Pretriangulated categories failing (TR4)
arXiv:2608.09777
Abstract
We introduce gemstones in a pretriangulated category, which are used to perturb pretriangulated structures. For a triangulated category whose stable category of coherent functors admits a gemstone, we construct pretriangulated structures on which fail the octahedral axiom (TR4). This confirms a long-standing conjecture on triangulated categories.