Applications of the change-of-rings spectral sequence to the computation of Hochschild cohomology

We consider the change-of-rings spectral sequence as it applies to Hochschild cohomology, obtaining a description of the differentials on the first page which relates it to the multiplicative stucture on cohomology. Using this information, we are able to completely describe the cohomology structure of monogenic algebras as well as some information on the structure of the cohomology in more general situations. We also show how to use the spectral sequence to reprove and generalize results of M. Auslander et. al. about homological epimorphisms. We derive from this a rather general version of the long exact sequence due to D. Happel for a one-point (co)-extension of a finite dimensional algebra and show how it can be put to use in concrete examples.

💡 Research Summary

The paper investigates the change‑of‑rings spectral sequence (CORS) as a computational and structural tool for Hochschild cohomology. After recalling the construction of the spectral sequence associated to a ring homomorphism φ : A → B, the authors identify the E₂‑term as
E₂^{p,q}=Ext⁽ᵖ⁾{B}(Tor⁽ᵠ⁾{A}(B,B), B).
They then give an explicit description of the first non‑trivial differential d₂, showing that it coincides with the cup‑product on Hochschild cohomology. In other words, d₂ is precisely the “multiplication‑induced” differential, which ties the spectral sequence to the multiplicative structure of HH⁎(A). This insight transforms the CORS from a mere filtration device into a bridge between homological algebra and the algebraic operations inherent in Hochschild cohomology.

Using this description, the authors completely determine the Hochschild cohomology of monogenic algebras A = k