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### Is the embedding dimension minus the dimension upper semicontinuous?

For a Noetherian local ring $R$ with maximal ideal $\mathfrak{m}$ and residue class field $K$, consider the invariant \operatorname{def}(R) := \operatorname{dim}_K(\mathfrak{m}/\mathfrak{m}^2) - \...
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### When a smooth algebra is regular?

Let $A \subseteq B$ be noetherian integral domains, $A$ regular (=every localization at maximal ideal is a regular local ring) and $B$ is a smooth $A$-algebra. For the definition of a smooth algebra, ...
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### Regularity of a tensor product

Let $A \subseteq B$ and $A \subseteq C$ be commutative noetherian domains. Assume that $A$ and $C$ are regular rings (=every localization at a maximal ideal is a regular local ring). Assume that $B$ ...
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### When every module is a scalar extension?

Let $A \subseteq B$ be commutative noetherian domains. Of course, if $M$ is an $A$-module, then $M \otimes_A B$ is a $B$-module. I am curious to know if there exist additional conditions on $A$ and $B$...
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Let $R$ be a complete DVR with fraction field $K$, $X$ be a regular scheme flat over $R$. Let $L$ be a finite field extension of $K$ and $Q$ be the integral closure of $R$ in $L$. Denote by $Y:=X \... 1answer 160 views ### Reducedness of a ring with prime nilradical Let$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$is an irreducible divisor ... 1answer 460 views ### Is every polynomial ring over any field regular? Is every polynomial ring over any field regular? For a field that is algebraically closed, it is true as any maximal ideal of$k[x_1,...,x_n]$corresponds to a point$(t_1,...,t_n)$in$\mathbb{A}^n$... 1answer 821 views ### geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings Let$R$be a local Noetherian ring. What is the geometric interpretation of: 1- Gorenstein rings 2- Complete intersections 3- Regular rings? and how can I realize differences by geometric ... 2answers 403 views ### Does regular field extension preserve regularity? Let$k$be an arbitrary field and suppose that$K/k$is a regular field extension. Let$V$be regular scheme of finite type over$\text{Spec }k$(not necessarily smooth). Is it true that$\text{Spec }...
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 ...
Basically, my question is whether this answer is correct. Here is the point. Let $R$ be a ring, and let $A$ and $B$ be $R$-algebras. Suppose that $A$ is regular and $B \otimes_R B$ is regular too. ...