An elementary description of K 1 (R) without elementary matrices
aa r X i v : . [ m a t h . K T ] O c t AN ELEMENTARY DESCRIPTION OF K ( R ) WITHOUT ELEMENTARYMATRICES
THOMAS HÜTTEMANN AND ZUHONG ZHANG
Abstract.
Let R be a ring with unit. Passing to the colimit with respect to the standardinclusions GL( n, R ) ✲ GL( n + 1 , R ) (which add a unit vector as new last row and col-umn) yields, by definition, the stable linear group GL( R ) ; the same result is obtained, up toisomorphism, when using the “opposite” inclusions (which add a unit vector as new first rowand column). In this note it is shown that passing to the colimit along both these families ofinclusions simultaneously recovers the algebraic K -group K ( R ) = GL( R ) /E ( R ) of R , givingan elementary description that does not involve elementary matrices explicitly. Let R be an associative ring with unit element , and let GL( n, R ) denote the group ofinvertible n × n -matrices with entries in R . The usual stabilisation maps i nn +1 : GL( n, R ) ✲ GL( n + 1 , R ) , A (cid:18) A
00 1 (cid:19) are used to define the stable general linear group
GL( R ) = S n ≥ GL( n, R ) , or, phrased incategorical language, GL( R ) = colim (cid:16) GL(3 , R ) i ✲ GL(4 , R ) i ✲ GL(5 , R ) i ✲ . . . (cid:17) . (1)The canonical group homomorphisms ι n : GL( n, R ) ✲ GL( R ) are injective and satisfy therelation ι n +1 ◦ i nn +1 = ι n . (2)There are other “block-diagonal” embedding i nj : GL( n, R ) ✲ GL( n +1 , R ) , for ≤ j ≤ n +1 ,characterised by saying that the j th row and j th column of i nj ( A ) are j th unit vectors, and thatdeleting these from i nj ( A ) recovers the matrix A . We will determine the result of stabilising overfirst and last embeddings simultaneously, that is, we identify the categorical colimit M of thefollowing group-valued infinite diagram: GL(3 , R ) i ✲ i ✲ GL(4 , R ) i ✲ i ✲ GL(5 , R ) i ✲ i ✲ · · · i n − ✲ i n − n ✲ GL( n, R ) i n ✲ i nn +1 ✲ GL( n + 1 , R ) i n +11 ✲ i n +1 n +2 ✲ · · · . (3)By the general theory of colimits, the group M comes equipped with canonical group homomor-phisms α n : GL( n, R ) ✲ M satisfying the relations α n +1 ◦ i nj = α n ( j = 1 , n + 1 ) . (4) Theorem.
The group M is canonically isomorphic to K ( R ) .Proof. First we observe that in M we have the commutation relation α n ( X ) α n ( Y ) = α n ( Y ) α n ( X ) for all X, Y ∈ GL( n, R ) . (5)Indeed, by (4) we can re-write α n ( X ) = α n (cid:0) i n − n i n − n − . . . i nn +1 ( X ) (cid:1) and α n ( Y ) = α n (cid:0) i n − i n − . . . i n ( Y ) (cid:1) , and the arguments of α n are block-diagonal matrices of the form (cid:18) X I n (cid:19) and (cid:18) I n Y (cid:19) which commute in GL(2 n, R ) ; hence their images under α n must commute as well. Date : October 11, 2019.2010
Mathematics Subject Classification.
Primary 19B99; secondary 16E20.
Key words and phrases. K -theory; invertible matrix; elementary matrix.
Let E ( n, R ) denote the subgroup of GL( n, R ) generated by the elementary matrices [Bas68,§V.1]. Since E ( n, R ) = (cid:2) E ( n, R ) , E ( n, R ) (cid:3) for all n ≥ , cf. [Bas68, Corollary V.1.5], thecommutation relation (5) implies E ( n, R ) ⊆ ker( α n ) . (6)Since the diagram (1) defining GL( R ) is contained in the diagram (3) defining M there is acanonical group homomorphism α : GL( R ) ✲ M described completely by α ◦ ι n = α n , thatis, α | GL( n,R ) = α n .Let E ( R ) = S n ≥ E ( n, R ) ⊆ GL( R ) be the stabilisation via the embeddings i nn +1 . In viewof (6) above we have the inclusion E ( R ) = [ n ≥ E ( n, R ) ⊆ ker α . (7)The group E ( R ) is normal in GL( R ) , cf. [Bas68, Theorem V.2.1], and K ( R ) = GL( R ) /E ( R ) is an abel ian group [Bas68, p. 229]. We write π : GL( R ) ✲ K ( R ) for the canonical projection.Let π n = π ◦ ι n denote the restriction of π to GL( n, R ) , and write [ X ] = π n ( X ) for the class of X ∈ GL( n, R ) in K ( R ) . By (7) we obtain a factorisation λ : K ( R ) ✲ M of α with λ ◦ π = α .Explicitly, λ is described by the formula λ : [ X ] = π n ( X ) α ◦ ι n ( X ) = α n ( X ) , for X ∈ GL( n, R ) . (8)We observe the relation π n +1 ◦ i nj = π n . Indeed, for X ∈ GL( n, R ) the matrices i nj ( X ) and i nn +1 ( X ) are related by the expression i nj ( X ) = P − i nn +1 ( X ) P for a permutation matrix P ∈ GL( n + 1 , R ) . It follows that said two matrices have the sameimage under π n +1 in the abel ian group K ( R ) whence, using (2), π n +1 ◦ i nj ( X ) = π n +1 ◦ i nn +1 ( X ) = π ◦ ι n +1 ◦ i nn +1 ( X ) = π ◦ ι n ( X ) = π n ( X ) . The various maps π n thus form a “cone” on the diagram (3) and induce a map ̺ : M ✲ K ( R ) such that π n = ̺ ◦ α n . (9)We verify the equality λ ◦ ̺ = id M . By the universal property of colimits, it is enough to showthat λ ◦ ̺ ◦ α n = α n for all n . But for X ∈ GL( n, R ) we calculate λ ◦ ̺ ◦ α n ( X ) = (9) λ ◦ π n ( X ) = (8) α n ( X ) , using relation (9) and the explicit description (8) of λ above.Using the same relations again, in opposite order, we finally verify that ̺ ◦ λ = id K ( R ) . Let X ∈ GL( n, R ) represent the element [ X ] ∈ K ( R ) as before; then ̺ ◦ λ (cid:0) [ X ] (cid:1) = (8) ̺ ◦ α n ( X ) = (9) π n ( X ) = [ X ] . (cid:3) With minor changes the argument also shows that the the colimit of the group-valued diagram
GL(3 , R ) i ✲✲✲ i ✲ GL(4 , R ) i ✲✲✲✲ i ✲ GL(5 , R ) i ✲✲✲✲✲ i ✲ · · · i n − ✲ ... i n − n ✲ GL( n, R ) i n ✲ ... i nn +1 ✲ GL( n + 1 , R ) i n +11 ✲ ... i n +1 n +2 ✲ · · · is canonically isomorphic to K ( R ) . References [Bas68] Hyman Bass.
Algebraic K -theory . W. A. Benjamin, Inc., New York-Amsterdam, 1968. Thomas Hüttemann, Queen’s University Belfast, School of Mathematics and Physics,Mathematical Sciences Research Centre, Belfast BT7 1NN, UK
E-mail address : [email protected] URL : https://t-huettemann.github.io/ Zhang Zuhong, School of Mathematics, Beijing Institute Of Technology,5 South Zhongguancun Street, Haidian District, 100081 Beijing, P. R. China
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