2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/210771In this paper, we introduce a notion of adjoint ideal sheaves along closed subvarieties of higher codimension and study its local properties using characteristic $p$ methods. When $X$ is a normal Gorenstein closed subvariety of a smooth complex variety $A$, we formulate a restriction property of the adjoint ideal sheaf $\adj_X(A)$ of $A$ along $X$ involving the l.c.i. ideal sheaf $\mathcal{D}_X$ of $X$. The proof relies on a modification of generalized test ideals of Hara and Yoshida.17 pages; v.2: minor changes, to appear in Crelles JournalAlgebraic GeometryCommutative Algebra13A35; 14B05text