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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6
votes
1answer
195 views

Existence of isomorphism mod every power of the maximal ideal

This problem is a continuation of 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. Assume $...
4
votes
0answers
129 views

Explicit description of injective hulls

Let $k$ be a field, let $R:=k[x_1,\ldots,x_n]$, and consider the $R$-module $M:=R/{(x_1,\ldots,x_n)}\cong k$. Then the injective hull $I_M$ of $M$ admits the following explicit description: $$ I_M = k[...
5
votes
0answers
421 views

Completion of a local ring of an arithmetic surface

Vaguely, an arithmetic surface is an algebraic curve with coefficients in some "ring of integers". More formally it is an integral normal scheme $\mathfrak{X}$ of dimension 2 of finite type over a ...
3
votes
1answer
93 views

Summing complete intersections

Suppose we have polynomials $f_1,\dots,f_r\in k[X_1,\dots,X_N]$ defining a complete intersection in $\mathbb{A}^N$. I suspect that it is then true that $f_1(X)+f_1(Y),\dots,f_r(X)+f_r(Y)\in k[X_1,\...
1
vote
0answers
160 views

Rank 2 vector bundle with trivial first chern class is self-dual

I saw a statement used in the paper that E of rank 2 with $c_1(E) = 0$ is self-dual. I was wondering, how does one prove this statement? If it makes a difference, let the underlying variety be ...
1
vote
0answers
55 views

What's the probability a random module element has prime annihilator?

I'm going to pose two versions of my question---an ill-defined version and a well-defined version. Ill-Defined Question (IDQ). Let $R$ be a (commutative, unital) Noetherian ring, $M$ a finitely ...
11
votes
3answers
595 views

Are archimedean subextensions of ordered fields dense?

Let $E$ be an ordered field and let $F$ be a real closed subfield. We say that $E$ is $F$-archimedean if for each $e\in E$ there is $x\in F$ such that $-x\le e\le x$. Is it true that if $E$ is $F$-...
1
vote
0answers
124 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 ...
6
votes
2answers
240 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}...
2
votes
1answer
157 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)...
14
votes
2answers
567 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 ...
8
votes
1answer
224 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
188 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
0answers
314 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
190 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 ...
11
votes
2answers
1k 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
1answer
96 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=...
3
votes
1answer
173 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
0answers
168 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
73 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 (...
10
votes
1answer
339 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
99 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 ...
3
votes
1answer
114 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
137 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
211 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
102 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
505 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
202 views

Algebraically independent vectors in tensor product

$\mathcal L$ and $\mathcal L'$ be full rank lattices in $\mathbb R^n$ with shortest vectors $v_1,\dots,v_n$ and $v_1',\dots,v_n'$ respectively where $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$ $$\|v_1'\|_2\...
2
votes
0answers
51 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
139 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
432 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
164 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
0answers
90 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 ...
7
votes
1answer
231 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
112 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
135 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
174 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
240 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
59 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 ...
9
votes
1answer
204 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
0answers
47 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
195 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
208 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
372 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
0answers
391 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
1answer
125 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 ...
10
votes
1answer
330 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
0answers
167 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
242 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
126 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 ``...