We investigate the relative cohomology and relative homology theories of $F$-Gorenstein modules, consider the relations between classical and $F$-Gorenstein (co)homology theories.

we investigate the relative cohomology and relative homology theories of $f$-gorenstein modules, consider the relations between classical and $f$-gorenstein (co)homology theories.

Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology, Bohmann–Gerhardt–Høgenhaven–Shipley–Ziegenhagen developed a coBökstedt spectral sequence to compute the homology $$\mathrm {coTHH}$$ for coalgebras over sphere spectrum. We construct relative study any commutative ring spectrum R. Further, we use algebraic structures in this complete some calcul...

We show that algebraically shifting a pair of simplicial complexes weakly increases their relative homology Betti numbers in every dimension. More precisely, let ∆(K) denote the algebraically shifted complex of simplicial complex K, and let β j (K, L) = dim k H j (K, L; k) be the dimension of the jth reduced relative homology group over a field k of a pair of simplicial complexes L ⊆ K. Then β ...