Purity of reciprocity sheaves (Q2309103)

Let \(\mathbf{PST}\) be the category of presheaves with transfers over a perfect field, defined by \textit{V. Voevodsky} [Ann. Math. Stud. 143, 87--137 (2000; Zbl 1019.14010)], and let \(\mathbf{HI}\) be the full subcategory of homotopy-invatiant presheaves with transfers, i.e. \(F\) such that for all smooth schemes \(X\), \(F(X)\to F(X\times \mathbb{A}^1)\) induced by the projection is an isomorphism. Let \(\mathbf{HI}_{\mathrm{Nis}}=\mathbf{HI}\cap\mathbf{NST}\). These categories and its deep properties play a fundamental role in the theory of triangualted category of mixed motives [\textit{V. Voevodsky}, Ann. Math. Stud. 143, 188--238 (2000; Zbl 1019.14009)]. In this paper, the author generalizes some of these properties to the bigger categories of reciprocity preheaves \(\mathbf{RSC}\) and reciprocity sheaves \(\mathbf{RSC}_{\mathrm{Nis}}\) constructed in [\textit{B. Kahn} et al., Compos. Math. 152, No. 9, 1851--1898 (2016; Zbl 1419.19001]. These categories contain many interesting non-homotopy invariant presheaves with transfers, like \(\mathbf{G}_a\), the sheaves of Kähler differentials \(\Omega^i_{/\mathbb{Z}}\) and \(\Omega^i_{/k}\) and the de Rham-Witt sheaves \(W_n\Omega^i\). Namely, the author proves the following two deep results, which for \(\mathbf{HI}\) and \(\mathbf{HI}_{\mathrm{Nis}}\) were proved in [\textit{V. Voevodsky}, Ann. Math. Stud. 143, 87--137 (2000; Zbl 1019.14010)]: \begin{itemize} \item[1.] Let \(a_{\mathrm{Nis}}:\mathbf{PST}\to \mathbf{NST}\) be the Nisnevich sheafification, then for \(F\in \mathbf{RSC}\) \(a_{\mathrm{Nis}}F\in \mathbf{RSC}_{\mathrm{Nis}}\). In particular \(\mathbf{RSC}_{\mathrm{Nis}}\) is an abelian subcategory of \(\mathbf{NST}\). \item[2.] Let \(F\in \mathbf{RSC}_{\mathrm{Nis}}\), \(\mathcal{X}\) the henselianization of a smooth scheme \(X\), with generic point \(\xi\), then the Cousin complex\[ F(X)\to F(\xi)\to \bigoplus_{x\in X^{(1)}}H^1_{x}(X_{\mathrm{Nis}},F)\to\ldots\to \bigoplus_{x\in X^{(k)}}H^k_{x}(X_{\mathrm{Nis}},F) \] \end{itemize} In order to attack these statements, the author uses the techniques of modulus presheaves with transfers, i.e. presheaves on a suitable category of correspondences for pairs \((X,D)\), where \(X\) is a scheme, \(D\subseteq X\) is an effective Cartier divisor with \(X-|D|\) smooth. In this category, the role of \(\mathbb{A}^1\) is taken by the modulus \(\overline{\Box}:=(\mathbb{P}^1,\infty)\). In order to avoid some pathologies, the author introduces the notion of semi-purity for a modulus presheaves with transfers \(F\): \(F\) is semi-pure if \(F(X,D)\to F(X-|D|,\emptyset)\) is injective. It is expected that the deep results contained in this paper play a crucial role in a future development of a theory of motives that encodes non-homotopy invariant phenomena.