  This paper provides a systematization of some recent results in homology of
algebras. Our main theorem gives criteria under which the homology of a diagram
algebra is isomorphic to the homology of the subalgebra on diagrams having the
maximum number of left-to-right connections. From this theorem, we deduce the
`invertible-parameter' cases of the Temperley-Lieb and Brauer results of
Boyd-Hepworth and Boyd-Hepworth-Patzt. We are also able to give a new proof of
Sroka's theorem that the homology of an odd-strand Temperley-Lieb algebra
vanishes, as well as an analogous result for Brauer algebras and an
interpretation of both results in the even-strand case. Our proofs are
relatively elementary: in particular, no auxiliary chain complexes or spectral
sequences are required. We briefly discuss the relationship to cellular
algebras in the sense of Graham-Lehrer.
