  Peterson varieties are subvarieties of flag varieties and their (equivariant)
cohomology rings are given by Fukukawa-Harada-Masuda in type A and soon later
the author with Harada and Masuda gives an explicit presentation of the
(equivariant) cohomology rings of Peterson varieties for arbitrary Lie types.
In this note we study the (equivariant) cohomology ring of the intersections of
Peterson variety with Schubert, opposite Schubert, and Richardson varieties in
more general. By the work of Goldin-Mihalcea-Singh, the intersections of
Peterson variety with Schubert varieties are naturally identified with smaller
Peterson varieties, so the problem reduces to the problem for opposite Schubert
intersections. In this note we provide a technical statement for (equivariant)
cohomology ring of a subvariety with some conditions of Peterson variety. By
using the statement, we calculate the (equivariant) cohomology rings for some
intersections of Peterson varieties with opposite Schubert varieties in type A.
We also explicitly present the (equivariant) cohomology rings for some
intersections of Peterson varieties with Richardson varieties in type A.
