Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,335
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Are there enough curves (to connect 'points' of f.g. algebras)?
(Intuition: any two points in a connected space may be connected by a path. I would like to know if something like this holds in certain category of `connected algebraic spaces'. I formulate the ...
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Primary decomposition with parameters
$\newcommand\QQ{\mathbb{Q}}$
Considering the polynomial
$$f = x^2 - a y$$
one notes, that it is irreducible in $\QQ[x,y]$ for all $a \neq 0 \in \QQ$ and factors for $a = 0$.
More generally, let $A=...
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Infinite Noetherian ring of dimension $1$ in which distinct non-zero ideals have distinct and finite index
Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an ...
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Behavior of Ext under base change
Let $x$ be a nonzerodivisor on a local Noetherian ring $(R,m).$ Let $M,N$ be finitely generated $R/xR$-modules.
How to show the existence of the following exact sequence
$\cdots\longrightarrow ...
- 1,703
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1
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A question on dominant morphism of affine schemes
Let $A \subseteq B$ be a ring extension where $A,B$ are both finitely generated $\mathbb C$-domain of the same Krull dimension. Also assume $A$ is regular (i.e. $A_{ \mathfrak p}$ is regular local ...
- 4,091
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Graded Betti numbers $\beta_{n,j}$ for points in $\mathbb{P}^n$
Let $S = \mathbb{C}[z_0, \dots, z_n]$, and let $X$ be a set of points in $\mathbb{P}^n$. I'm looking for references concerning results for the graded Betti numbers $\beta_{n,j}(S/I(X))$, i.e., the ...
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For every prime ideal $P$ of any Cohen-Macaulay ring $R$, is the sequence $\operatorname{depth}(R/P^n)$ eventually constant?
Let $P$ be a prime ideal of a Cohen-Macaulay ring $R$. Then is the sequence $\operatorname{depth}(R/P^n)$ eventually constant ?
- 1,283
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Counting the number of poles for rational functions in a coordinate ring of a curve
I'm working in an algebra of rational functions in compact Riemann surfaces with arbitrary genus. The idea I'm struggling is how to count the number $n$ of poles for the rational functions defined in ...
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Notions of connected components in a finite family fibration
Let $\Pi_0:\mathsf{FinFam}(\mathsf C)\to \mathsf{FinSet}$ be the fibration exhibiting the free finite coproduct completion of $\mathsf C$. Suppose $\mathsf C$ has finite limits so that the extensive $\...
- 10.3k
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Relation between lifts of simple roots and lifts of idempotents (Henselian property)
Let $f:A\to B$ be a morphism of commutative rings. Given a monic $\varphi\in A[x]$ write $Z(A,\varphi)$ for the set of simple roots of $\varphi$ in $A$. Consider the following properties of $f:A\to B$....
- 10.3k
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Existence of isomorphism mod every power of the maximal ideal
This problem is a continuation of Hensel lemma and rational points in complete noetherian local ring.
Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal.
Assume $...
- 6,117
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Module of Kahler differentials for manifolds [duplicate]
Let $A$ be a $k$-algebra and let $\mathcal{M}_A$ be the set of all $A$-modules. In $\mathcal{M}_A$, there exists a universal object $\Omega_{A/k}$, called the module of Kahler differentials, and a $k$...
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Explicit description of injective hulls
Let $k$ be a field, let $R:=k[x_1,\ldots,x_n]$, and
consider the $R$-module $M:=R/{(x_1,\ldots,x_n)}\cong k$.
Then the injective hull $I_M$ of $M$ admits the following explicit description:
$$
I_M = k[...
- 42.5k
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Are local, Noetherian rings with principal maximal ideal PIR?
A question asked by a friend. I believe it's false, but lack a decisive counterexample.
This question shows that it is true for valuation rings, but I know too little about them.
In the wider ...
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Summing complete intersections
Suppose we have polynomials $f_1,\dots,f_r\in k[X_1,\dots,X_N]$ defining a complete intersection in $\mathbb{A}^N$. I suspect that it is then true that $f_1(X)+f_1(Y),\dots,f_r(X)+f_r(Y)\in k[X_1,\...
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commutative ring satisfying descending chain condition on radical ideals
Let $R$ be a commutative ring with unity which satisfies d.c.c. on radical ideals. then does $R$ satisfy a.c.c. on radical ideals ? If this is not true in general, then what happens if we also assume $...
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What's the probability a random module element has prime annihilator?
I'm going to pose two versions of my question---an ill-defined version and a well-defined version.
Ill-Defined Question (IDQ). Let $R$ be a (commutative, unital) Noetherian ring, $M$ a finitely ...
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Are archimedean subextensions of ordered fields dense?
Let $E$ be an ordered field and let $F$ be a real closed subfield. We say that $E$ is $F$-archimedean if for each $e\in E$ there is $x\in F$ such that $-x\le e\le x$.
Is it true that if $E$ is $F$-...
- 2,367
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Number of simplicial polytopes with a given f-vector
Plenty of very nice literature is available on the characterization of f-vectors of simplicial complexes of diverse sorts (results by Billera, Bjoerner, Kalai, Stanley, among others). I mention, as an ...
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indecomposable semi local Bezout ring
A commutative Bezout ring with $1$ is a ring in which every finitely generated ideal is principal. Is there any charactrization for indecomposable semi local Bezout ring?
For example , for a ...
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Behavior of regularity under base change
Let $(R,\mathfrak{m})$ be a noetherian local domain. Let $(A,\mathfrak{n})$ be a regular local ring contains $R$ such that $\mathfrak{n}\cap R=\mathfrak{m}$ and $A$ is essential finitely generated as ...
- 554
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1
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On a special type of subring of $\mathbb C[x_0,...,x_{q-1}]$
Let $p,q$ be odd primes. Consider the polynomial ring $\mathbb C[x_0,...,x_{q-1}]$. For $m=0,1,...,p-1$, let
$$\sigma_m=\sum_{0\le j_0\le p;...;0\le j_{q-1}\le p; j_1+...+j_{q-1}=p; 1.j_1+...+(q-1)...
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Ideals invariant under ring automorphisms
I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:
$I$ is generated by two homogeneous elements;
$I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}...
- 30.3k
17
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2
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Is a matrix similar to its transpose over $\mathbb{Z}_p$?
Is every $n \times n$ matrix with entries in $\mathbb{Z}_p$ (or even $\mathbb{Z}$) conjugate to its transpose via a matrix in $GL_n(\mathbb{Z}_p)$?
On the one hand, I know the analogous fact is false ...
- 3,753
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Is $\dim_k M/xM$ a multiple of $\dim_k R/xR$ for $M$ finitely generated, torsion-free $R$-module?
Let $R$ be a one-dimensional, reduced and noetherian $k$-algebra (we may also assume that $R$ is a finite $k[x]$-algebra). Let $M$ be a finitely generated, torsion-free module over $R$, i.e. no ...
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Does every section of the map Gal$(\overline{k(\!(t)\!)}/k(\!(t)\!))\rightarrow$ Gal$(\overline{k}/k)$ stabilize a compatible system of roots of $t$?
There may be some technical issues with the question, but hopefully what I mean is clear...
Let $k$ be a number field (or maybe any finitely generated field over $\mathbb{Q}$ of characteristic 0)
...
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Hensel lemma and rational points in complete noetherian local ring
Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal.
If we have several polynomials $f_i \in A[X_1, \dots, X_m]$ which have a common zero $x_n$ in $A/\mathfrak{m}^n$ ...
- 6,158
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Is a scheme Noetherian if its topological space and its stalks are?
Is a scheme being Noetherian equivalent to the underlying topological space being Noetherian and all its stalks being Noetherian?
- 27.3k
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1
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Henselianizations over countable index sets
Let $A$ be a ring, $I\subset A$ a finitely generated ideal.
The henselianization $A^h$ of $A$ along $I$ is the universal $A$-algebra that is henselian along $I$ and can be presented as a direct limit ...
CommunityBot
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Relation between Hilbert function and complete intersection ideals
Consider $T=k[x_1,\ldots,x_n]$ ( $k$ alg. closed and of char $k=0$), and consider the ideal $$I=(x_1,x^{a_2}_2,\ldots,x^{a_n}_n)$$
with $2\leq a_2 \leq\ldots\leq a_n$. I want to prove that $$\sum_{i=...
- 6,197
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1
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Is a factorial scheme with Noetherian stalks locally Noetherian?
Motivation: The Oka's coherence theorem tells us that the structure sheaf of a complex manifold is coherent. Taking into account the fact that coherence is a local property stable under finite direct ...
- 6,670
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example of a non-finitely generated co-Hopfian module over a commutative QF ring
Can anyone provide an example of such a module? or show that no such module exists? For semisimple rings, we have co-Hopfian if and only if finitely generated. Perhaps the fact that QF rings (...
- 33.9k
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Maximal subalgebras in polynomial ring $\mathbb{R}[x]$ over the field $\mathbb{R}$ of real numbers
Question. What are the maximal subalgebras of polynomial ring $\mathbb{R}[x]$ over the field of real numbers?
13
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1
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Commutative algebraic version of algebraic geometric object
In my work, I have to understand certain objects in commutative algebra (for example Gorenstein rings, Cohen–Macaulay rings e.t.c). I have a reasonable background in commutative algebra (I suppose!) ...
- 30.3k
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1
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Classify commutative rings $R$ such that $A \otimes_{\Bbb Z} B = A \otimes _{R} B$
I have asked this in MSE but there was no reply. Feel free to close if inappropriate.
Let $R$ be commutative ring, what can we say about the rings $R$ such that $A \otimes_{\Bbb Z} B \cong A \...
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How do you quickly determine which coefficients are greater than zero when multiplying two univariate positive polynomials?
Suppose that I have two polynomials with a degree of $n$, $A(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0$ and $B(x) = b_nx^n + b_{n-1}x^{n-1} + ... + b_0$ and the coefficients of these polynomials are ...
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Is there an alternative formula for solving cubic equations?
It is known that Cardano's formula for solving cubic equations is not good in the case of positive discriminants. In this case it expresses the solution through cubic roots of complex numbers. ...
- 60.2k
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1
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An example of a local ring which is not CM and a MCM module over it
I am looking for an example of a commutative noetherian local ring $(A,m)$,
and a maximal Cohen-Macaulay module $M$ over $A$ (in particular $M$ is finitely generated over $A$), such that for some $p \...
- 1,283
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votes
1
answer
552
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Projective dimension of graded modules
Short version:
Why is the projective dimension of a graded module the same as the projective dimension of its underlying ungraded module?
Longer version:
Let $G$ be a commutative group, let $R$ ...
- 2,846
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1
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$2$-adic valuation on $\mathbb Q (\sqrt{-15})$
I was reading the book "The structure of groups of prime power order". In the book (page 226, Example 10.1.18) it is proved that
$-15$ has a square root is $\mathbb Z_2$ where $\mathbb Z_2$ ...
CommunityBot
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Are quotient varieties local complete intersections?
Let $G$ be a reductive group acting on the smooth affine variety $X$ such that the stabilizers are finite. Is it true that the quotient $X/G$ is a local complete intersection (LCI)? In particular, is ...
- 2,831
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Ring of invariants of some special type of subgroups of $GL_3(\mathbb C)$
For $\sigma \in \mathrm{GL}_n(\mathbb C)$ and $f(x_1,...,x_n)\in \mathbb C[x_1,...,x_n]$, let $f^ \sigma (x):=f(\sigma^{-1}x)$, for $x=(x_1,...,x_n)$.
For a subgroup $G$ of $\mathrm{GL}_n(\mathbb C)$...
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0
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Generating the annihilator ideal up to finite index
Let $\Lambda_2 = \mathbb{Z}_p[[T_1,T_2]]$ be the power series ring over $\mathbb{Z}_p$ in two variables, i.e., $\Lambda_2$ is a regular local ring of dimension 3. Let $M$ be the quotient of an ...
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Is there an adjoint to the inclusion of I-adically complete modules to all modules?
A module $M$ over a ring $R$ is $I$-adically complete with respect to the ideal $I$, if the canonical map $M \to \lim M/I^nM$ is an isomorphism. There exists a completion functor: $M \mapsto \lim M/I^...
- 15.2k
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modules whose every submodule is a homomorphic image
Let $R$ be a commutative ring with unity. Let us say that an $R$-module $M$ satisfies property $\mathcal P$ if every submodule of $M$ is a homomorphic image of $M$.
Can we characterize all ...
- 6,117
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1
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Tensoring with complex of finite flat dimension in derived category
Let $(R,m)$ be a Noetherian local ring, and $X$, $Y$ be complexes of finitely generated $R$ modules. Suppose $X$ is bounded above and $Y$ is bounded below. Let $S$ be an $R$-algebra of finite flat ...
- 1,204
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1
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Koszul-Tate Resolution for Subvarieties of $\mathbb P^n$
All varieties appearing below are assumed smooth projective over $\mathbb C$ and all vector bundles, sections etc are assumed to be algebraic/holomorphic. We use the word resolution to mean quasi-...
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Self-flat modules
(This is inspired by this question and asked out of pure curiosity.)
Let $R$ be a commutative ring. Let $M$ and $N$ be $R$-modules. Then, $N$ is called $M$-flat if whenever $M'\rightarrow M$ is a ...
- 6,670
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0
answers
61
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Efficient algorithm to prove that a polynomial ideal contains 1
I have the following problem:
Suppose to have an ideal $I\triangleleft k[x_1,...,x_n]$ defined by generators. There exists an efficient algorithm (perhaps more efficient than calculating the Groebner ...
- 297
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1
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435
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Injective homomorphism of modules and tensor product
Let $R$ be a commutative ring; let $M$, $N$ be $R$-modules and let $f\colon M\to N$ be an injective homomorphism of $R$-modules. Is $f\otimes {\rm id}\colon M\otimes_RN\to N\otimes_RN$ injective?
- 42.5k