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

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0
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11 views

Propreties of a subring of a 'completed' of k(X_1, X_2, …, X_n)

I'm looking for a reference in commutative algebra for the propreties of the ring made of polynoms in n indeterminate over a field k with "real exponents". I don't even know the name of this ring, ...
2
votes
0answers
40 views

'Noether normalization' for finite group schemes

Throughout let $p$ be a prime, and let $k$ be a field of characteristic $p$. Let $G$ be a compact Lie group. Such a $G$ can always be embedded as a closed subgroup of $SU(n)$ for some $n$. This ...
-3
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0answers
62 views

how to prove the Hom is not 0 [on hold]

$\mathrm Hom_\mathbb{Z}(\prod{Z_n},Q)\neq0$ how to find the map ,I reckon the map is from $(1,1,,\dotsc,1)$ to 1 what is the answer?
0
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0answers
111 views

Localisation of the formal power series ring

Let $A \colon= K[[X_1,...,X_d]]$ be a formal power series ring of $d$-variables over a field $K$. Let ${\frak a}$ be a height $r$ prime of $A$ given by ${\frak a} \colon= (f_1,...,f_r)$, where $f_1 ...
0
votes
1answer
77 views

Inverse Galois Problem ; Galois group of some branched cover of $P^{1}$ defined over $\mathbb{Q}$

I was trying to read a paper on Inverse Galois problem . I understands what the inverse Galois problem is. It asks if every finite group is the Galois group of some extension of the rationals. The ...
1
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0answers
63 views

Explicit construction of a bielliptic curve

Let $C$ be a (projective smooth complex) curve such that $K_C=2(D+p)$, with $D+p$ defining a $g_7^2$; $p$ is a base point and $D$ defines a 2-to-1 map $\varphi:C\rightarrow E\subset\mathbb{P}^2$ onto ...
2
votes
1answer
92 views

Generalized height of elements in abelian groups

In the book Infinite abelian groups Vol. I by L. Fuchs, on page 154, the notion of the generalized $p$-height of an element in an abelian group is defined, as follows: Let $A$ be an abelian group ...
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113 views

An ideal and its annihilator

Let $I$ be an arbitrary ideal of commutative ring $R$ with identity. If $ann (I)=AB$, where $A$ and $B$ are comaximal, how can we fine two ideal $I_1$ and $I_2$ with $I=I_1+I_2$ such that $I_1$ ...
2
votes
3answers
362 views

Krull-dimension of local domain

Let $(R,{\frak m}_R)$ be a local domain (not necessarily Noetherian). That is, $R$ is integral and ${\frak m}_R$ is the unique maximal ideal of $R$. Suppose that ${\frak m}_R$ is finitely generated. ...
2
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0answers
120 views

Inverse limit of Noetherian rings

Let $R_i$ be a noetherian regular local ring of Krull dimension being finite. Suppose we are given a surjective homomorphism $\phi_{i,j} \colon R_{j} \twoheadrightarrow R_i$ for each $i,j$ with $j >...
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78 views

Dirac functional embedding [on hold]

I got the following set up: Let $S \neq \emptyset$ equiped with the discrete topology and let $\ell_\infty(S) = \{f: S \to \mathbb C \mid f \text{ bounded}\}$. Not $\ell_\infty(S)$ with the pointwise ...
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0answers
68 views

A property of minimal prime ideals [on hold]

Let $R$ be a commutative ring with $1$, and let $p$ be a minimal prime ideal of $R$. If $p\subseteq I_1+ I_2$, where $I_1$ and $I_2$ are two ideals of $R$, can we deduce that $p\subseteq I_1 $ or $...
0
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0answers
135 views

Stalks of the sheaf

Let $X$ be a scheme. For $m < \infty$, given the surjection ${\cal O}_X^{\oplus m} \twoheadrightarrow {\cal F}$ between sheaves on $X$, where ${\cal O}_X$ is the structural sheaf of $X$. Choose a ...
0
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0answers
66 views

On semi-complete ring K[X_1,X_2,…,X_∞]] and Popescu theorem

Let $P_n \colon= K[X_1,...,X_n]$ be a $n$-variables polynomial ring. We define 'semi-complete' polynomial ring $P_{\infty}$ by the following$\colon$ $P_{\infty} = K[X_1,...,X_\infty]] \colon = \...
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0answers
80 views

Everywhere holomorphic functions over C [closed]

Let $f, g$ be two everywhere holomorphic functions over ${\Bbb C}$. We consider the local representation of $f, g$ at the origin of ${\Bbb C}$, i.e. $z = 0$. That is, we can consider $f, g \in {\Bbb C}...
2
votes
1answer
118 views

Milnor patching for general modules

The Milnor patching theorem for projective modules is the following statement. Given a pullback diagram of rings $$ \begin{array}{} R & \xrightarrow{f_2} & R_2 \\ \downarrow{f_1} & &...
3
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1answer
131 views

Characterizing the image of $D(A_f) \rightarrow D(A)$

Let $A$ be a ring and $f \in A$ an element. If $M$ is an $A$-module on which multiplication by $f$ is an isomorphism, then $M$ is in fact an $A_f$-module. Now suppose that $C \in D(A)$ is a complex ...
14
votes
2answers
358 views

Can you use Chevalley‒Warning to prove existence of a solution?

Recall the Chevalley‒Warning theorem: Theorem. Let $f_1, \ldots, f_r \in \mathbb F_q[x_1,\ldots,x_n]$ be polynomials of degrees $d_1, \ldots, d_r$. If $$d_1 + \ldots + d_r < n,$$ then the ...
4
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1answer
258 views

Can K[[T_1,…,T_∞]] be embedded into K[[X,Y]]?

In the MathOverflow question about common false beliefs, the following answer teaches us that there is an embedding $\iota_n \colon K[[T_1,...,T_n]] \hookrightarrow K[[X,Y]]$. Now let us define the ...
2
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1answer
72 views

Is there a characterization of r(M) by local cohomology instead of Ext

For a Noetherian local ring $(R,m)$, and a finite $R$-module $M$ with $\operatorname{depth} M=t,$ type of $M$ is defined to be $r(M):=dim_{R/m}Ext^t \ (R/m, M).$ Is there a characterization of $...
13
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1answer
606 views

Non-isomorphic rings that are localizations of each other

Do there exist commutative rings $A$ and $B$ and multiplicative subsets $S\subseteq A$, $T\subseteq B$ such that $A\not\simeq B$ but $S^{-1}A \simeq B$ and $T^{-1} B\simeq A$? This question comes ...
13
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1answer
416 views

Why sum of three squares of real polynomials is a sum of two squares?

If $f(x),g(x)$ are real polynomials, then $f^2+g^2+1$ is a sum of two squares of polynomials. This easily follows from Fundamental Theorem of Algebra, but is there an argument avoiding it? What are ...
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0answers
86 views

depth of ideal in polynomial ring

Let $R$ be a Noetherian local ring. Then the "depth" of an ideal $I$ measures the maximal length of regular sequence inside $I$. And $depth (R/I)$ measures the maximal length of regular sequence in ...
1
vote
1answer
141 views

Complete subring of F_p[[X]]

Pointed out on famous disbelief, I know now that there is an embedding $\iota_n \colon {\Bbb F}_p[[T_1,...,T_n]] \hookrightarrow {\Bbb F}_p[[X,Y]]$ for any $n \leq \infty$. Then I would like to ask ...
6
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0answers
177 views

On describing a sort of “well-behaved” subgroups of a free abelian group

I found this question when I tried to figure out what kind of subgroups of a free abelian group behave just as well as in the finitely generated case. Let $M$ be a free abelian group and $N$ a ...
1
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0answers
58 views

Macaulayfications and standard blow-ups

Let $X$ be an affine variety such that $X-p$ is Cohen-Macaulay, and let $\pi \colon \widetilde{X} \rightarrow X$ be a standard blow-up of $X$ with respect to $\mathcal{O}_X$, centered at $p$. It is ...
2
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0answers
87 views

Behaviour of an étale morphism under Galois action on points

Consider the following situation. Let $k$ be a characteristic $0$ field, and consider an étale morphism of $k$ schemes $f:X\rightarrow Y$. Moreover, let $K$ and $L$ be two extension fields of $k$ such ...
2
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0answers
191 views

Characterizing flatness in terms of pulling back relative schemes

All schemes are assumed to be noetherian (and maybe separated just in case) Here is a (possible) definition of flatness: Definition 1 (flatness): A morphism of schemes $f: X \to Y$ is flat iff ...
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0answers
65 views

Popescu-Neron Desingularization for K[[T_1,…,T_∞]]

Let $K[[T_1,...,T_n]]$ be a finitely many variables formal power series ring over a field $K$. Dorin Popescu proved that there are smooth algebras $A_{\lambda}$'s which are of finite type over $K$ ...
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0answers
55 views

Coherence of subrings of K[[X,Y]]

Let $K[[X,Y]]$ be a two-variables formal power series ring over a field $K$. Consider a sub-ring $\iota \colon A \subset K[[X,Y]]$. Q. Is A coherent? $\quad$ Or is it automatic that $\iota$ is ...
1
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1answer
194 views

Proj of some graded algebra

I was computing some GIT quotients and came up with the following question: to compute $\mathrm{Proj}(\mathbb C [f_1,f_2,f_3,f_4,f_5,f_6]/I)$ where $f_i$'s are homogeneous polynomials of same degree ...
2
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1answer
68 views

How a “sequentially Cohen–Macaulay” simplicial complex relates to “Cohen–Macaulay” simplicial complex?

Let $\Delta$ be a simplicial complex on $[n]$ of dimension $d − 1.$ Let $0\le i\le d-1.$ One defines the pure i_th skeleton of $Δ$ to be the pure subcomplex $\Delta(i)$ of $\Delta$ whose facets are ...
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1answer
152 views

Module-finiteness over the fixed subring

SETUP: Let $R$ be a finitely generated, noetherian, integral domain of characteristic $0$. Let $G$ be a finite group that acts on $R$ by ring automorphisms. QUESTION: Is $R$ a finitely generated ...
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87 views

Does every prime power generate a primary ideal?

Let $R$ be a commutative ring with identity and let $p\in R$ be a prime element (i. e. $(p)$ is a prime ideal). If $R$ is an integral domain, it can be shown that $(p^k)$ is a primary ideal for every $...
2
votes
1answer
282 views

Are there analogies between $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\Bbb Z$?

Much progress in understanding $\Bbb Z$ is made from analogies between $\Bbb F_q[x]$ and $\Bbb Z$. Can there be analogies between arithmetic in $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\...
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0answers
8 views

Why no prime number could appear as the length of a hypotenuse in more than one Pythagorean triangle? [migrated]

Why no prime number could appear as the length of a hypotenuse in more than one Pythagorean triangle? In other words, could any of you give me a algebraic proof for the following? given prime ...
7
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0answers
266 views

Capitulation of ideal classes in general Dedekind Domains

I’ve been working on a problem, and come across an issue with capitulation in Dedekind domains. Here is the set up: Let $D$ be a Dedekind domain, and $K$ its (perfect, but we’re willing to modify ...
3
votes
0answers
197 views

Equivariant sheaves over affine schemes

Let $k$ be a field, let $G$ be a linear algebraic group over $k$ and let $A$ be a commutative $k$-algebra which is acted on by $G$. We say that an $A$-module $M$ is a $(G,A)$-module if it satisfies ...
3
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2answers
422 views

These rings are isomorphic?

Consider the following rings: $A=\mathbb{C}\lbrace x,y,u \rbrace /(xy+x^3,y^2,xy^2+x^5) \ $ and $B=\mathbb{C}\lbrace x,y,u \rbrace /(xy+x^3,y^2+ux^4,xy^2+x^5)$ There is an isomorphism of $\mathbb{...
2
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0answers
60 views

Can we write an element in a super Grassmannian as a pair of matrices?

Super Grassmannians are introduced by Manin, see for example. Elements in a grassmannian can be written as matrices, see for example. Can we write an element in a super Grassmannian as a pair of ...
3
votes
1answer
114 views

Fixed points for action of finite group acting on Noetherian ring is a local Noetherian ring

Let $R$ be a local Noetherian ring which contains the field $\mathbb{Q}$ of rational numbers, let $G$ be a finite group acting on $R$, and let $R^G \subseteq R$ be the fixed points for the action of $...
8
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1answer
252 views

Vanishing of Kahler differentials vs. surjective Frobenius?

Let $A$ be an $\mathbf{F}_p$-algebra such that $\Omega_{A/\mathbf{F}_p}=0$. Is the Frobenius map on $A$ surjective? Some context: i. The converse is clearly true. ii. The answer is yes if $A$ is a ...
8
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1answer
218 views

$p$-adic completeness of the ring of Witt vectors

Let $R$ be a ring that is $p$-adically complete for a prime $p$ and let $W(R)$ denote the ring of $p$-typical Witt vectors. Is it true that $W(R)$ is $p$-adically complete? (A ring $A$ is $p$-adically ...
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0answers
89 views

Do we have super Plucker relations for a super Grassmannian?

Super Grassmannians are introduced by Manin, see for example. We have Plucker relation for Grassmannian. Are there some references about super Plucker relations for super Grassmannian? Thank you ...
3
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1answer
178 views

Geometric contractibility of noetherian rings

Let $A$ be a noetherian ring. Let us define $A$ to be $n$-contractible if All locally free sheaves of rank $\le n$ over $\text{Spec} A$ is trivial. There exist a non-trivial locally free sheaf of ...
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28 views

Reducing certain monomials away from a set of polynomial equations

Suppose I have six polynomials $F_1, G_1, F_2, G_2, F_3, G_3 \in \mathbb{Q}[x_1, ..., x_n]$. The exact situation I have is the following: We have $\deg f_i = \deg G_i = d_i$ and $d_3 > d_2 > d_1$...
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0answers
92 views

Projective dimension of modules

Let $M$, $N$ be two modules over commutative ring $R$. Suppose that they have finite projective dimension. Can we say something about the projective dimension of the $R-$module $Hom_{R}(M,N)$? Thanks!
3
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0answers
125 views

Weyl algebra acting on a polynomial ring

Let $\mathbb K$ be a characteristic-$0$ field, $R=\mathbb K[x_1,\ldots, x_n]$ be a polynomial ring, and $W=\mathbb K[x_1,\ldots,x_n,\partial_1,\ldots \partial_n]$ be the Weyl algebra. As usual $W$ ...
14
votes
1answer
352 views

Does the ring $R = \mathbb{Z}[X^{\pm1}]$ of Laurent polynomials over $\mathbb{Z}$ satisfy $SL_2(R) = E_2(R)$?

Let $R = \mathbb{Z}[X^{\pm1}]$ be the ring of Laurent polynomials on one indeterminate over $\mathbb{Z}$. Let $E_2(R)$ be the subgroup of $GL_2(R)$ generated by the matrices that differ from the ...
4
votes
0answers
156 views

Is the class of commutative generalized Euclidean rings stable under quotient and localization?

Let $R$ be an associative ring with indentity and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by matrices obtained from the identity matrix by replacing an off-diagonal entry by some $r∈R$. ...