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Bass’ \(NK\) groups and \(cdh\)-fibrant Hochschild homology. (English) Zbl 1238.19002

The \(K\)-theory of a polynomial ring \(R[t]\) contains the \(K\)-theory of \(R\) as a direct summand. The obstruction for the map \(K_n(R[t]) \rightarrow K_n(R)\) given by evaluation at \(t=0\) to being an isomorphism is often related to the singularities of \(\operatorname{Spec} R\).
Following a theorem of C. Traverso [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 24, 585–595 (1970; Zbl 0205.50501)] which asserts that \(\mathrm{Pic}(R[t_1]) = \mathrm{Pic}(R)\) implies \(\mathrm{Pic}(R[t_1,t_2]) = \mathrm{Pic}(R)\), Bass asked if \(K(R[t_1]) = K(R)\) implies \(K(R[t_1,t_2]) = K(R)\). He defined the following groups: \(NK_n(R) = \mathrm{Ker} (K_n(R[t]) \rightarrow K_n(R))\) and \(N^pK_n(R) = N(N^{p-1}K_n(R))\). With these notations, his question becomes: does \(NK_n(R)=0\) imply that \(N^2K_n(R)=0\)? After having recalled the fact that \(NK_n(R)\) has the structure of a module over the ring of big Witt vectors \(W(R)\), the authors show that there is an isomorphism of \(W(R)\)-modules \[ N^2K_n(R) \simeq (NK_n(R) \otimes t\mathbb{Q}[t]) \oplus (NK_{n-1}(R) \otimes \Omega^1_{\mathbb{Q}[t]}). \] Thus, the question of Bass can be reformulated as: does \(NK_n(R)=0\) imply that \(NK_{n-1}(R)=0\)? The proof relies on methods developed by the first three authors [Math. Ann. 344, No. 4, 891–922 (2009; Zbl 1189.19002)] and by J. F. Davis [“Some remarks on Nil groups in algebraic K-theory” (2007; arXiv:0803.1641)] which allow to compute \(NK_n(R)\) in terms of the Hochschild homology of \(R\) and of the \(cdh\)-cohomology of the higher Kähler differentials \(\Omega^p\).
These computations are used to prove that the answer to Bass’ question is positive when \(R\) is essentially of finite type over a field \(F\) of infinite transcendence degree over \(\mathbb{Q}\), but that the answer is negative in general. In [Proc. Am. Math. Soc. 139, No. 4, 1187–1200 (2011; Zbl 1255.19001)], the authors provide a counterexample to Bass’ question, namely \(R = F[x,y,z]/(z^2+y^3+x^{10}+x^7y)\) where \(F\) is any algebraic extension of \(\mathbb{Q}\).

MSC:

19D35 Negative \(K\)-theory, NK and Nil
13D15 Grothendieck groups, \(K\)-theory and commutative rings
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
19D55 \(K\)-theory and homology; cyclic homology and cohomology

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