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Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

1 vote
0 answers
51 views

Descent of $G$-invariant formal system of parameters using GAGF

Let $R=(R,\mathfrak{m})$ be a comm local regular ring of char $\neq 2$ (ie $2 \neq 0$ in $R$) with maximal ideal $\mathfrak{m}$ of (Krull) dimension $2$, ie $R$ admits system of parameters $x,y \in \m …
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2 votes
0 answers
164 views

Principle of degeneration as precursor of Zariski's connectedness theorem (geometric intuition)

I have following question about so-called "principle of degeneration" in algebraic geometry (which in modern terms is an immediate consequence of Zariski's main theorem and goes in it's original form …
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1 vote
1 answer
326 views

Proj construction and nilpotent homogenous elements in graded ring

Let $A= \bigoplus_{n \ge 0} A_n$ be a commutative Noetherian graded ring and $f \in A_d$ a nonzero homogeneous element of degree $d>0$. The natural ring map $q:A \to A/(f)$ induces a well defined map …
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0 votes
0 answers
176 views

Finite monomorphism $A \to B$ with reduced $A$ and special fiber implies $B$ reduced

I have a question about correctness of following statement claimed here in $\boxed{2} \ $: Let $k$ arbitrary field, let $f : X \longrightarrow Y$ be a finite dominant morphism between finite type $k$ …
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1 vote
0 answers
107 views

Inclusion between rings after localization

Let $\phi:A \to B $ an injective finite ring map between two noetherian integral domains $A,B$. Let $ C \subset B$ a subring of $B$ and assume that there exist a prime ideal $\mathfrak{p} \subset A$ , …
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1 vote
1 answer
178 views

Exactness of functor $ Et(B) \to \operatorname{(Ab)}, \ C \mapsto \mathcal{F}(C) $ (Etale Co...

I have question about a statement from Etale Cohomology and the Weil Conjecture by Freitag, Kiehl at the top of page 16. It seemingly uses the same notations as introduced at the bottom of page 15 and …
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0 votes
0 answers
296 views

Faithfully flat etale morphism from strictly Henselian ring (from Etale Cohomology and the W...

I have question about a statement found in Etale Cohomology and the Weil Conjecture by Freitag, Kiehl at the end of page 15. It starts with the Remark 1.18 : Let $A$ be a strictly Henselian ring (i.e. …
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3 votes
0 answers
155 views

Injective hulls of quotient rings $R/p$

Let $R$ be integral domain and $p \neq 0$ a prime ideal. It's well known that in category of $R/p$ modules the injective hull of $R/p$ is $K=\operatorname{Frac}(R/p)$. Is there a successful theory kno …
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5 votes
0 answers
1k views

Saturated ideals in computational algebra

Let $R$ a commutative ring with one and $I, J \triangleleft R$ two ideals. The saturated ideal $I^{sat}_J$ with respect $J$ is the ideal $$(I : J^\infty )= \cup_{n \geq 1} (I:J^n)$$ where $(I:J^n)= \ …
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0 votes
0 answers
189 views

When $K[X_1,X_2,...,X_n] \to K[Y_1,Y_2,...,Y_m]$ is a flat morphism

Let $K$ be a field and $\varphi: K[X_1,X_2,...,X_n] \to K[Y_1,Y_2,...,Y_m]$ a polynomial $K$-algebra morphism. Assume $n, m \ge 2$. By definition $\varphi$ endows $K[Y_1,Y_2,...,Y_m]$ with a $K[X_1,X_ …
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2 votes
0 answers
228 views

Necessary condition to extend a morphism of schemes

Consider two schemes $X,Y$ over a locally noetherian scheme $S$. Let $p \in X$ and assume that $X$ is irreducible and not affine spectrum of a semilocal ring. We assume moreover we have a morphism $ …
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2 votes
0 answers
400 views

Henselization and completions of local rings & schemes

That's the second part of my coarse becoming acquainted with Henselizations of fields and local rings. (in this question we focus on local rings as it is more algebro geometric motivated). So let $(R …
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4 votes
0 answers
752 views

Induced morphism of completions of local rings

Let $g: A \to B$ be a local ring morphism between local Noetherian (commutative) rings $A,B$ (so $g(m_A) \subset m_B$ for the unique maximal ideals of the corresponding rings). Assume that the induced …
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  • 6,018
1 vote
1 answer
636 views

Affine cone example

Consider the complete intersection ideal ${\displaystyle (f,g_{1},g_{2},g_{3})\subset \mathbb {C} [x_{0},\ldots ,x_{n}]}$ and let $X$ be a projective variety defined by the (product) ideal sheaf ${\di …
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1 vote
0 answers
120 views

Question about Local Henselian Rings

I have a question regarding properties/characterizations of local Henselian rings exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces": Here the relevant excerpt: Remark …
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