**1**

vote

**1**answer

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 ...

**2**

votes

**1**answer

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 ...

**0**

votes

**2**answers

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 ...

**1**

vote

**0**answers

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 ...

**4**

votes

**1**answer

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 $\Bbbk$, ...

**7**

votes

**0**answers

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 ...

**0**

votes

**2**answers

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 ...

**1**

vote

**0**answers

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 ...

**4**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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 ...

**2**

votes

**3**answers

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 ...

**0**

votes

**1**answer

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 ...

**1**

vote

**1**answer

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 ...

**9**

votes

**1**answer

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 ...

**0**

votes

**2**answers

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 ...

**2**

votes

**1**answer

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 ...

**3**

votes

**2**answers

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. ...

**0**

votes

**1**answer

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 ...

**0**

votes

**1**answer

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 $N \subseteq I$.
...

**13**

votes

**5**answers

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 ...

**2**

votes

**0**answers

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 ...

**8**

votes

**1**answer

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 ...

**2**

votes

**2**answers

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 ...

**2**

votes

**1**answer

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 ...

**4**

votes

**0**answers

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. ...

**7**

votes

**0**answers

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**

votes

**1**answer

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 ...

**6**

votes

**0**answers

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**

votes

**2**answers

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**

votes

**3**answers

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 ...

**5**

votes

**1**answer

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 ...

**1**

vote

**1**answer

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 ...

**1**

vote

**0**answers

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 + ...

**10**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**1**answer

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 $(X,Y)$?

**3**

votes

**2**answers

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, ..., ...

**1**

vote

**0**answers

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**

votes

**1**answer

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?

**1**

vote

**3**answers

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 ...

**0**

votes

**0**answers

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 ...

**2**

votes

**1**answer

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 $g: X \to Z$? ...

**5**

votes

**1**answer

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 ...

**1**

vote

**1**answer

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 ...

**1**

vote

**1**answer

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**

votes

**2**answers

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**

votes

**2**answers

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 ...

**6**

votes

**3**answers

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**

votes

**1**answer

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 ...

**0**

votes

**1**answer

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 ...