The Stacks project

Proof. This is more or less immediate from the definition. Namely, let $U = \mathop{\mathrm{Spec}}(A)$ be an affine open of $X$. Then $i^{-1}(U) = \mathop{\mathrm{Spec}}(A/I)$ for some ideal $I \subset A$ and $A/I$ has a finite resolution by finite projective $A$-modules by More on Algebra, Lemma 15.82.2. Hence $i_*\mathcal{O}_ Z|_ U$ can be represented by a finite length complex of finite locally free $\mathcal{O}_ U$-modules. This is what we had to show, see Cohomology, Section 20.49. $\square$