Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
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 ...