Extension of i-Modularity

Abstract: Let

K / k

be a purely inseparable extension of characteristic

p > 0

and of finite size. We recall that

K / k

is modular if for every

n i n m a t h b b N

,

K^{p^{n}}

and

k

are

k c a p K^{p^{n}}

-linearly disjoint. A natural generalization of this notion is to say that

K / k

is

l q

-modular if

K

is modular over a finite extension of

k

. Our main objective is to extend in definite form the results and definitions of the

l q

-modularity that have already been obtained in the case limited by the finiteness condition imposed on

[k : k^{p}]

in a rather general framework (framework of extensions of finite size called also

q

-finite extensions).First, by means of invariants, we characterize the

l q

-modularity of a

q

-finite extension. Next, we show that any intersection of a

q

-finite extensions covering

k

or

K

preserves the

l q

-modularity. We also prove that any

q

-finite extension

K / k

contains a greater

l q

-modular and relatively perfect sub-extension. In particular, this result is very useful for defining the modularity of order

i

linked to a

q

-finite extension

K / k

.Moreover, we give a necessary and sufficient condition for

K / k

to be

i

-modular. Certainly, the modularity level of

K / k

never exceeds the sizeof

K / k

. Notably, we explicitly describe the extension

K / k

whose degree of modularity is the size of

K / k

. In the end, we examine a particular decomposition of

K / k

defined by inverse chaining.