A Sufficient Criterion for Homotopy Cartesianess

In an abelian category, a commutative quadrangle is called bicartesian if its diagonal sequence is short exact, i.e. if it is a pullback and a pushout. A commutative quadrangle is bicartesian if and only if we get induced isomophisms on the horizontal kernels and on the horizontal cokernels. In a triangulated category in the sense of Verdier [3, Def. 1-1], a commutative quadrangle is called homotopy cartesian (or a Mayer-Vietoris square, or a distinguished weak square), if its diagonal sequence fits into a distinguished triangle. A homotopy cartesian square has a (non-uniquely) induced isomorphism on the horizontally taken cones [2, Lem. 1.4.4]. We consider the converse question : a commutative quadrangle that has an isomorphism induced on the horizontally taken cones, is it homotopy cartesian? We show this to be true if the endomorphism ring of the object in the terminal or initial corner satisfies a finiteness condition. This finiteness condition is for instance satisfied for the endomorphism rings occurring in D(A -mod), where A is a finite-dimensional algebra over some field; or in A -mod, where A is a finite-dimensional Frobenius algebra over some field. This finiteness condition, however, in general fails for the endomorphism rings occurring in K(Z -proj). We show by an example that the conclusion on our commutative quadrangle to be homotopy cartesian fails there as well.