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

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3
votes
1answer
26 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 ...
3
votes
1answer
341 views

Injective modules and torsion functors

(This is a related question.) Local cohomology is studied mostly over Noetherian rings. Parts of the machinery do in fact not rely on Noetherianness, but on some weaker properties, for example the ...
15
votes
2answers
602 views

injectivity of torsion submodules of injectives

Local cohomology with respect to an ideal $\mathfrak{a}$ is often studied over a Noetherian ring $R$. However, the proof of a lot of basic results does not rely on noetherianity of $R$, but rather on ...
-1
votes
0answers
30 views

Why do there is a unique continuous homomorphism? [migrated]

Is this a right place to ask help for an exercise? Let $n\geq 2$ be an integer and $D=\mathbb Z[1/n]$. Let $A$ be a complete commutative ring with unit for the $I$-adic topology, where $I$ is an ...
10
votes
1answer
413 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 ...
2
votes
1answer
156 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 ...
1
vote
1answer
67 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
0answers
58 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
2answers
109 views

Making idempotent element by a relation [on hold]

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?
5
votes
1answer
367 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 ...
5
votes
1answer
466 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
229 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 ...
3
votes
0answers
110 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 $S(M)$.)
6
votes
0answers
150 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 ...
1
vote
2answers
701 views

Existence of a minimal generating set of a module

Does a module (over a commutative ring) always possess a minimal generating set? When the module is not finitely generated, the typical Zorn's lemma type argument doesn't seem to work. More precisely, ...
0
votes
1answer
151 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
0answers
42 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
2answers
195 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
1answer
115 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 ...
0
votes
1answer
268 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
80 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 ...
0
votes
3answers
204 views

Example of indecomposable self injective ring

Is there any example of an indecomposable self-injective commutative ring with 4 or more maximal ideals?$$$$$$$$
2
votes
1answer
243 views

Castelnuovo-Mumford Regularity of Ideals of Maximal Minors

Hi, I have a $m \times 2m$ matrix of linear forms over $\mathbb{C}[x,y,z,w]$. It is of the form $M = ( x I - A z -B w | y I - C z - D w$. Here $A,B,C$ and $D$ are $m \times m$ scalar matrices. Let ...
44
votes
0answers
3k 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 ...
0
votes
0answers
72 views

A property of the semi-local ring of the normalization of a singular curve

I have two following questions. 1) Let $R$ be a local ring in an algebraic function field of one variable over an algebraic closed field $k$. Let $\bar{R}$ and $m$ be its integral closure and maximal ...
1
vote
0answers
71 views

Centers of Noetherian Algebras and K-theory

I'll start off a little vauge: Let $E$ be a noncommutative ring which is finitely generated over its noetherian center $Z$. Denote by $\textbf{mod}\hspace{.1 cm} E$ the category of finitely ...
2
votes
1answer
107 views

Examples of (non-normal) unibranched rings?

For a local integral domain $R$ the following are equivalent: a) The integral closure of $R$ in its fraction field (i.e., the normalization of $R$) is again local. b) The henselization of $R$ is ...
1
vote
1answer
119 views

Socle of Almost Complete Intersections

Let $(A,m)$ be a complete Artinian local ring over a field $K$. We focus on almost complete intersection ring $A$ of the form $A = K[[X_1,...,X_N]]/(f_1,...,f_{N+1})$. We assume that none of $f_i$ ...
6
votes
1answer
610 views

Maximal ideals of the rings of Baire-One Functions

A real function $f:X\rightarrow \mathbb{R}$ is called Baire-one function, if there is a sequence $(f_n)_{n=1} ^\infty$ of continuous functions $f_n:X\rightarrow \mathbb{R}$ on $X$ so that for all ...
0
votes
1answer
118 views

Is there a complete local analogue of the Artin-Tate lemma?

The Artin-Tate lemma states that if $A \subseteq B \subseteq C$ are commutative rings where $A$ and $C$ are Noetherian, $C$ is finitely generated as an $A$-algebra, and $C$ is finitely generated as a ...
17
votes
4answers
741 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 ...
3
votes
2answers
817 views

Free commutative magma over a set

BOURBAKI, inside his book on ALGEBRA defines and provides explicit constructions concerning the concepts of free magma, free monoid (and implicitly free semi-group) and free group, and as well free ...
3
votes
2answers
971 views

Projective & injective dimensions

$A$ a Noetherian local ring, $M\neq 0$ a finite $A$-module. I'm not quite sure about the relation between finiteness of projective and injective dimensions of $M$. Does the finiteness (or ...
0
votes
0answers
80 views

Elementary characterization of Krull dimension

I was reading the following paper: "A Short Proof for the Krull Dimension of a Polynomial Ring. Thierry Coquand and Henri Lombardi" and came across this corollary. (This is present with a better ...
-2
votes
0answers
56 views

Jacobson radical of an indecomposable commutative ring [migrated]

Let $R$ be a commutative indecomposable ring with identity which has infinit many maximal ideals. Can we deduce that $Jacobson$ radical, $J(R)$, (the intersection of all maximal ideals)of $R$ is th ...
6
votes
0answers
90 views

Augmentation ideal is finitely generated if and only if $A$ is finitely generated as a $k$-algebra?

Let $A \subset k[x_1, \dots, x_n]$ be a subalgebra, which is also a graded subspace $A = \oplus_{i \ge 0} A_i$. One can write $A = A_0 \oplus A_{> 0}$ where we have $A_0 = k^0[x_1, \dots, x_n] = k$ ...
19
votes
6answers
3k views

Expressing adj(A) as a polynomial in A?

Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\mathop{\mathrm{det}}(A-xI) = p_0 + p_1x + \dots + p_n ...
0
votes
1answer
223 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 ...
-1
votes
0answers
29 views

When $ht_SQ \le ht_Rf^{-1}(Q)$? When $ht_R P = ht_S f(P)$?

Let $R$ and $S$ be commutative rings with identity, and $f:R\to S$ be a homomorphism. Question 1. By what assumptions we can have $ht_SQ =ht_Rf^{-1}(Q)$, for every prime ideal $Q$ of $S$? What ...
5
votes
1answer
371 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 ...
5
votes
2answers
322 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
49 views

The trace ideal of a non zero $R$-module [migrated]

Let $R$ be a commutative ring with identity and $M$ be a cyclic $R$-module, we may define the ideal $tr(M)$ associated with $M$, the sum of the ideals $f(M)$, for all $R$-homomorphisms $f \in ...
5
votes
4answers
1k views

analysis over non-Archimedean ordered fields

Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at ...
1
vote
1answer
329 views

Completion and localisation on noetherian rings

Let $(A,m)$ be a commutative noetherian local ring such that $m$ is principal, say $m=(t)$. Let $(\hat A,\hat m)$ be its $m$-adic completion. Let $A\subset B\subset\hat A$ be any intermediate subring ...
2
votes
1answer
446 views

Krull Dimension

For all $n$, I need to find examples of rings $A\subset B$ such that: i) $\dim A-\dim B\gt n$ ii) $\dim B-\dim A\gt n$ (where $\dim$ is the Krull dimension)
0
votes
1answer
125 views

Is it possible to generalize a result of Wang?

Assume $A$ and $B$ are commutative algebras with $1$. There is a nice result of Wang, Corollary 8, which says the following: "Let $B = A[z] = A[Z]/(h(Z))$. Then $B$ is a separable algebra over $A$ if ...
0
votes
0answers
88 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 ...
2
votes
1answer
104 views

Projectivity of torsion-free modules over integral group rings

Let $G$ be a torsion-free group and assume that the integral group ring $\mathbb{Z}G$ is torsion-free as well. Let $M$ be a torsion-free, finitely generated module over $\mathbb{Z}G$. If we assume ...
2
votes
2answers
252 views

When $mB \neq B$? $m$ is a maximal ideal of $A$, $A \subseteq B$

The following is a question I have asked here without receiving any comments, therefore I post it here: Let $A \subseteq B$ be commutative rings, $m$ a maximal ideal of $A$. When $mB \neq B$? This ...
2
votes
0answers
124 views

Intuition behind if neither $D$ nor $K-D$ are equivalent to an effective divisor, then $\deg(D) = g-1$?

Is there any intuition behind the following fact? If neither $D$ nor $K-D$ are equivalent to an effective divisor, then $\deg(D) = g-1$. Here, $K$ is the canonical divisor. It means the degrees ...