We investigate local cooling of gauge field configurations in four-dimensional U(1) lattice gauge theory. Using the magnetic monopole currents identified in the lattice gauge fields, we compute the quark potential and extract the string tension after each cooling step. After many cooling steps we find that only a small fraction ( similar 10%) of the original monopole current remains. Even with this small amount of monopole current we are able to obtain a value of the string tension that agrees with the uncooled results. ((orig.)).