After OpenAI solved (it still has to be checked carefully!) the second Tachikawa, most of the important homological conjectures in our field might be wrong after all. Here I suggest some own interesting problems and conjectures related to homological algebra of finite dimensional algebras. (last updated 16.10.2026. I might add more conjectures in the future.)
Let us call a field torsion if every field element has finite multiplicative order. I conjecture that every K-algebra is weakly Gorenstein for torsion fields K, see https://arxiv.org/abs/1608.04212 for the notion of weakly Gorenstein and a first appearence of a related question. I plan to update this arxiv article soon with the interesting result that all K-algebras being weakly Gorenstein is equivalent to the condition that for all B-modules X,Y Ext_B^i(X,Y)=0 for all i>=m (for some m>=1) implies Ext_B^i(X,Y)=0 for all i>=1 for all symmetric K-algebras B. This would in particular imply the truth of the strong Nakayama conjecture for K-algebras with K torsion. Note that the counterexamples of OpenAI need fields that are not torsion!
Together with Daniel Owens we found in https://arxiv.org/abs/2505.12915 an alien-like local finite dimensional K-algebra with a non-trivial n-cluster tilting object for some n>=2. No other local algebra (given by quiver and admissible relations) with an n-cluster tilting object for some n>=2 is known at the moment. It would be interesting to find more examples, but even with the use of the best AI I did not found another example so far. For commutative local algebras, we conjecture there is no such algebra with an n-cluster tilting module for some n>=2.
Let A be a connected higher Auslander algebra with even global dimension g. Write \tau_g for the g-th higher Auslander--Reiten translate. Then in a conjecture with Tiago Cruz, we conjecture that \pdim_A \tau_g(DA) <= g-1 if and only if A is a QF-1 algebra. See https://arxiv.org/abs/2502.08422. This conjecture is perhaps not so important, but I like it since it would detect a complicated global property (being QF-1) by an easily computable property (\pdim_A \tau_g(DA) <= g-1). We proved this for Nakayama algebras in forthcoming work using a nice result for Nakayama algebras A that are higher Auslander algebras of even global dimension: Every indecomposable non-projective-injective A-module M has the property that the parity of its dominant dimension coincides with the parity of its injective dimension. Due to the high workload related to AI, we do not focus on writing up things related to Nakayama algebras at the moment.