All Questions

Let $(R,m)$ and $(S,n)$ be two local rings, $R \subseteq S$, $R$ regular, $S$ a finitely generated and flat $R$-algebra, and $mS=n$. In comments to this question it was claimed that in such situation ...

The answer to this question says the following: "The general statement is if $A \to B$ is finite and injective, and $A$ is noetherian and regular, then $B$ is CM if and only if $A \to B$ is flat. ...

Let $R \subseteq S$ be local rings with maximal ideals $m_R$ and $m_S$. Assume that: (1) $R$ and $S$ are (Noetherian) integral domains. (2) $\dim(R)=\dim(S) < \infty$, where $\dim$ is the Krull ...

If $ I $ is a homogeneous ideal of the ring of homogeneous polynomials $ \mathbb {C} [X_0, \dots, X_n] $ , under which conditions on the homogeneous ideal $ I $, and particularly on $ I_m $, the $m$ -...

Let $A$ be a regular ring and $\mathfrak q$ be an ideal, such that $\sqrt{\mathfrak q}$ is prime. Further assume that $\mathfrak q$ is locally principal (i.e. $\mathfrak q$ is an irreducible divisor ...

In Hartshorne's "Algebraic Geometry", the following statement is a weaker form of Theorem 8.21A (e), which he quotes from Matsumuura's book on commutative algebra: Proposition. Let $R$ be a regular ...