Let \(k\) be a commutative ring. For a \(k\)-algebra \(A\) one has Hochschild homology \(HH_* (A)\) and cyclic homology \(HC_* (A)\) which are related by Connes’ SBI-sequence. For a scheme \(X\) over \(k\) one may sheafify the Hochschild complex of \(A = {\mathcal O}_X (U)\), \(U \subset X\) open, to obtain a complex of sheaves \({\mathcal C}^h_*\). One defines the Hochschild homology of \(X\), \({\mathcal H} H_n (X) : = {\mathcal H}^{-n} (X, {\mathcal C}^h_*)\). S. C. Geller and the author [cf. “Étale descent for Hochschild and cyclic homology”, Comment. Math. Helv. 66, No. 3, 368-388 (1991; Zbl 0741.19007)] showed that \({\mathcal H} H_* (X)\) has the properties of a cyclic homology theory, i.e. (i) functoriality, (ii) Mayer-Vietoris, and (iii) the property that for an affine scheme \(X = \text{Spec} (A)\), one has \({\mathcal H} H_n (X) \cong {\mathcal H} H_n (A)\). In the underlying paper, the same result is established for cyclic homology of schemes. One sheafifies Connes’ double complex \((B_{**}, b, B)\) to obtain a double complex of sheaves \({\mathcal B}_{**}\). The cyclic homology of \(X\) is defined by \({\mathcal H} C_n (X) : = {\mathcal H}^{-n} (X, \text{Tot} {\mathcal B}_{**})\). The definition goes back to J. Loday, but the key property (iii) was still lacking. Here it is proved by a truncation procedure on the columns of the double complex \({\mathcal B}_{**}\). The hypercohomology \({\mathcal H}C_*\) becomes the limit of the hypercohomology of the truncated double complexes \(\tau_{q < r} {\mathcal B}_{**}\). The subtleties of hypercohomology (of not necessarily bounded below complexes) is explained in an appendix.