# On the $K$-theoretic Hall algebra of a surface

## TL;DR

This paper introduces a $K$-theoretic Hall algebra for 0-dimensional sheaves on surfaces, proves its associativity, and constructs a related shuffle algebra homomorphism, advancing algebraic geometry and representation theory.

## Contribution

It defines a new $K$-theoretic Hall algebra for surfaces, establishing its structure and linking it to shuffle algebra frameworks.

## Key findings

- The $K$-theoretic Hall algebra is associative.
- A homomorphism to a shuffle algebra is constructed.
- The algebra generalizes previous Hall algebra concepts.

## Abstract

In this paper, we define the $K$-theoretic Hall algebra for $0$-dimensional coherent sheaves on a smooth projective surface, prove that the algebra is associative and construct a homomorphism to a redefined shuffle algebra analogous to Negut.