  We relate finite generation of cones, monoids, and ideals in increasing
chains (the local situation) to equivariant finite generation of the
corresponding limit objects (the global situation). For cones and monoids there
is no analog of Noetherianity as in the case of ideals and we demonstrate this
in examples. As a remedy, we find local-global correspondences for finite
generation. These results are derived from a more general framework that
relates finite generation under closure operations to equivariant finite
generation under general families of maps. We also give a new proof that
non-saturated Inc-invariant chains of ideals stabilize, closing a gap in the
literature.
