Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
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Rings with terminating division chains of a given length
Let $R$ be an integral domain. Given $a,b\in R$, then a division chain for $(a,b)$ is a sequence where we take $r_{-1}=a$, $r_0=b$, and for each $n>0$ we take $r_n=r_{n-1}s_n+r_{n-2}$ for some $...
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Flat and algebraic (non-integral) local rings extension $R \subseteq S$ with $m_RS=m_S$
Let $R \subseteq S$ be two Noetherian local rings, not necessarily regular, which are integral domains,
with $m_RS=m_S$, namely, the ideal in $S$ generated by $m_R$ (= the maximal ideal of $R$) is $...
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Passing over $O_K \otimes_{\mathbb{Z}} A$ from $O_K$, how it affects the rank of a module?
This question was asked in MSE as well.
Let $K$ be a finite extension of the rationals $\mathbb{Q}$ with $O_K$ its the ring of integers.
Consider a $\mathbb{Z}$-algebra $A$ such that $|A|<\infty$.
...
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On the coherence of a Néron-ring
Let $A:= \underset{\lambda \in \Lambda}{\varinjlim} \,A_{\lambda}$ be an inductive limit of geometric regular local ring $(A_{\lambda}, {\frak m}_{\lambda})$, whose transition map $\phi_{\mu\lambda} \...
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Commutative algebra details on patching when proving $R = \mathbb{T}$ theorem (Calegari-Geraghty Paper)
I have originally posted this on math.SE and been suggested to post this here. I'm merely an undergraduate student and it is the first time for me to ask questions here. I'm sincerely sorry if these ...
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The Mumford-Tate conjecture
The Mumford-Tate conjecture asserts that, via the Betti-étale comparison isomorphism, and for any smooth projective variety $ X $, over a number field $ K $, the $ \mathbb{Q}_{ \ell } $-linear ...
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A similar construction to Ext, can we describe it better and does it have any use?
Let $R$ be a ring and $\text{Mod}\,R$ the category of $R$ modules. For two $R$-modules $X,Y$ one can define $\text{Ext}_R^n(X,Y)$ as follows. We take an injective resolution $0\rightarrow Y\rightarrow ...
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The universal multiset for a finite scheme - reference request
If $X$ is a finite set of size $n$, then by listing the elements of $X$ we get a canonical element of the symmetric power $X^n/\Sigma_n$, which we can call the universal multiset for $X$.
Now let $X$ ...
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Is the category of quasi-coherent sheaves not a topos?
There are two parts to my question:
Question #1: First a sanity-check: Am I right in that the category of quasi-coherent sheaves (over e.g. an affine scheme) is not a topos?
My reasoning is thus:
It ...
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Is every additive, left exact functor isomorphic to a hom functor?
Let $A$ be an Artin algebra, $\text{mod}\,A$ the category of finitely generated $A$-modules and $\text{Ab}$ the category of abelian groups. Is every additive, covariant, left-exact functor $F:\text{...
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Principal ideal domains with finitely many units
Question: What are the (in characteristic 0 if needed) principal ideal domains that have finitely many units?
Can such rings be classified?
(This is a more specialised version of the question in ...
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Betti numbers of a polynomial ideal after homogenization
Let $I \subset k[x_1,\ldots,x_n]$ be an ideal in a polinomial ring over a field $k$. Assume that $I$ is quasihomogeneous (that is, $I$ is not homogeneous with the usual grading, but it is homogeneous ...
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Number of rings with additive group $(\mathbb{Z}_{16})^2$. A341547(16) in OEIS
I would like to know if somewhere the number of non-isomorphic rings with additive group $(\mathbb{Z}_{16})^2$ is mentioned. If not, is someone able to calculate it?
And (easier) the commutative case? ...
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F-splitting and F-purity from commutative algebra viewpoint
First I define two terms:
Let $R$ be a commutative ring with identity,let char$R$ = $p$, let $F:R\rightarrow R$ be the Frobenius ring homomorphism. This makes $R$ into an $R$-module with respect to ...
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Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
The examples of rings not isomorphic to their opposite that I know of are not ...
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Is Koszul homology of a monomial ideal always generated by the "obvious" things?
Let $R = k[x_1 , \dots , x_n]$ be a polynomial ring over a field and $I$ a monomial ideal in $R$. Then, is it true that the Koszul homology of $R/I$ is always generated by elements of the form
$$r e_{...
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Given a unitary commutative ring $R$, what are the rings $R\langle x,y\rangle/(x^2-A,y^2-B,yx-a-bx-cy-dxy)$ called
We are studying the rings
$$
R \langle x, \, y \rangle\,\big/\left(x^2-A, \, y^2-B, \, yx-a-bx-cy-dxy \right)
$$
Do you know if they have a name?
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Class group of hypersurfaces of finite representation type
Let $k$ be an algebraically closed field of characteristic different from $2,3$ and $5$, and let $R=k[[x,y,x_2,\dots,x_d]]/(f)$, where $f\in(x,y,x_2,\dots,x_d)^2$, $f\neq0$. By results of Buchweitz-...
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On the degree of the Hilbert polynomial of a graded module over the Rees algebra
If $A=\oplus_{n=0}^\infty A_n$ is a Noetherian graded ring of finite dimension such that $A_0$ is local and $A=A_0[A_1]$, and if $M=\oplus_{n=0}^\infty M_n$ is a finitely generated graded $A$-module ...
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Can we write the isomorphism $F_i \otimes_R k \cong H_i (K_\bullet \otimes R/I)$ explicitly?
Let $R$ be a regular local ring (or polynomial ring over a field). Let $I$ be an ideal of $R$ and $F_\bullet$ a minimal free resolution of $R/I$. Let $K_\bullet$ denote the Koszul complex resolving ...
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The cokernel of an irreducible monomorphism is always the third term of an AR sequence with indecomposable middle term?
I recently studied the structure of the AR quiver of Dynkin type $\mathbb{A}_n/I$,
$\mathbb{D}_n/I$ with $I$ any admissable ideal, and found that the cokernel of an irreducible monomorphism is always ...
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Integral domains with finitely many units
Question: Is there a classification of (noetherian if needed) integral domains with finitely many units ? (of course we can exclude fields as trivial examples)
Probably there are many such domains ...
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Does a local ring with a separably closed residue field admit a universal homeomorphism from a local ring with algebraically closed residue field?
Let $A$ be a local ring with separably closed residue field. Under what conditions does there exist a local ring $B$ with an algebraically closed residue field and a local homomorphism $A\rightarrow B$...
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Quotient of ideal generated by regular sequence is a perfect module
I have recently started reading Bruns-Herzog's 'Cohen Macaulay rings' and this is problem 1.4.27 in it.
We say that a module $M$ over a Noetherian ring $R$ is perfect if the projective dimension of $M$...
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Normal simple ring extensions
Let $R$ be a $k$-algebra, $k$ is an algebraically closed field of characteristic zero.
Assume that $R$ is an integral domain or a UFD (being a UFD makes things easier),
$a$ is an algebraic element ...
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Passing to torsion of an exact sequence
If
$$
\Theta\colon\quad 0\to A\to B\to C\to 0
$$
is an exact sequence of abelian groups, and $n$ is an integer, then one obtains an exact sequence $$
0\to A[n] \to B[n] \to C[n] \stackrel{\delta_n(\...
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Nice small resolution and normality of blow-up
Let $X$ be a complex variety whose singular locus is a smooth variety $Z$.
Let $f:Y\rightarrow X$ be a small resolution of $X$ such that $f^{-1}(z)$ is smooth for any $z\in Z$ and $\dim(f^{-1}(z))$ is ...
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Is the Euler–Mascheroni constant an EL-number?
This question is based on Chow - What is a closed-form number?.
The author of the linked paper had proposed a plausible definition of "elementary numbers" (which he calls "EL-numbers&...
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Length of a module and Frobenius map
Let $(R,m)$ be a regular local ring of dimension $d$ and char $p>0.$ Let $F^e:R\longrightarrow R$ defined by $r\longrightarrow r^{p^e}$be the Frobenius map.
How to compute $l(R/m^{[p^e]})?.$
I ...
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Field whose absolute Galois group is $\mathbb{Z}_p$
Let $L$ be a perfect field. Assume there is an algebraic closure $L\subset \overline{L}$ and a prime $p$ such that $\mathrm{Gal}(\overline{L}/L)\cong \mathbb{Z}_p$ as topological groups.
Is there a ...
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Absolute Galois group with unique closed non-open subgroup
Is there an absolute Galois group that is not a subgroup of $\hat{\mathbb{Z}}$ and that has one and only one closed non-open subgroup?
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When do the kernels of module homomorphisms between rings whose kernels contain a given fixed ideal contain every prime ideal over it?
$\DeclareMathOperator{\Hom}{Hom}$All our rings are commutative with unity and, if necessary, we can suppose that they are actually polynomial rings over a field in finitely many variables where the ...
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Defining path on the prime spectrum
If $p$ and $q $ are two prime ideals of a commutative ring $R $ such that $p \subseteq q$, then we can easily define a continuous function (a path) $f$ from the unit interval $ [0,1]$ to the prime ...
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Integer valued polynomials and polynomials with integer coefficients
It is well known that the subring $S$ of integer valued polynomials ${\mathbb Q}[x]$ is generated by the binomial functions $P_n={x \choose n}$. One can ask a dual question: how to characterize the ...
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Making a generating set of a section of a graded polynomial $R$-module coming from a quotient into a basis of a quotient by higher degree polynomial
Denote the graded rings $R:=\mathbb{R}[x_{1},\dots x_{n}]$ and $S:=R[x_{0}]$ adding the homogenizing variable $x_{0}.$ Consider $h\in S$ a homogenous polynomial of degree $d$ with leading coefficient $...
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If the direct sum of $L$ and $M$ has a pseudoinverse, then do $L$ and $M$ have pseudoinverses?
Let $L$ and $M$ be matrices over a commutative ring $R$ equipped with an involution "$*$". Define $L \oplus M$ (the "direct sum" of $L$ and $M$) to be $\begin{bmatrix}L & 0 \\ ...
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When $K[X_1,X_2,...,X_n] \to K[Y_1,Y_2,...,Y_m]$ is a flat morphism
Let $K$ be a field and $\varphi: K[X_1,X_2,...,X_n] \to K[Y_1,Y_2,...,Y_m]$ a polynomial $K$-algebra morphism. Assume $n, m \ge 2$. By definition $\varphi$ endows $K[Y_1,Y_2,...,Y_m]$ with a $K[X_1,...
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Is the class of isomorphism types of a module category always a set?
Let $A$ be a ring and $\text{mod} A$ the category of finitely generated (right) modules over $A$. Is the class of isomorphism types of $\text{mod} A$ always a set? In particular, is it the case if $A$ ...
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Surjection from finite rank free $R$-module to finitely generated $R$-module and basis associated to generator set
Suppose the we have an epimorphism $s\colon M\to N,$ where $M$ is a free $R$-module of rank $r$ and $N$ is a finitely generated $R$-module, such that there exists a basis $B:=\{m_{1},\dots, m_{r}\}$ ...
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Must the inclusion of an indecomposable module in the direct sum of two copies always split?
We consider finitely generated modules over an Artin algebra. Let $X$ be an indecomposable module and let $f:X \longrightarrow X \oplus X$ a monomorphism. Must $f$ always be a split monomorphism?
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What is a good introduction to cluster algebras from surfaces?
What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory?
In my view, that means it should start off with unpunctured surfaces (and in fact,...
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Prime ideals in $R \subseteq \mathbb{C}[x,y]$, $\dim(R)=2$
Prime ideals in $\mathbb{C}[x,y]$ were listed here; they are:
(i) $(0)$.
(ii) $(f)$, where $f$ is an irreducible polynomial.
(iii) $(x-\lambda,y-\mu)$, where $\lambda,\mu \in \mathbb{C}$.
Now let $R \...
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Flatness of certain $R \subseteq \mathbb{C}[x,y]$
The two-dimensional Jacobian Conjecture over $\mathbb{C}$ says the following:
Let $p,q \in \mathbb{C}[x,y]$ satisfy $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$.
Then $\mathbb{C}[p,q]=...
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Creating prime ideals in rings
Are there different ways to create prime ideals in a ring other than taking quotients? I recently came across a construction of a prime ideal in a Noetherian ring $A$ given in the book on Algebraic ...
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Concerning a result of E. Formanek
A result of E. Formanek, in its two-dimensional version, says:
Let $k$ be a field of characteristic zero and let $R=k[x,y]$ be the polynomial ring in two variables.
Let $p,q \in R$ have invertible ...
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Indecomposable modules such that the radical is a submodule of the socle
We consider finitely generated modules over an Artin algebra. Let $X$ be an indecomposable module such that the radical $\text{rad} \,X$ is a submodule of the socle $\text{soc}\,X$. What can we say ...
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Preservation of finite presentation along restriction of scalars
Let $\theta \colon R \to S$ be a morphism of commutative rings (with unit). We assume that:
$R$ and $S$ are coherent.
$S$ is finitely presented as an $R$-module.
Now, let $M$ be an $S$-module, and ...
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Non-field example of a commutative, local, dual ring with nilradical $N$ such that $ann(N)\nsubseteq N$
I asked this question on math.stackexchange a month ago with no progress, even after a bounty. I hope to eliminate one if the other receives a satisfactory answer.
For an ideal $I\lhd R$ in a ...
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Algebraic closure of field of fractions of multivariate polynomial ring over $\mathbb{R}$
I am searching for good references on the topic of the behaviour of the elements in the algebraic closed field $(\mathbb{R}[x_{1},\dots,x_{n}])^{\operatorname{alg}}.$ I imagine that, when we try to ...
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The first female algebraist in US/Britain?
Recently I dug up some biographical details of Lindsay Burch, of Hilbert-Burch Theorem fame, whose few papers have had quite an impact on commutative algebra. This made me curious about the first ...
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