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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A little help with the unmixedness theorem?
I have two smooth subvarieties $Y$ and $Z$ of a smooth variety $X$. Their intersection $Y \cap Z$ has two irreducible components, both of the expected dimension and generically reduced. I want to ...
- 683
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2
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Solve for $A$ and $B$ in $AXB=Y$
Let $R = \mathbb{Z}[x_{1}, \dots, x_{r}]$.
Let $X$ be $n \times n$ matrix with entries in $R$.
Let $Y$ be $m \times m$ matrix with entries in $R$ formed from $\mathbb{Z}$-linear or $\mathbb{R}$-linear ...
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2 questions about "monogenic" coordinate rings of affine curves
Suppose $k$ is an algebraically closed field with characteristic 0 and let $X \subset \mathbb{A}^n$ be an irreducible curve.
If $f \colon X \to \mathbb{A}^1$ is finite of degree $d$, then the ...
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injectivity is a local property over noetherian rings [closed]
Let $R$ be a commutative noetherian ring, and let $M$ be an $R$-module. How I can show that if any localization $M_p$ at a prime ideal $p$ of the ring $R$ is injective over $R_p$, then $M$ is ...
- 15
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133
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Embedding of Gorenstein orbifold as a hypersurface
I am trying to understand if three complex dimensional orbifold singularity $\mathbb{C}^3 / \Gamma$ can be embedded as hypersurface in $\mathbb{C}^4$. The condition of being Gorenstein and having ...
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Finitely generated ordered monoids and noetherian subsets
(This question was asked a long time ago on MSE but got no answer so far...)
Let $E$ be an additively written cancellable commutative monoid with no non-trivial units. We furnish $E$ with the order ...
- 6,700
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A version of Peskine and Szpiro's theorem in vanishing of local cohomology
C. Peskine and L. Szpiro in "Dimension projective finie et cohomologie locale",(Proposition 4.1) proved the following vanishing theorem for local cohomology:
Let$R$ be a regular local ring of ...
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Krull-Schmidt Analogue for Complete / Graded Rings
Over the ring $\mathbb{Z}$, all finitely-generated modules decompose uniquely as a direct sum of indecomposable submodules; that's the Krull-Schmidt theorem.
I'm given to understand that if a (...
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First-order UFD (factorial ring) condition / pre-Schreier rings
All rings in this post are commutative and with $1$.
Everyone knows the definition of a factorial ring, a. k. a. unique factorization domain (UFD). I have been wondering about some variations ...
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$k[[x]]$ as a $(k[[x]])^p$ module for ugly fields
Suppose that $k$ is a field of characteristic $p$ such that $k$ is not a finite $k^p$-module. For example, $k = \mathbb{F}_p(x_1, x_2, x_3, ...)$.
Is it true that $k[[x]]$ is a free $(k[[x]])^p$-...
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Number of A Subset of Monomials
I need to count the number of monomials of degree $n$ in $k$ variables, $x_1,\ldots ,x_k$, that contain at least one variable with a power of 1. The monomials need not include all the variables. ...
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ideal of maximal minors is cohen-macaulay?
Let $k$ be an algebraically closed field.
Let $A$ be an $m \times n$ matrix with linear forms $a_{ij} \in k[x_1, \ldots, x_p]_1$ as entries. Let $I$ be the ideal generated by the maximal minors of $A$....
- 3,031
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Different term's contribution in zonal polynomial
I am very interested in the contribution of different terms in a zonal polynomial. Let's focus on the simplest case. In (20) in "Distributions of matrix variates and latent roots derived from normal ...
- 163
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Behavior of duality under pull-back
I have a technical question on commutative algebra. I am not an expert in the subject, and I would like to know if there are "typical conditions" making the following possible.
Let $\varphi:R\to S$ ...
- 2,406
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Symmetric algebra of an ideal and syzygies
Let $(R,\mathfrak{m})$ be a Cohen Macaulay local ring and $I=(a_1,\ldots,a_g,a_{g+1})$ an almost complete intersection ideal of codimension $g.$ Let $R^k\longrightarrow R^{g+1}\longrightarrow I\...
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Universal coefficient theorem for local ring
Let $R$ be a commutative local artin $k$-algebra,where $k$ is a field with characteristic $0$.I wonder whether universal coefficient theorem holds in this case.Namely,if $C$ is a chain of flat $R$-...
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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 ...
user78172
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A construction of Kähler differentials and Illusie cotangent complex as colimit over embeddings
Let $\Bbbk$ be a field, $X$ affine scheme of finite type over $\Bbbk$. Let $\mathcal C_X$ be the category of closed embeddings of $X$ into (say affine) smooth $Y$'s of finite type over $\Bbbk$, ...
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Which commutative rigs arise from a distributive category?
A rig is an algebraic object with multiplication and addition, such that multiplication distributes over addition and addition is commutative. However, instead of requiring that the set forms an ...
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UFD property descends?
Hi,
Let $k$ be a field and $A$ a local noetherian $k$-algebra. If its completion is a UFD,
is it true that $A$ is a UFD? Proof?
Thanks
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Formal criterion of flatness
Let $k$ be a field, $S$ and $R$ be local $k$-algebras with residue field $k$ and $\phi:S\to R$ be a local homomorphism. Then $\phi$ induces (obviously) a natural transformation of "functors of points" ...
- 16.2k
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Reversing the arrows-dual theorems
When one studies homological algebra one learns some basic stuff-diagram chasing, long exact sequences associated to short exact sequence of complexes and so on. Usually one works out the details with ...
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Doubt in this proof of Horrocks theorem
I'm beginning to study some research papers and I need right now to understand the solution of Vaseršteĭn of Serre's theorem (simplest proof of this theorem), to do so, I'm beginning to understand ...
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Can we express the degree 10 and degree 15 Galois resolvents of sextic binary forms in terms of its basic invariants?
Let $V_6$ denote the 7 dimensional $\mathbb{C}$-vector space of binary sextic forms. For $T = \begin{pmatrix} t_1 & t_2 \\ t_3 & t_4 \end{pmatrix} \in \operatorname{GL}_2(\mathbb{C})$, $T$ ...
- 26.9k
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Does every nontrivial sheaf of rings have a maximal ideal?
Let $R$ be a sheaf of rings on a topological space $X$. Assume $R \neq 0$. Does then $R$ have a maximal ideal? So this is a spacified analogon of the theorem, that every nontrivial ring has a maximal ...
- 63.1k
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Can height one maximal ideals in the normalization contract to non-height one primes in the base?
Let $R$ be a local (Noetherian) integral domain of dimension greater than one. Can the integral closure (i.e. normalization) of $R$ have a maximal ideal of height one?
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When is every "solid" perfect complex faithful?
Let $R$ be a noetherian commutative ring. Consider $D^{perf}(R)=K^b(R-proj)$ the category of bounded complexes of finitely generated projective $R$-modules, with maps of complexes up to homotopy.
...
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Support of a module over a polynomial algebra
In Atiyah and Bott's paper "The Moment Map and Equivariant Cohomology", they say that for any exact sequence of modules over $\mathbb{C}[u_1,...,u_l]$
$$D \to E \to F,$$
we have that
Supp $E \subset$ ...
- 502
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Intersection of localization with finitely generated subalgebra of fraction field
Let $R$ be a (commutative) noetherian integral domain. Let $I$ be a prime ideal of $R$. Let $S$ be a finitely generated $R$-subalgebra of $\mathrm{Frac}(R)$.
Is $S \cap R_I$ necessarily finitely ...
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Failure of Fin. Presented and Fin. Generated Modules to be Abelian Categories?
Let R be a ring. I'm trying to understand when the categories of finitely presented R-modules and finitely generated R-modules can fail to be abelian categories.
Poking around on the internet has ...
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Weakened conditions for étale + X implies faithfully flat.
Let $F:R \to S$ be an étale morphism of rings. It follows with some work that $f$ is flat.
However, faithful flatness is another story. It's not hard to show that faithful + flat is weaker than ...
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Uniform Artin-Rees bound for annihilators in Noetherian local rings
Let $(A,\mathfrak{m})$ be a Noetherian local ring. If $I$ is an ideal of $A$, then by (a weak version of) the Artin-Rees lemma, there exists $j \geq 0$ such that for all $i \geq j$,
$$\mathfrak{m}^i \...
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Division of multivariable polynomials by an ideal
Let $K$ be a commutative field and consider an ideal $I$ of $K[X_1,\dots,X_n]$.
Is there a well behaved "reduction modulo I", in the following sense :
Given a well-ordering $\leq$ on the set of ...
- 1,555
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A quaternion x generates a left ideal of rank 2 if and only if x, ix and jx are linearly dependent?
I am trying to understand the construction of Artin and Mumford of a non-rational unirational threefold in ([1], p.90).
Assume $S$ is a smooth projective surface over $\mathbb{C}$ with a smooth ...
- 1,025
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Are Chow groups a birational invariant?
Let us work in the category of smooth, projective varieties (say, over an algebraically closed field $k$). If $X$ and $X'$ are birational, then do they have the same Chow groups? Is there at least a ...
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Associated graded of double Koszul dual
Let $k$ be a field, and let $A$ be a graded, connected, augmented, locally finite $k$-algebra. If $\Omega^* A$ denotes the cobar complex of $A$ (i.e., the dual $Hom_k(B_*(A), k)$ of the bar complex ...
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A question about how polynomials simplify under substitution
This is a revised and more sensible version of the original question, thanks to the kind help of Anthony Quas and J. C. Ottem.
Fix polynomials $f_{1},\ldots, f_{n} \in \mathbb{C}[t]$.
Let $M_{k}$ be ...
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Regular commutative Banach algebras which are not closed under complex conjugate
Let $A$ be a semisimple commutative Banach algebra with the maximal ideal space $X$. Further, assume that $A$ is regular i.e. for every closed set $E\subseteq X$ and $x\in X\setminus E$, there is some ...
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Are there any neat algorithm to factor a homogenous polynomial, given we know this polynomial factors into linear forms?
Are there any neat algorithm to factor a multivariable (more than 2 variables) homogenous polynomial, given we know this polynomial factors into linear forms?
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Krull dimension of a completion
How does one study Krull dimension of some I-adic completion of a ring or, more generally, a module? I know that in case of Noetherian local ring Krull dimension of its completion equals Krull ...
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Classifying Algebras of Convolution
$L^1(\mathbb R^n)$, $L^1(\mathbb R_+)$, $C^0_c(\mathbb R_+)$, $C^\infty_c(\mathbb R_+)$ are algebras of convolution.
Question 1: is there a classification of subalgebras of convolution of $L^1(\...
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Geometry of componentially locally strongly separable algebras
Janelidze's categorical Galois theory yields, for nice adjunctions, a good notion of covering morphism.
The category of finitely affine schemes admits such an adjunction into the category of ...
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When does composing polynomials reduce the degree?
Let $\mathbb{F}$ be the field of size 2. For a function $f : \mathbb{F}^n \to \mathbb{F}$, let $d(f)$ be the smallest integer such that there exists a degree-$d(f)$, $n$-variate, multilinear ...
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finiteness dimension
$R$ is a local Noetherian ring. $f_I(M)$, the finiteness dimension of a module $M$ relative to $I$, is defined in ...
- 1,355
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Infinite domain with finite number of prime ideals(elements)
While trying to prove one property of commutative rings with units I can't prove one fact without assuming existence of infinitely many different prime ideals or elements. I tried to test if it was ...
- 606
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Multiplicity of eigenvalues in 2-dim families of symmetric matrices
Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of ...
- 1,452
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When does effective descent of modules hold?
Let $A$ be a commutative ring with identity. I denote by $\Delta_{\leq 1}$ the full subcategory of the simplex category $\Delta$ with objects $[0]$ and $[1]$. Let $B_{\cdot} : \Delta_{\leq 1}^{\mathrm{...
- 7,249
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Characterization of a finitely graded (almost) domain
Let $A= \bigoplus_{i=0}^{N}A_i$ be a finitely graded ring with the following property: if $x \in A_i$ and $y \in A_j$ and $i+j \leq N$, then
$$xy = 0 \text{ implies } x = 0 \text{ or } y = 0.$$
Hence ...
- 3,019
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Two-variable polynomials, irreducible as formal power series
Let $k$ be a field and $f\in k[a,b]$ an irreducible two-variable polynomial, $B := k[a,b]/(f)$ and $C$ the integral closure of $B$ in its fraction field.
I call $f$ good if it is irreducible in the ...
- 31
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Extensions of an ideal-theoretic criterion for a monoid to be BF
Let $H$ be a multiplicatively written, commutative monoid. We denote by $H^\times$ the set of units (or invertible elements) of $H$, and by $\mathcal A(H)$ the set of atoms (or irreducible elements) ...
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