**1**

vote

**1**answer

81 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 $U \mapsto H^1_{C \cap U}(\mathcal{O}_X|_U)$. ...

**4**

votes

**1**answer

124 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 ...

**-1**

votes

**0**answers

54 views

### Modules over polynomial algebras [on hold]

Let $M$ be a cyclic torsion module over the polynomial ring $k[x_1, \dots, x_n]$.
Let $M = K/J$, where $K$ denotes the polynomial ring $k[x_1, \dots, x_n]$.
Is it true that $J \cap k[x_i] \ne 0$ for ...

**0**

votes

**0**answers

22 views

### Relation between $reg(\langle I^2+m \rangle)$ and $reg(I^2)$

Let $R=k[x_1,\ldots,x_n]$ be a graded ring. Minimal free graded resolution
of $I$:
$\cdots \rightarrow \bigoplus_{j} R(-j)^ {\beta_{p,j}} \rightarrow \cdots \rightarrow ...

**2**

votes

**0**answers

80 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 ...

**1**

vote

**0**answers

73 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 ...

**8**

votes

**0**answers

130 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

**1**answer

208 views

### Bertini-type theorem in positive characteristic [on hold]

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 ...

**2**

votes

**1**answer

164 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 ...

**1**

vote

**1**answer

359 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 ...

**0**

votes

**0**answers

9 views

### Is the Jacobson radical of a ring with finite spectrum and nilpotent nilradical nilpotent? [migrated]

I tried to solve 1.3.1 in Bosch: Algebraic Geometry and Commutative Algebra.
I did not find a way to solve it. But I found this: Finitely many prime ideals ⇒ cartesian product of local rings.
And ...

**-4**

votes

**0**answers

82 views

### how to prove that $k[XY]/(XY)$ is a $k[Z]-$ module finitely generated? [closed]

I'm wonrking in this question: Let $A=k[X,Y]/(XY)$, and take the homomorphism $k[Z]\rightarrow A$ given by $Z\to X+Y$. Show that this homomorphism is finite and injective.
For the injectivity I can ...

**4**

votes

**1**answer

113 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 $\pi: X \rightarrow Y$ ...

**1**

vote

**0**answers

54 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) ?

**1**

vote

**0**answers

94 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 completion of $(L,w)$, $(K.v)$. Is the field extension $L_w/K_v$ finite?
For nonarchemidean ...

**4**

votes

**0**answers

172 views

### Castelnuovo -Mumford regularity properties

Let $R=k[x_1,\ldots,x_n]$ be graded ring and $I,J$ be quartic square free monomial ideals in $R$. Let $I=\langle n_1,\ldots,n_s \rangle$ and $J=\langle m_1,\ldots ,m_t\rangle$
Definition:
...

**4**

votes

**1**answer

151 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 ...

**4**

votes

**0**answers

194 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 ...

**1**

vote

**0**answers

55 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

**0**answers

115 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 ...

**2**

votes

**0**answers

65 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 ...

**1**

vote

**0**answers

36 views

### Intuition for morphisms of modules [migrated]

Let $A$ be a ring. We introduce the notation$$\text{Hom}(M, N) = \{A\text{-module morphisms }M \to N\}.$$Note that $\text{Hom}_A(M, N)$ is an abelian group (under pointwise addition of functions). If ...

**5**

votes

**0**answers

60 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 ...

**2**

votes

**1**answer

212 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 ...

**9**

votes

**2**answers

377 views

### Every finitely generated flat modules is projective over a commutative ring with a finite number of minimal primes?

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, ...

**5**

votes

**1**answer

149 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 ...

**0**

votes

**1**answer

101 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 ...

**0**

votes

**1**answer

74 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 ...

**-1**

votes

**0**answers

119 views

### When is the $I$- adic completion a faithfully flat ring extension?

Let $I$ be an ideal of a commutative ring with identity. It is well-known that the $I$-adic completion of $R$ when it is a Noetherian ring is a faithfully flat ring extension. Does this remain true if ...

**6**

votes

**1**answer

307 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 ...

**23**

votes

**2**answers

844 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 ...

**2**

votes

**1**answer

54 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 ...

**0**

votes

**2**answers

210 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 ...

**6**

votes

**1**answer

125 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 ...

**0**

votes

**1**answer

121 views

### Question regarding arbitrary chains of prime ideals

All rings considered are commutative with $1$. In my study of arbitrary chains of prime ideals of a ring $R$, I have noticed two possiblities for the cardinality of a chain when $\dim(R)=\infty$:
...

**-1**

votes

**0**answers

188 views

### Smoothness of $A \to A[T]/(h_1,\ldots,h_n)$

This is a follow-up of Smoothness of $A \to A[T]/(h)$.
Let $A$ be a commutative noetherian domain of characteristic zero, $T$ an indeterminate, $h_1,\ldots,h_n \in A[T]$, $B= A[T]/(h_1,\ldots,h_n)$ ...

**0**

votes

**0**answers

161 views

### Smoothness of $A \to A[T]/(h)$

Let $A$ be a commutative noetherian domain of characteristic zero, $T$ an indeterminate, $h \in A[T]$, $B= A[T]/(h)$ and assume $B$ is also a domain.
When $B$ is (formally) smooth over $A$?; namely, ...

**8**

votes

**0**answers

190 views

### comparison of completion and Henselization in class field theory

Given a ring $R$ with maximal ideal $\mathfrak{m}$, we can form the localization $R_\mathfrak{m}$, the completion $\hat{R}_\mathfrak{m}$ or the Henselization $\hat{R}^h_\mathfrak{m}$ of $R$ with ...

**2**

votes

**1**answer

80 views

### Integral domains equal to intersection of their height one localizations

Which integral domains have the property that $R = \bigcap R_P$, the intersection being taken over all height one prime ideals of $R$?
It is a standard fact that Krull domains, and thus noetherian ...

**1**

vote

**1**answer

89 views

### Projective dimension of a quotient ring

Assume $A$ and $B$ are commutative algebras with $1$, $B = A[z] = A[Z]/(h(Z))$,
$Z$ an indeterminate.
The first comment in this question says that, if $A$ is noetherian, then
$pd_{B\otimes_A B}(B) ...

**3**

votes

**0**answers

68 views

### Unibranch partial normalization

In a paper I recently read something about the "unibranch partial normalization" of a curve.
Say, $R$ is a local integral domain with maximal ideal $\mathfrak{m}$ and fraction field $K$. Is it ...

**0**

votes

**2**answers

132 views

### Making idempotent element by a relation [closed]

Let $R$ be a commutative ring with identity and let $a, b \in R$ such that $a=ab$. How can we make a non zero idempotent element of $R$ by this relation?

**10**

votes

**1**answer

466 views

### Is an irreducible ideal in $R$ also irreducible in $R[x]$?

Let $R$ be a commutative Noetherian ring and $I\subset R$ an ideal that is irreducible in the sense that if $I = J_1 \cap J_2$, then $I=J_1$ or $I=J_2$. Is (the ideal generated by) $I$ irreducible in ...

**0**

votes

**0**answers

232 views

### Algebraic characterization of commutative ring of Krull dimension zero [migrated]

Recently, I saw the following characterization of zero dimensional commutative rings, where can I find a verification for it?
Blockquote :
A commutative ring $R$ (with $1$) is $0$-dimensional ...

**3**

votes

**0**answers

120 views

### Annihilators of elements in symmetric algebras

Let $M$ be a module over a commutative ring, and $S(M)$ its symmetric algebra. What elements of $S(M)$ annihilate a given element $m\in M$ ? ($M$ is considered as a submodule of $S(M)$.)

**0**

votes

**1**answer

159 views

### Can you detect homological dimensions from homology?

Suppose you are given a bounded chain complex $M$ over a commutative ring $R$.
Is there a clear relation between homological dimensions of $M$ and homological dimensions of its cohomologies?
For ...

**0**

votes

**0**answers

45 views

### trying to understand particular modules, to contruct some nice free resolutions

Let $R$ be a local or graded ring, fix some morphism $\phi\in Hom_R (R^{\oplus n},R^{\oplus m})=:Mat(m,n;R)$.
In various applications one meets modules like $\frac{Mat(m,n;R)}{Span(U ...

**2**

votes

**2**answers

201 views

### Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theorem:
$(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...

**0**

votes

**1**answer

120 views

### perfect modules over polynomial algebra

This may be obvious. My question is short:
$R$ is the polynomial algebra $\mathbb{k}[X_{1},\dots , X_{n}]$. Is the $R$-module $\mathbb{k}$ perfect in the sense that $\mathbb{k}$ is a compact object ...

**6**

votes

**0**answers

155 views

### When is a polynomial ring free over a graded subalgebra?

Keep the setting of my previous question and let $I := k[x_1, \dots, x_n] \cdot A_{>0}$ be an ideal of the algebra $k[x_1, \dots, x_n]$ generated by the set $A_{>0}$. It is clear that $I$ is a ...