# Questions tagged [ac.commutative-algebra]

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

**1**

vote

**0**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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**

votes

**2**answers

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?

**0**

votes

**1**answer

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

**0**answers

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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 (...

**0**

votes

**0**answers

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**

votes

**1**answer

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

**1**answer

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

**0**answers

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}...

**0**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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?

**1**

vote

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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**

vote

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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**

votes

**0**answers

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**

vote

**1**answer

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

**1**answer

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 ...

**1**

vote

**0**answers

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

**0**answers

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

**0**answers

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**

votes

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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) ...