# 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.

## Key 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 $R$.