Bia{\l}ynicki-Birula decomposition for reductive groups

We generalize the Bia{\l}ynicki-Birula decomposition from actions of $G_m$ on smooth varieties to actions of linearly reductive group $\mathbf{G}$ on finite type schemes and algebraic spaces. We also provide a relative version and briefly discuss the case of algebraic stacks. We define the Bia{\l}ynicki-Birula decomposition functorially: for a fixed $\mathbf{G}$-scheme $X$ and a monoid $\bar{\mathbf{G}}$ which partially compactifies $\mathbf{G}$, the BB decomposition parameterizes $\mathbf{G}$-schemes over $X$ for which the $\mathbf{G}$-action extends to the $\overline{\bf G}$-action\mathrm{.}Thefreedomofchoiceof$\overline{\bf G}$ makes the theory richer than the $G_m$-case.