Abstract
Counterexamples to the first Finitistic Dimension Conjecture are constructed. In fact, for any fieldk and any integerm≧2, there exist finite dimensionalk-algebras Λ such that the little finitistic dimension of Λ ism, while the big finitistic dimension ism+1. Our examples are monomial relation algebras; within this class of algebras the big and little finitistic dimensions cannot differ by more than 1. The analysis of the examples is based on a sharp picture of arbitrary second syzygies over monomial relation algebras.
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