Stacks of algebras and their homology

For any increasing function $f:\mathbb{N}\to {\mathbb{N}}_{\ge 2}$ which takes only finitely many distinct values, a connected finite dimensional algebra $\mathrm{\Lambda }$ is constructed, with the property that $$\text{fin.dim}$_n, \Lambda = f(n)$forall$n$;here$$\text{fin.dim}$_n, \Lambda$isthe$n$-generatedfinitisticdimensionof$\Lambda$. The stacking technique developed for this construction of homological examples permits strong control over the higher syzygies of $\mathrm{\Lambda }$-modules in terms of the algebras serving as layers.