let $a_1$, $a_2$ be unital banach algebras and $x$ be an $a_1$-$a_2$- module. applying the concept of module maps, (inner) modulegeneralized derivations and  generalized first cohomology groups, wepresent several results concerning the relations between modulegeneralized derivations from $a_i$ into the dual space $a^*_i$ (for$i=1,2$) and such derivations  from  the triangular banach algebraof t...

For a smooth and proper scheme over an Artinian local ring with ordinary reduction the perfect residue field, we prove — under some general assumptions that relative de Rham–Witt spectral sequence degenerates crystalline cohomology, equipped its display structure arising from Nygaard complexes, has Hodge–Witt decomposition into direct sum of (suitably Tate-Twisted) multiplicative displays. As e...

In this paper, we construct a Hamiltonian Floer theory based invariant called relative symplectic cohomology, which assigns module over the Novikov ring to compact subsets of closed manifolds. We show existence restriction maps, and prove some basic properties. Our main contribution is identify natural geometric situation in cohomology two satisfy Mayer-Vietoris property. This tailored work und...