Smoothness of moduli space of stable torsion-free sheaves with fixed determinant in mixed characteristic

TL;DR

This paper proves the smoothness of the moduli space of Gieseker stable torsion-free sheaves with fixed determinant over a family of algebraic curves in mixed characteristic, extending understanding of moduli space deformation properties.

Contribution

It establishes the smoothness of the moduli space of stable torsion-free sheaves with fixed determinant in mixed characteristic, a significant advance in algebraic geometry.

Findings

Moduli space is smooth over the base ring R.

Results hold for sheaves of rank r ≥ 2.

Applicable to families of algebraic curves in mixed characteristic.

Abstract

Let $R$ be a complete discrete valuation ring with fraction field of characteristic $0$ and algebraically closed residue field of characteristic $p>0$. Let $X_R \to \mathrm{Spec}(R)$ be a smooth projective morphism of relative dimension $1$. We prove that, given a line bundle $\mathcal{L}_R$ the moduli space of Gieseker stable torsion-free sheaves of rank $r\geq 2$ over $X_R$, with determinant $\mathcal{L}_R$, is smooth over $R$.

Equations

$$0\to {\mathcal{E}}_n\to {\mathcal{E}}_{n-1}\to \cdots \to {\mathcal{E}}_0\to \mathcal{E}\to 0.$$

$$\operatorname{\mathcal{M}}_{X_{R},\mathcal{L}_{R}}(T):=\left\{\begin{array}[]{l}S\mbox{- equivalence classes of families of pure Gieseker }\\ \mbox{ semistable sheaves }\mathcal{F}\mbox{ on }X_{T}\mbox{ with the property that }\\ \mathrm{det}(\mathcal{F})\simeq\pi^{*}_{X_{R}}\mathcal{L}_{R}\otimes\pi^{*}_{T}{\mathcal{Q}},\mbox{ for some line bundle }\mathcal{Q}\mbox{ on }T\end{array}\right\}/\sim$$

$$\begin{diagram}$$

$$\theta (k):{\mathcal{M}}_R(k)\to {\mathrm{Hom}}_R(k,M_R).$$

$$\mathcal{D}_{[F_{k}]}(A):=\left\{\begin{array}[]{l}\mbox{ coherent sheaves }\mathcal{F}_{A}\mbox{ with Hilbert polynomial }P\\ \mbox{ on }X_{A}\mbox{ flat over}A\mbox{ such that its pull-back to }X_{k}\\ \mbox{ is isomorphic to }\mathcal{F}_{k}.\end{array}\right\}$$

$$\mathcal{D}_{[\mathrm{det}(F_{k})]}(A):=\left\{\begin{array}[]{l}\mbox{ coherent sheaves }\mathcal{F}_{A}\mbox{ with Hilbert polynomial the }\\ \mbox{ same as }\mathrm{det}(\mathcal{F}_{k})\mbox{ on }X_{A}\mbox{ flat over}A\mbox{ such that its }\\ \mbox{ pull-back to }X_{k}\mbox{ is isomorphic to }\mathrm{det}(\mathcal{F}_{k}).\end{array}\right\}$$

$$\mathcal{D}e{t}_A:{\mathcal{D}}_{[F_k]}(A)\to {\mathcal{D}}_{[\det (F_k)]}(A),{\mathcal{E}}_A\mapsto \det ({\mathcal{E}}_A).$$

$${\mathcal{D}}_{[{\mathcal{F}}_k],[\det {\mathcal{F}}_k]}(A):=\mathcal{D}e{t}_A^{-1}({\mathcal{L}}_A).$$

$${\varphi }_{ii}=\mathrm{Id},{\varphi }_{ij}:{\mathcal{F}}_i^{\prime }{|}_{U_i^{\prime }\cap U_j^{\prime }}={\mathcal{F}}_{A^{\prime }}{|}_{U_i^{\prime }\cap U_j^{\prime }}\stackrel{\mathrm{Id}+(({\varphi }_{ij}^{\prime \prime }\otimes a)\circ \pi )}{\to }{\mathcal{F}}_{A^{\prime }}{|}_{U_i^{\prime }\cap U_j^{\prime }}={\mathcal{F}}_j^{\prime }{|}_{U_i^{\prime }\cap U_j^{\prime }}$$

$${\varphi }_{jk}\circ {\varphi }_{ij}(s_t^i)$$

$${\varphi }_{jk}\circ {\varphi }_{ij}(s_t^i)=\pi (s_t^i)+a({\varphi }_{ik}^{\prime \prime }(\pi (s_t^i))={\varphi }_{ik}(s_t^i).$$

$${\varphi }_{ij}:{\mathcal{L}}_i{|}_{U_{ij}^{\prime }}\stackrel{\mathrm{Id}+(({\varphi }_{ij}^{\prime \prime }\otimes a)\circ \pi )}{\to }{\mathcal{L}}_j{|}_{U_{ij}^{\prime }}$$

$${\varphi }_1:H^1({\mathrm{Hom}}_{X_k}({\mathcal{F}}_k,{\mathcal{F}}_k){\otimes }_{k}I)\to {\mathcal{D}}_{[{\mathcal{F}}_k]}(A^{\prime }),\xi \mapsto {\mathcal{F}}_{A^{\prime }}(\xi )$$

$${\varphi }_2:H^1({\mathrm{Hom}}_{X_k}(\det ({\mathcal{F}}_k),\det ({\mathcal{F}}_k)){\otimes }_{k}I)\to {\mathcal{D}}_{\det ({\mathcal{F}}_k)}(A^{\prime }),{\xi }^{\prime }\mapsto \det ({\mathcal{F}}_A^{\prime })({\xi }^{\prime })$$

$${\overline{\varphi }}_{ij}:\det ({\mathcal{F}}_i^{\prime }){|}_{U_i\cap U_j}\to \det ({\mathcal{F}}_j^{\prime }){|}_{U_i\cap U_j},s_1^{(i)}\wedge ...\wedge s_r^{(i)}\mapsto {\varphi }_{ij}(s_1^i)\wedge ...\wedge {\varphi }_{ij}(s_r^i)$$

$${\overline{\varphi }}_{jk}\circ {\overline{\varphi }}_{ij}(s_1^i\wedge \cdots \wedge s_r^i)$$

$${\overline{\varphi }}_{ij}\circ {\overline{\psi }}_i(s_1^i\wedge \cdots \wedge s_r^i)$$

$${\mathrm{tr}}_U:\mathcal{H}om({\mathcal{F}}_k,{\mathcal{F}}_k)(U)\to \mathcal{H}om(\det ({\mathcal{F}}_k),\det ({\mathcal{F}}_k))(U),(\ast )$$

$$\varphi \mapsto {\mathrm{tr}}_U(\varphi ):=(s_1\wedge ...\wedge s_r\mapsto \sum _j{s}_1\wedge ..\wedge \varphi (s_j)\wedge ...\wedge s_r).$$

$$\mathrm{tr}:{\mathrm{Hom}}_{X_k}({\mathcal{F}}_k,{\mathcal{F}}_k)\to {\mathrm{Hom}}_{X_k}(\det ({\mathcal{F}}_k),\det ({\mathcal{F}}_k))$$

$${\mathrm{tr}}_U(f\varphi )$$

$${\mathrm{tr}}_x:{\mathrm{Hom}}_{X_k}({\mathcal{F}}_{k,x},{\mathcal{F}}_{k,x})\to {\mathrm{Hom}}_{X_k}(\det ({\mathcal{F}}_{k,x}),\det ({\mathcal{F}}_{k,x}))$$

$${\mathrm{tr}}^i:H^i(X_k,\mathcal{H}om({\mathcal{F}}_k,{\mathcal{F}}_k))\to H^i(X_k,\mathcal{H}om(\det ({\mathcal{F}}_k),\det ({\mathcal{F}}_k)))\cong H^i(X_k,{\mathcal{O}}_{X_k}).$$

$${\mathrm{tr}}^1:H^1(X_k,{\mathrm{Hom}}_{X_k}({\mathcal{F}}_k,{\mathcal{F}}_k))\to H^1(X_k,{\mathrm{Hom}}_{X_k}(\det ({\mathcal{F}}_k),\det ({\mathcal{F}}_k))$$

$$0\to \ker \mathrm{tr}\to {\mathrm{Hom}}_{X_k}({\mathcal{F}}_k,{\mathcal{F}}_k)\stackrel{\mathrm{tr}}{\to }{\mathrm{Hom}}_{X_k}(\det ({\mathcal{F}}_k),\det ({\mathcal{F}}_k))\to 0.$$

$$...\to H^1(X_k,{\mathrm{Hom}}_{X_k}({\mathcal{F}}_k,{\mathcal{F}}_k))\stackrel{{\mathrm{tr}}^1}{\to }{H}^1(X_k,{\mathrm{Hom}}_{X_k}(\det ({\mathcal{F}}_k),\det ({\mathcal{F}}_k)))\to H^2(\ker \mathrm{tr})\to ...$$

$${\det }_{ij}:\Gamma (U_i\cap U_j,\mathrm{Hom}({\mathcal{F}}_i^{\prime },{\mathcal{F}}_i^{\prime }))\to \Gamma (U_i\cap U_j,\mathrm{Hom}(\det ({\mathcal{F}}_i^{\prime }),\det ({\mathcal{F}}_i^{\prime })))$$

$${\varphi }_{ij}\in \Gamma (U_i\cap U_j,\mathrm{Hom}({\mathcal{F}}_i^{\prime },{\mathcal{F}}_i^{\prime }))\mapsto {\det }_{ij}({\varphi }_{ij}):=(s_1^i\wedge ...\wedge s_r^i\mapsto {\varphi }_{ij}(s_1^i)\wedge ...\wedge {\varphi }_{ij}(s_r^i))$$

$$\underset{ij}{\det }\circ (\mathrm{Id}+({\varphi }_{ij}^{\prime \prime }\otimes a)\circ \pi )=\mathrm{Id}+({\mathrm{tr}}_{U_{ij}}({\varphi }_{ij}^{\prime \prime })\otimes a)\circ \pi .$$

$$\begin{diagram}$$

Full text

1 Introduction

Notation 1.1**.**

Let $R$ be a complete discrete valuation ring with maximal ideal $\mathfrak{m}$. Denote by $K$ its fraction field of characteristic [math] and by $k$ its residue field of characteristic $p>0$. Assume $k$ is algebraically closed. Let $X_{R}\to\operatorname{\mathrm{Spec}}(R)$ be a smooth fibred surface and $X_{k}$ its special fibre. Fix a line bundle $\mathcal{L}_{R}$ on $X_{R}$. Let $P$ be a fixed Hilbert polynomial. Throughout this note, semistability always refers to Gieseker semistability (see [6, Definition $1.2.4$]).

In [8, Theorem $0.2$], Langer proves that the moduli functor of semi(stable)torsion-free sheaves with fixed Hilbert polynomial $P$ on $X_{R}$ is uniformly (universally) corepresented by an $R$-scheme $M_{X_{R}}(P)$ (respectively $M^{s}_{X_{R}}(P)$). Recall the definition of the moduli functor of flat families of (semi)stable torsion free sheaves with fixed Hilbert poynomial $P$ and determinant $\mathcal{L}_{R}$ on $X_{R}$ (see Definition 2.2). We denote this functor by $\operatorname{\mathcal{M}}^{s}_{X_{R},\mathcal{L}_{R}}$. In this note we prove the following:

Theorem 1.2** (see Proposition 2.3, Remark 2.4 and Theorem 4.5).**

We have the following:

The moduli functor $\operatorname{\mathcal{M}}_{X_{R},\mathcal{L}_{R}}$ is uniformly corepresented by a projective $R$-scheme of finite type denoted $M_{R,\mathcal{L}_{R}}$. The open subfunctor $\operatorname{\mathcal{M}}^{s}_{X_{R},\mathcal{L}_{R}}$ for stable sheaves is universally corepresented by a $R$-scheme of finite type, denoted $M^{s}_{R,\mathcal{L}_{R}}$. 2. 2.

The morphism $M^{s}_{R,\mathcal{L}_{R}}\to\operatorname{\mathrm{Spec}}(R)$ is smooth.

Part $1$ is proven analogously to [2, Theorem $3.1$]. For part $2$, we prove that the deformation functor at a point in the moduli space $M^{s}_{R,\mathcal{L}_{R}}$ is unobstructed (see Theorem 3.19).

Note that Theorem 1.2 is proven by Langer in the case when $R$ is a $k$-algebra (see [7, Proposition $3.4$]). However, the proof does not generalize to our setup. This is because it relies on [1, Proposition $1$], the proof of which does not hold in mixed characteristic. The main difficulty is that even in the case of vector bundles it uses the structure of $R$ as a $k$-algebra in a fundamental way (see [1, Section $3$]). We use the same philosophy as [1, Proposition $1$] (of using Cech cohomology) but take a more direct approach since we are working on a family of curves.

The setup is as follows: in §2 we recall the basic definitions and results needed for this note. We also prove the existence of the moduli space of stable torsion free sheaves with fixed determinant over $\operatorname{\mathrm{Spec}}(R)$. In §3 we show that the deformation functor at a point in the moduli space $M^{s}_{R,\mathcal{L}_{R}}$ is unobstructed. Finally in §4 we prove that this moduli space is smooth over $\operatorname{\mathrm{Spec}}(R)$.

Acknowledgements: The author thanks Prof. A. Langer for a discussion during the conference ’Topics in characteristic $p>0$ and $p$-adic Geometry ’. The author is grateful to the Berlin Mathematical School for financial support.

2 Basic Definitions and results

Keep Notations 1.1

In this section we define the moduli functor of (semi)stable sheaves with fixed determinant. We prove that it is uniformly corepresented by an $R$-scheme of finite type.

Definition 2.1**.**

Let $X_{R}\to\operatorname{\mathrm{Spec}}(R)$ be as in Notation 1.1.

Let $\operatorname{\mathcal{M}}_{X_{R}/\operatorname{\mathrm{Spec}}(R)}(P)$ (as in [2, Theorem $3.1$]) of pure Gieseker semistable sheaves. For simplicity we will denote this functor by $\operatorname{\mathcal{M}}_{R}$ and the corresponding moduli space by $M_{R}$. Denote by $\mathcal{P}ic_{X_{R}}$ the moduli functor for line bundles. By assumption $X_{R}\to\operatorname{\mathrm{Spec}}(R)$ is flat, projective with integral fibres, therefore by [3, Theorem $9.4.8$] the functor $\mathcal{P}ic_{X_{R}}$ is representable. We denote this moduli space by $\mathrm{Pic}(X_{R})$. 2. 2.

By assumption $X_{R}$ is smooth over $R$. By [6, Theorem $2.1.10$], every coherent sheaf $\mathcal{E}$ on $X_{R}$ admits a locally free resolution

[TABLE]

Then $\mathrm{det}(\mathcal{E}):=\otimes\mathrm{det}(\mathcal{E}_{i})^{(-1)^{i}}$.

Therefore we can define a natural transformation $\mathrm{det}:\operatorname{\mathcal{M}}_{R}\to\mathcal{P}ic_{X_{R}}$. This induces a morphism between the schemes corepresenting these functors $M_{R}\to\mathrm{Pic}(X_{R})$.

Now we define the moduli functor for families of pure Gieseker semistable sheaves with fixed determinant.

Definition 2.2**.**

Let $X_{R}\to\operatorname{\mathrm{Spec}}(R)$ be a smooth, projective morphism and $\mathcal{L}_{R}$ a line bundle on $X_{R}$. For $P$ a fixed Hilbert polynomial, we define the moduli functor $\operatorname{\mathcal{M}}_{X_{R},\mathcal{L}_{R}}(P)$, denoted $\operatorname{\mathcal{M}}_{{R},\mathcal{L}_{R}}$ for simplicity, on $X_{R}$ of sheaves with fixed determinant $\mathcal{L}_{R}$. Let $\operatorname{\mathcal{M}}_{X_{R},\mathcal{L}_{R}}:(\mathrm{Sch}/R)^{\circ}\rightarrow(\mathrm{Sets})$ be such that for an $R$-scheme $T$,

[TABLE]

where $\pi_{X_{R}}:X_{T}\rightarrow X_{R}$ and $\pi_{T}:X_{T}\to T$ are the natural projection maps and $\mathcal{F}\sim\mathcal{F^{\prime}}$, if and only if there exists a line bundle $\mathcal{L}$ on $T$, such that $\mathcal{F}\simeq\mathcal{F^{\prime}}\otimes\pi^{*}_{T}\mathcal{L}$.

We denote by $\operatorname{\mathcal{M}}^{s}_{X_{R},\mathcal{L}_{R}}$ the subfunctor for the stable sheaves.

We note that the moduli space $M^{s}_{R,\mathcal{L}_{R}}$ is a projective $R$-scheme.

Proposition 2.3**.**

The functor $\operatorname{\mathcal{M}}^{s}_{R,\mathcal{L}_{R}}$ is universally corepresented by a $R$-scheme of finite type. We denote this scheme by $M^{s}_{R,\mathcal{L}_{R}}$.

Proof.

We know from the proof of [2, Theorem $3.1$], there exists a subset of the Quot scheme denoted $\mathcal{R}^{s}$, such that $M^{s}_{R}$ is a universal categorical quotient of this subset by the action of a certain general linear group. Let $\alpha:\mathcal{R}^{s}\rightarrow M^{s}_{R}$ denote this quotient.

The natural transformation $\operatorname{\mathcal{M}}^{s}_{R}\rightarrow\mathcal{P}ic_{X_{R}}$ which induces the determinant morphism $\mathrm{det}:M^{s}_{R}\rightarrow\mathrm{Pic}(X_{R})$. By composing the morphism det with $\alpha$ we obtain, a morphism $\mathrm{det}_{\mathcal{R}^{s}}:\mathcal{R}^{s}\rightarrow M^{s}_{R}\rightarrow\mathrm{Pic}(X_{R})$.

Let $\mathcal{R}^{s}_{\mathcal{L}_{R}}:=\mathrm{det}_{\mathcal{R}^{s}}^{-1}(\mathcal{L}_{R})$ denote the fibre of the map $\mathrm{det}_{\mathcal{R}^{s}}$ at the point corresponding to $\mathcal{L}_{R}$ and let $N_{R,\mathcal{L}_{R}}:=\mathrm{det}^{-1}(\mathcal{L}_{R})$. Let $M^{s}_{R,\mathcal{L}_{R}}$ be a universal categorical quotient of $\mathcal{R}^{s}_{\mathcal{L}_{R}}$ by $\mathrm{GL}(V)$. By definition of categorical quotient, there exists a unique morphism from $\phi^{s}_{\mathcal{L}_{R}}:M^{s}_{R,\mathcal{L}_{R}}\rightarrow N_{R,\mathcal{L}_{R}}$. Since the quotients $\mathcal{R}^{s}\rightarrow M^{s}_{R}$ and $\mathcal{R}^{s}_{\mathcal{L}_{R}}\rightarrow M^{s}_{R,\mathcal{L}_{R}}$ are $\mathrm{PGL}(V)$-bundles in the fppf topology (see [9, Lemma $6.3$]), it implies $\phi^{s}_{\mathcal{L}_{R}}$ is an isomorphism. Therefore, we have the following diagram,

[TABLE]

Finally by [6, Theorem $4.3.1$] we conclude that the functor $\mathcal{M}^{s}_{R,\mathcal{L}_{R}}$ is universally corepresented by the $R$-scheme $M^{s}_{R,\mathcal{L}_{R}}$. ∎

Remark 2.4*.*

Note that the functor $\operatorname{\mathcal{M}}_{R,\mathcal{L}_{R}}$ is corepresented by a projective $R$-scheme, denoted $M_{R,\mathcal{L}_{R}}$ of finite type. Recall the proof of [2, Theorem $3.1$]. Since $X_{R}$ is smooth, using [6, Theorem $2.1.10$], we can define a morphism $\mathrm{det}^{\prime}:\mathrm{Quot}_{X_{R}}(\mathcal{H},P)\to\mathrm{Pic}(X_{R})$ mapping a coherent sheaf on $X_{R}$ to its determinant bundle. Denote by $A$ the (scheme-theoretic) intersection of $\mathrm{det}^{{}^{\prime}-1}(\mathcal{L}_{R})$ and $Q$, where $Q$ as in the proof of [2, Theorem $3.1$]. Then the statement follows after replacing $Q$ by $A$ in the proof of [2, Theorem $3.1$].

3 Deformation of moduli spaces with fixed determinant

Keep Notations 1.1. We have seen in the proof of Proposition 2.3 how $M^{s}_{R,\mathcal{L}_{R}}$ can be considered as the fiber of the determinant morphism $\mathrm{det}:M^{s}_{R}\to\mathrm{Pic}(X_{R})$ over the point corresponding to $\mathcal{L}_{R}$. Using the trace map (see Definition 3.13), we relate the obstruction theory of the deformation functor at a point in the moduli space $M^{s}_{R}$ to the obstruction theory of the deformation functor at a point in the moduli space $\mathrm{Pic}(X_{R})$. We use this (see Theorem 3.19) to show that the deformation functor at a point in the moduli space $M^{s}_{R,\mathcal{L}_{R}}$ is unobstructed.

We begin by recalling some basic definitions.

Notation 3.1**.**

We denote by $\mathrm{Art}/R$ the category of local artinian $R$-algebras with residue field $k$. Denote by $X_{k}:=X_{R}\times_{\operatorname{\mathrm{Spec}}(R)}\operatorname{\mathrm{Spec}}(k)$ and $X_{A}:=X_{R}\times_{\operatorname{\mathrm{Spec}}(R)}\operatorname{\mathrm{Spec}}(A)$. Let $[\mathcal{F}_{k}]$ denote a closed point of $M^{s}_{R}$. As $M^{s}_{R}\to\operatorname{\mathrm{Spec}}(R)$ is a morphism of finite type, the closed points of the moduli space $M^{s}_{R}$ are $k$-points. Since $k$ is algebraically closed, by [2, Theorem $3.1$] we have a bijection

[TABLE]

Therefore to a closed point of $M^{s}_{R}$ say $[\mathcal{F}_{k}]$, we can associate a Gieseker stable sheaf $\mathcal{F}_{k}$ on the curve $X_{k}$. Since the curve $X_{k}$ is smooth, the torsion-free sheaf is in fact locally free.

We define a covariant functor at the point $[\mathcal{F}_{k}]$ in $M^{s}_{R}$.

Definition 3.2**.**

We define the deformation functor $\mathcal{D}_{[\mathcal{F}_{k}]}:\mathrm{Art}/R\to(\mathrm{Sets})$, such that for $A\in\mathrm{Art}/R$

[TABLE]

Similarly, we define a covariant functor at the point $[\mathrm{det}(\mathcal{F}_{k})]$ of the moduli space $\mathrm{Pic}(X_{R})$.

Definition 3.3**.**

Let $\mathcal{D}_{[\mathrm{det}(F_{k})]}:\mathrm{Art}/R\to(\mathrm{Sets})$ be a covariant functor such that for $A\in\mathrm{Art}/R$

[TABLE]

The following theorem gives the obstruction theories of $\mathcal{D}_{[F_{k}]}$ and $\mathcal{D}_{[\mathrm{det}(F_{k})]}$.

Using this we prove the following corollary.

Remark 3.4*.*

By [5, Theorem $7.3$]] the functors $\mathcal{D}_{[\mathcal{F}_{k}]}$ and $\mathcal{D}_{[\mathrm{det}(F_{k})]}$ have obstruction theories in the groups $H^{2}(\operatorname{\mathcal{H}om}_{X_{k}}(\mathcal{F}_{k},\mathcal{F}_{k})\otimes_{k}I)$ and $H^{2}(\operatorname{\mathcal{H}om}_{X_{k}}(\det(\mathcal{F}_{k}),\det(\mathcal{F}_{k}))\otimes_{k}I)$ respectively. For $X_{k}$ a curve, by Grothendieck vanishing theorem, $H^{2}(\operatorname{\mathcal{H}om}_{X_{k}}(\mathcal{F}_{k},\mathcal{F}_{k})\otimes_{k}I)$ and $H^{2}(\operatorname{\mathcal{H}om}_{X_{k}}(\det(\mathcal{F}_{k}),\det(\mathcal{F}_{k}))\otimes_{k}I)$ vanish. Therefore, $\mathcal{D}_{[\mathcal{F}_{k}]}$ and $\mathcal{D}_{[\mathrm{det}(F_{k})]}$ are unobstructed.

Now we define a natural transformation between the two deformation functors.

Definition 3.5**.**

By assumption $\mathcal{F}_{k}$ is a locally-free $\mathcal{O}_{X_{R}}$ module. Moreover, by [5, Exercise $7.1$] any coherent sheaf $\mathcal{F}_{A}$ on $X_{A}$ satisfying the property $\mathcal{F}_{A}\otimes_{\mathcal{O}_{X_{A}}}\mathcal{O}_{X_{k}}\simeq\mathcal{F}_{k}$ is a locally free $\mathcal{O}_{X_{A}}$-module. Therefore, the notion of determinant is well-defined for any coherent sheaf on $X_{A}$ which pulls back to $\mathcal{F}_{k}$.

We define a natural transformation $\mathcal{D}et:\mathcal{D}_{[\mathcal{F}_{k}]}\to\mathcal{D}_{[\mathrm{det}(F_{k})]}$ such that for $A\in\mathrm{Art}/R$,

[TABLE]

Using this we define a deformation functor at a point in the moduli space $M^{s}_{R,\mathcal{L}_{R}}$.

Definition 3.6**.**

Let $\mathcal{L}_{R}$ be as in Notation 1.1. For $A$ a $R$-algebra, denote by $\mathcal{L}_{A}$ the pullback $p_{A}^{*}{\mathcal{L}_{R}}$ under the natural morphism $p_{A}:X_{A}\to X_{R}$.

We define a functor $\mathcal{D}_{[\mathcal{F}_{k}],[\mathrm{det}\mathcal{F}_{k}]}:\mathrm{Art}/R\to(\mathrm{Sets})$, such that for $A\in\mathrm{Art}/R$.

[TABLE]

3.7*.*

Group action on the torsors: By [5, Theorem $7.3$], the set $\mathcal{D}_{[F_{k}]}(A^{\prime})$ (respectively $\mathcal{D}_{[\mathrm{det}(F_{k})]}(A^{\prime})$) is a torsor under the action of $H^{1}(\operatorname{\mathcal{H}om}_{X_{k}}(\mathcal{F}_{k},\mathcal{F}_{k})\otimes_{k}I)$ (respectively $H^{1}(\operatorname{\mathcal{H}om}_{X_{k}}(\det(\mathcal{F}_{k}),\det(\mathcal{F}_{k}))\otimes_{k}I)$).

Since $X_{k}$ is noetherian, we can identify the sheaf cohomology $H^{1}(X_{k},\operatorname{\mathcal{H}om}(\mathcal{F}_{k},\mathcal{F}_{k})\otimes_{k}I)$ with the Cech cohomology $\check{H}^{1}(\mathcal{U},\operatorname{\mathcal{H}om}(\mathcal{F}_{k},\mathcal{F}_{k})\otimes_{k}I)$, where $\mathcal{U}$ is an affine open covering of $X_{k}$. Then an element, say $\xi$ of the cohomology group $H^{1}(\operatorname{\mathcal{H}om}(\mathcal{F}_{k},\mathcal{F}_{k})\otimes_{k}I)$ can be seen as a collection of elements $\{\phi^{\prime}_{ij}\}\in\Gamma(U_{i}\cap U_{j},\operatorname{\mathcal{H}om}(\mathcal{F}^{\prime}_{k},\mathcal{F}^{\prime}_{k}))$ satisfying the cocycle condition i.e. for any $i,j,k$, we have $\phi^{\prime}_{ik}|_{U_{ijk}}=\phi^{\prime}_{jk}|_{U_{ijk}}+\phi^{\prime}_{ij}|_{U_{ijk}}$. Since $I$ is a $k$-vector space, $\check{H}^{1}(\mathcal{U},\operatorname{\mathcal{H}om}(\mathcal{F}_{k},\mathcal{F}_{k})\otimes_{k}I)\simeq\check{H}^{1}(\mathcal{U},\operatorname{\mathcal{H}om}(\mathcal{F}_{k},\mathcal{F}_{k}))\otimes_{k}I$. Therefore, $\{\phi^{\prime}_{ij}\}_{i,j}$ is of the form $\{\phi^{\prime\prime}_{ij}\otimes a\}_{i,j}$ for $a\in I$ not depending on $i,j$ and $\phi^{\prime\prime}_{ij}\in\Gamma(U_{i}\cap U_{j},\operatorname{\mathcal{H}om}(\mathcal{F}_{k},\mathcal{F}_{k}))$ satisfying $\phi^{\prime\prime}_{ik}|_{U_{ijk}}=\phi^{\prime\prime}_{jk}|_{U_{ijk}}+\phi^{\prime\prime}_{ij}|_{U_{ijk}}$.

Let $\mathcal{F}_{A^{\prime}}$ be an extension of $\mathcal{F}_{A}$ on $X_{A^{\prime}}$ i.e an element of $\mathcal{D}_{[\mathcal{F}_{k}]}(A^{\prime})$. Since it is locally free, there exists a covering $\mathcal{U}^{\prime}=\{U^{\prime}_{i}\}$ of $X_{A^{\prime}}$ by such that $\mathcal{F}_{A^{\prime}}|_{U^{\prime}_{i}}$ is $\operatorname{\mathcal{O}}_{U^{\prime}_{i}}$-free. Denote by $\mathcal{U}:=\{U_{i}\}$ the cover of $X_{k}$ where $U_{i}:=U^{\prime}_{i}\cap X_{k}$. We know from the proof of [5, Theorem $7.3$] that $\mathcal{F}_{A^{\prime}}(\xi)$ is given by a collection of sheaves $\mathcal{F}^{\prime}_{i}:=\mathcal{F}_{A^{\prime}}|_{U^{\prime}_{i}}$ and isomorphisms $\phi_{ij}:\mathcal{F}^{\prime}_{i}|_{U_{i}\cap U_{j}}\to\mathcal{F}^{\prime}_{j}|_{U_{i}\cap U_{j}}$ such that

[TABLE]

where $\phi_{ij}^{\prime\prime},a$ are as above and $\pi$ is the natural restriction morphism $\mathcal{F}_{A^{\prime}}\to\mathcal{F}_{k}$. Then by [4, Ex. II.$1.22$], $\mathcal{F}_{A^{\prime}}(\xi)$ glues to a sheaf if the morphisms $\{\phi_{ij}\}$ satisfy the cocycle condition. In the following lemma we prove that this is indeed the case.

Lemma 3.8**.**

Let $\mathcal{F}^{\prime}_{i}$ and $\phi_{ij}$ be as above. The morphisms $\{\phi_{ij}\}$ satisfy the cocycle condition i.e. for any $i,j,k\ \phi_{ik}=\phi_{jk}\circ\phi_{ij}$.

Proof.

It suffices to prove this equality for the basis elements, say $s^{i}_{1},\dots,s^{i}_{r}$ generating $\mathcal{F}^{\prime}_{i}|_{U^{\prime}_{i}\cap U^{\prime}_{j}\cap U^{\prime}_{k}}$. For any basis element $s^{i}_{t}$,

[TABLE]

because $a^{2}=0$ in $A^{\prime}$. Since $\phi^{\prime\prime}_{ik}=\phi^{\prime\prime}_{ij}+\phi^{\prime\prime}_{jk}$, we have

[TABLE]

This shows that $\{\phi_{ij}\}_{i,j}$ satisfy the cocycle condition. ∎

Using this we conclude that $\mathcal{F}_{A^{\prime}}(\xi)$, obtained by glueing the sheaves $\mathcal{F}^{\prime}_{i}$ along the isomorphism $\phi_{ij}$ is a sheaf.

Similarly, an element say $\xi^{\prime}$ in $H^{1}(\operatorname{\mathcal{H}om}_{X_{k}}(\det(\mathcal{F}_{k}),\det(\mathcal{F}_{k}))\otimes_{k}I)$ acts on an element in $\mathcal{D}_{[\mathrm{det}(\mathcal{F}_{k})]}(A^{\prime})$, say $\mathrm{det}(\mathcal{F}_{A^{\prime}})$ to produce a line bundle $\mathrm{det}(\mathcal{F}_{A^{\prime}})(\xi^{\prime})$ given by a family of sheaves $\{\mathcal{L}_{i}:=\mathcal{L}_{A^{\prime}}|_{U_{i}}\}$ and isomorphisms

[TABLE]

where $\phi^{\prime\prime}_{ij}\in\Gamma(U_{i}\cap U_{j},\operatorname{\mathcal{H}om}(\mathrm{det}(\mathcal{F}_{k}),\mathrm{det}(\mathcal{F}_{k}))\otimes_{k}I)$ is the collection of sections corresponding to $\xi^{\prime}$ given by the ismomorphism $H^{1}(\operatorname{\mathcal{H}om}_{X_{k}}(\det(\mathcal{F}_{k}),\det(\mathcal{F}_{k}))\otimes_{k}I)\simeq\check{H}^{1}(\mathcal{U},\operatorname{\mathcal{H}om}(\mathrm{det}(\mathcal{F}_{k}),\mathrm{det}(\mathcal{F}_{k})))\otimes_{k}I$. Again by Lemma 3.8, $\mathrm{det}(\mathcal{F}_{A^{\prime}})(\xi^{\prime})$ is a sheaf.

Definition 3.9**.**

We have the following definitions.

We define a map

[TABLE]

which uniquely associates an extension $\mathcal{F}_{A^{\prime}}(\xi)$ of $\mathcal{F}_{A^{\prime}}$ (using Lemma 3.8) to an element $\xi$ of $H^{1}(\operatorname{\mathcal{H}om}_{X_{k}}(\mathcal{F}_{k},\mathcal{F}_{k})\otimes_{k}I)$. 2. 2.

Replacing $\mathcal{F}_{A^{\prime}}$ by $\mathrm{det}(\mathcal{F}_{A^{\prime}})$ and starting with $\mathrm{det}(\mathcal{F}_{A^{\prime}})$ we associate an extension say $(\mathrm{det}(\mathcal{F}_{A}^{\prime}))(\xi^{\prime})$ to an element $\xi^{\prime}$ of $H^{1}(\operatorname{\mathcal{H}om}_{X_{k}}(\det(\mathcal{F}_{k}),\det(\mathcal{F}_{k}))\otimes_{k}I)$. Hence we define a map

[TABLE]

Remark 3.10*.*

Note that by Corollary 3.4, there exist surjective morphisms $r_{1}:\mathcal{D}_{\mathcal{F}_{k}}(A^{\prime})\twoheadrightarrow\mathcal{D}_{\mathcal{F}_{k}}(A)$ and $r_{2}:\mathcal{D}_{\det(\mathcal{F}_{k})}(A^{\prime})\twoheadrightarrow\mathcal{D}_{\det(\mathcal{F}_{k})}(A)$. By [5, Theorem $7.3$], $r_{1}^{-1}(\mathcal{F}_{A})=\mathrm{Im}(\phi_{1})$, $r_{2}^{-1}(\mathrm{det}(\mathcal{F}_{A}))=\mathrm{Im}(\phi_{2})$.

The following lemma tells us that taking the determinant commutes with glueing of the sheaf.

Lemma 3.11**.**

The determinant of the sheaf $\mathcal{F}_{A^{\prime}}(\xi)$ is the line bundle obtained by glueing $\{\det(\mathcal{F}^{\prime}_{i})\}$ along the isomorphisms

[TABLE]

where $s_{1}^{i},...,s_{r}^{i}$ are the basis elements of $\mathcal{F}^{\prime}_{i}|_{U_{i}\cap U_{j}}$.

Proof.

By Lemma 3.8 for all $t=1,\dots,r$, we have $\phi_{ik}(s_{t}^{i})=\phi_{jk}(s^{i}_{t})\circ\phi_{ij}(s_{t}^{i})$. Then,

[TABLE]

Hence the morphisms $\{\overline{\phi}_{ij}\}$ satisfy the cocycle condition i.e $\overline{\phi}_{ik}=\overline{\phi}_{jk}\circ\overline{\phi}_{jk}$.

By Lemma 3.8, there exist isomorphisms $\psi_{i}:\mathcal{F}_{A^{\prime}}(\xi)|_{U^{\prime}_{i}}\simeq\mathcal{F}^{\prime}_{i}$ satisfying $\psi_{j}|_{U_{ij}}=\phi_{ij}\circ\psi_{i}|_{U_{ij}}$. We define $\overline{\psi}_{i}:\mathrm{det}(\mathcal{F}_{A^{\prime}}(\xi))|_{U^{\prime}_{i}}\simeq\mathrm{det}(\mathcal{F}^{\prime}_{i})$ as follows. Let $s^{i}_{1},\dots,s_{r}^{i}$ be the basis of $\mathcal{F}_{A^{\prime}}(\xi)|_{U_{i}}$. Then $\overline{\psi}_{i}(s_{1}^{i}\wedge\dots\wedge s_{r}^{i}):=\psi_{i}(s_{1}^{i})\wedge\dots\wedge\psi_{i}(s_{r}^{i})$. Therefore

[TABLE]

Then by the uniqueness of glueing mentioned in [4, Ex. II.$1.22$], $\{\mathrm{det}(\mathcal{F}_{i})\}$ glues along the isomorphisms $\{\overline{\phi}_{ij}\}_{i,j}$ to $\mathrm{det}(\mathcal{F}_{A^{\prime}})(\xi)$. ∎

3.12*.*

We relate the obstruction theory of $\mathcal{D}_{[F_{k}]}$ to that of $\mathcal{D}_{[\mathrm{det}(F_{k})]}$ by relating the action of the group $H^{1}(\operatorname{\mathcal{H}om}_{X_{k}}(\mathcal{F}_{k},\mathcal{F}_{k})\otimes_{k}I)$ on the vector bundle to the action of the group $H^{1}(\operatorname{\mathcal{H}om}_{X_{k}}(\det(\mathcal{F}_{k}),\det(\mathcal{F}_{k}))\otimes_{k}I)$ on the determinant of the vector bundle. This relation is given by the trace map which we recall here.

Definition 3.13**.**

Let $U$ be an affine open set on which $\mathcal{F}_{k}$ is free, generated by sections say $s_{1},...,s_{r}$ (for $r=\mathrm{rk}(F_{k})$). Recall the map,

[TABLE]

[TABLE]

Let $\mathcal{U}:=\{U_{i}\}$ be a small enough open cover of $X_{k}$ such that $\mathcal{F}_{k}$ is free on each $U_{i}$. Then the trace map is given by

[TABLE]

such that $\mathrm{tr}|_{U_{i}}=\mathrm{tr}_{U_{i}}$ for any affine open set $U_{i}$ of $X_{k}$.

Remark 3.14*.*

Note that the morphism $\mathrm{tr}_{U}$ is $\mathcal{O}_{X_{k}}$ linear. Let $f\in\mathcal{O}_{X_{k}}(U)$. Then

[TABLE]

Lemma 3.15**.**

The morphism $\mathrm{tr}$ is surjective.

Proof.

It suffices to prove surjectivity on the level of stalks. Let $x\in X_{k}$ be a closed point. Consider the induced morphism

[TABLE]

and basis $s_{1},...,s_{r}\in\mathcal{F}_{k,x}$. Since the map $\mathrm{tr}_{x}$ is $\operatorname{\mathcal{O}}_{X_{k},x}$ linear and $\operatorname{\mathcal{H}om}_{\operatorname{\mathcal{O}}_{X_{k}}}(\det(\mathcal{F}_{k,x}),\det(\mathcal{F}_{k,x}))\cong\operatorname{\mathcal{O}}_{X_{k},x}$, it suffices to show that $\mathrm{Id}\in\mathrm{Im}(\mathrm{tr}_{x})$. Let $\phi\in\operatorname{\mathcal{H}om}_{X_{k}}(\mathcal{F}_{k,x},\mathcal{F}_{k,x})$ defined as $\phi(s_{i})=s_{i}$ for $i=1$ and [math] otherwise. This concludes the proof. ∎

We can define the trace map cohomologically as follows:

Definition 3.16**.**

Let $\mathcal{U}:=\{U_{i}\}$ be a small enough open affine cover of $X_{k}$ such that $\mathcal{F}_{k}$ is free on each $U_{i}$. Using [4, III. Theorem $4.5$] we define Čech cocycle $\mathcal{C}^{p}(\mathcal{U},\mathcal{H}om(\mathcal{F}_{k},\mathcal{F}_{k}))$ (resp $\mathcal{C}^{p}(\mathcal{U},\mathcal{H}om(\det(\mathcal{F}_{k}),\det(\mathcal{F}_{k}))$), such that the corresponding Čech cohomology coincides with the sheaf cohomology $H^{i}(X_{k},\mathcal{H}om(\mathcal{F}_{k},\mathcal{F}_{k}))$ (resp $H^{i}(X_{k},\mathcal{H}om(\det(\mathcal{F}_{k}),\det(\mathcal{F}_{k})))$). The morphism $(\ast)$ of Definition induces a morphism on cohomologies

[TABLE]

As a corollary to Lemma 3.15 we have:

Corollary 3.17**.**

The morphism induced on cohomology

[TABLE]

is surjective.

Proof.

Consider the short exact sequence,

[TABLE]

We get the following terms in the associated long exact sequence,

[TABLE]

Since $X_{k}$ is a curve, by Grothendieck’s vanishing theorem, $H^{2}(\ker(\mathrm{tr}))=0$. Therefore, the morphism $\mathrm{tr}^{1}$ is surjective. ∎

The following proposition tells us that the determinant map ’commutes’ with the trace map.

Proposition 3.18**.**

Notation as in 3.7. Let

[TABLE]

be a morphism defined by

[TABLE]

where $s_{1}^{i},...,s_{r}^{i}$ are the basis elements of $\mathcal{F}^{\prime}_{i}|_{U_{i}\cap U_{j}}$. Then for any pair $i\neq j$, we have

[TABLE]

In other words, the following diagram is commutative:

[TABLE]

Proof.

Let $s_{1}^{i},...,s_{r}^{i}$ be the sections generating $\mathcal{F}^{\prime}_{i}|_{U_{i}\cap U_{j}}$.

Any section of $\operatorname{\mathcal{H}om}(\det(\mathcal{F}^{\prime}_{i}),\det(\mathcal{F}^{\prime}_{i}))$ is (uniquely) defined by the image of $s_{1}^{i}\wedge...\wedge s_{r}^{i}$. Hence it suffices to prove

[TABLE]

For $1\leq t\leq r$, $(\mathrm{Id}+(\phi_{ij}^{\prime\prime}\otimes a)\circ\pi)(s_{t}^{i})=s_{t}^{i}+a\phi_{ij}^{\prime\prime}(\pi(s_{t}^{i}))$ and since $I.m_{A^{\prime}}=0,a^{t}=0$ for $t>1$. Hence,

[TABLE]

[TABLE]

This completes the proof of the proposition. ∎

We end this section with the following theorem.

Theorem 3.19**.**

The functor $\mathcal{D}_{[\mathcal{F}_{k}],[\det(\mathcal{F}_{k})]}$ is unobstructed.

Proof.

Let $A^{\prime}\twoheadrightarrow A$ be a small extension in $\mathrm{Art}/R$ and $\phi_{1},\phi_{2}$ be as in Definition 3.9. Recall the surjective morphisms $r_{1},r_{2}$ from Remark 3.10. Then we have the following diagram.

[TABLE]

where the upper right square and the lower right square are commutative by definition and the lower left square is commutative by Proposition 3.18. To prove that $\mathcal{D}_{[\mathcal{F}_{k}],[\det(\mathcal{F}_{k})]}$ is unobstructed, we need to show that $\psi$ is surjective. Let $\mathcal{L}_{A}$ be the unique pull-back of $\mathcal{L}_{R}$ under the morphism $X_{A}\to X_{R}$ and $\mathcal{F}_{A}$ be an element in $\mathcal{D}_{[F_{k}],[\det(\mathcal{F}_{k})]}(A)$. Since $\mathcal{D}_{[\mathcal{F}_{k}],[\det(\mathcal{F}_{k})]}(A^{\prime})=\mathrm{det}(\mathcal{L_{A}^{\prime}})$ where $\mathcal{L}_{A^{\prime}}$ is $\pi^{*}\mathcal{L}_{R}$ for $\pi:X_{A^{\prime}}\to X_{R}$, we need to prove there exists a sheaf $\mathcal{F}_{A^{\prime}}$ on $X_{A^{\prime}}$ with determinant $\mathcal{L}_{A^{\prime}}$ which is an extension of $\mathcal{F}_{A}$.

By definition $r_{2}(\mathcal{L}_{A^{\prime}})=\mathcal{L}_{A}$. Since $\phi_{1}$ and $\phi_{2}$ are injective, $r^{-1}_{1}(\mathcal{F}_{A})=\mathrm{Im}(\phi_{1})$ and $r_{2}^{-1}(\mathcal{L}_{A})=\mathrm{Im}(\phi_{2})$. Therefore, there exists $t\in H^{1}(\operatorname{\mathcal{H}om}_{X_{k}}(\det(\mathcal{F}_{k}),\det(\mathcal{F}_{k}))\otimes_{k}I)$ such that $\phi_{2}(t)=\mathcal{L}_{A^{\prime}}$. By Corollary 3.17, $\mathrm{tr}^{1}\otimes\mathrm{Id}$ is surjective. Hence there exists $t^{\prime}\in H^{1}(\operatorname{\mathcal{H}om}_{X_{k}}(\mathcal{F}_{k},\mathcal{F}_{k})\otimes_{k}I)$ such that $\mathrm{tr}^{1}\otimes\mathrm{Id}(t^{\prime})=t$. Denote by $\mathcal{F}_{A^{\prime}}:=\phi_{1}(t^{\prime})$. By commutativity of the lower left square, $\det(\mathcal{F}_{A^{\prime}})=\mathcal{L}_{A^{\prime}}$. This concludes the proof of the theorem. ∎

4 Main results

In Theorem 3.19, we showed that the deformation functor $\mathcal{D}_{[\mathcal{F}_{k}],[\det(\mathcal{F}_{k})]}$ is unobstructed for any closed point $[\mathcal{F}_{k}]$ of the moduli space $M^{s}_{R,\mathcal{L}_{R}}$. In this section we prove that this functor is in fact prorepresented by the completion of the local ring at the point $[\mathcal{F}_{k}]$ (see Proposition 4.4). Using this we prove that the moduli space $M^{s}_{R,\mathcal{L}_{R}}$ of pure stable sheaves with fixed determinant $\mathcal{L}_{R}$ over $X_{R}$, is smooth over $\operatorname{\mathrm{Spec}}(R)$.

Notation 4.1**.**

Keep Notations 1.1 and 3.1. Let $[\mathcal{F}_{k}]$ be a $k$-rational point of $M^{s}_{R,\mathcal{L}_{R}}$ and denote by $\Lambda^{\prime\prime}:=\hat{\operatorname{\mathcal{O}}}_{M^{s}_{R,[\mathcal{F}_{k}]}}$, the completion of the local ring $\operatorname{\mathcal{O}}_{M^{s}_{R},[\mathcal{F}_{k}]}$. Under the determinant morphism $\mathrm{det}:M^{s}_{R}\to\mathrm{Pic}_{X_{R}}$, the line bundle $\mathrm{det}(\mathcal{F}_{k})$ is a $k$-point of $\mathrm{Pic}(X_{R})$. Denote by $\Lambda^{\prime}:=\hat{\operatorname{\mathcal{O}}}_{\mathrm{Pic}(X_{R}),[\mathrm{det}(\mathcal{F}_{k})]}$ and by $\Lambda:=\hat{\operatorname{\mathcal{O}}}_{M^{s}_{R,\mathcal{L}_{R}},[\mathcal{F}_{k}]}$.

Definition 4.2**.**

By ${\underline{\hat{\operatorname{\mathcal{O}}}}_{M^{s}_{R},[\mathcal{F}_{k}]}}$ we denote the covariant functor

[TABLE]

We define the functors $\underline{\hat{\operatorname{\mathcal{O}}}}_{\mathrm{Pic}(X_{R}),[\mathrm{det}(\mathcal{F}_{k})]}$ and $\underline{\hat{\operatorname{\mathcal{O}}}}_{M^{s}_{R,\mathcal{L}_{R}},[\mathcal{F}_{k}]}$ similarly.

Lemma 4.3**.**

The deformation functor $\mathcal{D}_{[\mathcal{F}_{k}]}$ (resp. $\mathcal{D}_{[\mathcal{L}_{k}]}$) are pro-representable by $\hat{\operatorname{\mathcal{O}}}_{M^{s}_{R,[\mathcal{F}_{k}]}}$ (resp. $\hat{\operatorname{\mathcal{O}}}_{\mathrm{Pic}_{X_{R}},[\mathrm{det}(\mathcal{F}_{k})]}$).

Proof.

Recall from the proof of [2, Theorem $3.1$], that for $m$ sufficiently large, $\mathcal{R}^{s}$ is the open subset of $\mathcal{Q}uot(\mathcal{H};P)$ where $\mathcal{H}:=\operatorname{\mathcal{O}}_{X_{R}}(-m)^{P(m)}$ parametrizing stable quotients. By [9, Lemma $6.3$], $\phi:\mathcal{R}^{s}\to M_{R}^{s}$ is an etale $\mathrm{PGL}(V)$-principal bundle . Therefore, $\hat{\operatorname{\mathcal{O}}}_{\mathcal{R}^{s},[\mathcal{F}_{k}]}\cong\hat{\operatorname{\mathcal{O}}}_{{M}^{s}_{R},[\mathcal{F}_{k}]}$.

Denote by $Q:=\mathcal{Q}uot(\mathcal{H};P)$ and by $\mathcal{D}_{Q,[\mathcal{F}_{k}]}$ the deformation functor corresponding to the Quot-scheme at the point $[\mathcal{F}_{k}]$. Recall that for any local Artin ring $A$, $\mathrm{Pic}(\operatorname{\mathrm{Spec}}(A))=0$, hence $\mathcal{D}_{[\mathcal{F}_{k}]}=\mathcal{D}_{Q,[\mathcal{F}_{k}]}$. Since the functor $\mathrm{Quot}$ is representable, the deformation functor $\mathcal{D}_{Q,[\mathcal{F}_{k}]}$ is pro-representable by $\underline{\hat{\operatorname{\mathcal{O}}}}_{Q,[\mathcal{F}_{k}]}$ i.e.,

[TABLE]

where the second isomorphism follows from the fact that $\mathcal{R}^{s}$ is an open subset of $Q$. Therefore, $\mathcal{D}_{[\mathcal{F}_{k}]}$ is isomorphic to $\underline{\hat{\operatorname{\mathcal{O}}}}_{M^{s}_{R},[\mathcal{F}_{k}]}$.

Using the same argument we can show that $\mathcal{D}_{[\mathrm{det}(\mathcal{F}_{k})]}\cong\underline{\hat{\operatorname{\mathcal{O}}}}_{\mathrm{Pic}_{X_{R}},[\mathrm{det}(\mathcal{F}_{k})]}$. This proves the lemma. ∎

Using this lemma we prove the following proposition.

Proposition 4.4**.**

The deformation functor $\mathcal{D}_{[\mathcal{F}_{k}],[\mathrm{det}(\mathcal{F}_{k})]}$ is pro-represented by the completion of the local ring $\mathcal{O}_{M^{s}_{R,\mathcal{L}_{R}},[\mathcal{F}_{k}]}$.

Proof.

By Lemma 4.3, $\mathcal{D}_{[\mathcal{F}_{k}]}$ (respectively $\mathcal{D}_{[\mathrm{det}(\mathcal{F}_{k})]}$) is pro-represented by $\hat{\operatorname{\mathcal{O}}}_{M^{s}_{R},[\mathcal{F}_{k}]}$ (respectively $\hat{\operatorname{\mathcal{O}}}_{\mathrm{Pic}(X_{R}),[\mathrm{det}(\mathcal{F}_{k})]}$). We have a natural transformation

[TABLE]

induced by the determinant morphism, $\mathrm{det}:M^{s}_{R}\to\mathrm{Pic}(X_{R})$ localized at the point $[F_{k}]$. Let $A\in\mathrm{Art}/R$ and $\mathcal{L}_{A}$ be the pullback of the line bundle $\mathcal{L}_{R}$ under the morphism $X_{A}\to X_{R}$. Recall the natural transformation ${\mathcal{D}et}_{A}$ defined in Definition 3.5. We have the following commutative diagram

[TABLE]

Hence the deformation functor $\mathcal{D}_{[\mathcal{F}_{k}],[\mathrm{det}(\mathcal{F}_{k})]}(A)\cong{\underline{\mathrm{det}}_{A}}^{-1}(\phi_{\mathcal{L}_{A}})$, where $\phi_{\mathcal{L}_{A}}:=\sigma(\mathcal{L}_{A})$. Therefore to prove that $\mathcal{D}_{[\mathcal{F}_{k}],[\mathrm{det}(\mathcal{F}_{k})]}$ is pro-represented by $\underline{\hat{\operatorname{\mathcal{O}}}}_{M^{s}_{R,\mathcal{L}_{R}},[\mathcal{F}_{k}]}$, we need to show that for any $A\in\mathrm{Art}/R$,

[TABLE]

By Lemma 4.3, $\mathcal{D}_{[\mathrm{det}(\mathcal{F}_{k})]}(A)\xrightarrow{\sim}\operatorname{\mathrm{Hom}}_{R}(\Lambda^{\prime},A)$. Hence for a fixed element $\mathcal{L}_{A}\in\mathcal{D}_{[\mathrm{det}(\mathcal{F}_{k})]}(A)$, the corresponding morphism from $\operatorname{\mathrm{Spec}}(A)\to\operatorname{\mathrm{Spec}}(\Lambda^{\prime})$ is unique and this is the morphism $\phi_{\mathcal{L}_{A}}$. This implies the commutativity of the following diagram

[TABLE]

where the morphism $\operatorname{\mathrm{Spec}}(R)\to\operatorname{\mathrm{Spec}}(\Lambda^{\prime})$ is the morphism corresponding to the line bundle $\mathcal{L}_{R}$. Then the bijection in (1) follows from the property of fibre product and the following diagram.

[TABLE]

Since $A$ was arbitrary, (1) holds for any $A\in\mathrm{Art}/R$. Hence $\mathcal{D}_{[\mathcal{F}_{k}],[\mathrm{det}(\mathcal{F}_{k}]}$ is pro-represented by $\hat{\mathcal{O}}_{M^{s}_{R,\mathcal{L}_{R}},[\mathcal{F}_{k}]}$. ∎

Using this we prove the following theorem.

Theorem 4.5**.**

The morphism $M^{s}_{R,\mathcal{L}_{R}}\to\operatorname{\mathrm{Spec}}(R)$ is smooth.

Proof.

Since the scheme $M^{s}_{R,\mathcal{L}_{R}}$ is noetherian and smoothness is an open condition, it suffices to check that the morphism $M^{s}_{R,\mathcal{L}_{R}}\to\operatorname{\mathrm{Spec}}(R)$ is smooth at closed points. Let $[\mathcal{F}_{k}]$ be a closed point of $M^{s}_{R,\mathcal{L}_{R}}$. Since the morphism $M^{s}_{R,\mathcal{L}_{R}}\to\operatorname{\mathrm{Spec}}(R)$ is of finite type, to prove that it is smooth at the point $[\mathcal{F}_{k}]$, we need to show that the functor $\underline{\hat{\operatorname{\mathcal{O}}}}_{M^{s}_{R,\mathcal{L}_{R}},[\mathcal{F}_{k}]}$ is unobstructed.

By Proposition 4.4, the completion of the local ring $\mathcal{O}_{M^{s}_{R,\mathcal{L}_{R}},[\mathcal{F}_{k}]}$ pro-represents the functor $\mathcal{D}_{[\mathcal{F}_{k}],[\mathrm{det}(\mathcal{F}_{k})]}$, i.e $\underline{\hat{\operatorname{\mathcal{O}}}}_{M^{s}_{R,\mathcal{L}_{R}},[\mathcal{F}_{k}]}\simeq\mathcal{D}_{[\mathcal{F}_{k}],[\mathrm{det}(\mathcal{F}_{k}]}$. By Theorem 3.19, the deformation functor $\mathcal{D}_{[\mathcal{F}_{k}],[\mathrm{det}(\mathcal{F}_{k})]}$ is unobstructed. Hence the functor $\underline{\hat{\mathcal{O}}}_{M^{s}_{R,\mathcal{L}_{R}},[\mathcal{F}_{k}]}$ is unobstructed. This implies $\hat{\mathcal{O}}_{M^{s}_{R,\mathcal{L}_{R}},[\mathcal{F}_{k}]}$ is unobstructed. Hence, the morphism $M^{s}_{R,\mathcal{L}_{R}}\to\operatorname{\mathrm{Spec}}(R)$ is smooth at the point $[\mathcal{F}_{k}]$. ∎