arXiv subjects
Publications and source records attributed to Vincent Gélinas.
We investigate the inequality ${\rm Findim}\ \! Λ^{op} \leq {\rm dell}\ \! Λ$ between the finitistic dimension and the delooping level of an Artin algebra $Λ$, and whether equality holds in general. We prove that equality ${\rm Findim}\ \! Λ^{op} = {\rm dell}\ \! Λ$ always holds for Artin algebras with radical square zero.
We investigate two invariants of Noetherian semiperfect rings, namely the depth and a new invariant we call the "delooping level". These give lower and upper bounds for the finitistic dimension, respectively. As first theorems, we give a necessary and sufficient criterion for the delooping level to be finite in terms of the splitting of the unit map of a related adjunction, and use this to give a sufficient torsionfreeness criterion for finiteness. We further relate these invariants to the Auslander-Bridger grade conditions for modules, which are vanishing conditions on double Ext duals. As main theorem, we prove that these bounds agree whenever the first non-trivial grade conditions are satified for simple modules, so that either invariant computes the finitistic dimension in this case. Over Artinian rings, we show that the delooping level also bounds the big finitistic dimension, and we obtain a sufficient cohomological criterion for the first finitistic dimension conjecture to hold. Over commutative local Noetherian rings, these conditions always hold and we obtain a new characterisation of the depth as the delooping level of the ring.
We show that a finite dimensional monomial algebra satisfies the finite generation conditions of Snashall-Solberg for Hochschild cohomology if and only if it is Gorenstein. This gives, in the case of monomial algebras, the converse to a theorem of Erdmann-Holloway-Snashall-Solberg-Taillefer. We also give a necessary and sufficient combinatorial criterion for finite generation.
We classify the Betti tables of indecomposable graded matrix factorizations over the simple elliptic singularity $f_λ= XY(X-Y)(X-λY)$ by making use of an associated weighted projective line of genus one.