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Jim Davis: A remark on the fundamental theorem of algebraic K-theory

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Christoph Winges

Jim Davis: A remark on the fundamental theorem of algebraic K-theory

If F is a functor from Rings to Abelian groups, then Bass defined the functor NF from Rings to Abelian groups as the kernel(F(R[x]) \to F(R)), induced by the ring homomorphism given by evaluation at 1. The Nil groups NK_q(R) appear in Bass/Quillen's Fundamental Theorem of algebraic K-theory. Since they vanish for regular rings, they measure regularity of the ring. Cortinas, Haesemeyer, and Weibel proved that if R is a commutative Q-algebra, that the iterated Nil group N^nK_q is an infinite direct sum of groups NK_{q-i}(R) where n-1 \geq i \geq 0.I later conjectured that this is true for any ring. A corollary is the computation of the K-theory of a polynomial ring K_q(R[x_1,\dots,x_n]) as K_q(R) \oplus_i (NK_{q-i}(R))^\infty.I recently proved this conjecture, using results of Davis-Quinn-Reich, the Farrell-Jones Conjecture, and a trick.