Homological and cohomological grading

Remark 6.1.8 (Cohomological grading of complexes). We grade chain complexes homologically, so that the differential \(d_n\colon C_n \to C_{n-1}\) lowers degree. Some references grade cohomologically instead, writing a complex as \((C^n)_{n \in \Z }\) with differential \(d^n\colon C^n \to C^{n+1}\) raising degree. The two conventions record the same data, related by negating the index: a homologically graded complex \((C_{\bullet }, d_{\bullet })\) becomes the cohomologically graded complex with \(C^n := C_{-n}\) and \(d^n := d_{-n}\), and conversely. We use homological grading throughout.