\(R\)-equivalence in spinor groups (Q2719029)

Let \(G\) be an algebraic group defined over a field \(F\) and let \(RG(F)\) be the normal subgroup of \(G(F)\) of \(R\)-trivial \(F\)-points \(g\), i.e. points for which there is a rational morphism \(f\colon\mathbb{A}_F^1\to G\) with \(f(0)=1\) and \(f(1)=g\). If the factor group \(G(E)/RG(E)\) is trivial for all field extensions \(E/F\), then \(G\) is said to be \(R\)-trivial. This property is useful in the study of the rationality problem for algebraic groups as \(R\)-triviality follows from \(G\) being stably rational over \(F\).NEWLINENEWLINENEWLINEIn the well written introduction, the authors recall the known results on the rationality and \(R\)-triviality problems for simply connected algebraic groups of classical types \(A_n\), \(C_n\). One of the main purposes of the present paper is to compute the group of \(R\)-equivalence classes in the remaining classical cases \(B_n\), \(D_n\), where the corresponding simply connected groups arise as twisted forms of the spinor groups of non-degenerate quadratic forms.NEWLINENEWLINENEWLINEOne of the main results concerns the construction of an isomorphism \(\alpha_F\colon G(F)/RH(F)\to A_0(X,K_1)\) where \(G\) is a reductive group, \(H=\text{ker}(\rho)\) for a character \(\rho\colon G\to\mathbb{G}_m\), \(X\) is a smooth projective variety subject to certain conditions, and \(A_0(X,K_1)\) is the kernel of the residue homomorphism \(\coprod_{x\in X_{(1)}}K_2F(x)\to\coprod_{x\in X_{(0)}}K_1F(x)\), where \(X_{(p)}\) denotes the points of dimension \(p\) in \(X\) and \(F(x)\) the residue field at \(x\in X_{(\cdot)}\). This is then applied to three examples.NEWLINENEWLINENEWLINEIn the first one, \(G=\text{GL}_1(A)\) for \(A\) a central simple algebra over \(F\), \(H=\text{SL}_1(A)\) (the kernel of the character induced by the reduced norm homomorphism), \(X\) the Severi-Brauer variety of \(A\). Here, the authors recover the isomorphism \(K_1(A)=\text{GL}_1(A)/R\text{SL}_1(A)\cong A_0(X,K_1)\) originally obtained by the second author and Suslin.NEWLINENEWLINENEWLINEIn the second example, \((V,q)\) is a non-degenerate quadratic space over \(F\) of dimension \(n\geq 2\), \(G=\Gamma^+(V,q)\) is the special Clifford group, \(H=\text{Spin}(V,q)\) is the spinor group (the kernel of the spinor norm homomorphism), \(X\) is the projective quadric associated to \(q\). In this example, the authors give a new definition of Rost's homomorphism \(\Gamma^+(V,q)\to A_0(X,K_1)\) and show that it gives rise to isomorphisms \(\Gamma^+(V,q)/R\text{Spin}(V,q)\cong A_0(X,K_1)\) and \(\text{Spin}(V,q)/R\cong\overline A_0(X,K_1)\), where \(\overline A_0(X,K_1)\) is the kernel of the norm homomorphism \(A_0(X,K_1)\to F^\times\). The rationality of \(\text{Spin}(V,q)\) thus implies the triviality of \(\overline A_0(X,K_1)\). Rationality is known if \(q\) contains a subform of codimension \(\leq 2\) which is a Pfister neighbor. For Pfister neighbors, the triviality of \(\overline A_0(X,K_1)\) (originally due to Rost) was used by Voevodsky in his proof of the Milnor conjecture.NEWLINENEWLINENEWLINEThe last example concerns the case where \(A\) is a central simple algebra and \((\sigma,f)\) is a so-called quadratic pair (an involution \(\sigma\) of the first kind on \(A\) and a certain linear map \(f\colon\text{Sym}(A,\sigma)\to F\)). Here, \(G=\Gamma(A,\sigma,f)\) is the Clifford group, \(H=\text{Spin}(A,\sigma,f)\) is the spinor group, and \(X\) is the so-called involution variety with respect to the quadratic pair \((\sigma,f)\). It is shown that if the index of \(A\) is \(\leq 2\) and \(A\) is of even dimension, then there are isomorphisms \(\Gamma(A,\sigma,f)/R\text{Spin}(A,\sigma,f)\cong A_0(X,K_1)\) and \(\text{Spin}(A,\sigma,f)/R\cong\overline A_0(X,K_1)\). This covers the case of all simply connected groups of type \(D_m\) with odd \(m\).