The tangent space to the space of 0-cycles

Let $S$ be a Noetherian scheme, and let $X$ be a scheme over $S$.

Under mild assumptions, one can construct the connected infinite symmetric power ${\rm Sym}^{\infty }(X/S)$, 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 $S$.

Viewing this completion as the space of relative $0$-cycles on $X/S$, we construct the sheaf of Kähler differentials $\Omega ^1_{{\rm Sym}^{\infty }(X/S)^+}$, and the tangent sheaf $T_{{\rm Sym}^{\infty }(X/S)^+}$.

We prove that the category of étale neighbourhoods at a point $P$ on the space of $0$-cycles is cofiltered.

Applying the stalk functor, we also obtain the stalk of the tangent sheaf at $P$, whose tensor product with the residue field is the needed tangent space to the space of $0$-cycles at $P$.