Finite ideal-power filtrations and coherent sheaves on the closed subscheme

Choose a finite affine cover of $U$. On one affine, write $M$ as a finite $A$-module and $I=(f_1,\dots ,f_r)$. Since its support is contained in $V(I)$, its localization at each $f_j$ is zero. Finitely many module generators then give an exponent $n_j\ge 1$ such that $f_j^{n_j}M=0$. Every monomial of degree $N=1+{\sum }_j(n_j-1)$ in the $f_j$ contains one such power, so $I^{N}M=0$. If $I=0$, take $N=1$. Taking the maximum of the finitely many affine exponents gives a global exponent: higher ideal powers are contained in lower ones.

Each ${\mathcal{I}}^{r}M$ and its quotient by ${\mathcal{I}}^{r+1}M$ is coherent by Noetherianity; the quotient is killed by $\mathcal{I}$ by multiplication. Thus step 1.1 supplies the stated finite filtration in the abelian category ${\mathcal{A}}_T$.

For any sheaf on the closed subspace, closed direct image has its given stalks on $T$ and zero stalks off $T$, so it is exact. A module sheaf annihilated by $\mathcal{I}$ descends uniquely to an ${\mathcal{O}}_T$-module sheaf on that subspace: locally this is simply restriction of scalars for $A\to A\mathrm{/}I$, with inverse giving the same module its $A\mathrm{/}I$ action. These local constructions and their maps glue because the action factors uniquely through the quotient sheaf of rings. Finiteness is equivalent over $A$ and $A\mathrm{/}I$, which are Noetherian, so this identifies precisely the coherent categories. Hence $i_{\ast }$ is fully faithful, exact and has image ${\mathcal{B}}_T$. Subobjects and quotients of a module killed by $\mathcal{I}$ are also killed by $\mathcal{I}$. This checks the asserted closure without imposing the generally false extension-closure condition.