general definitions of higher -groups.

It is also the kernel of the mapping . Indeed, there is an exact sequence

Equivalently, it is the Schur multiplier of the group of elementary matrices, so it is also a homology group: .

K3

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Gersten (1973) showed that .

References

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  • Gersten, S. M. (1973), " of a Ring is of the Steinberg Group", Proceedings of the American Mathematical Society, 37 (2), American Mathematical Society: 366–368, doi:10.2307/2039440, JSTOR 2039440
  • Milnor, John Willard (1971), Introduction to Algebraic -theory, Annals of Mathematics Studies, vol. 72, Princeton University Press, MR 0349811
  • Steinberg, Robert (1968), Lectures on Chevalley Groups, Yale University, New Haven, Conn., MR 0466335, archived from the original on 2012-09-10