statistics Auto update and norecompute will force recomplie

I have 1 Query:

I have 1 index on columns --> IX_A(a,b,c,d)

I set Auto update statistics On database level

set update statistics on a with norecompute while IX_A is set with recompute.

Will stats be updated on IX_A index on rebuild or there will any effect on recompliation of queries/sps etc.

Manish