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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3answers
123 views

Commutative von Neumann algebras and localizable measure spaces

This is not my subject so I apologize if my question is too obvious or understood from other pages. I read some pages such as Reference for the Gelfand-Neumark theorem for commutative von Neumann ...
0
votes
0answers
48 views

Intersections of ideals and nilpotence

Let $R$ be a polynomial ring over a field $k$, $R = k[x_1, \dots, x_n]$. Suppose $R'$ is an associative $R$-algebra and it has the property that there exists a degree $m<n$ monomial in the $x_i$'s ...
5
votes
2answers
67 views

Differential operators between modules, $\mathcal{D}_A(M, M)$ necessarily a filtered, almost commutative ring?

See here. Does it follow immediately that $\mathcal{D}_A(M, M)$ as defined in the link is a filtered, almost commutative ring? How can I visualize this geometrically?
15
votes
4answers
570 views

Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
14
votes
2answers
311 views

Equivalence of “Weyl Algebra” and “Crystalline” definitions of rings of differential operators between modules?

Let $B$ be a commutative $A$-algebra, and let $M$, $N$ be two $B$-modules. We can talk about the set of $A$-linear module homomorphisms $M \to N$, i.e. the set $\text{Hom}_A(M, N)$. Differential ...
-3
votes
0answers
76 views

Non-flat $R \subseteq S$, which is integral, separable, $R$ is a noetherian (not integrally closed) integral domain

On ramification theory in noetherian rings, of Auslander and Buchsbaum say: "Chapter 4 is devoted to showing that under various conditions if $S$ is unramified over $R$, then $S$ is $R$-projective. ...
3
votes
2answers
304 views

commutative algebra, diagonal morphism

can anyone help me with the following statement (it is part of a bigger proof where it is not explained). Let $B$ be a finite type commutative $A$-algebra (where $A$ is a commutative ring), and ...
0
votes
1answer
122 views

Change of grading used in the paper “The diagonal subring and the Cohen-Macaulay property of a multigraded ring” by Eero Hyry

I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I do not understand the following part in Lemma 1.1. here. Let ...
1
vote
0answers
68 views

A question about a specific inverse proposition of Combinatorial Nullstellensatz

From the Hilbert's Nullstellensatz, we have the following consequence which is usually called Combinatorial Nullstellensatz: Let $F$ be an arbitrary field, and let $f = f(x_1,x_2,\cdots,x_n)$ be a ...
2
votes
1answer
244 views

global sections of structure sheaf on the toric Calabi-Yau

Let P be a lattice polytope and lying in $ N \times {1} \subset N \times \mathbb{R}$. Let $\sigma$ be the cone over this polytope and $X_\sigma$ be the corresponding toric variety, which is an ...
0
votes
1answer
176 views

Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.

I would like to know a reference of the following statement (or counter example). Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is ...
49
votes
21answers
10k views

Errata for Atiyah-Macdonald

Is there a good errata for Atiyah-Macdonald available? A cursory Google search reveals a laughably short list here, with just a few typos. Is there any source available online which lists inaccuracies ...
3
votes
1answer
320 views

Valuations on tensor products

Let $A$ be a commutative ring, $B$ (resp. $C$) be a commutative $A$-algebra endowed with a valuation $v$ (resp. $w$), not necessarily of rank 1. Assume that $v$ and $w$ induce equivalent valuations on ...
0
votes
1answer
335 views

When is a power of an indeterminate in an ideal with 2 generators?

If I have an ideal ${\frak a} \colon= (f(X,S), g(X,S))$ of height $2$ in ${\Bbb C}[X, S]$, is it easy to know what power of $S$ is contained in $\frak a$? For example, what is the minimal number $m$? ...
4
votes
1answer
400 views

A Zero-Multiplicity Problem Related to Foulkes' Conjecture

I'm a combinatorialist that is interested in estimating multiplicities of irreps of $1^{S_{kn}}_{S_k \wr S_n}$ (the action of symmetric group on uniform partitions). I'm aware of the difficulty (or ...
5
votes
1answer
211 views

When two determinantal ideals together generate a power of the maximal ideal?

(A somewhat technical question, but maybe it is well known.) Consider matrices over the ring $k[[x_1,\dots,x_n]]$, whose entries vanish at the origin (i.e. belong to the maximal ideal ...
22
votes
6answers
4k views

Do convolution and multiplication satisfy any nontrivial algebraic identities?

For (suitable) real- or complex-valued functions f and g on a (suitable) abelian group G, we have two bilinear operations: multiplication - (f.g)(x) = f(x)g(x), and convolution - (f*g)(x) = ...
2
votes
2answers
153 views

Transitivity of discriminant for flat algebras

Sorry if the question doesn't feed this site, I'm reposting it from MSE. Nobody answered it there and I couldn't find the proof in general case(whenever it was mentioned the proof was referred to as a ...
1
vote
1answer
81 views

integral closure of m-primary ideals

I need help with this excercise Let $k[X_1,\ldots,X_d]$ be the polynomial ring in $X_1,\ldots,X_d$ over a field $k$, and let $F_1,\ldots,F_m$ be forms of degree $n$. Assume that ...
1
vote
2answers
104 views

A question about “large” indecomposable injectives over commutative rings

Here is my question: Does there exist an infinite commutative ring $R$ with identity with an indecomposable injective (unitary) $R$-module $M$ of larger cardinality than $R$ with the additional ...
0
votes
1answer
209 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 $Q(B)$, ...
0
votes
1answer
51 views

In what conditions every ideal is an extension ideal? Is every prime ideal extension of prime ideal?

Let $R$ and $S$ be commutative rings (with $1$), and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$). When $f$ is ...
9
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1answer
325 views

Excellent rings

If A is an excellent commutative ring and G is a finite group of automorphisms of A, is the invariant subring A^G still excellent ? I think this is false -- because if not it would probably be written ...
0
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0answers
98 views

book for help on problems with noetherian rings

Can you please introduce to me a book which would help me to prove the two following problems? In a noetherian ring, every integrally closed ideal is unmixed. Let $R$ be a noetherian ring, $P$ a ...
5
votes
1answer
239 views

Order of vanishing of an integer polynomial at a point

Let $f(x,y)$ be a polynomial with integer coefficients, and let $\alpha=(\alpha_1,\alpha_2)\in \mathbb{C}^2$ be a complex point. I want to show that $f$ cannot vanish at $\alpha$ to high order unless ...
0
votes
0answers
68 views

The universal property of ring of big Witt vectors

Let $X_1,X_2,\cdots$ be infinite many indeterminates. Define \begin{equation*} W_n = \sum_{d\mid n} d X_d^{n/d}. \end{equation*} A big Witt vector over a commutative ring $R$ is a sequence ...
0
votes
0answers
58 views

Can it occur that $q^{ce}$ is a prime ideal (of $S$), while $q^{ce}\neq q $?

Let $R$ and $S$ be commutative rings (with $1$) and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$) and for an ideal ...
23
votes
2answers
638 views

How slowly can a power of an ideal grow?

For a polynomial ideal $I\subset \mathbb{C}[x_1,x_2]$, let $D(I)$ be the smallest degree of any polynomial in $I$. How slowly can $D(I^n)$ grow as a function of $n$? For example, if $D(I^n)\leq ...
1
vote
1answer
129 views

M is an R-module which is not finitely generated. is it true that $\inf \{ i| H^i_I(M)\neq 0 \}\le ht_M I?$

Let $R$ be a commutative noetherian ring (with $1$), $I$ an ideal of $R$, and $M$, a finitely generated $R$-module such that $IM \neq M$. Then, by Theorem 6.2.7 of BRODMANN-SHARP's Local Cohomology ...
0
votes
0answers
84 views

If the direct sum of cyclic modules is cyclic, what happens to nontrivial extensions?

Let $R$ be any ring with unit over some field $k$ and let $M_1$ and $M_2$ be cyclic left $R$-modules with $dim_k(Ext^1_R(M_2,M_1))\geq 1$. Assume $M_1\oplus M_2$ is a cyclic left $R$-module. Given ...
1
vote
1answer
127 views

local cohomology and radical of ideal

Let $R$ be commutative ring with identity, $M$ an $R$-module, and $I$ an ideal of $R$ . One defines $I$-torsion functor $Γ_I$ as: $\Gamma_I(M)=\bigcup_{n\in N} (0:_MI^n).$ When $R$ is Noetherian, ...
6
votes
1answer
232 views

Definitions of the module $R/(x_0^\infty,x_1^\infty,\ldots,x_{n-1}^\infty)$

There are several constructions of the Prüfer group $\mathbb{Z}/p^\infty$; here are two that are relevant for this question. It can be constructed via the short exact sequence $$ 0 \to \mathbb{Z} ...
0
votes
1answer
73 views

How to find ideals of finite length in a power series ring with special properties?

Let $A$ be the power series ring $\mathbb{C}[[x,y]]$. Assume we are given two ideals $I,J$ of finite length in $A$ such that: $xJ\subseteq I\subseteq J$ Is it possible to find ideals of finite ...
1
vote
0answers
106 views

Separability of a simple ring extension

Assume $A=K[x,y]\subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...
0
votes
1answer
204 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 ...
5
votes
1answer
192 views

Generalization of Krull dimension for commutative rings

In the paper How to construct huge chains of prime ideals in power series rings by B. Kang and P. Toan the Krull dimension of a commutative ring with $1$ is defined as follows: Let $R$ be a ...
4
votes
1answer
335 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 ...
4
votes
0answers
70 views

Effective Nullstellensatz and bounds on the nilpotency index of reduced ideal together with linear forms

Let $K$ be an algebraically closed field of characteristic $p>0$ and let $I\subset K[x_{1},\dots,x_{n}]$ be an ideal generated by (homogeneous) polynomials of degree $d$. Assume that $I$ is ...
6
votes
0answers
225 views

Can every commutative ring of characteristic $p\in\mathbb P$ be written as the form $R/(p)$ with $R$ being a ring of characteristic $0$?

All rings here are associative, commutative and unital. By a ring of characteristic zero (resp. of characteristic $p$, for prime $p$) I mean a ring $A$ such that the canonical homomorphism $\mathbb ...
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0answers
52 views

An additive question on polynomials

Consider $S\cup T=\{0,1\}^n$ where $S\cap T=\emptyset$. Consider real multilinear (only monomials of form $x_ix_jx_k$) polynomials $P,Q$ such that: $$Q(S)=0\quad Q(T)\neq0\quad P(S)\neq0\quad ...
8
votes
3answers
651 views

Projective dimension of zero module

Is there any consensus on what the projective dimension of the zero module should be? Here are three statements one commonly encounters in textbooks, sometimes with or without the condition $M\neq 0$: ...
0
votes
0answers
80 views

Flatness of a simple ring extension

Assume $A \subseteq B=A[b]$ are integral domains, $b \in B$ is algebraic over $A$ (but not necessarily integral over $A$), and $A$ and $B$ have the same field of fractions. (Notice that $b=u/v$ for ...
1
vote
1answer
87 views

Conditions for the consistency of a system of affine polynomials

Let $f_1, f_2,\ldots,f_N$ be some affine polynomials. We consider the question if these polynomials have a common (affine) root. By homogenizing these polynomials, we can associate a projective ...
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vote
0answers
86 views

Are finitely presented modules finitely presentable? [closed]

Over a ring $R$ we have a notion of finitely presented module, namely: Definition 1 A module $F$ is finitely presented if there are $m$, $n$ positive integers such that $R^m\to R^n\to F\to 0$ is ...
0
votes
0answers
69 views

Finding a set of generators of an ideal with certain property in $k[x_1, …, x_n]$

I am interested in the following problem, and I would appreciate any comments, inputs, answers, references! Let $k$ be a field. For each $1 \leq j \leq n$, let $$ I_j = (x_j, u_{2}^{(j)}, ..., ...
5
votes
1answer
127 views

discrete valuation ring and ring of witt vectors

Given a perfect field $F$ of prime characteristic the ring of Witt Vectors $W(F)$ is a discrete valuation ring. For example, $W(\mathbb{F}_p)$ is the ring of $p$-adic integers. Is it possible to embed ...
1
vote
0answers
90 views

cohomlogy of Diagonal ring

Let $S=\bigoplus_{\underline n\in\mathbb N^r } S_{\underline n}$ be a standard multigraded ring over a local ring and M be a finitely generated $\mathbb N^r $-graded $S$-module. Let ...
3
votes
0answers
152 views

Kähler differentials, intuition behind $\text{div}(\omega)$, canonical divisor on algebraic curves?

See my two previous questions here: Intuition for thinking about R-module of Kähler differentials, universal receptacles, derivations? and Kähler differentials, define valuation? for background. If ...
15
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3answers
769 views

Is a retract of a free object free?

I wonder whether this is true in the categories of groups, monoids, commutative algebras, associative algebras, Lie algebras?
0
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0answers
100 views

Free resolutions of affine (non-projective!) varieties

Say, you have an ideal $I$ of a polynomial ring $R = K\lbrack X_1,\ldots,X_n \rbrack$ over an algebraically closed field $K$ (you can assume $K = \mathbb{C}$). What does a minimal free resolution of ...