Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,495 questions
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Restriction of a dominant map to a hypersurface
Suppose there is a dominant morphism $H: \mathbb{A}^1_k \times W \to X$ such that $H(0, -) \neq H(1, -)$ as morphisms from $W$ to $X$. Here $W$ and $X$ are varieties over a field $k$. Assume that $...
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Basepoints in the canonical system of algebraic surfaces
Let $X$ be a smooth projective variety defined over $\mathbb{C}$. In the context of the minimal model program it is often important to understand the geometry of the maps defined by the complete ...
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For a finite-type $\mathbb{Z}$-algebra $A$, is the intersection of all ideals $I$ such that $A/I$ is finite and local necessarily zero?
Background: A referee has suggested a shorter proof of one of my results, but I'm having trouble justifying one of their assertions. The setting is that $A$ is a commutative ring, and the referee's ...
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Extension of Tate's result regarding Tor
In a 1957 paper (Link), Tate shows that if $I \subset R$ is an ideal of the noetherian ring R then there is a graded commutative DGA $X$ over $R$ with $H_i X=0$ except $H_0 X= R/I$ (I guess R should ...
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Outlier absences of monomials in a group of inversion partition polynomials
Revamped and updated on Sep 12, 2022:
Given the complex coefficients $a_n$ of some generic formal power, Taylor, Laurent or other series, say the ordinary generating functions (o.g.f.) $f(z) = z +a_1 ...
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Higher Artin-Schreier homomorphism?
For the additive group over a characteristic $p$ field one has a short exact sequence of abelian algebraic groups
$$\{1\} \to {\mathbb Z}/p \to {\mathbb G}_a \to {\mathbb G}_a \to \{1\},$$
where the ...
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A particular family of symmetric functions (sums of powers of sums of subsets)
For any $m,k$ define
$$ f_{m,k}(x_1,\ldots,x_n) = \sum_{1\le i_1<i_2<\cdots<i_m\le n} (x_{i_1}+\cdots+x_{i_m})^k. $$
Do these symmetric polynomials have a name and any theory?
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Examples of integral ring extensions that $\operatorname{ht}P \lt \operatorname{ht}P\cap A$
$\DeclareMathOperator\ht{ht}$All rings are commutative Noetherian with identity.
Exercise 9.8 of Matsumura's book Commutative ring theory: Let $A$ be a ring and $A\subset B$ an integral extension. If $...
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Any ideal as an intersection of ideals primary to maximal ones
The Nullstellensatz says that any ideal $I \subset \mathbb{C}[x_1,\dots,x_n]$ has the property that
$\sqrt{I} = \bigcap_{\text{maximal } \mathfrak{m} \supset I} \mathfrak{m}$
Is it also true that we ...
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Are there more Nullstellensätze?
Over which fields $k$ is there a reasonable analogue of Hilbert's Nullstellensatz?
Here is a more precise formulation: let $k$ be an arbitrary field, $n$ a positive integer, and $R = k[t_1,..,t_n]$. ...
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A Krull-like Theorem and its possible equivalence to AC
A well known equivalent of the Axiom of Choice is Krull's Maximal Ideal Theorem (1929): if $I$ is a proper ideal of a ring $R$ (with unity), then $R$ has a maximal ideal containing $I$. The proof is ...
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Basic example of derived descent
I've been trying to understand the Adams spectral sequence, and I've gotten myself confused about how derived descent is supposed to work, so I would like to understand a simple example.
Given a ...
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Different generating sets for conjugation invariants of several matrices
There is a theorem of Procesi that the ring of polynomial functions on tuples $(A_1,A_2, \dots, A_m)$ of $n \times n$ matrices, which are invariant under simultaneous conjugation, is generated by ...
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Functorial criterion for local complete intersection morphisms?
Let me state the question for rings (rather than schemes) for simplicity. Let $R$ be a commutative ring with unit and $A$ an $R$-algebra of finite presentation. Recall that $R\to A$ is called a ...
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Generic Noether normalisation
Suppose that $M$ is a finitely generated module over $A=k[X_1,\ldots,X_n]$ of Krull dimension $m$ with $k$ an infinite field. Then one version of Noether normalisation says there is an $m$-dimensional ...
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Additivity of projective dimensions, or, help me lower my blood pressure
Sorry for the shameless title. I'm rather stuck on a lemma in commutative algebra - namely, I have both a proof and a counterexample! I have tried rather strenuously and frustratingly to find the ...
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Why is the natural map $\hom(A,\mathbb{R}/\mathbb{Z})\to K/A$ an isomorphism, $K/\mathbb{Q}_p$ unramified, $A=\mathcal{O}_K$?
While looking at an analogue of Pontryagin duality for compact Discrete Valuation Rings (DVRs), I came about the observation that generally one should have an isomorphism of $A$-modules
$$\hom_{\...
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Why does the Manin-Mumford conjecture over number fields imply the conjecture over arbitrary fields of characteristic 0?
The Manin-Mumford conjecture states that for an abelian variety A over a field F of characteristic 0 the torsion points are dense in an integral closed subvariety Z if and only if it is an abelian ...
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On the Artin-Rees Lemma for non-commutative rings
Consider a commutative noetherian ring $A$ with an ideal $I\subset A$. The Artin-Rees lemma implies that for f.g. modules $N\subset M$, the $I$-adic topology on $N$ agrees with the subspace topology ...
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Iterating exact triangles (particularly in Floer homology)
There are several different Floer-homological invariants of 3-manifolds (and knots). The most prominent of these are Heegaard Floer homology, monopole Floer homology, and instanton Floer homology. It ...
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Subtle examples of morphisms that are finite but not flat
Let $R$ be a ring (commutative noetherian with unit), and let $K(R)$ be its total ring of fractions (obtained by inverting all nonzerodivisors). Thus, $R \hookrightarrow K(R)$. Let $a \in K(R)$ be ...
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Constructibility of the locus of points where the fiber is an isomorphism modulo nilpotents
Let $f: X \rightarrow S$ be a finitely presented morphism of schemes and let $$E = \{s \in S \mid \text{ $X_s$ is a point with residue field $\kappa(s)$ } \}$$
Is $E$ a constructible set?
The basic ...
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Relative valuative criteria of properness for flat morphisms
Let $f: X\rightarrow S$ be a flat quasi-projective morphism, where $X$ is a smooth variety, and $S$ is a discrete valuation ring. Then we know that $f$ is proper morphism if and only if it satisfies ...
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Atomicity and BF-ness in monoids of integer points of a polyhedral cone of $\mathbb R^n$
Fix an integer $n \ge 2$ and let $H$ be the (additive) monoid of integer points of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_n\}$ is ...
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Factoring a polynomial into linear factors by ring extension
The following sounds so natural, I'm surprised I have never asked it before:
Question 1. Let $R$ be a commutative ring. Let $P \in R\left[X\right]$ be a polynomial. Can we find a commutative ring $S$ ...
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Local cohomology: Polynomial ring vs Power series ring
I study algebraic topology and am currently examining the applications of local (co)homology in algebraic topology. We have the canonical inclusion of rings $\mathbb{Z}[x_1,\cdots,x_n]\subset \mathbb{...
CommunityBot
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Which cluster algebras are coordinate rings of double Bruhat cells?
Background
A uselessly vague paragraph follows. A cluster algebra is a commutative algebra $A$ with a distinguished set of generators called cluster variables. These cluster variables are grouped ...
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Valuations of coefficients of minimal polynomials for tuples
Suppose you are given two valued fields $(K,v) \subseteq (L,w)$ and a tuple $a \in L^n$. What kind of restrictions do we have on the valuation of the coefficients of polynomials $q \in K[x_1,\dots x_n]...
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Injectivity of tuple of incomplete elementary symmetric polynomials
For $1\leq k,j \leq n$ and $a=(a_1,\ldots,a_n)\in {\mathbb R}^n$, denote by $s_{k,j}(a)$ the $k$-th symmetric polynomial in the $n-1$ variables obtained when $a_j$ is removed from $a_1,\ldots,a_n$. ...
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Does the rational power series ring $\mathbb{Q}[[X]]$ embed as a ring into the field of real numbers?
The title says it all. I'm wondering if the power series ring $\mathbb{Q}[[X]]$ (with rational coefficients) embeds as a ring into the field of real numbers. There are various topologies one might ...
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Noetherian but not strongly Noetherian
What are some examples of Tate rings $R$ (i.e. Huber rings with with topologically nilpotent units) which are Noetherian but not strongly Noetherian ($R$ is strongly Noetherian iff for all $n \in \...
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Homological dimensions of rings of smooth functions
What is the global dimension of the algebra $C^\infty\mathbb R$ of smooth functions $\mathbb R\to\mathbb R$? What is the global dimension of the algebra $(C^\infty\mathbb R)_0$ of germs of smooth ...
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Structure of reflexive modules over regular local rings
Let $R$ be a regular local ring and $M$ a finitely generated reflexive $R$-module. When $R$ has dimension 2, then $M$ is a free $R$-module. This is discussed in Reflexive modules over a 2-dimensional ...
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Surjectivity of natural map of rings
$\DeclareMathOperator\Hom{Hom}$Let $A$ be an integral domain and $P$ be a prime ideal in $A$. We denote $B=A/P$ then is the following natural map
$$\Hom_A(P,A)\otimes_A B\to \Hom_A(P,B)$$ surjective?
...
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Ideals whose quotient rings have a certain property
There are some well-known properties of ideals which are equally well-known to correspond to properties of their respective quotient rings. For example:
An ideal $p$ of a ring $R$ is prime iff $R/p$ ...
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Coherence of the I-adic completion of a local ring of a formal scheme
Let $K$ be a valued field of rank one and $K^+$ its valuation ring such that $K^+$ is $\varpi$-adically complete with respect to a pseudo-uniformizer $\varpi\in K^+$. Let $X$ be a smooth finite type $...
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DCC on the powers of ideals
My apologies if this question is below the level of MO. I posted the same question in MS about a week ago without an answer so far.
Let $R$ be a unital commutative ring. $R$ is called strongly $\pi$-...
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Local complete intersections which are not complete intersections
The following definitions are standard:
An affine variety $V$ in $A^n$ is a complete intersection (c.i.) if its vanishing ideal can be generated by ($n - \dim V$) polynomials in $k[X_1,\ldots, X_n]$. ...
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Quiver representations over any commutative ring
I'm reading a paper of Aidan Schofield "General Representations of Quivers" and he defines quiver representation over any commutative ring. See the below image.
Towards the end, he has this ...
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Characterizing atomicity in a commutative domain
In Proposition 1.1 of [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ...
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Rank 2 vector bundle with trivial first chern class is self-dual
I saw a statement used in the paper that E of rank 2 with $c_1(E) = 0$ is self-dual. I was wondering, how does one prove this statement? If it makes a difference, let the underlying variety be ...
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What are the main structure theorems on finitely generated commutative monoids?
I should read J. C. Rosales and P. A. García-Sánchez's book Finitely Generated Commutative Monoids and L. Redei's book The Theory of Finitely Generated Commutative Semigroups. I haven't. But here's ...
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Why are finitely generated modules over principal artin local rings direct sums of cyclic modules?
I am looking for a proof of the following fact:
If $R$ is a principal artin local ring and $M$ a finitely generated $R$-module, then $M$ is a direct sum of cyclic $R$-modules.
(Apparently such rings $...
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A local ring not a quotient of a regular local ring
In his book Commutative Ring Theory, Matsumura proves that if a local ring is equidimensional, and a quotient of a regular local ring, then its completion is equidimensional.
What is an example of a ...
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Does the set of ideals, whose Jacobson radical & nilradicals coincide, form a sublattice?
This question might be below the level of MO, so apologies in advance. I posted the same question in MS about a week ago without an answer so far.
Let $R$ be a unital commutative ring and $L(R)$ ...
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Completion of ring as direct limit
If $X$ is a variety and $x \in X$, there are several ways to look locally around the point $x$:
Localisation: taking the direct limit over open immersions around $x$.
Henselisation: taking the direct ...
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Which consequences can be deduced from this peculiar property of tetration?
Recently (assuming radix-$10$), I showed that, for any $a \in \mathbb{N}_{0}$ that is not a multiple of $10$, there exists a unique value $V(a) \in \mathbb{N}_{0}$ which corresponds to the number of ...
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Software to compute generators of a module over polynomial ring
Let $A=\mathbb{R}[x_1,\dots,x_n]$ be the algebra of real polynomials in $n$ variables. Fix polynomials $p_1,\dots,p_k\in A$.
Consider the subset
$$M:=\{(q_1,\dots,q_k)\in A^k|\, p_1q_1+\dots+p_kq_k=0\}...
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Duogenic quartic rings
Recall that a commutative, unital ring $R$ of finite rank which is isomorphic to $\mathbb{Z}^n$ for some $n \geq 1$ as $\mathbb{Z}$-module is said to be monogenic if there exists an element $\gamma \...
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Dependence of completion on the base ring
Let $M$ be a module over an $E_\infty$ ring, $A$. Let $I$ be an $A$-non unital commutative algebra together with an associative map $I \wedge_A M \to M$.
Define ${_A}(M/I^n)$ as the cofiber of $I^{\...
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