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Questions tagged [ac.commutative-algebra]

Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

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1 vote
1 answer
435 views

Is a ideal of direct limit of rings is itself a direct limit of ideal?

Suppose R is any commutative noetherian ring with 1 and R is a direct limit of rings rings $R_{i}$. Now suppose $I$ is any ideal in $R$. Does there direct system of ideals $ \lbrace I_{i} \rbrace$ of $...
1 vote
0 answers
89 views

Commutative square of module of differential is cartesian?

Is it true that the following square is Cartesian? $\require{AMScd}$ \begin{CD} R @>{d}>> \Omega^{1}_{R} \\ @VVV @VVV\\ \widehat{R} @>{ \widehat{d}}>> \Omega^{1}_{\widehat{R}} \end{...
4 votes
1 answer
195 views

Weighted blowup of a Cohen-Macaulay ring along a regular sequence is Cohen-Macaulay?

Let $(R, \mathfrak{m})$ be a Cohen-Macaulay ring, let $f_1, \dotsc, f_d \in \mathfrak{m}$ be a regular sequence, and let $n_1, \dotsc, n_d > 0$ be weights (feel free to assume that $n_1 = 1$ if it ...
5 votes
0 answers
493 views

ramification locus for finite morphism and Abhyankar's Lemma

I want to ask given a finite morphism between projective varieties $f:X\rightarrow Y$. What is exactly the ramification locus $\Delta(X/Y)$. If $X$, $Y$, $f$ are smooth, then I can more or less ...
3 votes
0 answers
991 views

Definition of Q gorenstein variety

I have a question about the definition of Q-Gorenstein variety. I saw a definition of Q-Gorenstein variety:for a normal variety $X$, it's Q-Gorenstein if the canonical divisor is Q Cartier. I wonder ...
7 votes
1 answer
748 views

Factorization in formal power series versus in convergent power series over the complexes

Let $R=\mathbb C\{x_1,...,x_n\}\subset S=\mathbb C [[x_1,...,x_n]]$ denote the ring of convergent, respectively formal, power series over $\mathbb C$. Suppose $f\in R$ is irreducible in $R$. Does it ...
15 votes
3 answers
567 views

Direct sum of Hopf algebras

I realise that this question might be rather basic but however I was unable to find the answer in any textbook nor manage to figure out the answer. The question is the following: given two Hopf ...
5 votes
1 answer
411 views

Trace ideal of a projective module

In his 1969 paper "On projective modules of finite rank", Wolmer Vasconcelos writes Let $M$ be a projective $R$-module... The trace of $M$ is defined to be the image of the map $M \otimes_R ...
18 votes
5 answers
2k views

Is a complete homogeneous symmetric polynomial irreducible?

Let $S=\mathbb{C}[x_1,x_2,\dots,x_n]$ be a polynomial ring. Let $n \geq 3$. Let $h_a$ denotes the complete homogeneous symmetric polynomial of degree $a$. $$ h_a=\text{ sum of all monomials of degree }...
5 votes
1 answer
248 views

Completion w.r.t. ideal generated by a part of regular system of parameter

Let $R$ be a $d$-dimensional Noetherian regular local $k$-algbera ($k$ any field of char($k$) = 0, $d \geq$ 2). Let $x, y$ be a part of regular system of parameter for $R$. Let $I = (x, y)$ be a ideal ...
2 votes
0 answers
288 views

Torsion freeness of direct image of structure sheaf?

I have the following question. Let $f:X\rightarrow Y$ be a surjective projective morphism between smooth projective varieties. I learned that if $\dim Y=1$, then $R^if_*\mathcal O_X$ is torsion free $\...
8 votes
2 answers
2k views

Original proof of Hilbert's syzygy theorem

Does anyone know an English reference for the original proof of Hilbert's syzygy theorem? The three proofs that I know use either: the theory of projective dimension and change of rings (plus a step ...
13 votes
0 answers
749 views

Rings whose Frobenius is flat

Let $R$ be a ring of characteristic $p>0$. The (absolute) Frobenius is the map of rings $F_R:R\rightarrow R$ defined by $x\mapsto x^p$. I am interested in rings for which $F_R$ is flat (hence ...
1 vote
1 answer
100 views

Big polynomial subalgebra of polynomials

Consider some algebraically independent polynomials $f_1,\ldots, f_n\in\mathbb{C}[x_1,\ldots, x_n]$. Is it possible that $I\subseteq\mathbb{C}[f_1,\ldots, f_n]\subsetneq\mathbb{C}[x_1,\ldots, x_n]$ ...
15 votes
1 answer
1k views

Countable Hom/Ext implies finitely generated

Today I learned this interesting fact from Jerry Kaminker: If $A$ is an abelian group such that $\mathrm{Hom}(A,\mathbb{Z})$ and $\mathrm{Ext}(A,\mathbb{Z})$ are both countably generated, then in fact ...
3 votes
0 answers
207 views

Question about derivations of $R[X]$

$\DeclareMathOperator\Der{Der}$Consider the polynomial algebra $R[X] = R[x_1,\ldots, x_n]$ over the ring $R = \mathbb{C}[t]$ and let $\Der_{R}(R[X])$ be the Lie algebra of derivations of $R$-algebra $...
2 votes
1 answer
246 views

Proving that finite, connected group schemes in characteristic 0 are trivial

I have a question about a specific proof that all finite group schemes in characteristic 0 are etale. The proof is here, Proposition 8 in lecture notes by Andrew Snowden. In his notation, let $A = k\...
5 votes
2 answers
277 views

Linkage and Cohen-Macaulay-ness

Suppose I have a reduced l.c.i. scheme with two irreducible components: $X = Y \cup Z$. I want to say that if $Y$ is Cohen-Macaulay then $Z$ is as well. I think this follows from Eisenbund Theorem 21....
6 votes
0 answers
159 views

Ring with different graded and ungraded global dimensions

Let $A$ be a $\mathbb N$-graded ring. One can consider the two categories $M_A^g$ and $M_A^u$ of graded and ungraded modules over $A$. Both have, say, enough projectives, hence one can compute various ...
0 votes
1 answer
193 views

Is it true that $g-t$ is divisible by $f$?

Assume $f\in k[x_1,\ldots, x_n]$ is irreducible. Let for $g\in k[x_1,\ldots, x_n]$, $\partial(g)$ is divisible by $f$ for each derivation $\partial$ with $f\in\ker\partial$. Is it true that $g-t$ is ...
2 votes
0 answers
174 views

When is there a path between two minimal prime ideals?

Recall a path from a point $x$ to a point $y$ in a topological space $X$ is a continuous function $f$ from the unit interval $[0,1]$ to $X$ with $f(0) = x$ and $f(1) = y$. Now let $R$ be a ...
14 votes
4 answers
6k views

When is an algebra of commuting matrices (contained in one) generated by a single matrix?

Let C be an nxn matrix, then the polynomials in C (with appropriate coefficients) form an algebra of commuting matrices. I feel that I should know if the converse is true but I do not. So my first ...
4 votes
0 answers
165 views

About Homotopy Transfer Lemma

If M, A are two differential graded complexes over a commutative ring R with the following data, $$(M,d_M) \overset{\Delta}{\longrightarrow} (A,d_A)$$ $$(A,d_A) \overset{f}{\longrightarrow} (M,d_M)$$ ...
4 votes
0 answers
178 views

Question about basis of $\text{Der}_{k}(k[X])$

Let $k[X] = k[x_1,\ldots, x_n]$ be the polynomial ring over a field of characteristic zero. Assume that $(D_1,\ldots, D_n)$ is a $k[X]$-basis of $\text{Der}_k(k[X])$. Suppose that the vector space $\...
0 votes
1 answer
114 views

On a sum of squares representation

We know $p a^2+q b^2+r ab$ can be represented as square (trivially) when $$p,q\geq0$$ $$r^2=|4pq|$$ holds and as a sum of squares (again trivially) of form $(m a+n b)^2$ under readily explainable ...
7 votes
0 answers
273 views

Is there a Swan-style description of topological K-homology?

A celebrated result of Swan [1] states that, on a compact Hausdorff space $X$, the category of finite rank complex vector bundles is equivalent to the category of finitely generated projective $\...
4 votes
0 answers
101 views

Associativity equation for topological rings and logarithms

Let $R$ be a topological ring of characteristic zero. Assume that $R$ is commutative. Let $G :R \times R \to R$ be a continuous function satisfying $G(G(x,y),z)=G(x,G(y,z))$ and $G(0,x)=x$, for all $x,...
1 vote
0 answers
90 views

Mod $N^2$ evaluation of a polynomial defined by first $N-1$ roots

Given a prime $N$ and integer $g$, where $g$ is able to generate the multiplicative subgroup $(\mathbb{Z}/N^2\mathbb{Z})^*$, I am interested in any results simplifying or evaluating $f\in (\mathbb{Z}/...
1 vote
0 answers
132 views

On the dual version of an isomorphism of spectral sequence term (from Cartan and Eilenberg)

I'm trying to take spectral sequences as a black box for application in commutative algebra and I admit that I haven't really gone through (or understand) all the proofs of all the isomorphisms ...
8 votes
0 answers
265 views

Chevalley restriction theorem: group vs lie algebra version

Let $G$ be a (split) reductive group over $k$, $T$ a split maximal torus, and W its Weyl group. I sometimes see the Chevalley restriction theorem stated as (1) $k[G]^G \xrightarrow{\sim} k[T]^W$ and ...
6 votes
1 answer
563 views

When annihilator of ideal and ideal is co maximal

Let $R$ be a commutative ring with identity. It is not always true that ideal $J$ and annihilator of ideal $ann(J)$ are co maximal (ex: integral domain) Is there a sufficient (necessary) condition( ...
6 votes
0 answers
220 views

Is $\mathrm{End}-\{0\}=\mathrm{Aut}$ for derivation Lie algebra?

Is it true that every nonzero endomorphism of Lie $\mathbb{C}$-algebra $\mathbb{C}[x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[x_1,\ldots, x_n]\partial_{x_n}$ is an automorphism? As I ...
1 vote
1 answer
759 views

Does every prime power generate a primary ideal?

Let $R$ be a commutative ring with identity and let $p\in R$ be a prime element (i. e. $(p)$ is a prime ideal). If $R$ is an integral domain, it can be shown that $(p^k)$ is a primary ideal for every $...
3 votes
0 answers
180 views

Units in group rings

Let $F$ be any field with $p$ elements and $G$ be any finite $p$-group, combining together they form a group ring $FG$. And $V(FG)$ denotes group of units of coefficient-sum equal to 1 in $FG$. We ...
5 votes
0 answers
256 views

Open covers by ind-affine ind-schemes

Many apologies if this is totally standard! I couldn't find it in the literature. Background definitions: A presheaf $X: \textbf{Aff}^\text{op} \to \textbf{Set}$ is an ind-scheme if it is a filtered ...
4 votes
1 answer
229 views

When is the intersection of two determinantal ideals equal to the product?

Let $S = k[x_{i,j}\mid 1\leq i\leq n, 1\leq j\leq m]$ be a polynomial ring over an arbitrary field $k$. Let $M$ be a generic $n\times m$ matrix of indeterminates in the ring $S$ where $n\leq m$. For ...
5 votes
0 answers
132 views

On a reference for computing global spectrum of $A_n$-curve singularities, by H.Dao and E.Faber

This question is about chasing down a reference in a paper relating to non-commutative crepant resolutions and Cohen-Macaulay representation theory. Allow me to first give a minor introduction. Let $(...
39 votes
2 answers
6k views

What is Serre's condition (S_n) for sheaves?

The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...
4 votes
0 answers
135 views

$K$-group of category of bounded chain complexes of Projective modules with finite length homologies

For a Commutative Noetherian local ring $(R, \mathfrak m)$, let $K_0^{\mathfrak m}(R)$ denote the abelian Group generated by isomorphism classes of bounded chain complexes of finitely generated free ...
3 votes
1 answer
2k views

Prime ideals of formal power series ring that are above the same prime ideal

Let $R$ denote a commutative ring with identity and let $R[[X]]$ denote the ring of formal power series over $R$ in an indeterminate $X$. If $I$ is an ideal of $R$, then $I[[X]]$, the set of power ...
2 votes
1 answer
158 views

Does $\mathcal{A}\otimes\mathbb{C}(t)\cong\mathcal{D}\otimes\mathbb{C}(t)$ imply an isomorphism of Lie algebras?

Assume that $\mathcal{A}$ is a Lie $\mathbb{C}$-algebra. Also denote the Lie $\mathbb{C}$-algebra $\mathcal{D}=\mathbb{C}[x_1,\ldots, x_n]\partial_{x_1}\oplus\ldots\oplus\mathbb{C}[x_1,\ldots, x_n]\...
7 votes
2 answers
822 views

Determining the kernel of the localization map when defining the localization by generators and relations à la Serre

All rings considered will be commutative and unitary. Let $A$ be a ring, $S \subseteq A$ a multiplicatively closed subset. The localization $\lambda_S : A \longrightarrow A[S^{-1}]$ can be ...
9 votes
1 answer
714 views

Curious anti-commutative ring

Has anyone seen the ring $\Lambda[x_0, x_1, x_2, \ldots]/(x_i x_j - (i+1) x_0 x_{i+j})$ in some natural context? Here $\Lambda[x_0, x_1, x_2, \ldots]$ is the (graded-)commutative algebra (either over ...
5 votes
1 answer
218 views

Closure of the product of subfunctors

Background: Let $X: \textbf{CRing} \to \textbf{Set}$ be a presheaf on the category of affine schemes and $Z \subseteq X$ a subfunctor. One defines $Z$ to be closed if for every ring $A$ and every ...
5 votes
0 answers
98 views

Rational functions with trivial Weil symbols at every point

Let $f, g$ be a pair of nonzero rational functions in $\mathbb{C}(t).$ For $\lambda\in \mathbb{C}$ let $a$ be multiplicity of $g(t)$ at $\lambda$ and $b$ - multiplicity of $f(t)$ at $\lambda.$ Weil ...
1 vote
1 answer
131 views

Question about Jacobian subalgebra

Assume that the algebraically independent polynomials $f, g\in\mathbb{C}[x, y]$ are such that the Jacobian matrix $\text{Jac}_{x, y}^{f, g}\in\mathbb{C}\setminus\{0\}$. Is it true that $\mathbb{C}[x, ...
3 votes
0 answers
285 views

Infinite sum in power series ring

Let $R$ be a commutative ring with $1$, $R[[x]]$ be the power series ring over $R$ and $A$ be an (prime) ideal of $R[[x]]$ with $x\not\in A$ and $\{f_i\}_{i=1}^\infty$ be a sequence of element of $A$. ...
132 votes
3 answers
21k views

When is the tensor product of two fields a field?

Consider two extension fields $K/k, L/k$ of a field $k$. A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. The answer is "no" and this answer is often ...
1 vote
1 answer
62 views

Continuations of derivations of Jacobian subring

Assume that the algebraically independent polynomials $f_1,\ldots, f_n\in\mathbb{C}[x_1,\ldots, x_n]$ are such that the Jacobian matrix $\text{Jac}_{x_1,\ldots, x_n}^{f_1,\ldots, f_n}\in\mathbb{C}\...
5 votes
0 answers
166 views

When do the spectra of overrings glue to a proper morphism?

This question is motivated by the construction of blowups. Let $A \subset K$ be a commutative domain and its fraction field, and let $\{A_i\}$ be some finite collection of overrings in between. Let $X ...

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