Motivic decomposition of certain special linear groups (Q2826771)

Let \(F\) be a perfect field, \(l\) be a prime number and let \(D\) be a central simple algebra over \(F\) of degree \(l\). The algebraic group \(\mathrm{SL}_1(D)\) of reduced norm 1 elements in \(D\) can be identified with \(\mathrm{SL}_l\) if \(D\) is a split algebra. The main result of the article is the description of the motive of \(\mathrm{SL}_1(D)\) in the Voevodsky's triangulated category of motives \(DM:=DM^{\mathrm{eff}}_{\mathrm{gm}}(F;\mathbb{Z}[\frac{1}{(l-1)!}])\) in terms of the motive of \(S\).NEWLINENEWLINEFrom the introduction, abridged: ``As a warm-up, consider the simplest case \(l=2\). The variety of \(G=\mathrm{SL}_1(D)\) is an open subscheme of a 3-dimensional projective isotropic quadric \(X\). The surface \(Y=X\setminus G\), given by \(Nrd=0\), is isomorphic to \(S\times S\). Computing the motives of \(X\) and \(Y\) (as in [\textit{E.~Shinder}, J. K-Theory 13, No. 3, 553--561 (2014; Zbl 1325.14037)]), we obtain an isomorphism \(M(G)\cong \mathbb{Z}\oplus N(2)[3]\), where the motive \(N\) is defined as the cone \(\mathbb{Z}(1)[2]\rightarrow M(S)\) with the first morphism dual to the canonical one \(M(S)\rightarrow \mathbb{Z}\).''NEWLINENEWLINEThe main result is a generalisation of this statement. Namely, for \(D\) as in the beginning, one defines \(N\) as the cone of the morphism \(\mathbb{Z}(l-1)[2l-2]\rightarrow M(S)\), and the following theorem is proved.NEWLINENEWLINETheorem. There is an isomorphism in \(DM\) between \(M(\mathrm{SL}_1(D))\) and \(\bigoplus_{i=0}^{l-1} \mathrm{Sym}^i(N(2)[3])\).NEWLINENEWLINEIn the split case, i.e. when \(\mathrm{SL}_1(D)=\mathrm{SL}_l\), \(S=\mathbb{P}^{l-1}\), the motive of \(G\) is a sum of pure Tate motives and was computed in [\textit{S.~Biglari}, Am. J. Math. 134, No. 1, 235--257 (2012; Zbl 1247.14050)]. In this case the motive \(N\) is isomorphic to a sum of Tate motives and the main theorem easily follows. Nevertheless, the author constructs a particular isomorphism between the motives in question using the higher Chern classes from algebraic K-theory to motivic cohomology constructed in [\textit{O.~Pushin}, K-Theory 31, No. 4, 307--321 (2004; Zbl 1073.14029)]. To prove the main theorem the author lifts this isomorphism (i.e. particular Chern classes) from the algebraic closure to the base field.NEWLINENEWLINEAn important role in the proof is played by a theory of compactifications of algebraic groups developed in [\textit{N.~Karpenko, A.~Merkurjev}, J. Reine Angew. Math., \url{doi:10.1515/crelle-2016-0015})]. To lift higher Chern classes authors consider the standard triangle of \(\mathrm{SL}_1(D)\) as an open subscheme in its compactifications and its various properties in \(DM\).NEWLINENEWLINEApplications of the main result are the calculation of the motivic cohomology groups \(H^{2i+1,i+1}\) of \(\mathrm{SL}_1(D)\) for all \(i\), and construction of certain rational elements in many other motivic cohomology groups of \(\mathrm{SL}_1(D)\).