**1**

vote

**3**answers

116 views

### ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$.
Let $f,g\in \mathbb{C}[x,y]$.
Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$.
...

**4**

votes

**0**answers

116 views

### On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes

Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...

**16**

votes

**1**answer

968 views

### Are Conway's omnific integers the Grothendieck group of the ordinals under commutative addition?

This is a question in two parts.
Say that $\mathbf{On}$ is the proper class of all ordinal numbers in ZFC. We can define a binary operator over $\mathbf{On}$ which corresponds to the commutative ...

**0**

votes

**0**answers

67 views

### Length of quotients and relations between $\ell(\mathrm{coker}\varphi),\ell(R/\det\varphi)$

Let $R$ be domain(not necessary local) with maximal ideal $\mathfrak{p}$ and $d \in R, d \neq 0$. $(R/(d))_{\mathfrak{p}} = 0 \iff (d) \not\subset \mathfrak{p}$(?). And if $ (d) \subset \mathfrak{p}$ ...

**6**

votes

**0**answers

367 views

### Find a polynomial not in any ideal generated by polynomials of total degree $o(n)$ [on hold]

Is there an explicit nontrivial (= not a constant) polynomial $p \in \mathbb{C}[x_1, \ldots, x_n]$ such that, for any ideal $I \not= \mathbb{C}[x_1, \ldots, x_n]$ generated by $f_1, f_2, \ldots, f_m$ ...

**1**

vote

**0**answers

110 views

### algebraic closedness in in residue field [on hold]

If $A\subseteq B$ are affine domains over an algebraically closed field of $k$ of characteristic zero, such that $Q(A)$ is algebraically closed in $Q(B)$, how can one show that $Q(A)$ is also ...

**0**

votes

**1**answer

121 views

### irreducibility of general fiber

I would like to get a reference of the following fact.
Let $A\subseteq B$ be affine domains over an algebraically closed field of characteristic zero. If $Q(A)$ is algebraically closed in $Q(B)$, ...

**2**

votes

**1**answer

205 views

### Can states on commutative Banach algebras be understood as probability measures?

Suppose $\mathcal{A}$ is a commutative Banach algebra. Is there always a measurable space $(\Omega,\mathcal{F})$ such that there is a bijective correspondence between states on $\mathcal{A}$ and ...

**0**

votes

**0**answers

68 views

### $\inf\{i\in \mathbb N \cup \{0\}\cup\infty\mid Ext^i_R(R/I,R)\neq 0\}=0 ?$

Let $R := k[x_1, \cdots, x_n, \cdots]/(x_1^1, \cdots , x^n_n, \cdots),$ where $k$ is a field. Set $I:=(x_1, \cdots, x_n, \cdots)$. the questions are:
Is $\inf\{i\in \mathbb N \cup \{0\}\cup ...

**1**

vote

**0**answers

72 views

### Syzygies in integral domains

Let $f$ be a homogeneous irreducible element in a graded commutative Noetherian ring.
What is the possible set of elements $f_1$ and $f_2$ such that $f=f_1g_1+ f_2g_2$?
Even in very particular cases ...

**1**

vote

**2**answers

371 views

### A Question About Free Resolutions

I would warmly appreciate it if someone could tell me whether the following question has an affirmative answer. I am new to the field of commutative algebra, so I am simply trying to fill in some ...

**2**

votes

**1**answer

308 views

### Castelnuovo Mumford Regularity

Consider the polynomial ring $S = C[x_1,\ldots,x_n]$. Let
$$
0 \rightarrow E_{n-1} \rightarrow \cdots \rightarrow E_1 \rightarrow E_0 \rightarrow I \rightarrow 0 ,
$$
be a ...

**-1**

votes

**0**answers

38 views

### Cohen-Macaulay rings and Normal rings [migrated]

is there an example that R is Cohen-Macaulay but not normal ring?
what about the converse example?

**-1**

votes

**0**answers

72 views

### Excercise of commutative rings, S. Balcerzyk and T. Józefiak [on hold]

Let $R=k[[x_1,...,x_n,y_1,...,y_n]]/(x_i y_j-x_j y_i)$, $i,j=1,\ldots,n$, where $k$ is a field. Prove that
(a) $R$ is a domain.
(b) $\dim R=n+1$.
(c) $R$ is Cohen-Macaulay.
(d) the type of $R$ ...

**1**

vote

**0**answers

162 views

### unfolding as resolution

Has anyone described 'unfolding' as used in mathematical physics (e.g. on-shell AND off-shell) as analogous to a resolution in algebra - higher derivatives are unfolded in terms of new variables?

**0**

votes

**0**answers

38 views

### To determine whether an ideal is prime using Macaulay 2 [closed]

I am new to Macaulay2, it seems to me that I cannot use it to determine whether an ideal in the polynomial ring over complex number is prime or not, because when I use the isPrime function, I got the ...

**1**

vote

**2**answers

213 views

### Ideals generated by two elements in the polynomial ring of two variables over a field

Let $k$ be a field. For example, $k=\mathbb{Q}$ or $\mathbb{Z}/p$, $p$ prime.
Let $k[x,y]$ be the polynomial ring.
Let $f,g\in k[x,y]$.
Let the ideal $I=(f,g)$ be the ideal of $k[x,y]$ ...

**10**

votes

**1**answer

730 views

### What is the mathematical structure called if we replace commutative group by commutative monoid in the definition of linear space?

Could anyone tell me what the mathematical structure is called if we replace commutative group by commutative monoid in the definition of linear space?
Also, are there any names for "commutative ...

**1**

vote

**2**answers

165 views

### Surjectivity of trace map

Let $R$ be a closed integral domain with its fraction field $F$. Let $K$ be a finite separable extension field of $F$, and let $A$ be the integral closure of $R$ in $K$.
It is well known that the ...

**2**

votes

**1**answer

171 views

### Uncountable cardinals and Prufer $p$-groups

Let $A$ be an elementary Abelian uncountable $p$-group. Is it known if there is an action of a Prufer $q$-group (here $q$ is a prime not necessarily distinct from $p$) $C_{q^{\infty}}$ onto $A$ such ...

**7**

votes

**4**answers

3k views

### When is an algebra of commuting matrices (contained in one) generated by a single matrix?

Let C be an nxn matrix, then the polynomials in C (with appropriate coefficients) form an algebra of commuting matrices. I feel that I should know if the converse is true but I do not. So my first ...

**6**

votes

**1**answer

656 views

### the direct sum of injective modules need not be injective

The Bass-Papp Theorem asserts that a commutative ring $R$ is Noetherian iff every direct sum of injective $R$-modules is injective. Thus every non-Noetherian ring carries a counterexample.
If
$$
I_1 ...

**0**

votes

**0**answers

75 views

### $K_1(R)$ and splitting

Let $R$ be a commutative ring with unit. Under what conditions does the following exact sequence split?
$1\to E(R)\to Gl(R)\to K_1(R)\to 1$.

**1**

vote

**0**answers

58 views

### Why is the polynomial relating the invariants of a binary polyhedral group fixed by an overgroup?

Let $G$ be a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$ and $N \triangleleft G$ a normal subgroup. Let $x, y, z$ be the fundamental invariants for the standard action of $N$ on $\mathbb{C}^2$, ...

**0**

votes

**1**answer

284 views

### Reference for a lemma on étale maps

The Stacks Project has the following really nice Lemma concerning étale maps of rings:
Let $A\rightarrow B$ be a finitely presented, étale morphism of rings. Then there exists a presentation
$$ ...

**1**

vote

**2**answers

145 views

### if $R$ is Noetherian local with a finite module of finite injective dimension and if “?” , then $R$ is “Gorenstein”

I know that if $R$ is Noetherian local with a finite module of finite injective dimension, then $R$ is Cohen-Macaulay.
Can one add assumptions on $M$, so that $R$ be Gorenstein or Complete ...

**0**

votes

**1**answer

164 views

### Extension of homomorphism to place in quotient field, or of local ring to valuation ring in quotient field

Let be given a domain $R$ (that can be supposed to be integrally closed if this can help), and $\varphi$ an homomorphism of $R$ into a field $F$.
$\varphi$ extends uniquely to a homomorphism ...

**4**

votes

**1**answer

287 views

### Valuation of an ideal in a two-dimensional regular local ring

Let $f,g$ be two coprime elements in the ring $K[[x,y]]$, with $K$ a field.
What is the smallest integer $n$ such that the inclusion of ideals $$(x^n)\subset (f,g)$$ holds in $K[[x,y]]$? Can we ...

**-1**

votes

**1**answer

96 views

### behavior of multiplicity in exact sequences

Bruns-Herzog define multiplicity When the ring and module are not necessarily graded as $e(M)=e(gr_m(M))$, see B-H 4.6. I have two questions:
Question1. Many concepts in commutative algebra have ...

**2**

votes

**1**answer

340 views

### Does this $\mathbb{Z}_p$-algebra morphism induce a closed immersion on the generic fiber?

Let $R$ be a local and smooth $\mathbb{Z}_p$-algebra and $B$ an $R$-algebra of finite type which is an integral domain with $\operatorname{dim}B\leq \operatorname{dim}R$ such that
$B/(p)$ is ...

**-1**

votes

**0**answers

23 views

### Property of free submodules for a module over a PID [migrated]

This question was asked here and remains without solution.
It's possible to produce an example of an integral domain $R$ and a free $R$-module $M$ with free submodules $L, L'$ such that $L+L'$ is ...

**4**

votes

**1**answer

150 views

### Number of generators of colon and intersection ideals of two finitely generated ideals in a Prufer domain

Hi!
I'm wanting to see why the following is true: Given 2 finitely generated ideals $B$ and $C$ in a Prufer domain $D$ with bases of $n$ and $m$ generators respectively, $B\cap C$ has a basis of ...

**3**

votes

**2**answers

283 views

### Can there be a non-trivial epimorphism (of rings) from a field? [closed]

I apologize if this question is trivial, but I just cant figure it out. Let $K$ be a field and let $K\longrightarrow A$ be an epimorphism of rings. Is it necessary that $A=K$?

**0**

votes

**0**answers

76 views

### Global dimension of graded Lie algebra

The rational global dimension of a graded algebra $A=(A_k)_{k\geq 0}$, with $A_0=\mathbb Q$, denoted here ${\rm gl}\dim A$ is defined to be the greatest integer $k$ (or $\infty$) such that ${\rm ...

**-4**

votes

**1**answer

203 views

### Integral closure of an ideal [closed]

Let $r^n+a_1r^{n-1}+\cdots+a_n=0$ be an equation of integral dependence of $r$ over an ideal $I$. Does exist a finitely generated ideal $J$, such that $J\subset I$ and $a_i\in J^i$ for all ...

**4**

votes

**1**answer

281 views

### Minimal Free resolution of the ideals

What is the motivation to study the minimal free resolution of the ideals? Which geometrical information can we get from $res(I)$ for The variety of $I$?

**1**

vote

**0**answers

40 views

### Minimal free resolution of sum of ideals

Let $S$ be the polynomial ring in $n$ variables, and let $I_1$ and $I_2$ be ideals in $S$. What can be said about the $\mathbb{Z}$-graded minimal free resolution of $I_1+ I_2$ in terms of the ...

**2**

votes

**1**answer

206 views

### Are schemes which agree on open set and its complement equal? - w/ applications to initial ideals/tropical basis

I appreciate the comments so far and am modifying based on something closer to the problem I'm interested in. I started out with something far too general.
This is probably easy, but I have been ...

**4**

votes

**0**answers

161 views

### Does integral closure commute with pushforward

Suppose that $\pi : Y \to X$ is a proper birational morphism between normal varieties (schemes, whatever). Suppose that $I$ is an ideal sheaf on $Y$.
One can form $\pi_* I$ and construct an ideal ...

**0**

votes

**0**answers

40 views

### Projective dimension of modules over non-discrete valuation rings

Let $\mathbb{C}_p$ be $p$-adic completion of an algebraic closure of $\mathbb{Q}_p$, $\mathcal{O}_{\mathbb{C}_p}$ be its valuation ring and $\mathfrak{m}$ its maximal ideal. Does anyone know the ...

**2**

votes

**1**answer

98 views

### Families of ideals with a given initial ideal

Assume a fintie set of monomials is given. Is there a way to find the family of ideals whose initial ideal (say w.r.t revlex order) is generated by that finite set? I'll appreciate any partial answer, ...

**4**

votes

**1**answer

203 views

### Reference request for division algebras, over $\mathbb{Q}_{p}((t))$

I was looking for a possible reference that would answer the following question,
Let $\mathbb{Q}_{p}$ be the $p$-adic numbers and $\mathbb{Q}_{p}((t))$ be the field of Laurent polynomials over ...

**1**

vote

**1**answer

189 views

### Necessary and sufficient condition for $can : A^X\otimes_A A^Y\rightarrow A^{X\times Y}$ to be an embedding

The two sets are, of course, supposed infinite.
This question is related to that one
Commutation of tensor products with inverse limits in a specific case
where it received a (partial) answer ($A$ ...

**24**

votes

**11**answers

7k views

### Elementary / Interesting proofs of the Nullstellensatz

Is there an easy proof of the Nullstellensatz that avoids the standard Noether-normalization techniques?
One proof I know proves first the 'weak' Nullstellensatz which ensures that maximal ideals ...

**3**

votes

**1**answer

143 views

### The complex of Kahler differentials and de Rham complex

Let $A$ be a unital commutative algebra (say over complex numbers). Consider the multiplication map $m:A \otimes A \to A$ and put $\Omega^1_u(A)=\ker m$ to be the space of universal differential ...

**46**

votes

**2**answers

4k views

### How would you solve this tantalizing Halmos problem?

1-ab invertible => 1-ba invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one?
Geometric series. In ...

**1**

vote

**1**answer

808 views

### Conjugate Matrix

Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix ...

**3**

votes

**1**answer

88 views

### A vector version of the Segre embedding: what is the kernel of the ring map?

TL;DR version.
Given a commutative ring $\mathbf{k}$ and $n+m$ "generic" vectors $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n, \mathbf{y}_1, \mathbf{y}_2, \ldots, \mathbf{y}_m$ in $\mathbf{k}^k$ ...

**3**

votes

**0**answers

70 views

### Intersection numbers on blow ups of toric varieties

Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...

**2**

votes

**1**answer

132 views

### Decide two indices of Ext functor

This question is from the proof of Theorem 11.34 in the book: Twenty-four Hours of Local Cohomology.
Let $R$ and $S$ be CM local ring and $R\to S$ a local homomorphism such that $S$ is a finite ...