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

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2
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68 views

Completion of Bezout Domain a Bezout Domain?

Let $R$ be a Bezout domain, and $I$ any ideal inside of $R$. Is the $I$-adic completion $$ \varprojlim_i R/I^i $$ necessarily a Bezout domain? If not, what conditions (on $R$ or $I$) might ensure that ...
0
votes
0answers
55 views

stability of discriminant curve of nonsingular nets of quadrics

The following are from this question: A pencil of quadrics in $\mathbb{P}^n$ is a line in $\mathbb{P}^N$, where $N=\frac{n(n+3)}{2}$. So the space of pencil of quadrics is the Grassmannian ...
4
votes
0answers
143 views

A doubt from the paper “The diagonal subring and the Cohen-Macaulay property of a multigraded ring”

I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I have a doubt about the first part of Theorem 2.5. In the proof $(1)\iff ...
0
votes
2answers
124 views

Ideals in the I-adic completion of a ring

Let $R$ be a commutative ring; does every ideal in the $I$-adic completion of $R$ $$ \varprojlim_i R/I^i $$ arise as the $I$-adic completion of some ideal inside of the original ring $R$?
3
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0answers
63 views

the annihilator of cokernel in a particular case

Let $A\in Mat(m,n;R)$ for $m\le n$ and $R$ a local ring. Consider the $mn\times(m^2+n^2)$ matrix $A\otimes 1_{nn}\oplus 1_{mm}\otimes A^T$, here $1_{mm}$, $1_{nn}$ are identity matrices. I'd like to ...
13
votes
3answers
489 views

Descent of functions along finite birational morphisms

Let $A\to B$ be a morphism of (unitary commutative) rings such that $B$ is module-finite over $A$ and there exists $f\in A$ which is a nonzerodivisor in $A$ and in $B$, with $A[1/f]\to B[1/f]$ an ...
2
votes
1answer
140 views

Vertices of a polytope as algebra generators

I am wondering if the following kind of objects has some name, or are there any studied examples. I apologize for perhaps too specific definition, this is an adoptation of a situation that arises in ...
21
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1answer
758 views

A short proof for $\dim(R[T])=\dim(R)+1$

For a commutative ring $R$ we clearly have $\dim(R[T]) \geq \dim(R)+1$. If $R$ is noetherian, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
6
votes
0answers
86 views

Finiteness for separated residually finite modules

Suppose that $A$ is a commutative noetherian Jacobson ring and $M$ is an $A$-module. Suppose in addition that $M$ is $\mathfrak{m}$-adically separated for every maximal ideal $\mathfrak{m}$, and that ...
3
votes
1answer
236 views

Reference for comparison of heart cohomology with standard cohomology

I'm looking for a reference for the following fact (which I believe to be true and should be easy for people who understand how spectral sequences arise from filtrations). Let A,B be two hearts of ...
7
votes
2answers
321 views

What is the probability that a random sequence of polynomials is regular?

Let $k$ be a finite field or a field with a height function, such as a number field. Consider the ring $k[[x_1,\dots, x_n]]$ and let $\mathfrak{m}$ be its maximal ideal. What is the asymptotic ...
4
votes
1answer
128 views

Interpretations of differentials in hypercohomology spectral sequences as Yoneda products

I would like to know whether the differentials in a particular hypercohomology spectral sequence can each be interpreted, in some natural way, as Yoneda products between extension groups. More ...
5
votes
1answer
170 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
1answer
439 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
0answers
103 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
1answer
225 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 ...
8
votes
1answer
728 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
0answers
207 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
2answers
251 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
2answers
250 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
0answers
66 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
1answer
175 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
1answer
138 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
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0answers
70 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
1answer
383 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
1answer
209 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
1answer
149 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
2answers
352 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
0answers
67 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
0answers
274 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
0answers
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
1answer
57 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
1answer
361 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
1answer
95 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
0answers
53 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
0answers
63 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
1answer
164 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
0answers
33 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
2answers
149 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
2answers
250 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
1answer
88 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
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0answers
46 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 ...
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vote
0answers
239 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
0answers
110 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
1answer
333 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
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0answers
264 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
1answer
162 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
1answer
89 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
1answer
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
1answer
65 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, ...