  Competition with magnetism is at the heart of high temperature
superconductivity, most intensely felt near a vortex core. To investigate
vortex magnetism we have developed a spatially resolved probe using nuclear
magnetic resonance. Our spin-lattice-relaxation spectroscopy is spatially
resolved both within a conduction plane as well as from one plane to another.
With this approach we have found a spin-density wave associated with the vortex
core in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$, which is expected from scanning
tunneling microscope observations of "checkerboard" patterns in the local
density of electronic states.[1] We determine both the spin-modulation
amplitude and decay length from the vortex core in fields up to H=30 T.
