Abstract: 

Suppose that $A$ and $B$ are unital Banach algebras with units
$1_A$ and $1_B$, respectively, $M$ is a unital Banach $A,B$-module, ${\cal
T}= \left [{A\ M\atop 0\ B}\right]$ is the triangular
Banach algebra, $X$ is a unital ${\cal T}$-bimodule, $X_{AA}=1_AX1_A$,
$X_{BB}=1_BX1_B$, $X_{AB}=1_AX1_B$ and $X_{BA}=1_BX1_A$. Applying two nice
long exact sequences related to $A$, $B$, ${\cal T}$, $X$, $X_{AA}$,
$X_{BB}$, $X_{AB}$ and $X_{BA}$ we establish some results on (co)homology of
triangular Banach algebras.

1. Ferdowsi University of Mashhad and
   Institute for Studies in
   Theoretical Physics and Mathematics (IPM)
   Iran
   , Home: http://www.um.ac.ir/~moslehian/