arXiv2026
Let $A$ be a finite-dimensional algebra and $e$ a rigid presentation. We give an explicit formula for the left adjoint to the inclusion $\operatorname{rep}(e^\perp)\hookrightarrow \operatorname{rep}(A)$ and construct a bijection between presentations in $\operatorname{rep}(e^\perp)$ and presentations compatible with $e$. The induced bijection on weights preserves generic $\operatorname{E}$-invariants and canonical decompositions. If $A$ is Jacobian, the reduction algebra is Jacobian for every rigid $e$. We prove mutation formulas for the signed completions and for the dimension vectors of the simple objects of $\operatorname{rep}(e^\perp)$, enabling an algorithm to find the projected quiver with potential when $e$ is extended-reachable. Additionally, we introduce a modified projection for quivers with potentials that preserves general presentations. For applications to cluster algebras, we establish a connection to the stabilization functors.