The tangent space to the space of 0-cycles

Abstract

Let $S$ be a Noetherian scheme, and let $X$ be a scheme over $S$, such that all relative symmetric powers of $X$ over $S$ exist. Assume that either $S$ is of pure characteristic $0$ or $X$ is flat over $S$. Assume also that the structural morphism from $X$ to $S$ admits a section, and use it to construct the connected infinite symmetric power ${\rm Sym}^{\infty }(X/S)$ of the scheme $X$ over $S$. This is a commutative monoid whose group completion ${\rm Sym}^{\infty }(X/S)^+$ is an abelian group object in the category of set valued sheaves on the Nisnevich site over $S$, which is known to be isomorphic, as a Nisnevich sheaf, to the sheaf of relative $0$-cycles in Rydh's sense. Being restricted on seminormal schemes over $\mathbb Q$, it is also isomorphic to the sheaf of relative $0$-cycles in the sense of Suslin-Voevodsky and Koll\'ar. In the paper we construct a locally ringed Nisnevich-\'etale site of $0$-cycles ${\rm Sym}^{\infty }(X/S)^+_{\rm {Nis-et}}$, such that the category of \'etale neighbourhoods, at each point $P$ on it, is cofiltered. This yields the sheaf of K\"ahler differentials $\Omega ^1_{{\rm Sym}^{\infty }(X/S)^+}$ and its dual, the tangent sheaf $T_{{\rm Sym}^{\infty }(X/S)^+}$ on the space ${\rm Sym}^{\infty }(X/S)^+$. Applying the stalk functor, we obtain the stalk $T_{{\rm Sym}^{\infty }(X/S)^+,P}$ of the tangent sheaf at $P$, whose tensor product with the residue field $\kappa (P)$ is our tangent space to the space of $0$-cycles at $P$.