Let \(k\) be a field. In [G. Cortiñas, J. Reine. Angew. Math. 503, 129-160 (1998; Zbl 0908.19004)] the author has introduced the “noncommutative infinitesimal cohomology” \(H^*_{\text{inf}}(A,-)\) of a nonunital \(k\)-algebra \(A\) as the sheaf cohomology of the site \(\inf(A/\text{alg})\) of surjective \(k\)-algebra epimorphisms \(B\twoheadrightarrow A\) with nilpotent kernel endowed with the indiscrete topology. In the paper under review it is shown that \(H^*_{\text{inf}}(A,{\mathcal O}/[{\mathcal O},{\mathcal O}])\) is isomorphic to the periodic homology \(HP_*(A)\), where \({\mathcal O}/[{\mathcal O},{\mathcal O}]\) is the sheaf sending \(B\twoheadrightarrow A\) to \(B/[B,B]\). This is parallel to Grothendieck’s original commutative result which states that in characteristic zero the infinitesimal cohomology of the structure sheaf is deRham cohomology.
The paper ends with a characterization when \(\text{char}(k)=0\) of the fiber of the Jones-Goodwillie character from K-theory to negative cyclic homology in terms of infinitesimal hypercohomology of K-theory. This follows from the results of [op. cit.] together with a result showing that a corresponding hypercohomology of negative cyclic homology vanishes.
In addition to these two results, the paper gives an isomorphism \(H^n_{\text{inf}}(A,K_1)\cong HP_n(A)\) for \(n\geq 2\) for \(\text{char}(k)=0\) and a sheaf-theoretic interpretation of the Chern character \(K_0(A)\to HP_0(A)\).