Questions tagged [regular-rings]
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23
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30
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On the definition of regular (non-noetherian, commutative) rings
All rings are commutative with unit. A ring $R$ is called regular if it satisfies
(Reg) Every finitely generated ideal of $R$ has finite projective dimension.
Clearly this gives the usual ...
14
votes
0
answers
1k
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A slick proof (?) of Zariski-Nagata purity in characteristic $p$
I am trying to understand the MathSciNet review written by Mark Kisin of the paper "Almost etale extensions" of Faltings. There Kisin illustrates Faltings' approach to the almost purity theorem with ...
11
votes
1
answer
2k
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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 ...
6
votes
2
answers
864
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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 }...
6
votes
1
answer
504
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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) - \...
5
votes
1
answer
489
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Tensor product of regular ring (with some conditions)
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. ...
4
votes
2
answers
2k
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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, ...
4
votes
1
answer
631
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The ring of global sections of a regular scheme
Let $X$ be a Noetherian regular scheme. Is $\mathcal{O}_X(X)$ a regular ring? For affine schemes this is true, see 02IU on the Stacks project.
3
votes
1
answer
252
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A particular Isomorphism of graded algebras over a regular local ring
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 ...
2
votes
1
answer
1k
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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 $\mathbb{A}^n$ ...
2
votes
1
answer
143
views
Geometric regularity for infinitely generated field extensions
Let $k$ be a field. Suppose that for a finite type $k$-algebra $A$, we define two following properties:
$A\otimes_k k'$ is a regular ring for all finitely generated field extensions $k\subset k'$.
$...
2
votes
0
answers
76
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A semigroup property related to von Neumann regularity
A very common and useful notion in rings is that of von Neumann regular elements: those elements $a\in R$ such that there exists $b\in R$ satisfying $aba=a$. As this is a property defined solely ...
1
vote
1
answer
224
views
Local rings $R \subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular
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 ...
1
vote
1
answer
109
views
Is the projective dimension of finite torsion-free modules over regular ring of dimension $n$ smaller that $n$?
Let $R$ be a Noetherian regular integral domain of Krull dimension $n$. Let $M$ be a finite torsion-free $R$-module. Is this true that $M$ has projective dimension $<n$ ?
This would be a ...
1
vote
1
answer
499
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 ...
1
vote
0
answers
182
views
Is the following local map unramified?
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 ...
1
vote
0
answers
111
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$A \to B$ with $A$ regular imply that $B$ is CM
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.
...
1
vote
0
answers
124
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Generating the annihilator ideal up to finite index
Let $\Lambda_2 = \mathbb{Z}_p[[T_1,T_2]]$ be the power series ring over $\mathbb{Z}_p$ in two variables, i.e., $\Lambda_2$ is a regular local ring of dimension 3. Let $M$ be the quotient of an ...
0
votes
2
answers
380
views
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 $B$ ...
0
votes
1
answer
312
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Under which conditions on the homogeneous ideal $ I $, the quotient ring $ \mathbb{C} [X_0, \dots, X_n]/I $ is a regular ring?
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 $m$ -...
0
votes
1
answer
447
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Base change of regular schemes [closed]
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 $Y:=X \...
0
votes
1
answer
99
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$S/I$-freeness of $I/I^2$ vs $I/I^{(2)}$, where $I$ is a radical ideal of regular local ring $S$
Let $I$ be a radical ideal of a regular local ring $S$. Put $R:=S/I$. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. It is well-known that $I^n \subseteq I^{(n)}$.
Is it true that $I/I^2$ is $R$-...
0
votes
1
answer
223
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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 $B$...