Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
1,707
questions with no upvoted or accepted answers
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A dimension condition on the cohomology of a homogeneous space
The rational cohomology of a homogeneous space $G/K$ admits a homomorphism from $H^*(BK)$ induced from the classifying map $G/K \to BK$ of the principal $K$-bundle $G \to G/K$. Assume the Lie group is ...
4
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195
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Shifts in the decomposition of Bott-Samelson bimodules
Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that $W\subset\...
4
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133
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Reference request for $R$-index
Let $R$ be a noetherian domain with field of fractions $F$, let $V$ be a finite-dimensional $F$-vector space, and let $M,N \subseteq V$ be $R$-lattices in $V$ (finitely generated $R$-submodules of $V$ ...
4
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153
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rings with 'flat functions'
Let $(R,\mathfrak{m})$ be a local ring over a field. Suppose the ring has flat elements, i.e. $\mathfrak{m}^\infty\neq\{0\}$. (The prototype is of course $C^\infty(\Bbb{R}^p,0)$, or a quotient of it, ...
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248
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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 ...
4
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110
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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 ...
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75
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Catenarity of monoid algebras
Let $R$ be a commutative ring, let $M$ be a commutative monoid, and let $R[M]$ denote the corresponding monoid algebra. Suppose further that $R$ is universally catenary. One may ask for conditions on $...
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211
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Generation of cohomology of graded algebras
Let $A$ be an unital, associative, graded algebra over a base ring $k$. I'm happy to assume that $k$ is a field if need be, and will insist that $A$ free and of finite rank in each degree (locally ...
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795
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Commutative algebra books representing the edge of research
Recently I have come across the books Combinatorial Commutative Algebra by Miller and Sturmfels along with Combinatorics and Commutative Algebra by Stanley. I will soon own a copy of each. I also ...
4
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75
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Minimal rank of a permutation resolution of a $G$-lattice
Let $G$ be a finite group.
By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$.
One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...
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216
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Intersection numbers on blow ups of toric varieties
Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...
4
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152
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On the computational complexity of the Hilbert polynomial of numerical semigroup rings
Let $(R, \mathfrak{m}) = k[[X^a, X^b, X^c]]$, $a<b<c$, $gcd(a, b, c) = 1$, be a semigroup ring. We have $R$ is a Cohen-Macaulay local ring of dimension one. It is well known that $\ell(R/\...
4
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186
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Categorical characterization of closed imbeddings
Let $f\colon X\to Y$ be a morphism of schemes.
Let $F_X$ and $F_Y$ be the contravariant functors from the category $Sch$ of schemes to the category of sets defined via the Yoneda construction, i.e. $...
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167
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Bound for the height of equations defining the singular locus of a variety
Fix positive integers $m, n, d$.
In what follows, the height of an algebraic number will mean the absolute multiplicative height.
Let $V \subset \bar{\mathbb{Q}}^n$ be an affine algebraic variety ...
4
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243
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When does a commutative DGA have a finitely generated quasi-free resolution?
Suppose that $A$ is a commutative dg-algebra (say over base $k$) which is bounded in non-positive degree (with cochain complex conventions). There exists a quasi-free resolution of $A$. My question is,...
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207
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A doubt from the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring"
I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I have a doubt about the first part of Theorem 2.5. In the proof $(1)\iff ...
4
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829
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A strong form of implicit function theorem (what happens when the derivative is degenerate?)
(this can be considered as some ad)
Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ ...
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254
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Degree bounds for Grobner Basis
Let $I= \langle f_1, \ldots f_n \rangle \subset K[x_1,\ldots, x_n]$ be a homogeneous ideal and $\operatorname{deg}(f_i) \leq d$ then it has Grobner Basis where degree of each generator is less than or ...
4
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128
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Is this duality operation on simplicial complexes/Stanley-Reisner rings previously known?
Let $K$ be an abstract simplicial complex on vertices $x_1,\ldots,x_n$, then there is the familiar construction of the face ideal $I_K=\langle x_{i_1}\cdots x_{i_r} | \{x_{i_1},\ldots,x_{i_{r}}\}\not\...
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344
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Domains with prime ideal theorems
Let $D$ be a domain, and for prime ideals $\frak P$ of $D$ the norm is $N({\frak P}):=|D/{\frak P}|$. The prime ideal counting function of $D$ is given by $\pi_D(x)=\#\{{\frak P}\in{\rm Spec}(D):N({\...
4
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309
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Dimension of a commuting nilpotent variety
Fix $k$ an algebraically closed field, $n$ a natural number, and $\lambda=(\lambda_1,\ldots,\lambda_m)$ a partition of $n$. Let $A$ be any $n\times n$ nilpotent matrix with entries in $k$ whose ...
4
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273
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First Fundamental Theorem for Alternating Group
I know it fails but is there an answer?
More precisely, let $V$ be the standard complex $n$-dimensional representation of the alternating group $A_n$, $kV$ the direct sum of its $k$ copies, $S(kV)$ ...
4
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689
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Is there fppf descent of locally free modules
Being locally free is a property of quasi-coherent modules which
does not descend in the fpqc topology (see Remark Tag 05VF). But what happens for fppf coverings? More precisely we ask:
Suppose $A \...
4
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413
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Ext groups of affine scheme
Let $A$ be a commutative ring. $\textbf{Spec}(A)$ is the the
spectrum of $A$. $M$ and $N$ are $A$-modules. $\tilde{M}$ and
$\tilde{N}$ are sheaves associated on $\textbf{Spec}(A)$
respectively.
I ...
4
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355
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What is known about this short exact sequence in Lie algebra cohomology?
In its most general form, I look at the following. Let $g$ be a dg Lie algebra and $Z$ be its center. The sequence $ Z \to g \to g/Z $ gives me a short exact
$ 0\to C(g,Z) \to C(g,g) \to C(g,g/Z) \to ...
4
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351
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Comparing different Euclidean algorithms on a Euclidean domain
I have posted this question at stackexchange (502413), without responses until now.
In the papers by T. Motzkin: The Euclidean Algorithm, Bull. AMS 55, 1949, pp. 1142--1146 and P. Samuel: About ...
4
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389
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Epimorphisms between external tensor products
Let $k$ be a field, $R,S$ commutative $k$-algebras. If $\mathsf{Mod}_{fp}$ denotes the category of finitely presented modules, the external tensor product $\mathsf{Mod}_{fp}(R) \otimes_k \mathsf{Mod}_{...
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164
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Nontrivial examples of rings of relative stable rank 1
Given a (commutative) ring $R$, one says that the stable rank $sr(R)\leq n$ if any unimodular row $(a_1,\ldots,a_{n+1})$ of length $n+1$ is stable, i.e. there exist $x_1,\ldots,x_n$ such that $(a_1+...
4
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290
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The support of a finite type module on an algebraic space
I'd like to ask this question to make sure I understand a very basic thing about supports. Let $X$ be an algebraic space and F a quasi-coherent sheaf on it of finite type.
In here the schematic ...
4
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199
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height of contracted prime ideals in power series rings
Let $K$ be a field of characteristic zero and let $R_n = K[[X_1,\ldots,X_n]]$ be the power series ring in $n$-variables. Let $P$ be a prime ideal in $R_n$. Let $Q = P \cap R_{n-1}$.
Then is $height(Q) ...
4
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402
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When $R/(f)$ is regular?
For R being a commutative regular excellent Noetherian ring of finite Krull dimension which conditions on $f\in R$ can ensure that the ring $R/(f)$ is regular (so, I want a sufficient condition)? I do ...
4
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107
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maximal degree of generators of graded ideals
Let $A$ be a commutative Noetherian ring, $\mathfrak{a}$ an ideal. Let $R(\mathfrak{a}) = \oplus_{n\geq 0} \mathfrak{a}^n$ be the Rees algebra of $\mathfrak{a}$. Let $I$ and $J$ be two graded ideals ...
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144
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Is a *complete ring complete in the graded category ?
The question concerns a definition of Bruns, Herzog: Cohen-Macaulay Rings (before Prop. 3.6.16):
The Noetherian *local ring $(R,m)$ is said to be *complete if $(R_0,m_0)$ is complete.
(a graded ...
4
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187
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A non-matroidal notion of dependence on a set of ideals
Assume we are given a set of ideals $I_1, \dots, I_s$ in a commutative polynomial ring. Let's define a subset indexed by $A\subseteq [s] = \{ 1,2,\dots, s\}$ as dependent if there exists an $a\in A$ ...
4
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252
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Software for Computations with Complexes
What software would you recommend for working with chain complexes? In particular I'd like to be able to compute cohomology of a finite complex of free modules over polynomial ring. Is it possible in ...
4
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211
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The ring generated by measures
Suppose $X$ is a space equipped with a $\sigma$-algebra $\mathcal{M}_X$. Then the set of measures on $X$ is closed under addition and scalar multiplication by elements of ${\mathbb R}$. Formally ...
4
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232
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Recovering syzygies of zero-dimensional schemes from those of their general linear projections.
Assume that I have a reduced zero-dimensional scheme $Z \subset \mathbb{P}^3$, not contained in any hyperplane, of degree $mn$ and having the following property:
For a general outer projection $\pi ...
4
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156
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Characterizing non-singularity of varieties through properties of their derivations
I am interested in knowing about the possible implications between the following properties of a commutative, complex algebra:
Its spectrum is non-singular.
Its derivation module is projective and ...
4
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298
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Connecting group ring, abelianization
For reasons arising in algebraic topology, I'm wanting to better understand the relations between two functors from groups to abelian groups, $\mathbb{Z}[\cdot]$ and $\operatorname{ab}$; group ring ...
4
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794
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$Ext$ functor, filtered complexes and spectral sequences
Let $\mathcal{A}$ an abelian category. Take $M$ an object of $\mathcal{A}$, and $K_*$ a bounded complex in $\mathcal{A}$ equipped with a bounded increasing filtration $F$. By using homological and ...
4
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1k
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An example of a noetherian ring in which the integral closure of a finite extension of its field of fractions is not noetherian
Similar to another question I posted. Does anyone know of an example of a noetherian ring in which the integral closure of a finite extension of it's field of fractions is not noetherian.
4
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318
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criteria for reduced fibres
I was wondering if it is foolish to ask if there is a criteria on a morphism $f: X \to Y$ between separated schemes of finite type over a perfect field which will assure that all the scheme theoretic ...
4
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206
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Kahler differentials and the m-adic filtration
Let $A$ be a comm. local $k$-algebra with max. ideal $m$. Let $gr(-)$ be the associated graded algebra or module for the $m$-adic filtration. The canonical derivation $d:A \to \Omega^1_A$ satisfies $...
4
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1k
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Commutative ring Notes by M. Artin
In 1966, Professor Michael Artin gave a course for first-year graduate students at MIT on commutative algebra. In that course he covered many classical topics, (the Spectrum of a commutative ring, ...
4
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336
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What to call the following variant of tame ramification
Suppose that $R \subseteq S$ is a generically separable extension of 1-dimensional normal domains (you can assume that $R$ is local if you'd like) of equal-characteristic $p > 0$ (for simplicity, ...
4
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350
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Artin approximation theorem for analytic functions over a field of zero characteristic
Artin's approximation theorem states: "if a system of locally analytic equations in several complex variables has a formal solution then it has a locally analytic solution".
(Artin 1968, "On the ...
4
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232
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When is the ring of invariants of a finite group generated by symplectic reflections a complete intersection ring?
Let V be a finite dimensional symplectic vector space over $\mathbb{C}$. Let $G$ be a finite subgroup of the symplectic group $Sp(V),$ which is
generated by symplectic reflections, i.e. by elements $g\...
4
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178
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Global dimensions of orders over non-Gorenstein centers
This question concerns the following Lemma 4.2 in this paper by Van den Bergh:
Lemma: Let $R$ be local, normal Gorenstein ring of dimension $d$. Suppose $M$ is a reflexive $R$ module such that $A=\...
4
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420
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Primary decomposition for non-affine schemes
I will call a (nonzero) ring primary if every zero divisor is nilpotent. (This implies that the prime spectrum is irreducible, although the converse does not hold.) An irreducible scheme I will call ...
4
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692
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Artin Schreier Theorem for Rings
This has been in my mind for quite some time. Looking at Artin Schreier Theorem for fields:
If L is a field and K its algebraic closure and if 1< [K:L] < infinity then L=K[i] and L is a real ...