Tag Archives: local rings

The nonvanishing K_2(Z/4)

We saw previously that $K_2(F) = 0$ for a finite field $F$, where $K_2$ is the second $K$-group of $F$. It may be helpful to refer to that post for the definitions of this functor. I thought that it might be disappointing because we did all that work to compute the second $K$-group of a […]

Solution: Kaplansky’s Commutative Rings 4.1.2

Problem. Let $R$ be a (commutative!) Noetherian local ring, $M\subset R$ its maximal ideal, and $A$ a finitely generated $R$-module. If ${\rm Ext}^1(A,R/M) = 0$ then $A$ is a free $R$-module. This problem will be a stepping stone to showing that a Noetherian local ring is regular if and only if the injective dimension of […]

Projective Modules over Local Rings are Free

Conventions and definitions: Rings are unital and not necessarily commutative. Modules over rings are left modules. A local ring is a ring in which the set of nonunits form an ideal. A module is called projective if it is a direct summand of a free module. Today I shall share with you the wonderful result […]