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

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1answer
55 views

Regularity of special monomial ideals

Let $R = k[x_1\ldots x_n]$, and $a,b$ are vectors with integer entries, whose all entries of $a$ are non-negative, and say sum of coordinates of $b$ is $0$. Let $I$ be a monomial ideal generated by ...
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1answer
339 views

Geometric interpretation of a (standard) commutative algebra fact

Which is your geometric interpretation (if any) of the following commutative algebra proposition? Proposition. Let $M$ be a finitely generated $A$-module, $I\subseteq A$ an ideal, and $\phi\in ...
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2answers
287 views

Pontryagin dual

Suppose $M$ is a $Z_p[[T]]$-module and $\widehat{M}$(the Pontryagin dual of $M$) is a finitely generated torsion $Z_p[[T]]$-module. How to prove that $\widehat{M}$ has $\mu$-invariant zero ...
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0answers
44 views

Morphisms preserving weak normality

I would like to find a class of morphisms for which weakly normality descends. The notion of seminormlaity is very close to the one of weakly normality and for seminormal schemes one has Theorem 5.8 ...
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1answer
238 views

A construction of Kähler differentials and Illusie cotangent complex as colimit over embeddings

Let $\Bbbk$ be a field, $X$ affine scheme of finite type over $\Bbbk$. Let $\mathcal C_X$ be the category of closed embeddings of $X$ into (say affine) smooth $Y$'s of finite type over $\Bbbk$, ...
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0answers
246 views

What is the state of art in Groebner bases

How big polynomial systems can we deal with? How do you know when you don't even have to try? Motivation: Recently I tried to solve a problem posed in another MO question and ultimately I got stuck ...
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2answers
179 views

0-dimensional Gorenstein local ring.

Assume the following condition for the ring T = F_p[[X,S]]/I: Condition 1. T is NOT a zero ring. Condition 2. I is generated by 3 elements of F_p[[X,S]], but NOT by 2 elements. Then, is T a ...
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0answers
136 views

Does the coordinate ring of affine variety admit a structure of infinite dimensional variety?

We work in the category of algebraic varieties over some algebraically closed field $k$. By infinite dimensional variety I mean a filtration: $$ V_0\subset V_1\subset V_2\subset\ldots $$ where each ...
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1answer
121 views

Which power of $2$ kills $W(k)$?

Is the following fact "well-known": if $-1$ is a sum of squares in a field $k$, then the Witt group $W(k)$ of quadratic forms is killed by multiplication by $2^N$ for some $N\ge 0$? What can one say ...
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63 views

Interpret some coefficients in algebras

Let $A$ be a real vector space equipped with a scalar product $\langle \,,\,\rangle$, and assume moreover that a multiplication is define on it so that becomes an algebra (e.g. polynomials with the ...
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3answers
263 views

Jacobson ring = a ring whose nilradical and Jacobson radical coincide?

In Wikepedia (Nilradical), It claims that "A ring is called a Jacobson ring if the nilradical coincides with the Jacobson radical." Here the word "ring" means a commutative ring. However, I remember ...
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1answer
252 views

Iwasawa invariants

Suppose $M$ is a finitely generated torsion $Z_p[[T]]$-module; the torsion comes from the $\mu$-invariant and the $\lambda$-invariant. Consider $M/(p)$ and $M[p]$ ($p$-torsion of $M$) which are ...
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1answer
182 views

Resolution of singularity of polynomials

Let $f$ be polynomial on a vector space $V$. Let $Z$ be the zero set of $f$ in $V$. Let $Z_{sing}$ be the singular part of $Z$. By Hironaka's desingularization theorem, there exists a birational map ...
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1answer
315 views

How is a descent datum the same as a comodule structure?

For a homomorphism of commutative rings $f:R\to S$, there are at least two notions of a descent datum for this map. One of these is to be an $S$-module $M$, with an isomorphism $M\otimes_R S\cong ...
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2answers
239 views

Rank of a $ \mathbb{Z}_{p}[[T]] $ module

Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p}$ vector spaces. So we can ...
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1answer
155 views

Checking flatness using radical ideals

Let $R$ be a commutative ring and $M$ a not necessary finitely presented $R$-module. I am looking for a prove or a counterexample to the following statement: $M$ is flat as an $R$-module if and only ...
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2answers
189 views

Kahler differentials on cluster varieties

On affine toric varieties there is a classical theorem of Danilov which gives some combinatorial ways to describe the global sections of an appropriate sheaf of Kahler differentials as a vector space. ...
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1answer
238 views

Does Noether normalization hold more general? [duplicate]

Noether normalization tells us that a finitely generated $k$-algebra is an integral extension of a polynomial algebra over the field $k$. My question is whether this still holds if we replace the ...
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1answer
153 views

$I/N$ is finitely presented module

Let $R$ be a commutative ring and $N = Nil(R)$ the set of its nilpotent elements. Suppose that $N$ is a divided prime ideal, i.e. for any ideal $I$ of $R$ either $I \subseteq N$ or $N \subseteq I$. ...
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5answers
687 views

is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?

This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent ...
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0answers
93 views

Universally catenary and all its formal fibers over minimal members are Cohen-Macaulay but it has a nonCohen-Macaulay formal fiber

Please help me to find a Noetherian local ring $R$ such that: $R$ is universally catenary and all its formal fibers over minimal members of $Spec(R)$ are Cohen-Macaulay but $R$ has a nonCohen-Macaulay ...
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1answer
392 views

Well founded induction attributed to Noether

What I know as well founded induction, namely the rule $$ \big(\forall y.(\forall z.z\lt y\Rightarrow\phi z)\Rightarrow\phi y\big)\Longrightarrow\big(\forall x.\phi x\big), $$ whose validity is the ...
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2answers
227 views

Does $\Gamma_*$ commute with tensor product?

Given a coherent sheaf $\mathcal{F}$ we denote by $\Gamma_*(\mathcal{F})=\oplus H^0(\mathcal{F}(d))$. Suppose, $\mathcal{F}_1$ and $\mathcal{F}_2$ are two coherent sheaves on $\mathbb{P}^n$. Denote by ...
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1answer
82 views

Example involving partially ordered Abelian groups

Definition 1: Let $(G,\leq)$ be a nonzero partially ordered Abelian group with order unit $u$. (Recall that $u\in G$ is a order unit if, for every $g\in G$, there exists $N\in\mathbb N$ such that ...
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0answers
405 views

A question on Castelnuovo-Mumford regularity

Consider a short exact sequence $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ of coherent sheaves on $\mathbb{P}^n$. Assume that $\mathcal{F}''$ (resp. $\mathcal{F}$) is $m-1$ (resp. ...
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0answers
147 views

Square of primary ideals

Is there any example of a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?
4
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1answer
194 views

Complexity of computing the Galois group

There has been some discussion of computing the Galois group of a polynomial over the integers, but I can't seem to find any results, or even a question of what the complexity of this might be. For ...
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0answers
77 views

Irreducibility testing and factoring

It is a result of van Hoeij and Novicin (Algorithmica, 2012) that factoring polynomials of degree $d$ over the integers can be done in $O(d^6 + d^4 \log^2 A)$ time, where $A$ is the coefficient bound. ...
6
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2answers
328 views

Pre-images of unipotent elements in $\operatorname{SL}_{n}(A)$

The starting point of this question is the (presumably) well-known theorem (the proof I know is from Abelian $\ell$-adic representations and elliptic curves from J-P.Serre in which it is a lemma for ...
5
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3answers
390 views

Exponentials in the opposite category of finite separable algebras

Let $K$ be a field and $G=Gal(K_s/K)$ is its absolute Galois group. Then, by Galois theory, the category of finite separable algebras over $K$ (denoted by $Sep(K)$) and the category of finite ...
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1answer
173 views

Characteristic polynomial of exterior power

Suppose $f$ is a linear map, and consider $\Lambda^k f$ as the usual exterior power of $f$ (if you prefer matrices, it is a matrix whose entries are the $k\times k$ minors of $f.$) The coefficients of ...
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1answer
75 views

Cohen-Macaulayness of the scheme of centralizer

Let $G$ be a simply connected group over an algebraically closed field $k$, and $I:=\{(g,\gamma)\in G\times G\vert~ g\gamma=\gamma g\}$ the scheme of centralizer. Is $I$ a Cohen-Macaulay scheme ...
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0answers
135 views

Ergod Theorem for $\mathbb{F}_{3}[[X,S]]$

Assume we have the automorphism on $2$-variable power series ring $\mathbb{F}_{3}[[X,S]]$ over finite field $\mathbb{F}_3$ as follows: $σ: S \longrightarrow S + S^3$ $σ : X \longrightarrow X + S + ...
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2answers
719 views

Fields aren't group objects in Ab, so what are they?

This might be a vague question, but I am troubled by the fact that fields do not admit a nifty categorical definition. An obvious attempt such a definition would be to say that fields are commutative ...
4
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1answer
174 views

Algebra structure $Tor(A,A)$

This is a question i asked on math.stackexchange but i didn't get any answer. Let $A$ be algebra over commutative ring $k$ and $P_{\bullet}=(P_i,d_i)\rightarrow A$, $k$ projective resolution. Then we ...
4
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1answer
126 views

algebraic multivariate power series over a field

Is there a "simple" proof that any power series in $\mathbb Q[[X,Y]]$ algebraic over $\mathbb Q(X,Y)$ is in the Henselization of $\mathbb Q[X,Y]$ localised in $(X,Y)$?
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2answers
231 views

Variety determined by interior product of the determinant?

Let $\Lambda^k(V)$ be the space of alternating $k$-linear tensors on $V$. Consider the map $f: \left(\mathbb{R}^n\right)^{n-k} \to \Lambda^k(\mathbb{R}^n)$ given by $\left(v_1,v_2, ..., ...
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0answers
42 views

Cohen-Macaulayness of inseparable isogeny k-algebras

Let $R$ and $S$ be 2 associated, commutative, and unita $k$-algebras where $k$ is an algebraically closed field of characteristic $p$. We call these algebras inseparable isogeny or $F$-isomorphism if ...
4
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1answer
113 views

Localizations of hereditary rings

It is known that if a commutative Noetherian ring $R$ is hereditary then for any maximal ideal $M$ the localization $R_M$ is also hereditary. Is the Noetherian assumption necessary?
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3answers
2k views

Multiplicative functions $\phi : M_n(F) \longrightarrow F$ with $\phi(I) = 1$

Let $F$ be an infinite field and let $f \in F[x_{11},x_{12},...,x_{nn}]$ be an arbitrary polynomial in $n^2$ variables. Consider the function $\phi : M_n(F)\longrightarrow F$ defined by ...
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0answers
142 views

A question on binary polynomials

This is probably a well-known result but I was not able to find a reference on my search. My question concerns general polynomials $f(x,y) \in \mathbb{Z}[x,y]$ such that $f$ cannot be written as a ...
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1answer
236 views

Etale descent of morphisms of schemes

Let $\pi: Y \to X$ be an etale morphism of finite-type schemes over a field $k$. Let $Z$ be a $k$-scheme. Suppose we have a morphism $f:Y \to Z$. When does $f$ descend to a morphism $g: X \to Z$? ...
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1answer
358 views

Resolution of a module as an $A_\infty$ module over resolution of an algebra

The following question looks like a basic question on resolutions over commutative rings, unfortunately I was not able to find a reference. Let $A$ be a regular commutative noetherian ring (and ...
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1answer
275 views

On exactness of the functors $M \mapsto \hat{M}$ and $M \mapsto \hat{A}\otimes_{A}M$

If $A$ is a Noetherian ring, $M$ is a finitely generated module, $I$ is an ideal of $A$, and $\hat{A}$ is the $I-adic$ completion of $A$, then we know that $\hat{A}\otimes_{A}M\cong\hat{M}$. Also in ...
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1answer
147 views

Polynomial analogue of “prime independence”

In number theory a well-known fact is that congruence modulo distinct primes are 'independent'. That is, to know that $n \equiv a \pmod{p}$ does not change the probability as to what $n \equiv x ...
5
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2answers
201 views

Degree of a projective scheme and its defining equations

Let $X \subset \mathbb{P}^n$ be any projective scheme. Denote by $I_X$ the (saturated) ideal of $X$. Suppose the degree of $X$ is $d$. Under what assumptions there exists a polynomial in $I_X$ of ...
4
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2answers
377 views

Linearisation of a group

If $G$ is a connected Lie group acting on a vector $\mathbb{C}$-space $V$ then it is well known that the algebra of invariants $\mathbb{C}[V]^G$ coincides with the algebra of invariants ...
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3answers
350 views

zero homology of augmented Koszul complex implies the sequence is regular?

Let $A$ be a Noetherian ring, $M$ a finite $A$-module and $I=(y_1,\cdots,y_n)$ an ideal of $A$ such that $M \neq IM$. Denote by $H_i(y_1,\cdots,y_n;M)$ the homology at dimension $i$ of the augmented ...
13
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1answer
371 views

History of Koszul complex

This is a question about history of commutative algebra. I'm curios why Koszul complex from commutative algebra is called Koszul complex? All Koszul's early papers are about Lie algebras and Lie ...
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1answer
114 views

Chain of Ideals of same height

I have been wondering about the following (and allready posted a similar question, see Dimension of ring completion wrt to a decreasing chain of ideals): Let $R$ be the ring of formal power series ...