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10.36 (1986)

Open

Is it true that $SL_n(\mathbb{Z}[x_1, \dots, x_r])$, for $n$ sufficiently large, is a group of type $(FP)_m$ (which is defined in the same way as $(FP)_\infty$ was defined in 6.3, but with that weakening that the condition of being finitely generated is not imposed on the terms of the resolution with numbers $> m$)?

Contributor: G. A. Noskov

Progress

An affirmative answer is known for $m = 0, n \geqslant 3$ (A. A. Suslin, Math. USSR Izvestiya, 11 (1977), 221–238) and for $m = 1, n \geqslant 5$ (M. S. Tulenbayev, Math. USSR Sbornik, 45 (1983), 139–154; U. Rehmann, C. Soulé, in: Algebraic K-theory, Proc. Conf., Northwestern Univ., Evanston, Ill., 1976, (Lect. Notes Math., 551), Springer, Berlin, 1976, 164–169).

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