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

learn more… | top users | synonyms (1)

9
votes
1answer
418 views

Is Max (R) a Hausdorff space?

I asked the following question at < http://math.stackexchange.com/questions/1508168/is-max-r-a-hausdorff-space >, but I pose it here for any help. Recall a space is totally disconnected if the ...
7
votes
0answers
160 views

Terminology for vanishing of Hochschild homology with symmetric coefficients?

In a title or abstract for a paper, if I say "Hochschild cohomology of this algebra $A$ vanishes in degrees two and above" then it should hopefully be understood by most readers as saying ...
1
vote
0answers
28 views

Projectivity of a faithfully balanced self-orthogonal bimodule

Let $_RT_S$ be a faithully balanced self-orthogonal bimodule over a pair of noncommutative rings $(R,S)$, if $_RT$ is projective as a left $R$-module, can we say $T_S$ is also projective as a right ...
0
votes
1answer
82 views

When can one infer degrees of generators of a ring from its hilbert series

I know that for a noetherian ring, it's hilbert series can be written as $$HS(t)=\frac{P(t)}{\prod_{i=1}^d{(1-t^{d_i})}}$$ where $P(t)$ is polynomial, and there are $d$ generators of degrees ...
3
votes
1answer
118 views

Simple proof that $(p_1x_1-q_1y_1,…,p_nx_n-q_ny_n)$ is a prime ideal

Is there a nice way to show that $$(p_1x_1-q_1y_1,\ldots ,p_nx_n-q_ny_n) \subseteq \mathbb{Z}[x_1,...,x_n,y_1,...,y_n]$$ is a prime ideal for coprime non-zero integers $p_i,q_i\,(i=1,...,n)$ ? I ...
0
votes
0answers
177 views

Name of some commutative ring akin to $p$-adics

I need help in identifying the naming convention of some commutative ring described below. Let $p$ be a prime, let $k$ be a positive integer, and let $$\textbf{e} = (e_0,\ldots,e_{k-1})$$ be a list ...
1
vote
0answers
37 views

Upper bound for the minimum number of generators of the canonical module

Let $P=k[x_1...x_n]$ be a poly over a field. Suppose that $R=k[x_1...x_n]/I$. The canonical module of $R$ is $\omega_R=Ext^{n-dim(R)}_P(R,P)$. The question is that is there any upper bound for the ...
3
votes
2answers
206 views

Connectedness of units in finite-dimensional commutative complex algebras

In the following, an algebra will always mean a finite-dimensional associative commutative unital algebra (over some field $k$). Let $A$ be a $\mathbb{C}$-algebra. I am trying to understand how its ...
1
vote
0answers
116 views

Are these quaternion algebras definite or indefinite?

By investigating a different problem I have ended up looking at Quaternion algebras and have a lot to learn about them. Before I do, however, I want to see if my idea has any hope of being useful. So ...
2
votes
1answer
92 views

projective module of rank one over notherian ring

Is finitely generated projective module M of rank one over regular commutative notherian ring free? Bass (Illinois Math J, 1963) showed that in case M is nonfinitely generated, it is free. I am ...
4
votes
0answers
193 views

Commutative algebraic version of algebraic geometric object

In my work, I have to understand certain objects in commutative algebra (for example Gorenstein rings, Cohen–Macaulay rings e.t.c). I have a reasonable background in commutative algebra (I suppose!) ...
5
votes
1answer
90 views

Flat dimension of injectives over a Gorenstein ring

Let $A$ be a Gorenstein noetherian local ring, and let $M$ be an $A$-module of finite injective dimension. If $M$ is a finite $A$-module, it is easy to show these assumptions imply that $M$ has ...
1
vote
0answers
92 views

A question from Hilbert's Nullstellensatz [closed]

From the Hilbert's Nullstellensatz, we know that for any algebraic closed field $K$ and any prime ideal $p$ of $K[X_1,X_2,\cdots,X_n]$, the intersection of all the maximal ideals containing $p$ is ...
6
votes
1answer
355 views

what are the finite etale covers of $\mathbb{Z}_p((x))$?

Let $R$ be the ring of integers of some $p$-adic field $K$ (finite over $\mathbb{Q}_p$) with uniformizer $\pi$ and residue field $k$. I'd like to understand the finite etale extensions of $R((x)) := ...
5
votes
1answer
112 views

What is the criterion for a matrix containing vectors and their permutations being invertible?

Consider the matrix $A\in\mathbb{R}^{m\times 2m}$. Let any arbitrary choice of $m$ columns of $A$ be linearly independent. Together with a permutation $P\in\mathcal{P_{2m}}$, one can build the matrix ...
4
votes
1answer
161 views

number of generators of maximal ideals in an order of a number field

let $K$ be a number field of degree $d$ over $\mathbb{Q}$), Let $\mathcal{O}\subset K $ be an order (i.e. a $\mathbb{Z}$-lattice of $K$ contained in the integer ring $\mathcal{O}_K$ of $K$). If $ ...
3
votes
1answer
111 views

A question about prime elements in a specific integral domain

Let $R,\Omega$ be two integral domains such that $R$ is Noetherian and $\Omega=R[\alpha]$ for some $\alpha\in \Omega$. If there are infinite prime elements in $R$, can we proof that there are ...
1
vote
0answers
107 views

Relation of primary decomposition of two ideals

I have a simple question: Let $R=\mathbb{C} \lbrace t,u \rbrace$ be the ring of formal series in two variables. Let $I,J \subset R$ ideals of heigth one, and $I \subset J$. What is the relation of the ...
0
votes
1answer
175 views

How to solve this system of equations? [closed]

I am trying to find a Poisson bracket on an algebra, and need to find a solution to a system of equations. The system of equations is very complicated, with more than 10000 equations and 60 variables. ...
3
votes
0answers
127 views

polynomial relations between modular functions

$\newcommand{\Qbar}{\overline{\mathbb{Q}}}$ We define a modular function to be a meromorphic modular form of weight 0 for some subgroup (not necessarily congruence) $\Gamma\le\text{SL}_2(\mathbb{Z})$ ...
7
votes
1answer
279 views

What are some examples of total derived functors that can't be computed from a functorial replacement?

(Or more generally, what are some examples of Kan extensions which are not pointwise?) By a total derived functor of a functor $F: C \to D$, I simply mean a (left or right) Kan extension of $F$ along ...
2
votes
0answers
66 views

quasi-ordinary singularities on a versal deformation?

Let $V$ be a variety over $\mathbb{C}$ and suppose $O$ is a singular point of $V$. Are there conditions on $(V,O)$ such that a versal deformation $W$ of $(V,O)$ has only quasi-ordinary singularities. ...
6
votes
1answer
245 views

Vector bundles on open (affine) curves

It is well-known by Grothendieck (or earlier by Dedekind-Weber) that every vector bundle on $\mathbb{P}^1_k$ for $k$ a field decomposes into a sum of the line bundles $\mathcal{O}(k)$. As ...
2
votes
1answer
119 views

On conflicting descriptions for tor of a local cohomology group

Let $X$ be a smooth projective surface and $C$ a Cartier divisor on $X$. Denote by $\mathcal{H}^1_C(\mathcal{O}_X)$ the sheaf associated to the presheaf $U \mapsto H^1_{C \cap U}(\mathcal{O}_X|_U)$. ...
5
votes
1answer
171 views

In a noetherian commutative ring with only one associated prime, is the nilradical locally free?

The title says it all. I suspect that the answer in general is no, although my intuition tells me that a jump in the dimension of the fibre of the nilradical at some point of Spec(A) can occur only ...
3
votes
0answers
107 views

Which sheaves on a projective bundle are flat over the base scheme?

Assume $X$ is a noetherian scheme over $\mathbb{C}$ and $E$ a locally free sheaf of finite rank on $X$. Denote the the associated projective bundle by $f: \mathbb{P}(E)\rightarrow X$. Are there any ...
2
votes
0answers
87 views

When does $R [x]/I $ have infinitely many idempotents in special case?

At < When does $R [x]/I $ have infinitely many idempotents? >, Er_Ro asked the following question. Let $R $ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking ...
13
votes
0answers
255 views

What are retracts of polynomial rings?

Is there a known example of a ring endomorphism $f: \mathbb{Z}[x_1, \ldots, x_n] \to \mathbb{Z}[x_1, \ldots, x_n]$ such that $f \circ f = f$ but whose image is not isomorphic to a polynomial ring? ...
1
vote
1answer
291 views

Bertini-type theorem in positive characteristic [closed]

Let $f:X \to Y$ be a morphism of finite type of irreducible schemes over an algebraically closed field of characteristic $0$. Assume that $Y$ is non-singular. Let $x \in X$ be a closed point and ...
3
votes
1answer
185 views

A basic question on local cohomology

I had posted this question on stackexchange but did not get any response, hence putting it up on mathoverflow. Let $X$ be a smooth, projective variety, $i:X \hookrightarrow \mathbb{P}^n$ a closed ...
2
votes
1answer
388 views

When does $R [x]/I $ has infinitely many idempotents?

Let $R$ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking for a ring with finitely many idempotents and an unextended ideal $I$ in $R[x]$ such that $R[x]/I$ has ...
5
votes
1answer
125 views

Can relative flatness of a sheaf be tested using (faithfully) flat morphisms?

Given a $\mathbb{C}$-scheme $S$, two $S$-schemes $X$ and $Y$ that are flat over $S$ and a coherent sheaf of $O_Y$-modules $F$. Assume we have a (faithfully) flat $S$-morphism $\pi: X \rightarrow Y$ ...
2
votes
0answers
68 views

Torsion ideal in symmetric algebra

Let D be a a commutative domain, M be a D-module without torsion and S(M) its symmetric algebra. Is the D-torsion ideal of S(M) the prime ideal of S(M) ?
4
votes
1answer
264 views

Completion of a finite field extension is also finite?

Let $(L,w)/(K,v)$ be a finite extension of valuation fields, and let $L_w$, $K_v$ be the respective completions of $(L,w)$, $(K,v)$. Is the field extension $L_w/K_v$ finite? For nonarchimedean ...
5
votes
1answer
216 views

Basic questions about simplicial commutative rings

I am trying to learn about simplicial commutative rings, and would be grateful if one can help with some basic facts about them. Basically, I would like to understand how to do homological algebra ...
5
votes
0answers
208 views

A roadmap to learn about finite-dimensional commutative associative real or complex unital algebras

I've always been secretly fascinated with the rich structure and applications of finite-dimensional associative unital algebras over complete fields. In particular, I am very much interested in the ...
2
votes
0answers
75 views

conditions for a subfield of a rational function field to be algebraically closed

Let's start with the following general question. Let $k$ be the ground field. Let $K=k(x_1,\cdots, x_n)$ be a rational function field and let $L$ be a subfield of $K$. Is there a condition to ...
1
vote
0answers
140 views

Does the functor of taking invariants commute with tensor products? [closed]

Suppose that $G$ is a group acting on a commutative ring $R$, inducing an action on each $R$-module. For any $R$-module $M$, let $M^G$ denote the collection of elements of $M$ invariant under the ...
3
votes
0answers
84 views

The Geometric Intuition behind Minimal Primary Decompositions versus an arbitrary Primary Decomposition

Given an ideal in a Noetherian domain, it has a primary decomposition. This decompositions may not be minimal. I am interested in the relationship of primary ideals that are not in a minimal primary ...
6
votes
0answers
88 views

Constructing the normal sheaf for the plucker embedding in MAGMA (or a similar programming language)

How would one construct the normal sheaf $N_{G(2,6)/\mathbb P^{14}}$ to the plucker embedding of the grassmannian $G(2,6) \rightarrow \mathbb P^{14}$ as a sheaf in MAGMA (or another programming ...
3
votes
1answer
265 views

Connections to physics, geometry, geometric probability theory of Euler's beta integral (function)

Euler"s integral for the beta function $B(s,\alpha) = $ (with $x = 1$) $$ \frac{(s-1)!(\alpha-1)!}{(s+\alpha-1)!} x^{s+\alpha-1} = \int_0^\infty t^{s-1}\; H(x-t) \; (x-t)^{\alpha-1} dt = \int_0^x ...
13
votes
2answers
527 views

Every finitely generated flat module over a ring with finitely many minimal primes is projective

Over a commutative ring $R$, a finite type locally free (weak sense) module for which the rank function is locally constant is projective. If we notice that for each minimal prime $p$ of the ring, ...
8
votes
1answer
294 views

q-Integer-valued polynomials

For $n \in \mathbb{Z}_{\geq 0}$, let $[n]_q := (1-q^n)/(1-q) = (1+q+...+q^{n-1})$ as is customary, with $[0]_q=0$. Let $R$ be the subring of $\mathbb{Q}(q)[x]$ consisting of all $f$ such that ...
1
vote
1answer
137 views

A characterization for the ideals of $A+XB[X]$ and $A+XB[[X]]$

Let $A \subseteq B$ be an extension of commutative rings with identity. Then $A+XB[X]$ and $A+XB[[X]]$ are the polynomial and power series rings over $B$ whose constant terms are in $A$. Is there any ...
1
vote
1answer
102 views

Castelnuovo- Mumford regularity properties

Let $R=k[x_1,\ldots,x_n]$ be a graded ring and $S,T,U$ be monomials ideals. $reg(S)=max\{j-i \backslash \beta_{i,j}(S) \neq 0\}$. Assume $S+T=U$ prove \disprove : $reg(S+T^2) \leq reg(U^2)$. We can ...
7
votes
1answer
350 views

Local cohomology groups and linearity

I am reading local cohomology and am confused on a silly point. Let $U$ be an affine, non-singular variety and $Z \subset U$ a hypersurface section on $U$ (i.e., complete intersection in $U$ of ...
25
votes
2answers
976 views

Why is there a duality between spaces and commutative algebras?

1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated algebras over $\mathbb{C}$. The equivalence associates to an affine variety its ...
5
votes
2answers
156 views

The finiteness of the associated primes of $Ext^i_R(R/I, M)$

Let $(R,m)$ be a Noetherian local ring and $M$ an $R$-module. If $M$ is a finite $R$-module, then we know that $Ext^i_R(R/I,M)$ is a finite $R$-module for all $i\geq 0$; Now suppose $supp(M)\subseteq ...
1
vote
2answers
221 views

Smoothness of a subalgebra of a smooth algebra

Given commutative rings $A \subseteq B \subseteq C$ with $C$ a smooth $A$-algebra, I am interested to know if there are some "mild" conditions that make $B$ a smooth $A$-algebra. For example: Assume ...
7
votes
1answer
156 views

Littlewood-Richardson-Type Rule for Restriction from $S_{2n}$ to $S_{2(n-t)} \times (S_2 \wr S_t)$

It is well-known that the Littlewood-Richardson coefficient $c^{\nu}_{\lambda \mu}$ is the number of times the irreducible representation $V_\lambda \bigotimes V_\mu$ of the product of symmetric ...