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Hodge decompositions of Loday symbols in \(K\)-theory and cyclic homology. (English) Zbl 0824.19002

Let \(A\) be a commutative ring with unit supposed to be of finite type over some field \(\ell\) of characteristic zero. Denote by \(HH_ *(A)\), \(HC_ *(A)\) and \(K_ *(A)\) the Hochschild homology, cyclic homology and the \(K\)-theory of \(A\), respectively. All three of them admit direct sum decompositions (Hodge decompositions) according to the action of the Adams operations \(\psi^ k\) and \(\lambda^ k\), e.g. \(HH_ n(A)= \bigoplus^ n_{i= 0} HH^{(i)}_ n(A)\), such that \(\psi^ k\) and \(\lambda^ k\) act as multiplication by \(k^{i+ 1}\) and \((- 1)^ k k^ i\), respectively, on \(HH^{(i)}_ n(A)\). Similarly for \(HC^{(i)}_ n(A)\). On \(K^{(i)}_ *(A)\), \(\psi^ k\) and \(\lambda^ k\) act as multiplication by \(k^ i\) and \((- 1)^ k k^{i- 1}\), respectively. In its greatest generality the above Hodge decompositions are not well understood. In the paper under consideration the Hodge decomposition for special elements in \(K\)-theory (and also in Hochschild and cyclic homology), the so-called Loday symbols, is explored. Hochschild homology, cyclic homology and \(K\)-theory are not unrelated: one has the SBI sequence for Hochschild homology and cyclic homology, the Dennis trace map from \(K\)-theory to Hochschild homology, and the Chern character from \(K\)-theory to cyclic homology. In particular, if \(A\) is a \(\mathbb{Q}\)-algebra the Dennis trace map factors through \(K_ m(A)\otimes \mathbb{Q}\) and one can compare the Hodge decomposition on \(K\)-theory with the Hodge decompositions on Hochschild and cyclic homologies, e.g. for such algebras the Dennis trace map preserves the Hodge decomposition of Loday symbols. If \(A= \bigoplus_{i\geq 0} A_ i\) is a graded \(\ell\)-algebra and \(H\) is any functor from \(\ell\)-algebras to abelian groups, let \(\widetilde H(A)\) denote the kernel of the augmentation map \(H(A)\to H(A_ 0)\). With this notation the Dennis trace map \(D\) factors as \(\widetilde K_ m(A)@> \nu>> HC_{m- 1}(A)@> B>>\widetilde{HH}_ m(A)\). It is shown that \(\nu\) commutes with \(\psi^ k\) and \(\lambda^ k\), and that for \(x\in \widetilde K_ m(A)\) one has \(\psi^ k(D(x))= kD(\psi^ k(x))\). Furthermore, \(D\) preserves the Hodge decomposition of \(\widetilde K_ *(A)\). Several results on the various Loday symbols and their images in the various Hodge components are proved.
Let \(\ell\) be a commutative ring containing \(\mathbb{Q}\). Then a commutative ring \(A= \ell[x_ 1,\dots, x_ m]/{\mathfrak I}\), where \(\mathfrak I\) is an ideal generated by monomials, is called a discrete Hodge algebra over \(\ell\). It is believed (proved?) that in this situation \(\nu\) is injective, and for \(i< n\), \(\nu: \widetilde K^{(i)}_ n(A)\cong \widetilde{HC}^{(i- 1)}_{n- 1}(A)\); for \(i= n\), the cokernel of \(\nu: \widetilde K^{(n)}_ n(A)\to \widetilde{HC}^{(n- 1)}_{n- 1}(A)\) is effectively computable; for \(i> n\), \(\widetilde K^{(i)}_ n(A)= 0\).
In low dimensions one can compute the Hodge decomposition of various Loday symbols in cyclic homology and \(K\)-theory. E.g. for \(A= \ell[x, y, z]/(x, y, z)^ 2\) one shows that the difference of Loday symbols \(\langle\langle x, x, y, y, z\rangle\rangle- \langle\langle y, y, x, x, z\rangle\rangle\) is a non-zero element in \(K^{(2)}_ 5(A)\), thus disproving a conjecture of Beilinson and Soulé which says that \(K^{(i)}_ n(A)= 0\) for \(i< n/2\), or, equivalently, in terms of motivic cohomology, \(H^{(j)}_{\mathcal M}= 0\) for \(j< 0\). One can construct other counterexamples, but in all these cases the rings are singular, so the Beilinson-Soulé conjecture may still hold for regular rings.
For (locally) complete intersection rings one has the result that for a graded locally complete intersection ring \(A\) over a field of characteristic zero every Loday symbol in \(K_ n(A)\) belongs to the subspace \(\bigotimes_{n/2< i\leq n} K^{(i)}_ n(A)\).
For graded algebras which are not complete intersections one may use Hanlon’s generating functions to determine the Hodge indices of Loday symbols. For Loday symbols involving only few (two or three) variables one can give explicit examples of special ones and their corresponding Hodge indices, in particular when they are pure. Among the cases that are studied in detail are \(A= \ell[x, y]/(x^ 2, xy, y^ 2)\) and the Loday symbol \(\langle\langle y, y, x,x,\dots, x\rangle\rangle\in HC_ n(A)\), and \(A= \ell[x, y, z]/(x, y, z)^ 2\) and (alternating sums of) Loday symbols \[ \langle\langle z, y, x, \dots, x\rangle\rangle,\dots, \langle\langle z, x, \dots, x, y\rangle\rangle\in K_ n(A). \]

MSC:

19D55 \(K\)-theory and homology; cyclic homology and cohomology
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