# Stratified bundles on the Hilbert Scheme of $n$ points

Saurav Holme Choudhury  The Institute of Mathematical Sciences, (HBNI),
Chennai 600113<EMAIL_ADDRESS>
###### Abstract.

Let $k$ be an algebraically closed field of characteristic $p>3$ and $S$ be a
smooth projective surface over $k$ with $k$-rational point $x$. For $n\geq 2$,
let $S^{[n]}$ denote the Hilbert scheme of $n$ points on $S$. In this note, we
compute the fundamental group scheme $\pi^{\textrm{alg}}(S^{[n]},\tilde{nx})$
defined by the Tannakian category of stratified bundles on $S^{[n]}$.

##  1\. Introduction

For a variety $X$ over $\mathbb{C}$, one has the classical notion of the
fundamental group $\pi_{1}(X^{\textrm{an}},x)$ defined using the analytic
topology on $X$. Over arbitrary base fields $k$, one has several analogues of
the fundamental group defined in terms of algebro-geometric information.

In [SGA1], Grothendieck introduced the notion of étale fundamental group
$\pi^{\textrm{\'{e}t}}(X,x)$, where $X$ is a scheme and $x$ is a geometric
point of $X$, in terms of the finite etale covers of $X$. In [N76], Nori
defined the Nori fundamental group scheme $\pi^{N}(X,x)$, where $X$ is a
connected, reduced and complete scheme over a perfect field $k$ and $x$ is a
$k$-rational point, via Tannakian reconstruction using the category of
essentially finite vector bundles on $X$. The definition of $\pi^{N}(X,x)$ was
extended to the case of connected and reduced $k$-schemes in [N82]. Another
analogue, the S-fundamental group scheme $\pi^{S}(X,x)$ was introduced and
studied by Langer in [Lan11] and [Lan12] for smooth projective varieties $X$
over an algebraically closed field $k$. It is defined via Tannakian
reconstruction using the category of numerically flat vector bundles on $X$.
The S-fundamental group scheme for a smooth projective curve $C$ over an
algebraically closed field $k$ was already introduced and studied in [BPS06].

The variant of the fundamental group scheme which is of prime importance in
this note is the algebraic fundamental group $\pi^{\textrm{alg}}(X,x)$. In
[Gie75], Gieseker defined $\pi^{\textrm{alg}}(X,x)$ as the fundamental group
scheme corresponding to the Tannakian category of $\mathcal{D}_{X}$-modules,
where $\mathcal{D}_{X}$ is the sheaf of differential operators on $X$. For $X$
smooth over a field of positive characteristic, Gieseker introduced the notion
of stratified bundles and showed that the category of
$\mathcal{D}_{X}$-modules is tensor equivalent to the category of stratified
bundles on $X$. Stratifed bundles were further studied in [dS07] and [BHdS21].
Precise definitions and statements will be given in the next section.

Let $S$ be a smooth projective surface over $k$. For $n\geq 2$, let $S^{[n]}$
denote the Hilbert scheme $n$ points on $S$. It is well known that $S^{[n]}$
is a smooth projective variety of dimension $2n$. In [PS20], the authors show
that for char $k>3$ and $n\geq 2$, there is an isomorphism of affine group
schemes over $k$

$\pi^{\dagger}(S,x)_{\textrm{ab}}\to\pi^{\dagger}(S^{[n]},\tilde{nx})$

where $\dagger=S,N\textrm{ or \'{e}t}$.

In this note, we extend their results to the case of $\pi^{\textrm{alg}}$ and
prove the following theorem.

######  Theorem.

Let char $k>3$ and $n\geq 2$. There is an isomorphism of affine group schemes
over $k$

$f:\pi^{\textrm{alg}}(S,x)_{\textrm{ab}}\to\pi^{\textrm{alg}}(S^{[n]},\tilde{nx})$

In section 2, we recall the definition of stratified bundles and some of their
basic properties. The formalism of Tannakian reconstruction is recalled in
section 3 and used to define the algebraic fundamental group
$\pi^{\textrm{alg}}(X,x)$.

The geometrical properties of the Hilbert scheme of $n$ points on a smooth
projective surface are in section 4. In section 5, we prove a result about
descent of stratified bundles which allows us to define the homomorphism $f$
by defining the associated functor of Tannakian categories. The concluding
section 6 establishes the main theorem by showing that $f$ is an isomorphism.

### Acknowledgements

We would like to thank Indranil Biswas and Ronnie Sebastian for their comments
on earlier drafts of this note.

##  2\. Stratified bundles

Let $k$ be a field of characteristic $p$ and $X$ be a noetherian scheme over
$k$. Stratified bundles on $X$ are sequences of coherent sheaves on $X$
satisfying infinite Frobenius descent. More precisely, the category of
stratified bundles on $X$, denoted $\mathcal{S}(X)$, consists of

  * •

Objects $(\mathcal{E}_{i},\alpha_{i})$ are sequences of coherent
$\mathcal{O}_{X}$-modules $\mathcal{E}_{i}$, $i\in\mathbb{N}$ along with
isomorphisms

$\alpha_{i}:F^{*}\mathcal{E}_{i+1}\to\mathcal{E}_{i}$

for all $i\in\mathbb{N}$, where $F$ is the absolute Frobenius on $X$.

  * •

Morphisms $\phi:(\mathcal{E}_{i},\alpha_{i})\to(\mathcal{F}_{i},\beta_{i})$
consists of a sequence of $\mathcal{O}_{X}$-module morphisms
$\phi_{i}:\mathcal{E}_{i}\to\mathcal{F}_{i}$ such that
$\phi_{i}\circ\alpha_{i}=\beta_{i}\circ F*(\phi_{i+1})$

Let $f:Y\to X$ be a morphism and $(\mathcal{E}_{i},\alpha_{i})$ be a
stratified bundle on $X$. Then we can define the pullback along $f$, denoted
$f^{*}(\mathcal{E}_{i},\alpha_{i})$, as consisting of the sequence of $O_{Y}$
coherent sheaves $f^{*}E_{i}$ and isomorphisms are given by the composite maps

$F^{*}f^{*}\mathcal{E}_{i+1}\xrightarrow{\gamma_{\mathcal{E}_{i+1}}}f^{*}F^{*}\mathcal{E}_{i+1}\xrightarrow{f^{*}(\alpha_{i})}f^{*}\mathcal{E}_{i}$

where $\gamma:F^{*}f^{*}\to f^{*}F^{*}$ is the natural isomorphism of
functors.

Thus $\mathcal{S}(X)$ is contravariant functor in $X$. One also has a tensor
product on $\mathcal{S}(X)$ defined by taking term by term tensor product.
Also $\mathcal{S}(X)$ is an abelian category [cf. [BHdS21], Proposition 4.4].

We recall some well known results about stratified bundles [cf [dS07],
[Gie75]].

######  Proposition.

If $(\mathcal{E}_{i},\alpha_{i})$ is a stratified bundle on $X$, then
$\mathcal{E}_{i}$ is a locally free $\mathcal{O}_{X}$-module for all
$i\in\mathbb{N}$.

This allows us to define duals of stratified bundles, making $\mathcal{S}(X)$
into an abelian rigid tensor category.

The rank of a stratified bundle $(\mathcal{E}_{i},\alpha_{i})$ is defined to
be the rank of $\mathcal{E}_{0}$. The trivial stratified bundles on $X$ are of
the form $\oplus(\mathcal{O}_{X},...;F^{*},...)$.

Let $\mathcal{D}_{X}$ be the sheaf of differential operators on $X$. The
category of $\mathcal{D}_{X}$ modules consists of

  * •

Objects coherent $\mathcal{O}_{X}$ modules $\mathcal{E}$ equipped with a
$\mathcal{D}_{X}$ action i.e a morphism of $\mathcal{O}_{X}$-algebras

$\mathcal{D}_{X}\to\mathcal{E}\textrm{nd}_{k}(\mathcal{E})$

  * •

Morphisms $\mathcal{O}_{X}$-linear maps $\mathcal{E}\to\mathcal{F}$ compatible
with the $\mathcal{D}_{X}$ action

A theorem of Katz [[Gie75], Theorem 1.3] shows that for $X$ smooth over $k$,
then the category of stratified bundles on $X$ and the category of
$\mathcal{D}_{X}$ modules are tensor equivalent to each other.

We close this section with the definition of $G$ equivariant stratified
bundles on a variety $X$ admitting action of a group $G$ on it.

######  Definition.

A stratified bundle $(\mathcal{E}_{i},\alpha_{i})$ is said to be a
$G$-equivariant stratified bundle if $\mathcal{E}_{i}$ are $G$-equivariant
vector bundles and $\alpha_{i}$ are $G$-equivariant $\mathcal{O}_{X}$ module
morphisms.

##  3\. Tannakian categories and fundamental group schemes

In this section we recall the definition and basic properties of Tannakian
categories. We then recall Gieseker’s definition of the fundamental group
scheme $\pi^{\textrm{alg}}$ using the Tannakian formalism.

###  3.1. Tannakian Categories and affine group schemes

Tannakian categories were defined and studied in [DM82] to formalize the
properties of $\textrm{Rep}_{k}(G)$, the category of finite dimensional
$k$-representations of $G$, an affine group scheme over $k$.

######  Definition (Neutral Tannakian Categories).

A rigid abelian tensor category $\mathcal{C}$ with End $\mathbb{I}=k$ is a
neutral Tannakian category if it admits an exact faithful $k$-linear tensor
functor $\omega:\mathcal{C}\to\textrm{Vec}_{k}$. Any such functor is said to
be a fiber functor for $\mathcal{C}$.

Given a neutral Tannakian category $(\mathcal{C},\otimes,\omega,\mathbb{I})$,
we define the functor
$\textrm{Aut}^{\otimes}(\omega):k-\textrm{algebra}\to\textrm{Sets}$ such that
for $k$-algebra $R$, $\textrm{Aut}^{\otimes}(\omega)(R)$ consists of the
families $(\lambda_{X})$ for $X\in\textrm{ob}(\mathcal{C})$, where
$\lambda_{X}$ is a $R$-linear automorphism of $X\otimes R$ such that
$\lambda_{X_{1}\otimes X_{2}}=\lambda_{X_{1}}\otimes\lambda_{X_{2}}$,
$\lambda_{\mathbb{I}}=id_{R}$, and

$\lambda_{Y}\circ(\alpha\otimes 1)=(\alpha\otimes 1)\circ\lambda_{X}:X\otimes
R\to Y\otimes R$

for all morphisms $\alpha:X\to Y$.

######  Theorem (Main theorem for neutral Tannakian categories, [DM82],
Theorem 2.11).

Let $(\mathcal{C},\otimes)$ be a rigid abelian tensor category such that
$k=\textrm{End}(\mathbb{I})$ and let $\omega:\mathcal{C}\to\textrm{Vec}_{k}$
be an exact faithful tensor functor. Then

  * •

The functor $\textrm{Aut}^{\otimes}(\omega)$ of $k$-algebras is represented by
an affine group scheme $G$.

  * •

The functor $\mathcal{C}\to\textrm{Rep}_{k}(G)$ is an equivalence of tensor
categories.

######  Theorem.

Let $(\mathcal{C},\otimes,\omega,\mathbb{I})$ and
$(\mathcal{C}^{\prime},\otimes,\omega^{\prime},\mathbb{I^{\prime}})$ be
neutral Tannakian categories which correspond to the representation categories
of the affine $k$ group schemes $G$ and $G^{\prime}$ respectively. Then any
functor of Tannakian categories from $\mathcal{C}\to\mathcal{C}^{\prime}$ is
induced by a unique morphism of affine $k$ group schemes $G^{\prime}\to G$.

This theorem allows us to define many variants of fundamental groups of a
scheme $X$ by considering different Tannakian categories naturally associated
with $X$. The following result is very useful in establishing a given morphism
between affine group schemes is an isomorphism.

######  Theorem ([DM82], Theorem 2.21).

Let $f:G\to G^{\prime}$ be a homomorphism of group schemes over $k$ and
$\textrm{Rep }(f):\textrm{Rep }(G^{\prime})\to\textrm{Rep }(G)$ be the
corresponding functor of Tannakian categories. Then

  * •

$f$ is faithfully flat if and only if $\textrm{Rep }(f)$ is fully faithful and
has essential image closed under subobjects i.e for $V^{\prime}\in\textrm{Rep
}(G^{\prime})$ and suboject $W\subset\textrm{Rep }(f)(V^{\prime})$, there is a
subobject $W^{\prime}\subset V^{\prime}$ in $\textrm{Rep }(G^{\prime})$ such
that $\textrm{Rep }(f)(W^{\prime})\simeq W$ in $\textrm{Rep }(G)$

  * •

$f$ is closed immersion if and only if every object of $\textrm{Rep }(G)$ is a
subquotient of some object in the essential image of $\textrm{Rep }(f)$.

We finish by recalling a basic result on affine group schemes (we refer to
section $4.1$ in [PS20] for details).

Let $G$ be a affine group scheme over $k$, $G_{\textrm{ab}}$ be its
abelianization (i.e the maximal abelian quotient of $G$) and $\alpha:G\to
G_{\textrm{ab}}$ be the (faithfully flat) quotient morphism . We can then
define the composite morphism

$\phi:G^{n}\xrightarrow{\alpha^{n}}G_{\textrm{ab}}^{n}\xrightarrow{m}G_{\textrm{ab}}$

where $m$ is the multiplication homomorphism. As $\mathfrak{S}_{n}$ acts on
the $k$-group scheme $G^{n}$, we can define the notion of a
$\mathfrak{S}_{n}$-invariant group morphism $\psi:G^{n}\to H$ for any
$k$-group scheme $H$.

######  Lemma 3.1.

Let $G$ and $H$ be two group schemes over $k$. For an integer $n\geq 2$, the
set of $\mathfrak{S}_{n}$-invariant group morphisms $G^{n}\to H$ is in
bijective correspondence with the set of group morphism $G_{\textrm{ab}}\to H$
i.e any morphism of $k$-group schemes $\phi:G^{n}\to H$ which is
$\mathfrak{S}_{n}$-invariant factors uniquely through a morphism
$\psi:G_{\textrm{ab}}\to H$ such that $\phi=\psi\circ h$

###  3.2. The group scheme $\pi^{\textrm{alg}}(X,x)$

Classically, over $\mathbb{C}$, the Riemann-Hilbert correspondence identifies
the category of vector bundles equipped with integrable connections on a
smooth connected projective variety $X$/$\mathbb{C}$ with the category of
representations of the topological fundamental group $\pi^{\textrm{top}}(X,x)$
for some chosen base point $x$. Via GAGA, this gives a purely algebraic
description of the category of representations of the topological fundamental
group $\pi(X,x)$. This category (equipped with the fiber functor
$(E,\nabla)\to E_{x}$) is a neutral Tannakian category and can be identified,
via the Tannakian formalism, with the representation category of the
proalgebraic completion of the topological fundamental group, denoted as
$\pi^{\textrm{top}}(X,x)^{\textrm{alg}}$.

Over a field $k$ of characteristic $0$, the category of flat connections on a
smooth variety $X$ is tensor equivalent to the category of
$\mathcal{D}_{X}$-modules. However over a field of characteristic $p$, the
category of flat connections on $X$ is not as well behaved as the category of
$\mathcal{D}_{X}$-modules and one defines a fundamental group scheme for $X$
by Tannakian formalism using the category of $\mathcal{D}_{X}$-modules. By
Katz’s theorem mentioned before, the fundamental group coincides with the one
defined using $\mathcal{S}(X)$ below.

Let $x\in X(k)$ be a $k$-rational point. Then the abelian rigid tensor
category $\mathcal{S}(X)$ is neutralized by the fiber functor

$T_{x}:\mathcal{S}(X)\to Vec_{k}$

The fundamental group scheme defined by the neutral Tannakian category
$(\mathcal{S}(X),\otimes,T_{x},(\mathcal{O}_{X},F^{*}))$ is called the
algebraic fundamental group of $X$ based at $x$ and is denoted by
$\pi^{\textrm{alg}}(X,x)$.

The following basic properties of $\pi^{\textrm{alg}}$ are well known.

  * •

(Independence of basepoint) Let $X$ be a geometrically connected, smooth
projective $k$-scheme. Then for all $x_{1},x_{2}\in X(k)$, one has

$\pi^{\textrm{alg}}(X,x_{1})\simeq\pi^{\textrm{alg}}(X,x_{2})$

  * •

(Product rule) For $X_{1},X_{2}$ geometrically connected and smooth over $k$
and $x_{i}\in X_{i}(k)$, there is an isomorphism

$\pi^{\textrm{alg}}(X_{1},x_{1})\times\pi^{\textrm{alg}}(X_{2},x_{2})\to\pi^{\textrm{alg}}(X_{1}\times
X_{2},(x_{1},x_{2}))$

  * •

For $X$ smooth and open immersion $U\xrightarrow{i}X$ such that the complement
of $U$ in $X$ has codimension $\geq 2$ and $x\in U(k)$, then the homomorphism

$\pi^{\textrm{alg}}(U,x)\to\pi^{\textrm{alg}}(X,x)$

associated to the restriction functor $i^{*}:\mathcal{S}(X)\to\mathcal{S}(U)$
is an isomorphism.

##  4\. Geometry of Hilbert Scheme of points

Let $S$ be a smooth projective surface over $k$. We fix notation as follows

  * •

$S^{n}$ denotes the $n$-fold cartesian product of $S$ with itself.

  * •

$S^{(n)}$ denotes the $n$th symmetric product of $S$ defined as the quotient
$S^{n}/\mathfrak{S}_{n}$, where $\mathfrak{S}_{n}$ denotes the symmetric group
on $n$ letters.

  * •

$S^{[n]}$ denotes the Hilbert scheme of $n$ points on $S$.

Let $\rho:S^{n}\to S^{(n)}$ be the quotient map and $h:S^{[n]}\to S^{(n)}$ be
the Hilbert-Chow morphism. We write $S^{(n)}_{\circ}$ for the open subset of
$S^{(n)}$ consisting of distinct points with
$S^{[n]}_{\circ}:=h^{-1}(S^{(n)}_{\circ})$ and
$S^{n}_{\circ}:=\rho^{-1}(S^{(n)}_{\circ})$. The map
$h_{n,\circ}:S^{[n]}_{\circ}\to S^{(n)}_{\circ}$ is an isomorphism. We have
the diagram:

${{S^{[n]}}}$${{S^{n}}}$${{S^{(n)}}}$$\scriptstyle{h_{n}}$$\scriptstyle{\rho_{n}}$

In general, Hilbert schemes of points on a projective variety display a lot of
pathological features. But in [Fog68] the author shows that, in the case of
smooth projective surface $S$, $S^{[n]}$ is a smooth projective variety. Thus,
in this case, the Hilbert-Chow morphism $h:S^{[n]}\to S^{(n)}$ is a resolution
of singularities.

One can consider $S^{(n)}$ as the set of effective $0$-cycles of degree $n$ on
$S^{(n)}$. In this case it is easy to see that $S^{(n)}$ admits a
stratification by type, where the type of a $0$-cycle $y$ of degree $n$ is a
tuple $(n_{1},\dots,n_{r})$ where $y$ can be written as

$y=\Sigma_{j=1}^{r}n_{j}x_{j}$

where $x_{j}$ are distinct points of $S$ with multiplicities $n_{1}\geq
n_{2}\geq\dots\geq n_{r}$, where $n_{j}$ are positive integers.

Let $C(n_{1},\dots,n_{r})$ denote the subset of $S^{(n)}$ consisting of points
of the type $(n_{1},\dots,n_{r})$. Let $S^{(n)}_{*}=C(1,\dots,1)\bigcup
C(2,\dots,1)$ denote the open subset of $S^{(n)}$ consisting of points of type
$(1,\dots,1)$ and $(2,\dots,1)$. Let $S^{[n]}_{*}$ and $S^{n}_{*}$ denote the
preimage of $S^{(n)}_{*}$ under $h$ and $\rho$ respectively.

We recall some basic properties below which we will need later (we refer to
[Fog68], [PS20] for details).

  * •

The subsets $C(n_{1},\dots,n_{r})$ are nonsingular of dimension $2r$.

  * •

The closed subset $S^{(n)}\setminus S^{(n)}_{*}$ is of codimension $\geq 2$ in
$S^{(n)}$.

  * •

The closed subset $S^{[n]}\setminus S^{[n]}_{*}$ is of codimension $2$ in
$S^{[n]}$.

  * •

The closed subset $S^{n}\setminus S^{n}_{*}$ is of codimension $\geq 4$ in
$S^{n}$.

  * •

The closed subset $S^{(n)}_{*}\setminus S^{(n)}_{\circ}$ is of codimension $2$
in $S^{(n)}_{*}$.

  * •

When characteristic of $k\neq 2$, for $y\in C(2,1,\dots,1)$, the scheme
theoretic fiber $h^{-1}(y)$ is isomorphic to $\mathbb{P}^{1}_{k}$. In fact,
$S^{[n]}_{*}$ is the blowup of $S^{(n)}_{*}$ along $C(2,1,\dots,1)$.

We end this section by recalling a result of Fogarty ([Fog77], Proposition
3.6).

######  Proposition.

If $L$ is a $\mathfrak{S}_{n}$-invariant line bundle on $S^{n}$, there exists
a line bundle $L^{\prime}$ on $S^{(n)}$ such that $h^{*}L^{\prime}\simeq L$.

It follows that $L^{\prime}$ in the proposition is isomorphic to
$\sigma_{*}(L)^{\mathfrak{S}_{n}}$

##  5\. The functor between Tannakian categories

Let $S$ be a smooth projective surface over $k$ and
$(\mathcal{E}_{i},\alpha_{i})$ be a stratified bundle on $S^{[n]}$.
Restricting to $S^{[n]}_{*}$ gives us a functor

$i^{*}:\mathcal{S}(S^{[n]})\to\mathcal{S}(S^{[n]}_{*})$

which is a equivalence of categories as $S^{[n]}_{*}$ is the complement of a
codimension $2$ closed subset of $S^{[n]}$.

Next we show that a stratified bundle on $S^{[n]}_{*}$ can be pushed forward
under $h$ to get a stratified bundle on $S^{(n)}_{*}$. First we begin by a
result on descent of vector bundles along the morphism $h:S^{[n]}_{*}\to
S^{(n)}_{*}$. Similar results have been established by authors in [Ish83] and
[PS20].

######  Proposition 1.

Assume char $k\neq 2$. Let $\mathcal{E}$ be a vector bundle on $S^{[n]}_{*}$
which restricts to trivial vector bundles on the fibers of $h$ over
$S^{(n)}_{*}$. Then $h_{*}\mathcal{E}$ is a locally free
$\mathcal{O}_{S^{(n)}_{*}}$-module. Moreover the natural map

$h^{*}h_{*}(\mathcal{E})\to\mathcal{E}$

is an isomorphism.

###### Proof.

Let $x\in S^{(n)}_{*}$ be a point of type $(2,1,\dots,1)$. Then by assumption,
the fiber of $h$ over $x$ is isomorphic to $\mathbb{P}^{1}_{k}$. Let
$\mathcal{J}$ be the ideal sheaf of the closed subscheme $h^{-1}(x)$ and
$\mathcal{I}_{x}$ be the ideal sheaf of the closed point $x$. We have

$\mathcal{J}=\mathcal{I}_{x}\mathcal{O}_{S^{[n]}_{*}}$

For all $n\geq 1$, let $Y_{n}$ denote the closed subscheme of $S^{[n]}_{*}$
corresponding to the ideal sheaf $\mathcal{J}^{n}$. Consider the following
short exact sequence of sheaves on $S^{[n]}_{*}$

$0\to\mathcal{J}\otimes\mathcal{E}\to\mathcal{E}\to\mathcal{E}|_{Y_{1}}\to 0$

Pushing forward by $h$, we get the following exact sequence of sheaves on
$S^{(n)}_{*}$

$h_{*}\mathcal{E}\to H^{0}(Y_{1},\mathcal{E}|_{Y_{1}})\to
R^{1}h_{*}(\mathcal{J}\otimes\mathcal{E})$

We claim that the completion of $R^{1}h_{*}(\mathcal{J}\otimes\mathcal{E})$ at
the maximal ideal $m_{x}$ in $\mathcal{O}_{S^{(n)}_{*},x}$ is $0$. The proof
uses the theorem of formal functions which says that

$(R^{1}h_{*}(\mathcal{J}\otimes\mathcal{E}))^{\wedge}\simeq\varprojlim
H^{1}(Y_{n},\mathcal{J}\otimes\mathcal{E}\otimes\mathcal{O}_{S^{[n]}_{*}}/\mathcal{J}^{n})$

We prove by induction that
$H^{1}(Y_{n},\mathcal{J}\otimes\mathcal{E}\otimes\mathcal{O}_{S^{[n]}_{*}}/\mathcal{J}^{n})=0$.
As $Y_{1}\simeq\mathbb{P}^{1}_{k}$, the sheaves
$\mathcal{J}^{n}/\mathcal{J}^{n+1}$ are locally free. These sheaves are also
globally generated over $Y_{1}$ as we have the surjection

$m_{x}^{n}/m_{x}^{n+1}\otimes_{\mathcal{O}_{S^{(n)}_{*},x}}\mathcal{O}_{S^{[n]}_{*}}\simeq\mathcal{I}_{x}^{n}/\mathcal{I}_{x}^{n+1}\otimes_{\mathcal{O}_{S^{(n)}_{*}}}\mathcal{O}_{S^{[n]}_{*}}\twoheadrightarrow\mathcal{J}^{n}/\mathcal{J}^{n+1}$

As $\mathcal{J}^{n}/\mathcal{J}^{n+1}$ is locally free on
$Y_{1}\simeq\mathbb{P}^{1}_{k}$ and globally generated, it is a direct sum of
line bundles each of which has degree $\geq 0$. Thus one gets the base case of
induction from degree considerations, as

$H^{1}(Y_{1},\mathcal{J}\otimes\mathcal{E}\otimes\mathcal{O}_{S^{[n]}_{*}}/\mathcal{J}=H^{1}(Y_{1},\mathcal{J}/\mathcal{J}^{2}\otimes\mathcal{E}_{Y_{1}})=0$

Assume that the claim is true for $n$. Then the proof for $n+1$ follows from
the long exact sequence in cohomology attached to the short exact sequence of
sheaves on $Y_{n+1}$

$0\to\mathcal{J}^{n+1}/\mathcal{J}^{n+2}\otimes\mathcal{E}\to\mathcal{J}/\mathcal{J}^{n+2}\otimes\mathcal{E}\to\mathcal{J}/\mathcal{J}^{n+1}\otimes\mathcal{E}\to
0$

which gives us the exact sequence

$H^{1}(Y_{n+1},\mathcal{J}^{n+1}/\mathcal{J}^{n+2}\otimes\mathcal{E})\to
H^{1}(Y_{n+1},\mathcal{J}/\mathcal{J}^{n+2}\otimes\mathcal{E})\to
H^{1}(Y_{n+1},\mathcal{J}/\mathcal{J}^{n+1}\otimes\mathcal{E})$

We know
$H^{1}(Y_{n},\mathcal{J}^{n+1}/\mathcal{J}^{n+2}\otimes\mathcal{E})=H^{1}(Y_{1},\mathcal{J}^{n+1}/\mathcal{J}^{n+2}\otimes\mathcal{E})=0$
(by degree consideration) and
$H^{1}(Y_{n+1},\mathcal{J}/\mathcal{J}^{n+1}\otimes\mathcal{E})=H^{1}(Y_{n},\mathcal{J}/\mathcal{J}^{n+1}\otimes\mathcal{E})=0$
(by induction hypothesis), thus we get

$H^{1}(Y_{n+1},\mathcal{J}/\mathcal{J}^{n+2}\otimes\mathcal{E})=0$

Thus the stalk of $R^{1}h_{*}(\mathcal{J}\otimes\mathcal{E})$ at $x$ is $0$.

This shows that the natural map $h_{*}\mathcal{E}\to
H^{0}(Y_{1},\mathcal{E}|_{Y_{1}})$ is surjective in a neighbourhood of $x$.
Let $f_{1},...,f_{r}$ be a basis of $H^{0}(Y_{1},\mathcal{E}|_{Y_{1}})$. Let
$\textrm{Spec}(R)$ be an affine neighbourhood of $x$ where the natural map is
surjective and let
$\tilde{f_{i}}\in\Gamma(\textrm{Spec}(R),h_{*}\mathcal{E})=\Gamma(h^{-1}(\textrm{Spec}(R)),\mathcal{E})$
be lifts of $f_{i}$. Using $\tilde{f}_{i}$ one defines a homomorphism

$\mathcal{O}_{S^{[n]}_{*}}^{\oplus
r}|_{h^{-1}(\textrm{Spec}(R))}\to\mathcal{E}$

on $h^{-1}(\textrm{Spec}(R)$ which is a surjection (and hence an isomorphism)
on $Y_{1}$. As $h$ is proper, there exists a smaller affine neighbourhood $U$
of $x$ over which there is an isomorphism

$\mathcal{O}_{V}^{\oplus r}\simeq\mathcal{E}$

where $V=h^{-1}(U)$. Applying $h_{*}$, we get

$(h_{*}\mathcal{O}_{V})^{\oplus r}\simeq h_{*}\mathcal{E}$

As $S^{(n)}_{*}$ is normal and $h:S^{[n]}_{*}\to S^{(n)}_{*}$ is birational
with connected fibers, by a form of Zariski’s main theorem [cf [Har77],
Corollary 11.3 and 11.4], we have that
$h_{*}\mathcal{O}_{V}\simeq\mathcal{O}_{U}$ and thus $h_{*}\mathcal{E}$ is
locally free. The natural morphism

$h^{*}h_{*}(\mathcal{E})\to\mathcal{E}$

is clearly an isomorphism.

∎

Let $\mathbb{VB}_{S^{(n)}_{*}}$ be the category of locally free sheaves on
$S^{(n)}_{*}$ and $\mathbb{VB}^{h}_{S^{[n]}_{*}}$ be the category of locally
free sheaves on $S^{[n]}_{*}$ which restrict to trivial vector bundles on the
fibers of $h$. Proposition 1 above gives us an equivalence of categories.

######  Proposition 2.

Assume char $k\neq 2$. The pushforward functor

$h_{*}:\mathbb{VB}^{h}_{S^{[n]}_{*}}\to\mathbb{VB}_{S^{(n)}_{*}}$

is an equivalence of categories with the inverse given by

$h^{*}:\mathbb{VB}_{S^{(n)}_{*}}\to\mathbb{VB}^{h}_{S^{[n]}_{*}}$

###### Proof.

We observe that if $\mathcal{E}^{\prime}\simeq h^{*}(\mathcal{E})$, then
$\mathcal{E}\simeq h_{*}\mathcal{E}^{\prime}$. This shows that $h_{*}$ is
essentially surjective. The natural map

$\textrm{Hom}_{S^{(n)}_{*}}(h_{*}\mathcal{E},h_{*}\mathcal{F})\to\textrm{Hom}_{S^{[n]}_{*}}(\mathcal{E},\mathcal{F})$

is bijective. Thus $h_{*}$ is an equivalence of categories. ∎

######  Corollary.

For all $\mathcal{E}\in\mathbb{VB}^{h}_{S^{[n]}_{*}}$, the natural map

$F^{*}h_{*}(\mathcal{E})\to h_{*}F^{*}(\mathcal{E})$

is an isomorphism over $S^{(n)}_{*}$.

###### Proof.

As $F^{*}\mathcal{E}$ is also an object of $\mathbb{VB}^{h}_{S^{[n]}_{*}}$,
thus both sheaves are locally free of the same rank. Thus it suffices to show
that the natural map

$F^{*}h_{*}(\mathcal{E})\to h_{*}F^{*}(\mathcal{E})$

is surjective. As $F$ is faithfully flat on the smooth locus of $S^{(n)}_{*}$,
the claim holds on the smooth locus. Let $x\in S^{(n)}_{*}$ be of type
$(2,1,\dots,1)$. Then the restriction of $F^{*}h_{*}(\mathcal{E})$ to $x$ is
naturally isomorphic to $H^{0}(Y_{1},\mathcal{E}|_{Y_{1}})$ and the
restriction of $h_{*}F^{*}(\mathcal{E})$ to $x$ is
$H^{0}(Y_{1},F^{*}(\mathcal{E}|_{Y_{1}})$. The restriction of the natural map
to $x$ is the map

$F^{*}:H^{0}(Y_{1},\mathcal{E}_{1})\to H^{0}(Y_{1},F^{*}\mathcal{E}_{1})$

which is surjective. ∎

By Theorem $2.2$ of [Gie75], we have that every stratified bundle on
$\mathbb{P}^{1}_{k}$ is trivial. Thus the above results give us

######  Proposition 3.

Assume char $k\neq 2$. Let $(\mathcal{E}_{i},\alpha_{i})$ be a stratified
bundle on $S^{[n]}_{*}$. Then $h_{*}(\mathcal{E}_{i})$ is locally free
$\mathcal{O}_{S^{(n)}_{*}}$-module for all $i\in\mathbb{N}$. Moreover the
natural map

$h^{*}h_{*}(\mathcal{E}_{i})\to\mathcal{E}_{i}$

is an isomorphism. Furthermore the natural map

$F^{*}h_{*}(\mathcal{E}_{i})\to h_{*}F^{*}(\mathcal{E}_{i})$

is an isomorphism over $S^{(n)}_{*}$.

This allows us to define the pushforward of a stratified bundle
$(\mathcal{E}_{i},\alpha_{i})$ on $S^{[n]}_{*}$. The pushforward denoted
$h_{*}(\mathcal{E}_{i},\alpha_{i})$ is given by the sequence of vector bundles
$h_{*}\mathcal{E}_{i}$ for all $i\in\mathbb{N}$ and the isomorphisms are given
by the composite

$F^{*}h_{*}(\mathcal{E}_{i+1})\xrightarrow{\eta_{\mathcal{E}_{i+1}}}h_{*}F^{*}(\mathcal{E}_{i+1})\xrightarrow{h_{*}(\alpha_{i})}h_{*}(\mathcal{E}_{i})$

where $\eta:F^{*}h_{*}\to h_{*}F^{*}$ is the natural transformation.

Thus we get a functor

$h_{*}:\mathcal{S}(S^{[n]}_{*})\to\mathcal{S}(S^{(n)}_{*})$

$h_{*}$ is additive tensor functor as on the smooth locus $S^{(n)}_{\circ}$ we
have the isomorphisms

$h_{*}((\mathcal{E}_{i},\alpha_{i})\oplus(\mathcal{F}_{i},\beta_{i}))|_{S^{(n)}_{\circ}}\simeq
h_{*}(\mathcal{E}_{i},\alpha_{i})|_{S^{(n)}_{\circ}}\oplus
h_{*}(\mathcal{F}_{i},\beta_{i}))|_{S^{(n)}_{\circ}}$

$h_{*}((\mathcal{E}_{i},\alpha_{i})\otimes(\mathcal{F}_{i},\beta_{i}))|_{S^{(n)}_{\circ}}\simeq
h_{*}(\mathcal{E}_{i},\alpha_{i})|_{S^{(n)}_{\circ}}\otimes
h_{*}(\mathcal{F}_{i},\beta_{i}))|_{S^{(n)}_{\circ}}$

which extend to $S^{(n)}_{*}$ due to codimension reasons.

The following commutative diagram shows that
$h^{*}h_{*}(\mathcal{E}_{i},\alpha_{i})$ is isomorphic to
$(\mathcal{E}_{i},\alpha_{i})$ as stratified bundles with the isomorphism
given by the natural morphisms $h^{*}h_{*}\mathcal{E}_{i}\to\mathcal{E}_{i}$.

${{F^{*}h^{*}h_{*}\mathcal{E}_{i+1}}}$${{h^{*}F^{*}h_{*}\mathcal{E}_{i+1}}}$${{h^{*}h_{*}F^{*}\mathcal{E}_{i+1}}}$${{h^{*}h_{*}\mathcal{E}_{i}}}$${{F^{*}\mathcal{E}_{i+1}}}$${{F^{*}\mathcal{E}_{i+1}}}$${{\mathcal{E}_{i}}}$$\scriptstyle{h^{*}\eta_{\mathcal{E}_{i+1}}}$$\scriptstyle{h_{*}h^{*}\alpha_{i}}$$\scriptstyle{\alpha_{i}}$$\scriptstyle{\gamma_{h_{*}\mathcal{E}_{i+1}}}$$\scriptstyle{=}$

Consider the pullback functor

$\rho^{*}:\mathcal{S}(S^{(n)}_{*})\to\mathcal{S}(S^{n}_{*})$

which takes values in the category of $\mathfrak{S}_{n}$-equivariant
stratified bundles on $S^{n}_{*}$. Also we have the extension functor

$j_{*}:\mathcal{S}(S^{n}_{*})\to\mathcal{S}(S^{n})$

which is an equivalence of categories. Composing these functors together, we
get a functor

$T:\mathcal{S}(S^{[n]})\to\mathcal{S}(S^{n})$

given by

$T=j_{*}\circ\rho^{*}\circ h_{*}\circ i^{*}$

Clearly $T$ is an additive tensor functor. Note that $h_{*}$ is fully
faithful, $\rho^{*}:\mathcal{S}(S^{(n)}_{*})\to\mathcal{S}(S^{n}_{*})$ is
fully faithful (as $\rho:S^{n}_{\circ}\to S^{(n)}_{\circ}$ is finite étale)
and $j_{*}:\mathcal{S}(S^{n}_{*})\to\mathcal{S}(S^{n})$ is an equivalence of
categories (due to codimension reasons). Thus $T$ is fully faithful.

###  5.1. The homomorphism

Fix $n$ distinct $k$-valued points $x_{1},\dots,x_{k}\in S(k)$. Let
$\tilde{x}\in S^{[n]}$ such that $h(\tilde{x})=\sigma(x_{1},\dots,x_{n})=z\in
S^{(n)}_{\circ}$. Then the categories $\mathcal{S}(S^{[n]})$ and
$\mathcal{S}(S^{n})$ are neutralized by the respective fiber functors

$\tau_{\tilde{x}}:\mathcal{S}(S^{[n]})\to\textrm{Vec}_{k}$
$(\mathcal{E}_{i},\alpha_{i})\mapsto(\mathcal{E}_{0})_{\tilde{x}}$

$\tau_{(x_{1},\dots,x_{n})}:\mathcal{S}(S^{n})\to\textrm{Vec}_{k}$
$(\mathcal{F}_{i},\beta_{i})\mapsto(\mathcal{F}_{0})_{(x_{1},\dots,x_{n})}$

If $T((\mathcal{E}_{i},\alpha_{i}))=(\mathcal{F}_{i},\beta_{i})$ that we have
natural isomorphisms $(\mathcal{E}_{0})_{\tilde{x}}\simeq
h_{*}(\mathcal{E}_{0})_{z}\simeq(\mathcal{F}_{0})_{(x_{1},\dots,x_{n})}$.

Thus we have a functor of Tannakian categories

$T:(\mathcal{S}(S^{[n]}),\otimes,\tau_{\tilde{x}},(\mathcal{O}_{S^{[n]}},d))\to(\mathcal{S}(S^{n}),\otimes,\tau_{(x_{1},\dots,x_{n})},(\mathcal{O}_{S^{n}},d))$

which by the independence of basepoint property of $\mathcal{S}$ induces a
functor of Tannakian categories

$T:(\mathcal{S}(S^{[n]}),\otimes,\tau_{\tilde{nx}},(\mathcal{O}_{S^{[n]}},d))\to(\mathcal{S}(S^{n}),\otimes,\tau_{(x,\dots,x)},(\mathcal{O}_{S^{n}},d))$

and hence a morphisms of the associated fundamental group schemes

$\tilde{f}:\pi^{\textrm{alg}}(S^{n},(x,\dots,x))\to\pi^{\textrm{alg}}(S^{[n]},\tilde{nx})$

Note that by proposition • ‣ 3.2 we have

$\pi^{\textrm{alg}}(S^{n},(x,\dots,x))\simeq\pi^{\textrm{alg}}(S,x)^{n}$

.

As

$T:(\mathcal{S}(S^{[n]}),\otimes,T_{\tilde{nx}},(\mathcal{O}_{S^{[n]}},d))\to(\mathcal{S}(S^{n}),\otimes,T_{(x,\dots,x)},(\mathcal{O}_{S^{n}},d))$

takes stratified bundles on $S^{[n]}$ to $\mathfrak{S}_{n}$-equivariant
stratified bundles on $S^{n}$ and a $\mathfrak{S}_{n}$-equivariant stratified
bundles on $S^{n}$ corresponds to a $\mathfrak{S}_{n}$-invariant
representation of $\pi^{\textrm{alg}}(S,x)^{n}$, by 3.1, $\tilde{f}$ factors
uniquely through

$f:\pi^{\textrm{alg}}(S,x)_{\textrm{ab}}\to\pi^{\textrm{alg}}(S^{[n]},\tilde{nx})$

##  6\. Isomorphism of fundamental group schemes

In this section, we show that $f$ is an isomorphism of affine group schemes.
We begin by proving a result about $\mathfrak{S}_{n}$-equivariant stratified
line bundles on $S^{n}$.

######  Proposition 4.

Let $(L_{i},\alpha_{i})$ be a $\mathfrak{S}_{n}$-equivariant stratified line
bundles on $S^{n}$. Then there exists a stratified line bundle
$(\mathcal{L}_{i},\beta_{i})$ such that
$\rho^{*}(\mathcal{L}_{i},\beta_{i})\simeq(L_{i},\alpha_{i})$

###### Proof.

By Fogarty’s result mentioned above, for any $\mathfrak{S}_{n}$-equivariant
line bundle $L_{i}$ there exists line bundle
$\mathcal{L}_{i}\simeq\rho_{*}L_{i}^{\mathfrak{S}_{n}}$ such that
$\rho^{*}\mathcal{L}_{i}\simeq L_{i}$. Pushing forward $\alpha_{i}$ and taking
$\mathfrak{S}_{n}$ invariants we get the isomorphism

$\rho_{*}(F^{*}L_{i+1})^{\mathfrak{S}_{n}}\xrightarrow{\rho_{*}(\alpha_{i})^{\mathfrak{S}_{n}}}\rho_{*}(L_{i})^{\mathfrak{S}_{n}}$

We show that the natural homomorphism

$F^{*}(\rho_{*}(L_{i})^{\mathfrak{S}_{n}})\to(F^{*}\rho_{*}(L_{i})^{\mathfrak{S}_{n}})\to(\rho_{*}F^{*}(L_{i})^{\mathfrak{S}_{n}})$

is an isomorphism. Pulling back the morphism under $\rho$, we get the
commutative diagram

${{\rho^{*}F^{*}((\rho_{*}L_{i})^{\mathfrak{S}_{n}})}}$${{\rho^{*}((\rho_{*}F^{*}L_{i})^{\mathfrak{S}_{n}})}}$${{F^{*}L_{i}}}$${{F^{*}L_{i}}}$$\scriptstyle{=}$

where the vertical morphisms are the natural morphism which are isomorphisms
by Fogarty’s theorem. By pushing forward under $\rho$ and taking
$\mathfrak{S}_{n}$ invariants we get that

$F^{*}(\rho_{*}(L_{i})^{\mathfrak{S}_{n}})\to(\rho_{*}F^{*}(L_{i})^{\mathfrak{S}_{n}})$

is an isomorphism. We define $\beta_{i}$ to be the composite isomorphism

$F^{*}(\rho_{*}(L_{i})^{\mathfrak{S}_{n}})\to(\rho_{*}F^{*}(L_{i})^{\mathfrak{S}_{n}})\xrightarrow{\rho_{*}(\alpha_{i})^{\mathfrak{S}_{n}}}\rho_{*}(L_{i})^{\mathfrak{S}_{n}}$

The commutative diagram also gives us that
$\rho^{*}(\mathcal{L}_{i},\beta_{i})\simeq(L_{i},\alpha_{i})$ ∎

###  6.1. Faithfully flat

Next we show that the morphism $f$ is faithfully flat

######  Proposition 5.

The homomorphism

$f:\pi^{\textrm{alg}}(S,x)_{\textrm{ab}}\to\pi^{\textrm{alg}}(S^{[n]},\tilde{nx})$

is faithfully flat.

###### Proof.

By [[DM82] Theorem 2.21], this is equivalent to showing that the functor

$T:\mathcal{S}(S^{[n]})\to\mathcal{S}(S^{n})$

is fully faithful and the essential image of $T$ is closed under taking
subobjects. We already know that $T$ is fully faithful. Let
$\mathcal{E}_{\bullet}=(\mathcal{E}_{i},\alpha_{i})$ be a stratified bundle on
$S^{[n]}$ and $\mathcal{F}_{\bullet}:=T(\mathcal{E}_{\bullet})$ be the
corresponding $\mathfrak{S}_{n}$-equivariant stratified bundle on $S^{n}$. If
$\mathcal{F}^{\prime}_{\bullet}\subset\mathcal{F}_{\bullet}$ is a
$\mathfrak{S}_{n}$-equivariant stratified subbundle, then we need to show
there exists $\mathcal{E}^{\prime}_{\bullet}\subset\mathcal{E}_{\bullet}$ such
that $T(\mathcal{E}^{\prime}_{\bullet})=\mathcal{F}^{\prime}_{\bullet}$.

The proof proceeds by induction on the rank of $\mathcal{E}_{\bullet}$. If
rank $\mathcal{E}_{\bullet}=1$, the proof is immediate. Let rank
$\mathcal{E}_{\bullet}\geq 2$

Then the stratified bundles $\mathcal{F}_{\bullet}$ and
$\mathcal{F}^{\prime}_{\bullet}$ correspond to the representations

$\pi^{\textrm{alg}}(S^{n},(x,\dots,x)\to\pi^{\textrm{alg}}(S,x)_{\textrm{ab}}\to
GL(V)$

and

$\pi^{\textrm{alg}}(S^{n},(x,\dots,x)\to\pi^{\textrm{alg}}(S,x)_{\textrm{ab}}\to
GL(V^{\prime})$

respectively.

As $\pi^{\textrm{alg}}(S,x)_{\textrm{ab}}$ is an abelian affine group scheme
over $k$, all its irreducible representations are one dimensional. Thus one
gets that the $\pi^{\textrm{alg}}(S,x)_{\textrm{ab}}$-module $V/V^{\prime}$
has a one dimensional quotient $W$. Thus there is a
$\pi^{\textrm{alg}}(S,x)_{\textrm{ab}}$-module surjection $V\to W$ such that
the kernel contains $V^{\prime}$. Let $\mathcal{L}_{\bullet}$ be the
$\mathfrak{S}_{n}$-equivariant stratified bundle corresponding to $W$. Thus we
have a short exact sequence of $\mathfrak{S}_{n}$-equivariant stratified
bundles

$0\to\mathcal{K}_{\bullet}\to\mathcal{F}_{\bullet}\to\mathcal{L}_{\bullet}\to
0$

where $\mathcal{F}^{\prime}_{\bullet}\subset\mathcal{K}_{\bullet}$.

By proposition 1 above, we know that
$L_{i}:=\rho_{*}\mathcal{L}_{i}^{\mathfrak{S}_{n}}$ is a line bundle on
$S^{(n)}$ and $\rho^{*}L_{i}=\mathcal{L}_{i}$

We claim that the following complex of sheaves on $S^{(n)}_{*}$ is exact for
all $i\in\mathbb{N}$

(1)
$0\to(\rho_{*}\mathcal{K}_{i})^{\mathfrak{S}_{n}}|_{S^{(n)}_{*}}\to(\rho_{*}\mathcal{F}_{i})^{\mathfrak{S}_{n}}|_{S^{(n)}_{*}}\to(\rho_{*}\mathcal{L}_{i})^{\mathfrak{S}_{n}}|_{S^{(n)}_{*}}\to
0$

It is enough to show that
$(\rho_{*}\mathcal{F}_{i})^{\mathfrak{S}_{n}}|_{S^{(n)}_{*}}\to(\rho_{*}\mathcal{L}_{i})^{\mathfrak{S}_{n}}|_{S^{(n)}_{*}}$
is surjective. We note that
$(\rho_{*}\mathcal{F}_{i})^{\mathfrak{S}_{n}}|_{S^{(n)}_{*}}=h_{*}(\mathcal{E}_{i}|_{S^{[n]}_{*}})$.
Let $C$ be the cokernel

$h_{*}(\mathcal{E}_{i}|_{S^{[n]}_{*}})\to(\rho_{*}\mathcal{L}_{i})^{\mathfrak{S}_{n}}|_{S^{(n)}_{*}}\to
C\to 0$

Pulling back under $\rho$, we get the following commutative diagram on
$S^{n}_{*}$

${{\rho^{*}h_{*}(\mathcal{E}_{i}|_{S^{[n]}_{*}})}}$${{\rho^{*}((\rho_{*}\mathcal{L}_{i})^{\mathfrak{S}_{n}}|_{S^{(n)}_{*}})}}$${{\rho^{*}C}}$${0}$${{\mathcal{F}_{i}}}$${{\mathcal{L}_{i}|_{S^{n}_{*}}}}$${0}$$\scriptstyle{=}$$\scriptstyle{=}$

The rows are exact and hence $\rho^{*}C=0$. As $\rho$ is surjective, this
implies $C=0$. Thus
$K_{i}:=(\rho_{*}\mathcal{K}_{i})^{\mathfrak{S}_{n}}|_{S^{(n)}_{*}}$ is
locally free on $S^{(n)}_{*}$

Pulling back the exact sequence (1) under $h$, we get a short exact sequence
of locally free sheaves on $S^{[n]}_{*}$

$0\to
h^{*}K_{i}|_{S^{[n]}_{*}}\to\mathcal{E}_{i}|_{S^{[n]}_{*}}\to\tilde{L}_{i}|_{S^{[n]}_{*}}\to
0$

where $\tilde{L}_{i}:=h^{*}L_{i}$.

As the complement of $S^{[n]}_{*}$ in $S^{[n]}$ is of codimension $\geq 2$ and
$\mathcal{E}_{i},L$ are locally free, the surjective morphism

$\mathcal{E}_{i}|_{S^{[n]}_{*}}\to\tilde{L}_{i}|_{S^{[n]}_{*}}$

extends to a unique morphism $\tau_{i}:\mathcal{E}_{i}\to\tilde{L}_{i}$. This
is surjective as $L$ is of rank $1$ and $\tau:=(\tau_{i})$ give a nonzero
morphism of stratified bundles

$\mathcal{E}_{\bullet}\to\tilde{L}_{\bullet}$

where
$\tilde{L}_{\bullet}:=h^{*}(\rho_{*}(\mathcal{L}_{\bullet})^{\mathfrak{S}_{n}})$.
Let $\kappa_{\bullet}$ be the kernel of the morphism
$\mathcal{E}_{\bullet}\to\tilde{L}_{\bullet}$. Then
$T(\kappa_{\bullet})=\mathcal{K}_{\bullet}$. Thus, by the induction hypothesis
on rank, there exists a stratified subbundle
$\mathcal{E}^{\prime}_{\bullet}\subset\kappa_{\bullet}\subset\mathcal{E}_{\bullet}$
such that $T(\mathcal{E}^{\prime}_{\bullet})=\mathcal{F}^{\prime}_{\bullet}$.

∎

###  6.2. Closed immersion

We begin by recalling a result from [PS20].

Let $p\in S^{(n)}$ be a point of type $(n_{1},n_{2},\dots,n_{r})$. Let
$p^{\prime}_{i}$, for $i=1,2,\dots m$ be the points in the fiber $h^{-1}(p)$.
Let $A$ be the local ring $\mathcal{O}_{S^{(n)},p}$ and $B$ be the semilocal
ring $\mathcal{O}_{S^{n}}\otimes_{\mathcal{O}_{S^{(n)}}}A$. Then $B$ is a
finite $A$ module and $B^{\mathfrak{S}_{n}}=A$.

######  Lemma 6.1.

When char $k>n_{1}$, any $\mathfrak{S}_{n}$-equivariant surjective $B$-module
homomorphism $f:M\to N$ of finitely generated $B$ modules descends to
surjective $A$-module homomorphism of the $\mathfrak{S}_{n}$-invariants
$M^{\mathfrak{S}_{n}}\to N^{\mathfrak{S}_{n}}$

This allows us to prove the following analogue of Proposition 5.3.6 in [PS20].

######  Proposition.

Let $\mathcal{E}_{\bullet}=(\mathcal{E}_{i},\alpha_{i})$ be a
$\mathfrak{S}_{n}$-equivariant stratified bundle on $S^{n}$

  1. (1)

Let $p\in S^{(n)}$ be a point of type $(n_{1},n_{2},\dots,n_{r})$. If char
$k>n_{1}$, then the sheaf $\rho_{*}\mathcal{E}_{i}^{\mathfrak{S}_{n}}$ is
locally free in a neighbourhood of $p$ for all $i$.

  2. (2)

Let $U$ denote the largest open subset where
$\rho_{*}\mathcal{E}_{i}^{\mathfrak{S}_{n}}$ is locally free, then on
$\rho^{-1}(U)$, the natural morphism

$\rho^{*}\rho_{*}\mathcal{E}_{i}^{\mathfrak{S}_{n}}\to\mathcal{E}_{i}$

is an isomorphism for all $i\in\mathbb{N}$

###### Proof.

The first assertion is proved by induction on the rank of
$\mathcal{E}_{\bullet}$. If $\mathcal{E}_{\bullet}$ is a
$\mathfrak{S}_{n}$-equivariant stratified bundle of rank $1$, then by
proposition 1, $\rho_{*}\mathcal{E}_{i}^{\mathfrak{S}_{n}}$ is locally free on
$S^{(n)}$ for all $i$. In general, as $\mathcal{E}_{\bullet}$ corresponds to a
representation of the abelian group scheme
$\pi^{\textrm{alg}}(S,x)_{\textrm{ab}}$, there exists a
$\mathfrak{S}_{n}$-equivariant short exact sequence of locally free sheaves on
$S^{n}$

$0\to\mathcal{K}_{\bullet}\to\mathcal{E}_{\bullet}\to\mathcal{L}_{\bullet}\to
0$

Pushing forward by $\rho$ and taking $\mathfrak{S}_{n}$-invariants we get the
exact sequence for all $i$

$0\to\rho(\mathcal{K}_{i})^{\mathfrak{S}_{n}}\to\rho(\mathcal{E}_{i})^{\mathfrak{S}_{n}}\to\rho(\mathcal{L}_{i})^{\mathfrak{S}_{n}}$

We claim that the homomorphism on the right is surjective in the neighbourhood
of a point $p$ of type $(n_{1},n_{2},\dots,n_{r})$. Surjectivity can be
checked after passing to a formal neighbourhood of $p$ and thus reduces to
lemma 6.1. By induction hypothesis on rank, both
$\rho(\mathcal{K}_{i})^{\mathfrak{S}_{n}}$ and
$\rho(\mathcal{L}_{i})^{\mathfrak{S}_{n}}$ are locally free on a neighbourhood
of $p$ and hence so is $\rho(\mathcal{E}_{i})^{\mathfrak{S}_{n}}$.

The second assertion follows from the observation that the natural
homomorphism

$\rho^{*}\rho_{*}\mathcal{E}_{i}^{\mathfrak{S}_{n}}\to\mathcal{E}_{i}$

is an isomorphism on $\rho^{-1}(S^{(n)}_{\circ})$ as as $\rho:S^{n}_{\circ}\to
S^{(n)}_{\circ}$ is finite étale. As the complement of $S^{n}_{\circ}$ in
$\rho^{-1}(U)$ is of codimension $\geq 2$ and both sheaves are locally free on
$\rho^{-1}(U)$, thus the natural morphism is an isomorphism. ∎

######  Proposition 6.

Let char $k>3$. The homomorphism

$f:\pi^{\textrm{alg}}(S,x)_{\textrm{ab}}\to\pi^{\textrm{alg}}(S^{[n]},\tilde{nx})$

is faithfully flat.

###### Proof.

By [[DM82], Theorem 2.21], it is enough to show that the functor

$T:\mathcal{S}(S^{[n]})\to\mathcal{S}(S^{n})$

is essentially surjective. Thus we want to show that for any
$\mathfrak{S}_{n}$-equivariant stratified bundle $\mathcal{E}_{\bullet}$ on
$S^{n}$, there exists a stratified bundle $\mathcal{F}_{\bullet}$ on $S^{[n]}$
such that $T(\mathcal{F}_{\bullet})=\mathcal{E}_{\bullet}$.

Let $U$ be the open subset of $S^{(n)}$ consisting of points of type
$(1,1,\dots,1),(2,1,\dots,1),(3,1,\dots,1)$ and $(2,1,1,\dots,1)$. By
assumption on characteristic of $k$ and the previous proposition, we get that
$\rho_{*}\mathcal{E}_{i}^{\mathfrak{S}_{n}}$ is locally free on $U$. Also we
have on $\rho^{-1}(U)$, the natural morphism

$\rho^{*}\rho_{*}\mathcal{E}_{i}^{\mathfrak{S}_{n}}\to\mathcal{E}_{i}$

is an isomorphism. Imitating proposition 1 above, this allows us to define a
stratified bundle $(\rho_{*}\mathcal{E}_{i}^{\mathfrak{S}_{n}},\beta_{i})$ on
$U$ such that
$\rho^{*}(\rho_{*}\mathcal{E}_{i}^{\mathfrak{S}_{n}},\beta_{i})\simeq\mathcal{E}_{\bullet}$.
Pulling back under $h$ to $h^{-1}(U)$ (whose complement in $S^{[n]}$ has
codimension $\geq 3$) and extending to $S^{[n]}$, we get a stratified bundle
$\mathcal{F}_{\bullet}$ such that
$T(\mathcal{F}_{\bullet})=\mathcal{E}_{\bullet}$ ∎

As $f$ is both faithfully flat and a closed immersion, we get the following
theorem

######  Theorem 6.2.

Let char $k>3$. The homomorphism

$f:\pi^{\textrm{alg}}(S,x)_{\textrm{ab}}\to\pi^{\textrm{alg}}(S^{[n]},\tilde{nx})$

is an isomorphism.

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