Syzygy-Quiver Formulas and Finitistic Dimensions of Special Biserial Algebras

Let $Λ$ be a finite dimensional algebra over a field $\mathbf{k}$. We introduce the reachable syzygy quiver of $Λ$, a directed multigraph whose vertices are the isomorphism classes of indecomposable nonprojective direct summands of iterated syzygies of simple modules, and whose arrows record nonprojective summands of first syzygies. The projective dimension of a vertex is one plus its directed path height, allowing the global and finitistic dimensions of $Λ$ to be read off from this height function. This formula holds for every finite dimensional algebra, but its states are arbitrary modules. We show that when the first syzygy of an indecomposable module admits a closed combinatorial description from the bound quiver, this state space reduces to a smaller combinatorial quiver, making the global and finitistic dimensions exact computable invariants. We carry this out in three stages. Over a special biserial algebra, the Liu--Morin closure theorem identifies the reachable syzygy quiver with a string-syzygy quiver whose vertices are string modules. Over a string algebra, this compresses to the relation-chain quiver of Xin and Zhang, recovering their global dimension formula along with a parallel formula for the finitistic dimension. Over an arbitrary biserial algebra, the universal formula holds, but uniform compression to paths or strings is impossible, with the Skowroński--Waschbüsch algebra providing a witness. Finally, we show two string algebras may share the same Gabriel quiver but have distinct global dimensions, proving the ideal of relations is indispensable for these invariants, and we prove that a pointwise homological reduction to a string algebra on the same quiver preserving the monomials of the relations is impossible.