**0**

votes

**1**answer

62 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

370 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

101 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

71 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

174 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

166 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

284 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

91 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

48 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

286 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

115 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

347 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

271 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

166 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

137 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

173 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

71 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

97 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

32 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

52 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

132 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

557 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

184 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

87 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

172 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

177 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 ...

**1**

vote

**0**answers

164 views

### Surjectivity of $f\colon\Gamma_Z(M)\rightarrow\bigoplus_{p\in Z\backslash Z'}\Gamma_{pR_p}(M_p)$

Suppose $Z'\subseteq Z\subseteq\text{Spec} R$ such that every element in $Z\backslash Z'$ is a minimal element (with respect to inclusion as ideals) in $Z$. Assume further that both $Z$ and $Z'$ are ...

**1**

vote

**0**answers

90 views

### Graded-irreducible ideals are irreducible?

One knows that graded ideals in polynomial rings over a field are primary iff they are graded-primary. What about the irreducible ideals?
Let $I$ be a graded ideal in a polynomial ring over a ...

**10**

votes

**1**answer

260 views

### Homological criteria for finite generation and finite presentation of modules?

(I'm new here; if I'm doing something wrong please help me out.)
In short, my question is: There are some results on the behavior of finite generation and finite presentation in exact sequences (of ...

**3**

votes

**1**answer

295 views

### Can non-isomorphic field extensions be isomorphic fields?

This is related to my earlier question on isomorphism of general quotients of $\:F\hspace{.02 in}[x]\:$.
Let $F$ be a field, let $p$ and $q$ be (non-zero) monic irreducible polynomials, let $I$ and ...

**1**

vote

**1**answer

144 views

### Universal coefficient theorem for local ring

Let $R$ be a commutative local artin $k$-algebra,where $k$ is a field with characteristic $0$.I wonder whether universal coefficient theorem holds in this case.Namely,if $C$ is a chain of flat ...

**3**

votes

**2**answers

334 views

### An affine singular surface

Let $n$ be a positive integer and let $A$ be the subring of ${\mathbb C}[x,y]$ generated
by $x,xy,...,xy^n$. Let $S=Spec(A)$. This is an affine surface, which is clearly singular if
$n\neq 1$. Is ...

**0**

votes

**2**answers

245 views

### Computing the minimal free resolution of a coherent sheaf on projective space

Most books on commutative algebra explain Grobner bases in the non graded case and minimal free resolutions in the local case. I like projective geometry and want to compute the minimal free ...

**29**

votes

**2**answers

1k views

### Is every Noetherian Commutative Ring a quotient of a Noetherian Domain?

This was an interesting question posed to me by a friend who is very interested in commutative algebra. It also has some nice geometric motivation.
The question is in two parts. The first, as stated ...

**0**

votes

**0**answers

87 views

### Relation between dimension of Proj(S) and dimension of S

Let $S$ be Noetherian standard ${\mathbb{N}}^r$ graded ring where $S_{\underline{0}}$ is an Aritinian local ring.
$$Proj(S)=\lbrace{P\in Spec S | S_{++}\not\subseteq P, P\hspace{0.1cm} ...

**-3**

votes

**1**answer

243 views

### Doubt in this proof of Horrocks theorem

I'm beginning to study some research papers and I need right now to understand the solution of Vaseršteĭn of Serre's theorem (simplest proof of this theorem), to do so, I'm beginning to understand ...

**1**

vote

**1**answer

83 views

### composition of Puiseux series?

Formal series over a field (or ring) k can be composed in the following sense: given $y \in k[[x]]$, $y$ lying in the maximal ideal, there exists a unique map of topological rings $k[[x]] \to k[[x]]$ ...

**1**

vote

**0**answers

109 views

### Two questions regarding $f$-adic completions of (non noetherian) rings

Lately I've looked at $f$-adic completions of commutative rings. I had posted two questions regarding the topic on math.SE which didn't receive any attention and I think they might be fit for ...

**0**

votes

**1**answer

206 views

### reduction of an admissible filtration

Let $(R,m)$ be a local ring and $I$ an $m$-primary ideal of $R.$ $\lbrace I_n\rbrace_{n\in\mathbb{Z} }$ is called $I$-admissible filtration
1) if $m\geq n$ then $I_m\subset I_n.$
2) for all $m,n,$ ...

**2**

votes

**2**answers

253 views

### local cohomology mayer-vietoris sequence

(I originally asked this question on Math.SE here. As suggested on meta.MathOverflow (posting an unanswered Math.SE question on MathOverflow), I've waited about a week before reposting it here. Note ...

**1**

vote

**1**answer

135 views

### local cohomology of Buchsbaum ring

Let $(R,m)$ be a Buchsbaum ring of dimension d. Can we say that $d$-th local cohomology $H_{m}^d(R)$ has finite length?

**0**

votes

**1**answer

95 views

### Relation between local cohomology and koszul cohomology of multigraded ring

Let $R$ be a ${\mathbb {Z}}^k$-graded Noetherian ring, $J=(x,y)$ an ideal of $R$ where $x,y\in R_{(1,\ldots,1)}.$ Is this following true $$H_{J}^i(R)=\underset{n}\varinjlim{H^i((x^n,y^n),R)},$$ where ...

**6**

votes

**1**answer

775 views

### A naive algebraic geometry question

Suppose $X$ is a scheme over a ring $A$, $B$ is an $A$-algebra, and $X\times_AB$ is affine. I am looking for conditions on $A$ and $B$ (and perhaps the structure morphism of $X$ over $A$) that will ...

**3**

votes

**1**answer

195 views

### Endomorphism Ring of Indecomposable MCM Modules

Let $R = k[[x, y]]/(f)$, where $k$ is algebraically closed of characteristic zero. I'm particularly interested in studying the endomorphism ring of indecomposable MCM (maximal Cohen-Macaulay) modules ...

**17**

votes

**2**answers

400 views

### Rings for which no polynomial induces the zero function

For any commutative ring $R$ let $R[x]$ denote the ring of polynomials with coefficients in $R$. Any polynomial $p \in R[x]$ naturally induces a function $\hat{p} :R \rightarrow R$. In some cases, a ...