Level structure over \(\widehat{E(n)}\) and stable splitting by Steinberg idempotent (Q1585434)

Let \(A\) be a finite abelian \(p\)-group and \(D(A^*)\) the representing ring of the character group \(A^*=\text{Hom}(A,\mathbb Q/\mathbb Z),\) and let \(\widehat{E(n)}\) be the \(I_n\)-adic complete Johnson-Wilson spectrum. The author constructs a spectrum \(F(A)\) and proves the main theorem of this paper which asserts that \[ \widehat{E(n)}^0(F(A))\otimes\mathbb Q\cong D(A^*)\otimes\mathbb Q \] showing the relation between the \(\widehat{E(n)}\)-cohomology of the constructed spectrum and the level structures. The author extends in this way the similar results obtained by Greenlees and Strickland about chromatic group cohomology rings.