Proof: Apply Statement 3 to N = ΣiRmi. If one assumes instead that R is complete and M is separated with respect to the I-adic topology for an ideal I in R, this last statement holds with I in place of J(R) and without assuming in advance that M is finitely generated. Here separatedness means that the I-adic topology satisfies the T1 separation axiom, and is equivalent to Consequences Local rings In the special case of a finitely generated module over a local ring with maximal ideal , the quotient is a vector space over the field . Statement 4 then implies that a basis of lifts to a minimal set of generators of . Conversely, every minimal set of generators of is obtained in this way, and any two such sets of generators are related by an invertible matrix with entries in the ring. Geometric interpretation In this form, Nakayama's lemma takes on concrete geometrical significance. Local rings arise in geometry as the germs of functions at a point. Finitely generated modules over local rings arise quite often as germs of sections of vector bundles. Working at the level of germs rather than points, the notion of finite-dimensional vector bundle gives way to that of a coherent sheaf. Informally, Nakayama's lemma says that one can still regard a coherent sheaf as coming from a vector bundle in some sense. More precisely, let be a coherent sheaf of -modules over an arbitrary scheme . The stalk of at a point , denoted by , is a module over the local ring and the fiber of at is the vector space . Nakayama's lemma implies that a basis of the fiber lifts to a minimal set of generators of . That is: Any basis of the fiber of a coherent sheaf at a point comes from a minimal basis of local sections. Reformulating this geometrically, if is a locally free -module representing a vector bundle , and if we take a basis of the vector bundle at a point in the scheme , this basis can be lifted to a basis of sections of the vector bundle in some neighborhood of the point. We can organize this data diagrammaticallywhere is an n-dimensional vector space, to say a basis in (which is a basis of sections of the bundle ) can be lifted to a basis of sections for some neighborhood of