Tensor structures in the theory of modulus presheaves with transfers
Consider the diagram \[ \begin{tikzcd} \textbf{Cor} \arrow[r, equal] \arrow[d, "\mathbb{Z}_{tr}"] & \textbf{Cor} \arrow[d, "\mathbb{Z}_{tr}"] & \textbf{MCor} \arrow[l, "\omega"] \arrow[d, "\mathbb{Z}_{tr}"] \\ \textbf{PST} \arrow[d, shift left= 1ex, "h_0^{\mathbb{A}^1}"] \arrow[r, equal] & \textbf{PST} \arrow[r, shift left= 1ex, "\omega^*"] \arrow[d, shift left= 1ex, "\rho"] & \textbf{MPST} \arrow[l, shift left= 1ex, "\omega_!"] \arrow[d, shift left= 1ex, "h_0^{\overline{\boxempty}}"] \\ \textbf{HI} \arrow[u, hookrightarrow, shift left= 1ex, "\iota^{\mathbb{A}^1}"] \arrow[r, hookrightarrow, shift left= 1ex] & \textbf{RSC} \arrow[u, hookrightarrow, shift left= 1ex, "\iota^{\natural}"] \arrow[l, shift left= 1ex, "h_0^{rec}"] \arrow[r, shift left= 1ex, "\omega^{\textbf{CI}}"] & \textbf{CI} \arrow[u, hookrightarrow, shift left= 1ex, "\iota^{\overline{\boxempty}}"] \arrow[l, shift left= 1ex, "\omega_{\textbf{CI}}=\omega_!"] \end{tikzcd} \] where: \begin{itemize} \item \(\textbf{Cor}\) is Voedosky's category of finite correspondences whose objects are smooth varieties over a perfect base field \(k\), \item \(\textbf{PST}\) the category of presheaves over \(\textbf{Cor}\), \item \(\textbf{HI}\) is the full subcategory of \(\textbf{PST}\) consisting of \(\mathbb{A}^1\)-invariant presheaves, \item \(\textbf{MCor}\) is the category whose objects are proper modulus pairs \(M=(\overline{M}, M^{\infty})\), \item \(\textbf{MPST}\) the category of presheaves over \(\textbf{MCor}\), \item \(\textbf{CI}\) the full subcategory of \(\textbf{MPST}\) consisting of \(\overline{\boxempty}\)-invariant presheaves (with \(\overline{\boxempty} = ( \mathbb{P}^1, (\infty))\)), \item \(\textbf{RSC}\) is the full subcategory of \(\textbf{PST}\) consisting of presheaves having SC-reciprocity, \item and where the functors are the obvious ones. \end{itemize} The column on the left consists of well-known categories defined by \textit{V. Voevodsky} [Ann. Math. Stud. 143, 87--137 (2000; Zbl 1019.14010)]. The other categories were developed in the work of Bruno Kahn, Hiroyasu Miyazaki, Shuji Saito, and Takao Yamazaki, see e.g. [\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 (such as \(\mathbb{G}_a\) or \(\Omega^i_{/\mathbb{Z}}\)). The present paper focuses on generalizing the tensor product of \(\mathbb{A}^1\)-invariant presheaves to reciprocity sheaves via the theory of modulus presheaves with transfers. One of the main result is that, for any \(F_i \in \textbf{HI}\) and any regular local \(k\)-algebra \(R\), we have an isomorphism \[ \omega_! h_0^{\overline{\boxempty}} ( \omega^{\textbf{CI}}F_1 \otimes^{\textbf{MPST}} \dots \otimes^{\textbf{MPST}} \omega^{\textbf{CI}}F_n ) (R) \simeq (F_1 \otimes^{\textbf{HI}} \dots \otimes^{\textbf{HI}} F_n) (R). \] In particular, this result says that the product \[ \textbf{RSC}^{\times n} \to \textbf{RSC}, (F_1,\dots, F_n) \mapsto \omega_! h_0^{\overline{\boxempty}}(\omega^{\textbf{CI}}F_1 \otimes \dots \otimes \omega^{\textbf{CI}}F_n) \] is an extension to \(\textbf{RSC}\) of the tensor structure on \(\textbf{HI}\). \par An important ingredient in the proofs is Saito's result that Nisnevich sheafification preserves \(\textbf{RSC}\), which generalizes the analogue statement for \(\textbf{HI}\) by Voevodsky, see [\textit{S. Saito}, Adv. Math. 366, Article ID 107067, 70 p. (2020; Zbl 1437.19001)]. Another key point is the injectivity property of reciprocity sheaves proved in [\textit{B. Kahn} et al., Compos. Math. 152, No. 9, 1851--1898 (2016; Zbl 1419.19001)] which implies that a morphism between reciprocity sheaves is an isomorphism if it is so on all function fields. As an application of these methods, the authors obtain that certain higher Chow groups of zero-cycles with modulus condition defined via either the sum or the ssup convention do agree.
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