. Going up and going down The going up theorem is essentially a corollary of Nakayama's lemma. It asserts: Let be an integral extension of commutative rings, and a prime ideal of . Then there is a prime ideal in such that . Moreover, can be chosen to contain any prime of such that . Module epimorphisms Nakayama's lemma makes precise one sense in which finitely generated modules over a commutative ring are like vector spaces over a field. The following consequence of Nakayama's lemma gives another way in which this is true: If is a finitely generated -module and is a surjective endomorphism, then is an isomorphism. Over a local ring, one can say more about module epimorphisms: Suppose that is a local ring with maximal ideal , and are finitely generated -modules. If is an -linear map such that the quotient is surjective, then is surjective. Homological versions Nakayama's lemma also has several versions in homological algebra. The above statement about epimorphisms can be used to show: Let be a finitely generated module over a local ring. Then is projective if and only if it is free. This can be used to compute the Grothendieck group of any local ring as . A geometrical and global counterpart to this is the Serre–Swan theorem, relating projective modules and coherent sheaves. More generally, one has Let be a local ring and a finitely generated module over . Then the projective dimension of over is equal to the length of every minimal free resolution of . Moreover, the projective dimension is equal to the global dimension of , which is by definition the smallest integer such that Here is the residue field of and is the tor functor. Inverse function theorem Nakayama's lemma is used to prove a version of the inverse function theorem in algebraic geometry: Let be a projective morphism between quasi-projective varieties. Then is an isomorphism if and only if it is a bijection and the differential is injective for all . Proof A standard proof of the Nakayama lemma uses the following technique due to . Let M be an R-module generated by n elements, and φ: M → M an R-linear map. If there is an ideal I of R such that φ(M) ⊂ IM, then there is a monic polynomial with pk ∈ Ik, such that as an endomorphism of M. This assertion is precisely