In this note we give a minimal presentation of spectral sequences following EGA. We cover essentially only that part of the theory needed in algebraic geometry. In Section 2 we start with a filtration of a complex, and show how the various pieces of a spectral sequence arise. Applying these observations in Section 3 to two natural filtrations of the total complex of a bicomplex, we deduce two s...

The notion of symmetry is the main property a metric function. area fixed point theory has suitable structure for in mathematics. goal this paper to find and common results bicomplex valued b-metric space mixed type rational contractions with control functions. Some well-known literature findings were generalized our findings. We provide an example strengthen validate results. As example, conte...

We introduce a geometric framework needed for a mathematical understanding of the BRST symmetries and chiral anomalies in gauge field theories. We define the BRST bicomplex in terms of local cohomology using differential forms on the infinite jet bundle and consider variational aspects of the problem in this cohomological context. The adjoint representation of the structure group induces a repr...

We introduce a bicomplex which computes the triple cohomology of Lie– Rinehart algebras. We prove that the triple cohomology is isomorphic to the Rinehart cohomology [13] provided the Lie–Rinehart algebra is projective over the corresponding commutative algebra. As an application we construct a canonical class in the third dimensional cohomology corresponding to an associative algebra.

Let H be a bialgebra. An H-module-algebra is an associative algebra A that is also an H-module such that the multiplication map on A becomes an H-module morphism. This algebraic structure arises often in algebraic topology, quantum groups [11, Chapter V.6], Lie and Hopf algebras theory [3, 16, 19], and group representations [1, Chapter 3]. For example, in algebraic topology, the complex cobordi...