**5**

votes

**1**answer

174 views

### Injective flat module

Let $R$ be a (right noetherian) ring. Is there always a right $R$-module which is both flat and injective? If $R$ is an integral domain, then the answer is indeed yes, as the quotient field is such.
...

**4**

votes

**1**answer

454 views

### A weak version of Bass' conjecture

Let $A$ be a finitely generated $\mathbb{Z}$-algebra which is a UFD. Then (a special case of) the Bass conjecture states that $K_0(A)$ is a finitely generated abelian group. As far as I am aware, this ...

**2**

votes

**0**answers

113 views

### A strong form of implicit function theorem (what happens when the derivative is degenerate?)

(this can be considered as some ad)
Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...

**2**

votes

**1**answer

226 views

### Are Abelian varieties (sometimes) globally $F$-split?

As defined by Karen Smith here, beginning of section 3? If $E$ is an elliptic curve, then it is when $E$ is ordinary. I wonder about higher dimension cases. Any references would be greatly ...

**9**

votes

**1**answer

749 views

### Descent of regularity under a faithfully flat morphism: Where does my proof fail?

While having lunch today with my advisor I tried to come up with a proof of the following fact:
EGA 0-IV (17.3.3): Let $\phi : (A,\mathfrak{m}) \to (B,\mathfrak{n})$ be a flat local ...

**1**

vote

**0**answers

208 views

### Global dimension of a subalgebra with all units

(All rings here are always assumed to be unital and associative).
Setup
Let $R$ be a ring, and $A$ and $B$ be $R$-algebras, with $A$ a commutative subalgebra of $B$ satisfying:
If $u$ is a unit ...

**1**

vote

**2**answers

259 views

### Let $f \colon X \to Y$ be an étale morphism of schemes. If $Y$ is integral, then is $X$ integral? [closed]

Let $f \colon X \to Y$ be an étale morphism of schemes.
We know:
(1) if $Y$ is normal, then $X$ is normal.
(2) if $Y$ is regular, then $X$ is regular.
(3) if $Y$ is reduced, then $X$ is ...

**3**

votes

**2**answers

271 views

### Are quotients of affine schemes by finite groups faithfully flat?

Let $R$ be a (Noetherian) ring, and $G$ a finite group acting on $R$. Consider the subring $R^G$. Is the map $R^G\rightarrow R$ faithfully flat?
If not, does this become true if we restrict to ...

**2**

votes

**0**answers

69 views

### Semidirect products of semigroups [closed]

Let $S,T$ be two semigroups. A function $f:S\to T$ is called multiplicative if for any $x,y\in S$ we have $f(xy)=f(x)f(y)$. If $T=S$ then $f:S\to S$ is called automorphism on $S$.
A function ...

**5**

votes

**1**answer

201 views

### formally smooth functor

Let $p$ be a prime number, $\mathcal{O}$ the integers of a finite extension of $\mathbb{Q}_p$ with residue field $k$. Let $\mathcal{C}$ be the category of complete, local, noetherian ...

**5**

votes

**1**answer

147 views

### Is a Gorenstein ring a quotient of a local complete intersection

The title says it all - Suppose you are given a noetherian Gorenstein local ring $(A,m,k)$ of finite Krull dimension.
Does there exist a local complete intersection ring $B$ such that $A$ is a ...

**4**

votes

**0**answers

76 views

### Degree bounds for Grobner Basis

Let $I= \langle f_1, \ldots f_n \rangle \subset K[x_1,\ldots, x_n]$ be a homogeneous ideal and $\operatorname{deg}(f_i) \leq d$ then it has Grobner Basis where degree of each generator is less than or ...

**13**

votes

**1**answer

384 views

### A weird question about two weird decompositions of $\mathbb{R}$ as a $\mathbb{Q}$-vector space

While working in a question about the affine group $\text{Aff}(\mathbb{R})$, I have come up with the following strange question about the real numbers:
Question: Do there exist a non-trivial ...

**4**

votes

**1**answer

215 views

### Circulant matrix with integer entries and determinant 1 or -1

CONJECTURE
Let $A= (c_0,c_1,\ldots,c_n)$ be a circulant matrix, i.e if $(c_0,c_1,\ldots,c_n)$ is the first column of $A$ then the $i$th column of $A$ is obtained by applying the permutation ...

**6**

votes

**1**answer

153 views

### Problem with Eisenbud's Lemma “Symmetry of Diagonalization”?

This question was first asked on MathSE but nobody answered.
In his proof of Lemma A2.5 in his book Commutative Algebra with a View towards Algebraic Geometry, Prof. Eisenbud writes something like ...

**6**

votes

**2**answers

355 views

### Powers of elements in an Artinian Ring

Let $R$ be an local Artinian ring, with maximal ideal $\mathfrak{m}$.
Let $e$ be the smallest positive integer for which $\mathfrak{m}^e=(0)$.
Let $t$ be the smallest positive integer for which ...

**0**

votes

**0**answers

68 views

### Depth of conormal sheaf of a quotient singularity in a smooth variety

Let $U= \mathbb{C}^n/ \mathbb{Z}_r$ be a cyclic quotient singularity by the finite cyclic group $\mathbb{Z}_r$ of order $r$.
Assume that the group action is free outside a closed subset $Z \subset ...

**10**

votes

**0**answers

278 views

### (When) is isomorphism on differentials enough to guarantee that a map is étale?

I'm sorry if this is too easy for MO.
Let $S$ be a locally noetherian scheme, flat over $\mathrm{Spec}\,\mathbb{Z}$, $X$ and $Y$ be flat $S$-schemes locally of finite presentation, and let $f:X\to Y$ ...

**6**

votes

**0**answers

63 views

### Injective dimension over enveloping algebra

Let $k$ be a field, and let $A$ be a commutative noetherian $k$-algebra.
If a finitely generated $A$-module $M$ has finite injective dimension over $A$, does this imply that $M\otimes_k M$ has finite ...

**0**

votes

**1**answer

61 views

### Is it possible for a MCM module over a hypersurface to have infinite injective dimension?

Let $(R,\frak{m})$ be a hypersurface (i.e., $R=Q/(f)$, where $Q$ is a regular local ring, $0\not=f\in Q$). If $M$ is a MCM $R$-module, is it possible for the injective dimension of $M$ over $R$ to be ...

**3**

votes

**1**answer

366 views

### Is every element of $\mathrm{SL}(n,R)$ of finite order diagonalizable?

Let $k>0$ be an integer, let $R$ be a ring (commutative, unital), which contains $\mathbb{Q}$ (i.e. with a ring homomorphism $\mathbb{Q}\to R$) and all $k$-roots of unity. The examples I have in ...

**0**

votes

**1**answer

100 views

### Simultaneous triangularizability over a commutative ring

Let $R$ be a commutative ring with unity and $A,B\in M_n(R)$ satisfying the property
(*) All elements of the two-side ideal, in $M_n(R)$, generated by $AB-BA$, are nilpotent.
McCoy showed that, if ...

**2**

votes

**0**answers

57 views

### Is this duality operation on simplicial complexes/Stanley-Reisner rings previously known?

Let $K$ be an abstract simplicial complex on vertices $x_1,\ldots,x_n$, then there is the familiar construction of the face ideal $I_K=\langle x_{i_1}\cdots x_{i_r} | ...

**1**

vote

**0**answers

67 views

### Are these rings Cohen Macaulay?

Let $R=k[M]$ be a monoid algebra, where $M$ is a fine and saturated monoid. If $k$ is a field, it is a theorem of Hochster that $R$ is Cohen-Macaulay. What if $k$ is a Dedekind domain? Is $R$ ...

**4**

votes

**1**answer

169 views

### Vector Spaces of Symmetric Matrices of Low Rank

Let $K$ be a field, if necessary algebraic closed or of characteristic zero. Let $k$ be a positive integer. I am interested in linear subspaces $M \subseteq \textrm{Sym}_n(K)$, where ...

**2**

votes

**0**answers

36 views

### Metabolic vs stably metabolic

Let $A$ be a commutative ring with unit. A non-degenerate symmetric bilinear form $\phi$ on a finitely generated projective $A$-module $P$ is called metabolic if there is a direct summand $L$ of $P$ ...

**1**

vote

**2**answers

162 views

### “Exactness” of groupify functor

For each commutative monoid $M$, there exists a "groupification" $\widehat{M}$, i.e. an abelian group that satisfies an obvious universal property.
I tried to prove the following: If in the diagram ...

**1**

vote

**2**answers

274 views

### In this special situation, does $M \otimes B=0$ imply $M=0$?

Let $\Phi:A \rightarrow B$ be a flat morphism of commutative rings. Let $f \in A$, not a unit and $A/fA \cong B/fB$ induced by $\Phi$.
Let $M$ be an $A_f$-module. Is it true that $M \otimes_A B = 0 ...

**0**

votes

**1**answer

89 views

### Colon operation after adjoint variables

Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. Let $I$ an ideal of $R$. We have
$$0:_MI = \cap_x(0:_Mx),$$
where $x$ runs a set of generators of $I$.
Now set $S = ...

**0**

votes

**0**answers

47 views

### Let $(R, m)$ be noetherian local, $\dim(R)=1$. Show $CH^1(R)=\mathbb{Z}/(\gcd([k_i, R/m]))$, where $k_i$ are residue fields of normalization

Let $R$ be a $1$-dimensional noetherian local domain. Then we have that $CH^1(R)=\mathbb{Z}/(\gcd([k_i, k]))$ where the $k_i$ are residue fields of the normalization and $k$ is the residue field of ...

**1**

vote

**0**answers

259 views

### Kunneth spectral sequence

In Rotman's Homological Algebra, 1st edition, there is written:
Is every detail of 11.31-11.35 correct? Isn't the spectral sequence in 11.35 1st quadrant and not 3rd quadrant? Do 11.34-35 also ...

**4**

votes

**0**answers

114 views

### interpretation of homology of “non-commutative Koszul complex”

Let $A = Sym^*(V)$ be a polynomial ring. The Koszul complex
$\cdots \to \wedge^2 V \otimes A \to V \otimes A \to A$
gives a resolution of the residue field $k$, so for any $A$-module $M$, the ...

**13**

votes

**1**answer

336 views

### If the tensor power $M^{\otimes n} = 0$, is it possible that $M^{\otimes n-1}$ is nonzero?

This question was asked earlier on math.stackexchange: click here. See the comments and the answer by Jack Schmidt there.
Let $M$ be a module over a commutative ring $R$.
It is possible that $M ...

**15**

votes

**0**answers

267 views

### Are dualizable modules finitely generated?

Let $A$ be a commutative noetherian ring, and assume that $A$ has a dualizing complex $R$. Let $D(-) := \operatorname{RHom}_A(-,R)$ be the associated dualizing functor, and let $M$ be an $A$-module.
...

**3**

votes

**1**answer

164 views

### A strengthened version of Noether's normalisation lemma?

Noether's normalisation lemma says that if $R$ is an integral domain, finitely generated over a field $k$, with transcendence degree $n$ over $k$, then there exist elements $x_{1}, x_{2}, \ldots x_{n} ...

**0**

votes

**1**answer

121 views

### Theorem 2.5 in “Castelnuovo-Mumford regularity of products of ideals” by Conca & Herzog

Theorem 2.5 in Conca and Herzog, Castelnuovo-Mumford regularity of products of ideals http://arxiv.org/abs/math/0210065, says that if $R$ is a polynomial ring over a field $k$, $I$ a homogeneous ideal ...

**4**

votes

**1**answer

172 views

### Constructing a ring whose spectrum is given by order ideals of Z with generic point

Put the following topology on $\mathbf{Z}_{>0} \cup \infty$: the closed sets are the initial invervals $\{1,\dots,n\}$ for all $n$. If I understood Hochster's characterization of the underlying ...

**1**

vote

**1**answer

67 views

### primary ring and primary module [closed]

I want to know that in the commutative rings, the definition of the primary module is coincide with the definition of the primary ring? And where i can find it?
The primary ring is a ring when ab=0, ...

**1**

vote

**1**answer

96 views

### Original sources for two theorems by Bass, Matlis and Papp

It is an interesting fact that a commutative ring $R$ is noetherian if and only if direct sums of injective $R$-modules are injective, and if and only if every injective $R$-module is a direct sum of ...

**0**

votes

**0**answers

31 views

### Bigraded analogue of Ratliff-Rush closure filtration

Consider the filtration $\lbrace{I^rJ^s}\rbrace_{r,s\in\mathbb{Z}}.$
What will be the bigraded analogue of Ratliff-Rush closure filtration $\tilde{{I}^n}=\cup_{k\geq1}({I}^{n+k}:{I}^k)$?
Will it be ...

**3**

votes

**0**answers

105 views

### Triangulations of special polyhedra

Let $A_1,A_2,A_3 \in \mathbb{N}^3$ be three points in space all lying in some plane $x+y+z=d$ where $d$ is a positive integer. If $\{e_1,e_2,e_3\}$ is the standard basis in $\mathbb{R}^3$, we can ...

**0**

votes

**0**answers

81 views

### Profinite Local Ring inside Polynomial Ring

This is a "technical" question that I came across in my research.
Let $A = \textbf{Z}_{p}[\![t_1, \cdots, t_a ]\!]<z_1, \cdots, z_b>$ be the $(p, t_1, \cdots, t_a)$-adic completion of the ...

**0**

votes

**0**answers

49 views

### Depth of multigraded modules

Can any one please give me some references on depth of multigraded module over a standard multigraded ring.

**3**

votes

**1**answer

130 views

### Idempotent fractional ideals of a Noetherian domain

Let $R$ be a commutative Noetherian domain, $K$ its fraction field, and $J$ a fractional ideal (i.e. a finitely generated sub-$R$-module of $K$) such that $J^2=J$. Is it true that $J=0$ or $J=R$? If ...

**6**

votes

**1**answer

533 views

### Algebraic Closure of a Ring is Not a Ring?

I'm trying to motivate the notion of integrality in a ring extension. It seems that the following would be a good motivation, because it would show that the notion of algebraic elements over a ring is ...

**7**

votes

**1**answer

183 views

### Semiring of vector bundles on $\mathbb{C}\mathbb{P}^1$

Consider the semiring
$$\mathbb{N}[H,H^{-1}]/(H^p+H^q = H^{p+q}+1)_{p,q \in \mathbb{Z}}.$$
Is it finitely presentable? Is there any simplification of the relations (except for $p \geq q \geq 0$)?
...

**-1**

votes

**1**answer

86 views

### Variety of commutative semi group [closed]

V is a variety of commutative semi group satisfying the identity $x^2 = x^3$.
I need to prove that:
$|F_V(\{x_1\dots,x_n\})|$ = $3^n -1$.
Any hints on this ?
$F_V$ is V-free algebra.

**1**

vote

**2**answers

170 views

### Is there an intuitionistic generalized boolean algebra (of Stone)?

A "boolean algebra without the greatest element" was called by Stone "generalized boolean algebra" and he axiomatized it. Is there any publication about "preudo-boolean algebras without the greatest ...

**1**

vote

**1**answer

175 views

### Can the property of essential finite type checked at a point?

Let $k$ be a field, and let $A$ be a commutative $k$-algebra which is noetherian.
Suppose that for each prime ideal $p$ of $A$, it holds that the field $k(p)$, the field of fractions of $A/p$ has ...

**4**

votes

**6**answers

1k views

### Algebraic Geometry for non-mathematician [closed]

I think I sound stupid but I have heard a lot about Algebraic Geometry as a subject and wish to study it without actually studying abstract algebra. I have never studied abstract algebra since I am a ...