Let $k$ be a field. Is the morphism $k[x_1,\ldots]\to k[[x_1,\ldots]]$ (countably infinite number of variables) flat? For a Noetherian ring $R$, the map $R\to \hat{R}$ to its completion is flat (see e.g. Atiyah-MacDonald 10.14) but my intuition breaks down for non-Noetherian rings.