Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
1
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0answers
93 views
Behavior of regularity under base change
Let $(R,\mathfrak{m})$ be a noetherian local domain. Let $(A,\mathfrak{n})$ be a regular local ring contains $R$ such that $\mathfrak{n}\cap R=\mathfrak{m}$ and $A$ is essential finitely generated as ...
4
votes
2answers
177 views
Ideals invariant under ring automorphisms
I am looking for ideals $I\subset \mathbb{F}_2[x,y]$ with the following properties:
$I$ is generated by two homogeneous elements;
$I$ is invariant under the $SL_2(\mathbb{F}_2)$-action on $\mathbb{F}...
3
votes
1answer
116 views
On a special type of subring of $\mathbb C[x_0,…,x_{q-1}]$
Let $p,q$ be odd primes. Consider the polynomial ring $\mathbb C[x_0,...,x_{q-1}]$. For $m=0,1,...,p-1$, let
$$\sigma_m=\sum_{0\le j_0\le p;...;0\le j_{q-1}\le p; j_1+...+j_{q-1}=p; 1.j_1+...+(q-1)...
13
votes
2answers
438 views
Is a matrix similar to its transpose over $\mathbb{Z}_p$?
Is every $n \times n$ matrix with entries in $\mathbb{Z}_p$ (or even $\mathbb{Z}$) conjugate to its transpose via a matrix in $GL_n(\mathbb{Z}_p)$?
On the one hand, I know the analogous fact is false ...
6
votes
1answer
181 views
Is $\dim_k M/xM$ a multiple of $\dim_k R/xR$ for $M$ finitely generated, torsion-free $R$-module?
Let $R$ be a one-dimensional, reduced and noetherian $k$-algebra (we may also assume that $R$ is a finite $k[x]$-algebra). Let $M$ be a finitely generated, torsion-free module over $R$, i.e. no ...
11
votes
1answer
163 views
Does every section of the map Gal$(\overline{k(\!(t)\!)}/k(\!(t)\!))\rightarrow$ Gal$(\overline{k}/k)$ stabilize a compatible system of roots of $t$?
There may be some technical issues with the question, but hopefully what I mean is clear...
Let $k$ be a number field (or maybe any finitely generated field over $\mathbb{Q}$ of characteristic 0)
...
13
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0answers
240 views
Hensel lemma and rational points in complete noetherian local ring
Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal.
If we have several polynomials $f_i \in A[X_1, \dots, X_m]$ which have a common zero $x_n$ in $A/\mathfrak{m}^n$ ...
4
votes
1answer
167 views
Henselianizations over countable index sets
Let $A$ be a ring, $I\subset A$ a finitely generated ideal.
The henselianization $A^h$ of $A$ along $I$ is the universal $A$-algebra that is henselian along $I$ and can be presented as a direct limit ...
12
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2answers
799 views
Is a scheme Noetherian if its topological space and its stalks are?
Is a scheme being Noetherian equivalent to the underlying topological space being Noetherian and all its stalks being Noetherian?
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1answer
85 views
Relation between Hilbert function and complete intersection ideals
Consider $T=k[x_1,\ldots,x_n]$ ( $k$ alg. closed and of char $k=0$), and consider the ideal $$I=(x_1,x^{a_2}_2,\ldots,x^{a_n}_n)$$
with $2\leq a_2 \leq\ldots\leq a_n$. I want to prove that $$\sum_{i=...
0
votes
0answers
42 views
Restriction of fractional ideal sheaf to irreducible component is torsion-free
I translate the question into commutative algebra:
Let $R$ be a one-dimensional, reduced ring (which is also finite free over some PID since the considered curve corresponding lies finitely over the ...
3
votes
1answer
147 views
Is a factorial scheme with Noetherian stalks locally Noetherian?
Motivation: The Oka's coherence theorem tells us that the structure sheaf of a complex manifold is coherent. Taking into account the fact that coherence is a local property stable under finite direct ...
6
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0answers
135 views
Maximal subalgebras in polynomial ring $\mathbb{R}[x]$ over the field $\mathbb{R}$ of real numbers
Question. What are the maximal subalgebras of polynomial ring $\mathbb{R}[x]$ over the field of real numbers?
3
votes
1answer
49 views
example of a non-finitely generated co-Hopfian module over a commutative QF ring
Can anyone provide an example of such a module? or show that no such module exists? For semisimple rings, we have co-Hopfian if and only if finitely generated. Perhaps the fact that QF rings (...
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0answers
62 views
Localization of Hom module in an advanced setting (0-dualizing module)
Let $A,\,B$ be noetherian rings such that and let $M$ be an $A$-module. Let $g:A \to B$ be a ring homomorphism which makes $B$ into a finite free $A$-algebra. Now we can regard the $A$-module $\...
8
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1answer
262 views
Classify commutative rings $R$ such that $A \otimes_{\Bbb Z} B = A \otimes _{R} B$
I have asked this in MSE but there was no reply. Feel free to close if inappropriate.
Let $R$ be commutative ring, what can we say about the rings $R$ such that $A \otimes_{\Bbb Z} B \cong A \...
2
votes
1answer
92 views
How do you quickly determine which coefficients are greater than zero when multiplying two univariate positive polynomials?
Suppose that I have two polynomials with a degree of $n$, $A(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0$ and $B(x) = b_nx^n + b_{n-1}x^{n-1} + ... + b_0$ and the coefficients of these polynomials are ...
0
votes
0answers
94 views
is the homology of a free complex again free?
Let $k$ be a field and $R$ be a finitely generated graded $k$-algebra (e.g. a polynomial ring in some variables).
Let $M$ be a finitely generated graded vector space so the graded module $\widetilde{M}...
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0answers
82 views
A reference for studying special ring
A topological space $X$ is called profinite if it is compact, Hausdorff, and has a basis of open–closed sets. Also a commutative ring $R$ with 1 is called a topological ring it there is a topology on ...
3
votes
1answer
74 views
Projective dimension of graded modules
Short version:
Why is the projective dimension of a graded module the same as the projective dimension of its underlying ungraded module?
Longer version:
Let $G$ be a commutative group, let $R$ ...
2
votes
1answer
123 views
An example of a local ring which is not CM and a MCM module over it
I am looking for an example of a commutative noetherian local ring $(A,m)$,
and a maximal Cohen-Macaulay module $M$ over $A$ (in particular $M$ is finitely generated over $A$), such that for some $p \...
2
votes
1answer
148 views
$2$-adic valuation on $\mathbb Q (\sqrt{-15})$
I was reading the book "The structure of groups of prime power order". In the book (page 226, Example 10.1.18) it is proved that
$-15$ has a square root is $\mathbb Z_2$ where $\mathbb Z_2$ denotes ...
1
vote
1answer
97 views
Tensoring with complex of finite flat dimension in derived category
Let $(R,m)$ be a Noetherian local ring, and $X$, $Y$ be complexes of finitely generated $R$ modules. Suppose $X$ is bounded above and $Y$ is bounded below. Let $S$ be an $R$-algebra of finite flat ...
5
votes
1answer
411 views
Is there an adjoint to the inclusion of I-adically complete modules to all modules?
A module $M$ over a ring $R$ is $I$-adically complete with respect to the ideal $I$, if the canonical map $M \to \lim M/I^nM$ is an isomorphism. There exists a completion functor: $M \mapsto \lim M/I^...
2
votes
0answers
45 views
Efficient algorithm to prove that a polynomial ideal contains 1
I have the following problem:
Suppose to have an ideal $I\triangleleft k[x_1,...,x_n]$ defined by generators. There exists an efficient algorithm (perhaps more efficient than calculating the Groebner ...
7
votes
0answers
130 views
Self-flat modules
(This is inspired by this question and asked out of pure curiosity.)
Let $R$ be a commutative ring. Let $M$ and $N$ be $R$-modules. Then, $N$ is called $M$-flat if whenever $M'\rightarrow M$ is a ...
7
votes
1answer
312 views
Koszul-Tate Resolution for Subvarieties of $\mathbb P^n$
All varieties appearing below are assumed smooth projective over $\mathbb C$ and all vector bundles, sections etc are assumed to be algebraic/holomorphic. We use the word resolution to mean quasi-...
5
votes
1answer
149 views
Injective homomorphism of modules and tensor product
Let $R$ be a commutative ring; let $M$, $N$ be $R$-modules and let $f\colon M\to N$ be an injective homomorphism of $R$-modules. Is $f\otimes {\rm id}\colon M\otimes_RN\to N\otimes_RN$ injective?
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0answers
73 views
Generating the annihilator ideal up to finite index
Let $\Lambda_2 = \mathbb{Z}_p[[T_1,T_2]]$ be the power series ring over $\mathbb{Z}_p$ in two variables, i.e., $\Lambda_2$ is a regular local ring of dimension 3. Let $M$ be the quotient of an ...
5
votes
1answer
155 views
Relationship between Hilbert-Samuel multiplicity and polar multiplicity
Let $f \in \mathbb{C}[[x,y]]$ be the germ of an isolated plane curve singularity. Then the Hilbert-Samuel multiplicity $e_f$ of $f$ is given as follows:
$$e_f = \lim_{s \to \infty}\frac{1}{s} \cdot \...
1
vote
1answer
96 views
modules whose every submodule is a homomorphic image
Let $R$ be a commutative ring with unity. Let us say that an $R$-module $M$ satisfies property $\mathcal P$ if every submodule of $M$ is a homomorphic image of $M$.
Can we characterize all ...
3
votes
1answer
100 views
commutative ring satisfying descending chain condition on radical ideals
Let $R$ be a commutative ring with unity which satisfies d.c.c. on radical ideals. then does $R$ satisfy a.c.c. on radical ideals ? If this is not true in general, then what happens if we also assume $...
2
votes
1answer
158 views
Is there a “minimal” center of a blowup?
Let $X$ be a scheme, let $i : Z \to X$ be a closed subscheme, let $Y := \mathrm{Bl}_{Z}(X)$ be the blowup of $X$ at $Z$ with projection $\pi : Y \to X$. Suppose $U \supseteq X \setminus Z$ is an open ...
1
vote
2answers
206 views
The Jacobian Conjecture over a commutative $\mathbb{Q}$-algebra which is not an integral domain
Let $R$ be a commutative $\mathbb{Q}$-algebra which is not an integral domain,
for example: $R=\frac{\mathbb{Q}[t]}{(t^2-1)}$.
Let $k$ be an algebraically closed field of characteristic zero, and let ...
1
vote
0answers
50 views
Monoidal structure on left dg-modules over a brace algebra
Relating to my other question: Modules over Hopf Algebras and $E_2$-algebras
Preliminary: Let $A$ be an associative dg-algebra that is also an algebra over the brace operad. Let $M$ and $N$ be left ...
7
votes
1answer
172 views
The Locus of Complete Intersection Points
Let $X$ be an algebraic variety over an algebraically closed field. Consider the two subsets $X_0\subseteq X_1 \subseteq X$:
$$X_0 = \{a\in X| a \mbox{ is a scheme-theoretic complete intersection in }...
1
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0answers
44 views
Hilbert series of filtered algebras
Let $R=\mathbb{Q}[x_1,x_2,\ldots,x_n]$ and $I$ a non-homogeneous ideal of $R$. Then the algebra R/I is filtered. It has an associated graded algebra gr(R/I).
Let $I_1$ be the initial ideal of $I$, ...
0
votes
0answers
137 views
Example of variety which is not a complete intersection with respect to any projective embedding
Suppose $X$ is a smooth projective variety. Whether $X$ is a complete intersection or not when viewed as a subvariety of some projective space $\mathbb P^n$ is dependent on the specific choice of the ...
3
votes
1answer
149 views
Is the annihilator of a minimal prime ideal principal?
My setup is as follows: $X$ is a projective, reduced curve (which is not integral) with a finite morphism onto $\mathbb{P}_k^1$.
$\DeclareMathOperator{\Ann}{Ann}$
Let $R$ be a coordinate ring of $X$ ...
3
votes
2answers
363 views
Ring of invariants of some special type of subgroups of $GL_3(\mathbb C)$
For $\sigma \in \mathrm{GL}_n(\mathbb C)$ and $f(x_1,...,x_n)\in \mathbb C[x_1,...,x_n]$, let $f^ \sigma (x):=f(\sigma^{-1}x)$, for $x=(x_1,...,x_n)$.
For a subgroup $G$ of $\mathrm{GL}_n(\mathbb C)$...
12
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0answers
361 views
Emanuel Lasker, Max Noether, and Emmy Noether
In 1900, Emanuel Lasker (world chess champion from 1894 to 1921) received his Ph.D. under Max Noether. In 1905, Lasker published a theorem that Emmy Noether generalized in 1921, now well known as the ...
1
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1answer
118 views
How much can a map $R^n\to R^n$, $R$ a DVR, increase the valuation?
Let $R$ be a DVR, and $f:R^n\to R^n$ a map. Suppose $f(r_1,\dots,r_n)=(s_1,\dots,s_n)$, and write $d=\min(v(s_1),\dots,v(s_n))$, where $v$ is the valuation on $R$. Knowing $d$, what is the best bound ...
9
votes
1answer
304 views
Noetherian spectral space comes from noetherian ring?
Let $X$ be a spectral space (en.wikipedia.org/wiki/Spectral_space), i.e. a space of the form $\textrm{Spec}(A)$ for some commutative ring $A$. If $X$ is noetherian, does there also exist a noetherian ...
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0answers
146 views
Skew-symmetric multi-derivations of $k[x_1,…,x_n]/I$
Let $I = \langle f_1, \ldots f_r \rangle$ be an ideal in $R=k[x_1,\ldots,x_n]$ where $k$ is a field, and put $A = R/I$.
(If $I$ is prime then $A$ is the coordinate ring of an irreducible affine ...
3
votes
0answers
220 views
Coordinate ring of complete intersection Calabi Yau (CICY)
EDIT: If the question is for SE level just delete from here as it is also posted there. In fact I have seen some questions in SE regarding the coordinate rings of product of projective varieties but ...
2
votes
0answers
116 views
tangent space to a (not necessarily algebraic/Lie/..) group
Are there some standard ways to define the tangent space to a group $G$ at its unit element, $e$, when the group is not (pro)algebraic/(pro)Lie, not necessarily over a field, does not have the ``...
0
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0answers
73 views
Squarefree monomial ideals, locally complete intersections
Let $I\subset S=\mathbb{C}[x_0,\dots,x_n]$ be an ideal generated by squarefree monomials of the same degree. Is it true that $S/I$ is
a locally complete intersection?
4
votes
0answers
68 views
Support of Matlis dual
Let $(A,m)$ be a commutative noetherian local ring,
$E$ the injective hull of $A/m$, and $M$ a finitely generated $A$-module.
What is the connection between the support of $M$ and the support of the ...
9
votes
2answers
268 views
Divided power algebra is artinian as a module over the polynomial ring
I already asked this on math.stackexchange.com, but did not receive much responses. I hope this is also appropriate for mathoverflow.
In the paper Homological algebra on a complete intersection, with ...
5
votes
1answer
200 views
$f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$?
Is it possible to find $f,g \in \mathbb{Z}[x,y]$ (with $\deg(f),\deg(g) \geq 1$) such that the following two conditions are satisfied:
(1) $\operatorname{Jac}(f,g)=f_xg_y-f_yg_x = 0$.
(2) ...
