# Tagged Questions

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

**4**

votes

**0**answers

57 views

### Support of Matlis dual

Let $(A,m)$ be a commutative noetherian local ring,
$E$ the injective hull of $A/m$, and $M$ a finitely generated $A$-module.
What is the connection between the support of $M$ and the support of the ...

**5**

votes

**0**answers

93 views

+100

### Divided power algebra is artinian as a module over the polynomial ring

I already asked this on math.stackexchange.com, but did not receive much responses. I hope this is also appropriate for mathoverflow.
In the paper Homological algebra on a complete intersection, with ...

**4**

votes

**1**answer

160 views

### $f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$?

Is it possible to find $f,g \in \mathbb{Z}[x,y]$ (with $\deg(f),\deg(g) \geq 1$) such that the following two conditions are satisfied:
(1) $\operatorname{Jac}(f,g)=f_xg_y-f_yg_x = 0$.
(2) ...

**7**

votes

**2**answers

180 views

### Noetherian local ring and the growth of $\dim_k \operatorname{Ext}^i(k,k)$

Let $A$ be a noetherian local ring with residue field $k$, one can consider $\operatorname{Ext}^i(k,k)$ for every natural number $i$. If it is zero for large $i$, then $A$ is regular and the converse ...

**2**

votes

**1**answer

223 views

### A peculiar operation on $M_2(\mathbb Z)$ which along with the usual matrix addition, makes $M_2(\mathbb Z)$ into a commutative ring with unity [closed]

For $A=\begin{pmatrix} a_1 & b_1 \\ c_1&d_1 \end{pmatrix}, B=\begin{pmatrix} a_2 & b_2 \\ c_2&d_2 \end{pmatrix}\in M_2(\mathbb Z)$, define
$A*B:=a_1L_1BR_1+b_1L_1BR_2+c_1L_2BR_1+...

**7**

votes

**0**answers

83 views

+50

### Cellular and primary binomial ideals

An ideal $I$ is called cellular if every variable is either a nonzerodivisor modulo $I$ or is nilpotent modulo $I$.
An ideal $I$ is called primary if whenever $fg \in I$ then or $f \in I$ or $g$ is ...

**8**

votes

**0**answers

87 views

### Is there a homological interpretation for the cokernel of the kernel of a map between complexes induced by tensor product?

Let $A$ be a free abelian group of rank 2, and let $S = \mathbb{Z}[A]\cong\mathbb{Z}[a_1^{\pm1},a_2^{\pm1}]$ the group algebra for $A$.
Let $t : S\times S\rightarrow S$ be the $S$-module map given by ...

**0**

votes

**0**answers

100 views

### Is it possible to generalize a result of Katsylo-Zhang concerning the two-dimensional JC?

Let $p=p(x,y), q=q(x,y) \in \mathbb{C}[x,y]$ be a Jacobian pair, namely, $Jac(p,q)=p_xq_y-p_yq_x \in \mathbb{C}^{\times}$. Denote the total degrees of $p$ and $q$ by $\deg(p)$ and $\deg(q)$, and ...

**8**

votes

**1**answer

83 views

### Injective indecomposable modules over Laurent polynomial rings

What does the injective envelope of $\mathbb C[x,x^{-1}]/(p(x,x^{-1}))$ as a $\mathbb C[x,x^{-1}]$-module look like where $p(x,x^{-1})$ is an irreducible element? I’m sure this is well known, but ...

**37**

votes

**2**answers

842 views

### The roots of unity in a tensor product of commutative rings

For $i\in\{1,2\}$ let $A_i$ be a commutative ring with unity whose additive group is free and finitely-generated. Assume that $A_i$ is connected in the sense that $0$ and $1$ are unique solutions of ...

**2**

votes

**1**answer

111 views

### $(x + y + z)…(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$ To find $P$

$$(x + y + z)(x + y\omega_n + z\omega_n^{n-1})(x + y\omega_n^2 + z\omega_n^{n-2})....(x + y\omega_n^{n-1} + z\omega_n) = x^n + y^n + z^n - P(x,y,z)$$ where $\omega_n$ is an nth root of unity.
The ...

**4**

votes

**0**answers

83 views

### Compute the closure of graph of function from complement of hypersurface in $\mathbb{A}^n$

I'm hoping someone can give me some tips to help speed up computation on the following problem:
Suppose I have a map $G=(g_1/f,\dots,g_m/f):\mathbb{A}^n\setminus{V(f)}\to \mathbb{A}^m$. I'm ...

**0**

votes

**0**answers

84 views

### gcd of polynomial values

Suppose that $f$ and $g$ are two coprime polynomials in $\mathbb Z[x]$. I'm interested in any sort of upper bound on $gcd(f(a),g(a))$, in terms of the integer $a$. Are there any results of this type?...

**8**

votes

**1**answer

375 views

### Do commutative rings without unity have the IBN property?

Let $R$ be a commutative rng, i.e. a commutative ring without an identity element.
Does $R$ still have the Invariant Basis Number (IBN) property?
Recall that a ring is said to have the IBN ...

**6**

votes

**0**answers

142 views

### Fraction fields of strict henselizations of DVRs

Let $A_{1},A_{2}$ be discrete valuation rings whose fraction fields are isomorphic. Let $A_{i}^{\mathrm{sh}}$ be the strict henselization of $A_{i}$, and let $K_{i}$ be the fraction field of $A_{i}^{\...