  We introduce a derived enhancement of local Galois deformation rings that we
call the "spectral Hecke algebra", in analogy to a construction in the
Geometric Langlands program. This is a Hecke algebra that acts on the spectral
side of the Langlands correspondence, i.e. on moduli spaces of Galois
representations. We verify the simplest form of derived local-global
compatibility between the action of the spectral Hecke algebra on the derived
Galois deformation ring of Galatius-Venkatesh, and the action of Venkatesh's
derived Hecke algebra on the cohomology of arithmetic groups.
