부산대학교 도서관

A.~A. Suslin [in {\it Algebraic $K$-theory}, 53--74, Adv. Soviet Math., 4, Amer. Math. Soc., Providence, RI, 1991; MR1124626] computed the $K$-cohomology groups of the (split) special linear group $\bold{SL}_n$ and the symplectic group $\bold{Sp}_{2n}$ using higher Chern classes in $K$-cohomology. In [$K$-Theory {\bf 31} (2004), no.~4, 307--321; MR2068875], O. Pushin constructed higher Chern classes in motivic cohomology and computed the motivic cohomology of the group $\bold{GL}_n$. In [Amer. J. Math. {\bf 134} (2012), no.~1, 235--257; MR2876145], S. Biglari computed the motives of split reductive groups over $\Bbb{Q}$. In particular, he showed that $$ {M(\bold{SL}_n)}_{\Bbb{Q}}\simeq \bigsqcup_{i=0}^{n-1} \roman{\bf Sym}^i\left (\Bbb{Q}(2)[3]\oplus\Bbb{Q}(3)[5]\oplus \cdots \oplus \Bbb{Q}(n)[2n-1] \right ). $$ In [Compos. Math. {\bf 142} (2006), no.~4, 907--936; MR2249535], A. Huber-Klawitter and B. Kahn determined the motives over $\Bbb{Z}$ of split reductive groups. In [J. K-Theory {\bf 13} (2014), no.~3, 533--561; MR3214391], E. Shinder computed the slice of the slice filtration of the motive $M(\bold{GL}_1(D))$ for a division algebra $D$ of prime degree. \par In this article, the author studies the motive of the group $G=\bold{SL}_1(D)$ where $D$ is a central simple algebra of a prime degree $l$ over a perfect field $F$. Since the group $G$ and its motive are split over a field extension of degree $l$, the torsion part of the motivic cohomology of $G$ is $l$-torsion. The author works over the coefficient ring ${\Bbb{Z} [\frac{1}{(l-1)!} ] }$. Let $S$ be a Severi-Brauer variety of the algebra $D$ and let $N$ be the motive defined by the exact triangle $$ \Bbb{Z}(l-1)[2l-2]\to M(S)\to N\to \Bbb{Z}(l-1)[2l-1] $$ in $\Cal{D}\coloneq {\roman{DM}_{\roman{gm}}^{\roman{eff}}(F)}_{\Bbb{Z} [\frac{1}{(l-1)!}]}$. As $(l-1)!$ is invertible in the coefficient ring, one can define the symmetric powers $\bold{Sym}^i(M)$ and the alternating powers $\bold{Alt}^i(M)$ of any motives $M$ for $0\leq i\leq l-1$. The main theorem in this paper is that in $\Cal{D}$, there exists an isomorphism $$ M(G)\simeq \bigsqcup_{i=0}^{l-1}\bold{Sym}^i(N(2)[3])= \bigsqcup_{i=0}^{l-1}(\bold{Alt}^iN)(2i)[3i]. $$ As applications, the author computes the motivic cohomology $H^{p,q}(G)$ with $2q-p\leq 1$ and certain differentials in the motivic spectral sequence of $G$.