KLR-Schur algebra of coherent sheaves on the projective line: Tilting and PBW bases

We begin the study of Khovanov-Lauda-Rouquier type algebras associated to moduli stacks of coherent sheaves on smooth projective curves. We consider the case of $\mathbb{P}^1$ and define, for any pair $(r,d)$ of a rank and a degree, the KLR and Schur algebras $A_{r,d}, \mathcal{R}_{r,d}$ as suitable convolution algebras in the Borel-Moore homology of an analog of the Steinberg stack built from the stacks $Coh_{r,d}(\mathbb{P}^1)$. We use the tilting equivalence and Bridgeland stability conditions to construct an interpolation between the KLR or Schur algebras of the categories of coherent sheaves on $\mathbb{P}^1$ and the KLR or Schur algebras of the categories of representations of the Kronecker quiver. We also introduce a stratification of the Steinberg stacks into cohomologically pure pieces and use this to construct a PBW basis of the corresponding algebra.