# The tangent space to the space of 0-cycles

Research paper by **Vladimir Guletskii**

Indexed on: **07 Mar '18**Published on: **07 Mar '18**Published in: **arXiv - Mathematics - Algebraic Geometry**

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#### 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 $P$.