One can use classical varieties to attack the problem of classifying finitely-generated modules over finite-dimensional algebras. Given such an algebra, one can write down a number of varieties which parameterize modules with certain isomorphism invariants. Furthermore, these varieties come with morphic actions by algebraic groups whose orbits are in one-to-one correspondence with isomorphism classes of such modules. Using path algebras modulo relations, we can exploit the quiver structure to learn about the structure of these varieties. We use this to give a proof of rationality of one such variety parameterizing graded modules.

The work analyzes various ring-theoretic properties of the endomorphism rings $\tilde\Lambda^{op}$ of strong tilting modules in the category of left modules of some truncated path algebra $\Lambda=KQ/\langle\text{all paths of length }L\rangle$. The indecomposable summands $T_i$ of the strong tilting module $T$ are described combinatorially and graphically. The endomorphism ring $\tilde\Lambda^{op}$ is analyzed by means of the homomorphisms between the $T_i$. A bound for both the Loewy length of $\tilde\Lambda$ and for the right finitistic dimension of $\tilde\Lambda$ are obtained in terms of the quiver $Q$ and the number $L$.