Low Dimensional Homology of Linear Groups over Hensel Local Rings
نویسنده
چکیده
We prove that if R is a Hensel local ring with infinite residue field k, the natural map Hi(GLn(R),Z/p) → Hi(GLn(k), Z/p) is an isomorphism for i ≤ 3, p 6= char k. This implies rigidity for Hi(GLn), i ≤ 3, which in turn implies the Friedlander–Milnor conjecture in positive characteristic in degrees ≤ 3. A fundamental question in the homology of linear groups is that of rigidity: given a smooth affine curve X over an algebraically closed field k and closed points x, y on X, do the corresponding specialization homomorphisms sx, sy : H∗(G(k[X]),Z/p) −→ H∗(G(k),Z/p) coincide? Here, G is a reductive algebraic group and p is a prime not equal to the characteristic of k. The answer is yes when X is the affine line and G = SLn, GLn, PGLn since the inclusion G(k) → G(k[t]) induces an isomorphism H∗(G(k),Z) −→ H∗(G(k[t]),Z) (see [6]) and the map is split by evaluation at any x ∈ A1. Rigidity in algebraic K-theory has spectacular consequences, including the calculation of the K-theory of algebraically closed fields and the solution of the Friedlander–Milnor conjecture for the stable general linear group GL. Similarly, a proof of rigidity forG(k[X]) would imply the Friedlander–Milnor conjecture for G (see Section 5). Rigidity would follow if one could prove the following stronger result. Let X be a smooth curve over an algebraically closed field k and let x be a closed point on X. Denote by Oh x the henselization of Ox. Conjecture. The inclusion G(k) → G(Oh x) induces an isomorphism H∗(G(k),Z/p) −→ H∗(G(O h x),Z/p)
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