4 problems

Let $A=\mathbb{C}\oplus\mathfrak{m}$ be a finite-dimensional local algebra of dimension $n$, where $\mathfrak{m}$ is its maximal ideal, and let $\operatorname{Aut}(A)^{\circ}$ deno…

Let $A$ be a local commutative algebra with an $n$-cluster tilting module. Rigidity conjecture. Then $n=1$ and there is an integer $m\geq 1$ such that … The conjecture is one of th…

Let $A$ be the local finite-dimensional algebra described in the paper, and let $M$ be its 2-precluster tilting module. Marczinzik's conjecture. The module $M$ is 2-cluster tilting…

Let $k[x]/x^2$ and $k[y]/y^2$ be the two local $k$-algebras, and let $\mathcal{L}$ denote the category of local $k$-algebras with algebra homomorphisms as morphisms. A coproduct of…