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

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5
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1answer
290 views

automorphisms of local rings vs local change of coordinates

Let $R$ be a local (commutative, associative) ring over a field of zero characteristic. (My typical examples are $k[[x_1,..,x_p]]/I$, $k\{x_1,.,x_p\}/I$, $C^\infty(\Bbb{R}^p,0)$. If it helps one can ...
3
votes
0answers
113 views

On the computational complexity of the Hilbert polynomial of numerical semigroup rings

Let $(R, \mathfrak{m}) = k[[X^a, X^b, X^c]]$, $a<b<c$, $gcd(a, b, c) = 1$, be a semigroup ring. We have $R$ is a Cohen-Macaulay local ring of dimension one. It is well known that ...
3
votes
1answer
230 views

Non-commutative normalization

Let $A$ be a (non-commutative) associative algebra with 1. Assume that $A$ contains a cental subalgebra $Z$ such that a) $Z$ is a noetherian domain b) $A$ is a finitely generated module over $Z$. ...
9
votes
3answers
709 views

Completion of a local ring of a curve

Let $X$ be a smooth projective irreducible curve defined over an algebraically closed field $\mathbb{K}$ (of arbitrary characteristic), and let $p\in X$ be a closed point. Denote by $\mathcal{O}_p(X)$ ...
2
votes
1answer
192 views

Hochschild cohomology of commutative quotients

Notation: Let $k$ be a commutative local ring and let $HH^{i}(A,N)$ denote the $i^{th}$ Hochschild cohomology $k$-module of a $k$-algebra A with coefficients in an $(A,A)$-bi-module $N$. If ...
3
votes
1answer
189 views

Classical deformation of algebras

Given a complex manifold (or a smooth scheme) $X$, the classical (infinitesimal) deformation theory is parametrized by the first cohomology with coefficients in the tangent sheaf $H^1 (X, T_X)$. ...
0
votes
1answer
135 views

if $ \lambda (I)= \dim R$, can one claim that $I$ is an $m$-primary ideal?

definition from Bruns-Herzog: It is easy to see that if $I$ is a $m$-primary ideal of $R$ then $ \lambda (I)= \dim R$. I wonder if the converse is true: if $ \lambda (I)= \dim R$, can one ...
0
votes
1answer
111 views

Connected curve

Assume we have a normal,connected quasi projective scheme $Y:=X\backslash D$ where $X$ is a quasi projective scheme over field $k$, not necessarily char zero and also $D$ is a simple divisor, not ...
2
votes
0answers
66 views

variants of ramification groups - need terminology and sources

I've asked this question in several more elementary forums, and haven't get any answer. So I presume this is not so an elementary question. Let $L/K$ be a Galois extension, and $w$ be a valuation of ...
1
vote
1answer
175 views

Bounded dg algebra vs unbounded dg algebras

1)Let $Cd_{\geq 0}ga$ be the category of non negatively commutative cochain dg algebra over a field $\Bbbk$ of charachteristic zero. Let $w\: : \: Cd_{\geq 0}ga\to dg_{\geq 0}Mod$ be the forgethfull ...
3
votes
3answers
255 views

Injective map between two schemes

Assuem we have a finite surjective map between two irreducible, separated schemes, $f:X \rightarrow Y$, and for a dense open $U \subset Y$ and for any $y \in U$, $|X_y| =1$, then can we say $f$ is ...
0
votes
1answer
204 views

Extension of homomorphism to place in quotient field, or of local ring to valuation ring in quotient field

Let be given a domain $R$ (that can be supposed to be integrally closed if this can help), and $\varphi$ an homomorphism of $R$ into a field $F$. $\varphi$ extends uniquely to a homomorphism ...
1
vote
1answer
240 views

intuitive interpretation of analytic spread

I am studying analytic spreads from Bruns-Herzog's book. The definition is clear but calculation of the analytic spread of an ideal is hard for me in practice. I wonder if it is hard for you too. ...
2
votes
1answer
102 views

Any two bivariate algebraically dependent polynomials are always in the same ring generated by some bivariate polynomial?

If $f(x,y)$ and $g(x,y)$ are two algebraically dependent polynomials over some field $k$, is it true that there exists a bivariate polynomial $p(x,y)$ such that both $f(x,y)$ and $g(x,y)$ are in the ...
2
votes
1answer
91 views

On transforming pair of bivariate polynomials to pair of univariate polynomials by applying polynomial map

We know that a polynomial map $f(x,y), g(x,y)$ is polynomial automorphism if there exists polynomials $p(x,y)$ and $q(x,y)$ such that $f(p,q)$=x and $g(p,q)=y$. Jacobian conjecture tries to ...
1
vote
0answers
144 views

On Prüfer domains

Is there any Prüfer domain $R$ that has a prime ideal $P$ that is not finitely generated but $xP$ is subset of a finitely generated ideal $I$,for some $x$ in $R-P$ and $I$⊂$P$?
1
vote
0answers
119 views

Criterion for normality of a schematic image

Consider a projective flat morphism $$ f\colon X\to Y $$ between normal varieties. Let's say over the complex numbers. The geometric fibers of $f$ are all irreducible. I would like a criterion to ...
2
votes
1answer
281 views

A perfect domain that is not integrally closed?

Does there exist an integral domain $R$ of characteristic $p > 0$ that is perfect (i.e., $x \mapsto x^p$ is bijective on $R$) but not integrally closed in its field of fractions?
6
votes
1answer
206 views

Purely noncommutative algebra-Morita equivalence

Morita equivalence of algebras certainly don't preserve commutativity: even if $A$ is commutative there are plenty of noncommutative algebras which are Morita equivalent with $A$---for example all ...
2
votes
1answer
154 views

what are the possible approximations for ideals

(Fix some local ring $(R,\mathfrak{m})$ over a field of zero characteristic.) Suppose an ideal $J$ is defined by some complicated formula/procedure. And there is no hope of computing it/or writing ...
0
votes
1answer
153 views

Codimension in zero and positive characteristic

Let $F_0,\ldots,F_m\in\mathbb{Z}[x_0,\ldots,x_n]$ be polynomials with integer coefficients and let $p$ be a prime integer. Consider the two ideals: $$I_0:=(F_0,\ldots,F_m)\subset ...
1
vote
0answers
149 views

Geometric (or intuitive) interpretation of Almost Gorenstein and Cohen-Macaulay rings

This question is related to This one: Darius Math in his good answer added that Cohen-Macaulay ring's singularities is nice. So I'd like to complete that question and ask: Let R be a local ...
7
votes
0answers
391 views

What is the etale fundamental group of Spec Z((x))?

I know the etale fundamental group of $\mathbb{Z}$ is trivial. For algebraically closed fields $K$, the etale fundamental group of $K((x))$ is $\hat{\mathbb{Z}}$, since all covers in this case are ...
0
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0answers
103 views

The Euler characteristic of Hilbert series

The Hilbert series of a graded vector space $V=\bigoplus_{n\mathbb Z}V_n$ is the (ordinary) generating function of the dimensions of its homogeneous components, $h_V(t)=\sum_{n\in\mathbb Z}t^n\dim ...
0
votes
1answer
74 views

Homologue of the Inertia group and of the Frobenius theorem for the group of values of a valuation

As I said previously, I have some problems in the theory of valuations and places. Let L/K be a finite (say) Galois extension, F a place of L, and v a valuation of L. I denote by l and k the residue ...
0
votes
1answer
146 views

Valuations and places - decomposition and inertia group

I feel very uncomfortable with some aspects of the theory of valuations, places, and valuation rings. Here is one of my problems : Assume that L/K is a finite Galois extension of fields, and that F is ...
6
votes
3answers
451 views

How to prove that two univariate polynomials are always algebraically dependent?

How to prove that two univariate polynomials(over any field) are always algebraically dependent? Also, how to prove the generalization of this question i.e if number of polynomials are more than ...
4
votes
1answer
129 views

Minimal length of quotient by parameter ideals

Consider a commutative noetherian local ring $R$ of dimension $d$ and define $$c_R\colon=\min_{(x_1,\ldots,x_d)} \{\mathrm{length}\ R/(x_1,\ldots,x_d)R\mid (x_1,\ldots,x_d)\ \mathrm{is\ a\ system\ of ...
9
votes
0answers
115 views

How useful is knowing every torsionfree $\mathcal O(D)$ module is flat?

One of the corollaries of Weiertrass' factorization theorem plus the theorem of Mittag Leffler is that $\mathcal O(\Bbb C)$, more generally $\mathcal O(D)$ for some region $D$ is such that every ...
2
votes
2answers
249 views

Irreducibility after substitution

I would like to show that when $f(x,y)$ is irreducible over $\mathbb{C}[x,y]$ then $f(x^2,y)$ is irreducible over $\mathbb{C}[x,y]$. I know that this is not true in general, for example, $f(x,y) = ...
5
votes
0answers
142 views

Link between abelian groups and endomorphisms

When teaching Algebra, I try to share my fascination about two apparently unrelated questions, which turn out to involve the same theory: classifying the finitely generated abelian groups, ...
3
votes
1answer
166 views

Extending descent data from the special fiber of an extension of DVR's

My question is about the proof of Lemma D.3 on p. 147 of the book "Neron models" by Bosch, Lutkebohmert, and Raynaud. Namely, towards the end of that proof there is the sentence "That $\varphi$ ...
7
votes
1answer
152 views

Is there a ring which is not Hermite but is coherent?

Call a commutative unital ring $R$ Hermite if for all $m, n\in \mathbb{N}$ with $m<n$, and all $f\in R^{m\times n}$ such that transpose($f$) is left invertible (with a matrix with entries from ...
5
votes
2answers
217 views

Compute adjugate matrix over commutative ring

Let $A$ be a $n\times n$ matrix over a commutative ring. I'm looking for a good method to compute its adjugate matrix. My current approach is to use the Cayley-Hamilton theorem: $$\text{adj}(A) = ...
1
vote
1answer
178 views

Regular rings and formally smooth algebras

Let $A\rightarrow B$ be a commutative $A$-algebra. If $A$ is a field and $B$ Noetherian and formally smooth over $A$, then it is known that $B$ must be a regular ring. Is there a partial converse of ...
0
votes
1answer
172 views

Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.

I would like to know a reference of the following statement (or counter example). Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is ...
4
votes
1answer
232 views

Transcendence degree of the surreals over the subfield generated by the ordinals

Consider the Grothendieck ring $K[\Omega]$ of the semiring $\Omega$ of all ordinals under the operations of natural sum and product. Its quotient field $K(\Omega)$ is naturally a subfield of the ...
1
vote
0answers
86 views

Computing the bourbaki ideals

By virtue the Griffith's paper and subsequently e.g. Goto's paper several examples of several desired class of Noetherian normal domains with specific finite length local cohomologies are constructed ...
0
votes
0answers
254 views

Affine communication lemma and finite limits in the category of rings

Let $X$ be a scheme and $\mathrm{Spec}(B) = V \subseteq X$ be an open affine subset. When using the affine communication lemma (c.f. Theorem 6.3.2, Vakil's notes, Foundations of Algebraic Geometry), ...
5
votes
2answers
229 views

Integral domains with totally ordered spectra

In my research I ended up trying to prove some properties of integral domains such that their spectrum is a totally ordered poset. Are there some nice (ubiqitous/natural) examples of such domains, ...
1
vote
1answer
145 views

ideal of maximal minors is cohen-macaulay?

Let $k$ be an algebraically closed field. Let $A$ be an $m \times n$ matrix with linear forms $a_{ij} \in k[x_1, \ldots, x_p]_1$ as entries. Let $I$ be the ideal generated by the maximal minors of ...
0
votes
1answer
144 views

When a proper morphism of schemes is a closed imbedding?

Let $X$ and $Y$ be finitely presented schemes over $\mathbb{C}$. Let $f\colon X\to Y$ be a proper morphism. Let us assume that for any finitely presented scheme $S$ the induced map $$Mor_{Sch}(S,X)\to ...
4
votes
1answer
313 views

Intersection of nonzero prime ideals is zero — does it have a name?

The Rabinowitch trick (in Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, page 132) says that $R$ (commutative unital ring) is Jacobson if and only if for every prime ideal $P ...
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vote
0answers
93 views

Criterion for global dimension of subring

All rings are assumed to be associative and unital. If $B$ is a commutative sub-ring of $A$ (which itself needs not be commutative) then what properties of $B$ are both necessary and sufficient for ...
0
votes
0answers
94 views

Residual Intersections of a complete intersection

Let $R$ be a Cohen-Macaulay local ring and $I=(b_1,\dots,b_s)$ be a complete intersection generated by a regular sequence $\underline{b}$. Let $\mathfrak{a}\subseteq I$ such that ...
1
vote
2answers
221 views

Computing the nonsingular projective model of a plane curve

Is there an implemented algorithm available in standard software systems (Sage, Magma, Macaulay, etc.) that will compute the nonsingular projective model of a plane curve over $\mathbb Q$?
1
vote
0answers
126 views

Questions on prime integral ideal congruences

Suppose that we are given a fixed pair $a_1, a_2$ of non-zero irrational algebraic integers in some number field $K$ which are independent over $\mathbb{Q}$. Suppose that $\mathcal{P}$ is a prime ...
0
votes
1answer
173 views

A condition on isolated singularity

Suppose $F: {\mathbb C}^N \to {\mathbb C}$ defines a singularity at the origin (for simplicity one can assume that $F$ is a quasi-homogeneous polynomial). Suppose it is nondegenerate, i.e., $dF(z) = ...
1
vote
0answers
77 views

Does this condition imply a polynomial is a product of linear factors

Let $\Lambda$ be a lattice (i.e. $\Lambda \simeq \mathbb{Z}^n$) with a positive subcone $\Lambda^+$. Let $H: \Lambda^+ \rightarrow \mathbb{C}$ be a function such that $\forall\mu \in \Lambda^+$, ...
4
votes
0answers
145 views

How to check whether a scheme of finite type over Spec Z is regular or not [duplicate]

Let $f_1, f_2, \ldots f_k$ be a set of polynomials in $n$ variables, with integer coefficients. These define an affine scheme $X$ of finite type over $Spec \mathbb{Z}$. (We could also consider ...