Finite generation of relative log canonical algebras of semi-dlt pairs

We prove that the relative log canonical algebra of a semi-dlt pair $(X,{\mathrm{\Delta }}_X)$ projective over a scheme $T$ is finitely generated, equivalently, that $(X,{\mathrm{\Delta }}_X)$ admits a relative stable model, provided that the normalization admits a log canonical model over $T$, that $(X,{\mathrm{\Delta }}_X)$ admits a stable model over an open dense subscheme $T^0\subset T$ whose exceptional locus does not contain any stratum of the conductor, and that the log centers contained in the conductor have images meeting $T^0$. Our main contribution is a direct proof that avoids Kollár’s gluing theory entirely. As a consequence, we recover demi-normal versions of results of Hacon–Xu and Birkar. Furthermore, we derive the existence of certain MMP steps for slc pairs, recovering results of Ambro and Kollár. Finally, these results lay the groundwork for a streamlined proof of the properness of the Kollár–Shepherd-Barron–Alexeev moduli space of stable pairs, to be completed in forthcoming work.