On the homotopy invariant presheaves with Witt-transfers (Q2925755)

Let \(k\) be a field of characteristic not equal to 2, and let \(Sm_k\) be the category of smooth affine varieties over \(k\). Let \(Wor_k\) be the additive category with the same objects as \(Sm_k\). Given two objects \(X\) and \(Y\), the set of morphisms \(Wor_k(X,Y)\) between \(X\) and \(Y\) in \(Wor_k\) is the Witt group of the exact category of \(k[X \times Y]\)-modules that are finitely generated and projective over \(k[X]\). The composition of morphisms from \(X\) to \(Y\) and from \(Y\) to \(Z\) is defined as the tensor product over \(k[Y]\) of the corresponding quadratic spaces. An abelian presheaf with Witt-transfers is an arbitrary presheaf of abelian groups on the category \(Wor_k\).NEWLINENEWLINEIn [Ann. Math. Stud. 143, 188--238 (2000; Zbl 1019.14009)], \textit{V. Voevodsky} constructed the triangulated category of motives over a field. In the paper under review, the author states a theorem that ``The Nisnevich sheafification \({\mathcal{F}}_{Nis}\) of a homotopy invariant presheaf \(\mathcal{F}\) with Witt-transfers is a homotopy invariant''. The author claims that this is the first step of constructing the triangulated category of Witt-motives. The proof of this theorem is based on some lemmas in recent Russian preprints of the author [``Preservation of the homotopy invariance of presheaves with Witt-transfers under sheafification in the Zariski topology'', Preprint 07/2014, St. Petersburg Branch of the Steklov Mathematical Institute, St. Petersburg, 10 p. (2014); ``Preservation of the homotopy invariance of presheaves with Witt-transfers under sheafification in the Nisnevich topology'', Preprint 08/2014, St. Petersburg Branch of the Steklov Mathematical Institute, St. Petersburg, 14 p. (2014)].