  To distinguish between the effects of strain and magnetism on the
superconductivity in $c$-axis
La$_{2/3}$Ca$_{1/3}$MnO$_3$/YBa$_2$Cu$_3$O$_{7-\delta}$ (LCMO/YBCO)
heterostructures, we study perovskite/YBCO/perovskite thin films using either
ferromagnetic LCMO or paramagnetic LaNiO$_3$ (LNO) as perovskite. For a
lattice-symmetry matched comparison, we also use orthorhombic
PrBa$_2$Cu$_3$O$_{7-\delta}$ (PBCO) in place of the pseudocubic perovskites.
Unlike PBCO/YBCO/PBCO, both LCMO/YBCO/LCMO and LNO/YBCO/LNO trilayers show
strong attenuation of the superconducting $T_c$ as YBCO layer thickness is
reduced from 21.4 to 5.4 nm. Our results indicate that heteroepitaxial strain,
rather than long-range proximity effect, is responsible for the long length
scales of $T_c$ attenuation observed in $c$-axis LCMO/YBCO heterostructures.
