Determinantal ring

History

The definition that we are using follows Commutative algebra with a view towards algebraic geometry by David Eisenbud.

Definition

A commutative unital ring $R$ is said to be a determinantal ring over the commutative unital ring $S$ if it can be written as $S/I$ where $I$ is the ideal generated by the $r\times r$ minors of a $p\times q$ matrix $M$, for some $p,q,r$, such that that codimension of $I$ in $S$ is exactly $(p-r+1)(q-r+1)$.

Facts

• A determinantal ring of type $(1,1)$ over a regular ring is termed a complete intersection
• Any determinantal ring over a Cohen-Macaulay ring is Cohen-Macaulay