Let k be a field, H be a Hopf algebra, A be a right H-comodule algebra and C be a right H-module coalgebra. We extend to the category of (H, A, C)-Doi–Hopf modules a result of Doi on projectivity of every relative (A, H)-Hopf module as an A-module. We also extend the Fundamental Theorem of [C, H]-Hopf modules due to Doi to the category of (H, A, C)-Doi–Hopf modules. Then we discuss relative projectivity and relative injectivity in this category.
The expansion of a function
f(θ)
of an angle
θ
varying between 0 and π in terms of a series proceeding by the sines of the multiples of
θ
depends on the fundamental theorem, ∫
π
0
sin
pθ
sin
qθ dθ
= 0, where
p
and
q
are integer numbers not equal to each other.
On Landesman-Lazer conditions and the fundamental theorem of algebra
