0
votes
0answers
52 views

Change of relative base

If $k$ is a commutative ring, $A$ a $k$-algebra and $\phi: k \rightarrow k'$ is a morphism of rings then how (/under what conditions) can the relative homology functors $Ext_{A/k}(-,-)$ and ...
9
votes
2answers
254 views

Existence of a ring with specified residue fields

Given a finite set of fields $k_1, \ldots, k_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m_i$ such that $R/m_i \cong k_i$? To prevent things from being too easy, I ...
3
votes
2answers
170 views

Minimal fields of isomorphism for varieties

Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...
11
votes
2answers
626 views

Are there only finitely many associative algebras of fixed dimension?

Given an algebraically closed field $F$, for any positive integer $n$, are there always only finitely many non-isomorphic (noncommutative) associative algebras (possibly without identity) with ...
3
votes
2answers
319 views

Degrees of factors of polynomial $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$

I’m facing the problem of factoring polynomials of type $f(x)=x^q-(ax^2+bx+c)\in \mathbb{F}_q[x]$ and the degrees of factors seem to be quite special. For example, according to my experimental results ...
4
votes
0answers
266 views

The logarithm over $\mathbb F_1$

In 'Cyclotomy and analytic geometry over F1', Manin proposes a version of the notion of `analytic function' over the 'field with one element $\mathbb F_1$'. Question 1: can somebody explain or give ...
2
votes
0answers
97 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 ...
7
votes
0answers
159 views

Square of primary ideals

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

when does one want to use the Reynolds operator in GIT?

The role of Reynolds operator in GIT has always been a little mystery to me. Actually I see that in some proofs it gets used in an efficient way, but what I cannot grasp is the general philosophy. I ...
4
votes
0answers
223 views

Ext groups of affine scheme

Let $A$ be a commutative ring. $\textbf{Spec}(A)$ is the the spectrum of $A$. $M$ and $N$ are $A$-modules. $\tilde{M}$ and $\tilde{N}$ are sheaves associated on $\textbf{Spec}(A)$ respectively. I ...
2
votes
0answers
103 views

How do I compute the Azumaya locus?

I have an algebra $A$ that is finite over its centre $Z(A)$ and I want to compute the Azumaya locus of $Z(A)$ (or equivalently, its ramification locus). Looking in McConnell-Robson Noncommutative ...
3
votes
2answers
222 views

on a characterisation of regular D-modules

Let $X$ be a smooth variety over a field of characteristic zero. Let $M$ be in the derived category of holonomic $\mathcal{D}_{X}$-modules, $D^{b}_{h}(\mathcal{D}_{X})$. We know that if we assume ...
11
votes
1answer
507 views

Counting representations of $k[x,y]$ when $k$ is finite

$\newcommand{\GFq}{\mathbf F_q}$ Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial ...
2
votes
1answer
207 views

Symmetric algebra of an ideal and syzygies

Let $(R,\mathfrak{m})$ be a Cohen Macaulay local ring and $I=(a_1,\ldots,a_g,a_{g+1})$ an almost complete intersection ideal of codimension $g.$ Let $R^k\longrightarrow R^{g+1}\longrightarrow ...
0
votes
1answer
224 views

Finite extension of a field [closed]

Is it true, that if $A$ is finitely generated commutatative algebra over a field $k$, not necessary algebraically closed, then prime ideal $p \subset A$ is maximal if and only if $k \subset Quot(A/p)$ ...
4
votes
2answers
521 views

Example of a reduced ring whose completion is not reduced

Let $(R,\mathfrak{m})$ be a local Noetherian ring. Give an example such that R is reduced but $\mathfrak{m}$-adic completion of R is not reduced.
3
votes
1answer
120 views

something like a kleene algebra of rational functions?

There's a procedure that control engineers (used to) do to calculate the transfer function of a linearized system, gradually reducing a block diagram to a rational function of s. It's justified by ...
9
votes
4answers
532 views

zeros of a homogeneous polynomial

Hi All, Let $F$ be a finite field, $\lambda\in F$, and $$p_\lambda (x,y,z)=\left|\begin{array}{ccc}x & y & z \\ y & z & x +\lambda z \\ z& x+\lambda z & y+\lambda x+\lambda ...
1
vote
0answers
110 views

group of automorphisms of the Lie algebra of vector fields on affine variety

Hello, Let $X$ be an affine variety and $A(X)$ be a ring of regular functions on $X$. Consider a Lie algebra of derivations of $A(X)$ which we denote as $Der(A(X))$. It is known that $Aut(X)$ (group ...
3
votes
1answer
165 views

degree of polynomial in Gröbner basis

Let $f(x, y) = \sum_{m=0}^{M-1}\sum_{n=0}^{N-1} a_{m,n} x^m y^n$ and $g(x, y) = \sum_{m=0}^{M-1}\sum_{n=0}^{N-1} b_{m,n} x^m y^n$. Computing the Gröbner basis, we get an univariate polynomial ...
0
votes
1answer
155 views

Embedded associated prime

$\underline{\textbf{Embedded associated prime}}$ I am reading the book "Joins and Intersections". In the proof of Rees theorem I have some doubt. Let $\mathbf M$ be a finitely generated $\mathbf ...
1
vote
0answers
103 views

Standard equivalences and non-vanishing maps

EDIT : I edited the question according to Prof. Rickard's suggestions Let $Y$ be an affine variety over $\mathbb{C}$ and $A$ and $B$ be $2$ algebras with finite homological dimension over $Y$ such ...
2
votes
2answers
394 views

Is there an almost-direct product decomposition for disconnected reductive algebraic groups?

$\textbf{Some definitions:}$ Let $G$ be an algebraic group (for me that is the complex points of an affine algebraic group). We say $G$ is reductive if its unipotent radical (maximal connected normal ...
3
votes
3answers
311 views

Support of a module over a polynomial algebra

In Atiyah and Bott's paper "The Moment Map and Equivariant Cohomology", they say that for any exact sequence of modules over $\mathbb{C}[u_1,...,u_l]$ $$D \to E \to F,$$ we have that Supp $E \subset$ ...
1
vote
2answers
322 views

Are hensel valuation rings N2?

This seems like the kind of thing an expert should be able to answer off the top of their head: Recall that a valuation ring is an integral domain $A$ such that for every $a \in Frac(A)$ we have ...
1
vote
0answers
198 views

bivariate polynomial

Hello, Let $p(x,y) = \sum_{m=1}^M\sum_{n=1}^N a_{m,n}x^{m-1}y^{n-1}$ be a bivariate polynomial where $\{a_{m,n}\}$ are complex. If $(x_k, y_k), k=1,2,\cdots, MN-1$ are roots of $p(x,y)=0$ where ...
11
votes
1answer
912 views

Deformations of the punctured affine plane

Let $k$ be some field, algebraically closed and of characteristic $0$, if you like. Let $U= \mathbb{A}^2_k \setminus \{ (0,0) \}$ be the punctured affine plane over $k$. Write $U$ as the union of ...
7
votes
1answer
354 views

Constructive proof of “Projective implies proper”

For every ring $A$, the structural morphism of schemes $\pi_A : {\bf P}^n_{A} \to {\rm Spec}{A}$ is a closed map. The usual proof of this fact is not constructive : given equations of a closed subset ...
1
vote
1answer
150 views

fixed point scheme in caracteristic p

Let X\rightarrow A^{n} a smooth affine scheme over an affine space. Everything is defined over a field k. Let G a finite group acting on X and suppose that his order is divisible by the caracteristic ...
0
votes
2answers
208 views

Projectives in the category of coherent sheaves on a projective variety

Hi all I thought that the following was true but was unable to think of a proof and after browsing the internet now I am not so sure. My understanding of graded rings and modules is not great so I'm ...
28
votes
2answers
990 views

What do cluster algebras tell us about Grassmannians?

One of the first examples of a cluster algebra given in Fomin and Zelevinsky's original paper is the homogeneous coordinate ring $\mathbb{C}[G_{2,n}]$ of the Grassmannian of planes in $\mathbb{C}^n$. ...
8
votes
4answers
548 views

Which concept of dimension of a ring of functions on a manifold, gives the dimension of the manifold?

Let $R$ be a ring of (smooth?) functions on a (connected?) manifold of dimension $n$. What concept of dimension (of the ring $R$) gives the dimension of the manifold? To what class of rings does this ...
4
votes
2answers
359 views

When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau?

I would like to ask a simple question. Let $A=\mathbb{C}\langle x_{1},\dots,x_{n} \rangle/I$, where $I$ is the two-sided ideal generated by $x_{i}x_{j}=a_{ij}x_{j}x_{i}$ for $1\le i,j\le n$. We say a ...
16
votes
4answers
1k views

What is the geometric object corresponding to a subalgebra in a polynomial ring

Many introductory texts on algebraic geometry set up some sort of algebra-geometry dictionary in which radical ideals correspond to varieties, and so on. I am wondering if there is a geometric way to ...
9
votes
1answer
326 views

Koszulness of the cohomology ring of moduli of stable genus zero curves

Let $n \geq 3$. The ring $H^\bullet(\overline{M}_{0,n},\mathbf Q)$ was determined by Sean Keel. It is generated by the cohomology classes of boundary divisors $D_{A,B}$ corresponding to partitions $A ...
14
votes
1answer
435 views

Affine “real algebraic geometry” of hyperbolic space?

Real algebraic geometry, at least to start with, traditionally studies the zero-sets of real polynomials in a given set of variables. But treating, say, the Euclidean plane as an uncoordinatized ...
0
votes
0answers
249 views

Passing from Regular sequence to Prime ideal, for power sum symmetric polynomial

Let $S=\mathbb{C}[x_1,x_2,x_3,x_4]$ be a polynomial ring. Let $p_i=x_1^i+\cdots+x_4^i$ be the power sum symmetric polynomial in $\mathbb{C}[x_1,x_2,x_3,x_4]$. Let $I=(p_1,p_2)$ be an Ideal of ...
0
votes
1answer
221 views

Regular sequence of power sum symmetric polynomials in polynomial ring.

Let $S=\mathbb{C}[x_1,\dots,x_n]$ be a polynomial ring and $p_a=x_1^a+\cdots+x_n^a$ be a power sum symmetric polynomial in $S$. Let $n \geq 3$. Question: To show $p_m,p_{2m}, \dots,p_{nm}$ forms a ...
2
votes
0answers
104 views

Algorithms for “Ideals” in polynomial algebras over the max-plus semi-ring

I'm a beginner in tropical geometry, and I'm running into the following question: In the usual polynomial ring over a field, one has algorithms (i.e. using a Groebner basis) for determining whether ...
1
vote
0answers
189 views

level of rings and stable range of rings

The level $s(A)$ of a ring $A$ with unity $1$ is the smallest natural number $s$ such that $-1$ is a sum of $s$ squares in $A$. (If $-1$ is not a sum of squares in $A$, we say that $s(A) =\infty$.). ...
18
votes
2answers
438 views

non-isomorphic stably isomorphic fields

Q1: What is the simplest example of two non-isomorphic fields $L$ and $K$ of characteristic $0$ such that $L(x)\simeq K(x)$ (here $x$ is an indeterminate)? Q2: Do we have a sufficient criterion for ...
11
votes
1answer
548 views

Explicit Bijection between Central Simple Algebras and twists of $\mathbb P^n$

The automorphism group of the algebra of $n$-dimensional matrices over a field $K$ is $PGL_n(K)$. The automorphism group of $n-1$-dimensional projective space over $K$ is also $PGL_n(K)$. Therefore, ...
6
votes
2answers
659 views

Generalization of finitely generated, finitely presented modules?

Let $R$ be a commutative ring and $M$ an $R$-module. The module $M$ is finitely generated iff there is an exact sequence $R^{k_0} \to M \to 0$. Similarly, $M$ is finitely presented iff there is an ...
2
votes
2answers
323 views

What is the original statement of Jung-Abhyankar theorem?

I can find many modification of the Jung-Abhyankar theorem. I can even find a new proof of the theorem (by K. Kiyek and J. L. Vicente). But I cannot find the original statement. Does any one know ...
1
vote
1answer
170 views

Ring of a Spectral Space

It is said, as far as I can tell that an arbitrary spectral space, i.e. a space that is $T_0$, sober and quasi-compact whose collection of quasi-compact open sets forms a basis and is closed under ...
4
votes
2answers
445 views

Is the normalisation of an integral noetherien dimension one ring a finite morphism?

This feels like something I should know but I can't find an answer in Liu or in Atiyah-MacDonald, or a counter-example. To state the question again: let $A$ be an integral Noetherien ring of Krull ...
2
votes
1answer
314 views

Maximal Ideals and Kahler Differentials

For an algebraic variety $V$, denote its ring of regular functions by ${\cal O}(V)$. The Kahler differentials of $V$ are the quotient of the kernel $M$ of the multiplication map $$ m: {\cal O}(V) ...
2
votes
2answers
351 views

About the ring used in the definition of drinfeld modules. Why is that ring a dedekind domain?

Let $K/F$ be a function field with exact field of constants $F$ ($F$ is a finite field of characteristic $p$ prime). A prime in $K$ is a discrete valuation in $K$ containing $F$. It has a unique ...
5
votes
1answer
229 views

Radical of group algebra

Let $G$ be a finite group of size $p^a\cdot r$. Does there exist a simple way to calculate radical of the group algebra $F_p[G]$?
2
votes
2answers
398 views

(non-trivial) isotrivial family of elliptic curves over C^{\times}

So How does one prove (rigorously) that $$ Frac(\mathbb{C}[x,y,t]/(y^2-x^3-t)) \not\simeq Frac(\mathbb{C}[t][x,y]/(y^2-x^3-1))? $$ So here $Frac$ denotes the fraction field of an integral domain. ...