# Tag Archives: projective dimension

## Solution: Kaplansky’s Commutative Rings 4.1.01

If $R$ is a commutative ring and $M$ an $R$-module, a regular sequence on $M$ is a sequence $x_1,\dots,x_n\in R$ such that $(x_1,\dots,x_n)M \not=M$ and for each $i$, the element $x_{i+1}$ is not a zero divisor on the module $M/(x_1,\dots,x_{i})$. The term regular sequence in $R$ just refers to a regular sequence on $R$ as […]

## A Case of No Positive Finite Projective Dimension

A commutative Noetherian local ring $R$ with maximal ideal $M$ is called a regular local ring if the Krull dimension of $R$ is the same as the dimension of $M/M^2$ as a $R/M$-vector space. In studying regular local rings one often uses the following lemma in inductive arguments: if \$ […]