Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,316
questions
6
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1
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Morita equivalence and isomorphisms in cohomology theories
Let $A,B$ be two unital algebras. We say that $A,B$ are Morita equivalent if there are $A-B$ and $B-A$ bimodules $P,Q$ such that
$$P \otimes_{B} Q \cong A, Q \otimes_A P \cong B$$
(as $A-A$ and $B-B$ ...
12
votes
1
answer
720
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Determinant is to Pfaffian as resultant is to what?
This is an irresponsible question: I do not have done any thinking on it, or even literature search.
I just became curious whether there is some modification of the notion of a common root of two ...
2
votes
0
answers
74
views
Analytic spread of an ideal after reduction
Let $(R,m)$ be a local ring and $I$ an ideal in $R.$ Let $l(I):=\dim \bigoplus_{n\geq 0}(I^n/mI^n)$ and $x\in R\setminus I.$
My question is
what is the relation between $l(I)$ anf $l(I+(x)/(x))?$
3
votes
0
answers
349
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Transverse intersection of two divisors
Let $X$ be a smooth variety and $D_1$ and $D_2$ are two smooth divisors which intersect transversely. Assume also that $D_1\cap D_2$ is irreducible. Is it true that $\mathcal{O}(-D_1)|_{D_2}\cong \...
4
votes
0
answers
142
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Deminormal and Gorenstein
Let X be an irreducible deminormal variety such that the normalisation is Gorenstein. Does it follow that X is also Gorenstein?
for deminormal definition, see https://arxiv.org/pdf/1506.02002.pdf
3
votes
1
answer
525
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On integral domains over which special kind of modules are projective
For an integral domain $R$ let $\mathrm{Frac}(R)$ denote its field of fractions. Then $R$ is embedded in $\mathrm{Frac}(R)$ and we can consider $\mathrm{Frac}(R)$ as an $R$-module.
Can we ...
3
votes
1
answer
178
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Definability of nilradical in the model theory of rings
I am looking for a reference dealing with the first-order definability of the nilradical of a commutative ring. The only thing I have found so far is an exercise in Wilfrid Hodges' book Model Theory (...
0
votes
0
answers
100
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Solutions of the linear equation from K[[X_1,X_2,X_3]] to K[[X_1,X_2]]
Let $A_3 := K[[X_1,X_2,X_3]]$ be a three-variable formal power series ring over a field $K$. We consider a linear equation
$(\sharp) \phantom{aa} a_1(X_1,X_2,X_3)Y_1 + \ldots + a_n(X_1,X_2,X_3)Y_n = ...
2
votes
1
answer
257
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Union of Cohen-Macauley components
Let $X=X_1\cup X_2\cup X_3$ be a variety with three irreducible equidimensional components which are normal and Cohen-Macauley. Suppose $X_1$ and $X_3$ intersects trasversally and $X_2$ and $X_3$ ...
4
votes
0
answers
474
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Endomorphisms of free modules and extension of scalars
I asked this question on Mathematics Stackexchange, but got no answer.
Let $B$ be a commutative ring with $1$, let $A$ be a subring such that any unit of $B$ which belongs to $A$ is a unit of $A$, ...
5
votes
2
answers
170
views
T-nilpotency and quasinilpotency of ideals
Let $R$ be a commutative ring and let $\mathfrak{a}\subseteq R$ be an ideal. The ideal $\mathfrak{a}$ is called T-nilpotent if for every sequence $(r_i)_{i\in\mathbb{N}}$ in $\mathfrak{a}$ there ...
1
vote
1
answer
383
views
Equal degree factoring of homogeneous polynomials over $\Bbb Q[x_1,\dots,x_n]$?
Given $f(x_1,\dots,x_n)\in\Bbb Q[x_1,\dots,x_n]$ of form $\prod_{i=1}^df_i(x_1,\dots,x_n)$ where each of $f,f_i$ are homogeneous and each $f_i$ are irreducible and of equal degree what is the best ...
8
votes
1
answer
316
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Which ideals have standard Hilbert series?
Let $m$ and $d$ be two positive integers.
Consider the polynomial ring $R = \mathbb{C}[x_1 , \dots , x_m]$. Let $I$ be an ideal of $R$ generated by a finite family of polynomials of degree $d$, and ...
2
votes
0
answers
135
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Ext over a certain commutative algebra
Let $A$ be the algebra $K[x,y]/(x^2,y^2,xy,yx)$. Then $A$ is a 3-dimensional commutative algebra and $Ext^i(M,A) \neq 0$ for any indecomposable non-projective module $M$ for some $i>0$. Namely: $...
4
votes
1
answer
568
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Dualizing complex definition ubiquity
The following definition is the one, I found here http://stacks.math.columbia.edu/tag/0A7A. But let me recall it (out of consistency's sake):
Definition
For $A$ a Noetherian ring, a dualizing complex ...
1
vote
0
answers
57
views
Finite presentation for left-exact endofunctors
For a commutative unital ring R and a left-exact endofunctor L on the category of R-modules, what is, or should be, the definition of finite presentation for L ? Possibilities:
L( R ) is ...
1
vote
1
answer
342
views
singular locus of semi-normal variety
Let X be a seminormal variety and S be the singular locus of X. Is Blow$_S X=$ the normalisation of X?
Is the singular locus given by the conductor ideal?
11
votes
1
answer
846
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Detailed modern references for basic properties of Pfaffians over commutative rings
Pfaffians are important to algebraic combinatorics, at least.
This is to propose the making of a 'wiki' list, more modern, precise and compressed than e.g. the relevant Wikipedia page (nothing against ...
1
vote
0
answers
126
views
Nielsen--Schreier for fields
Is it true that a subextension of a purely transcendental extension is itself purely transcendental?
In symbols, suppose we have field extensions $M/L/K$, with $M/K$ purely transcendental. Must $L/K$ ...
3
votes
1
answer
2k
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Tensor product of field extensions
Let $K$ be a field of characteristic 0 and $L$ a finite extension of $K$. Denote by $m$ the natural multiplication map from $L \otimes_K L$ to $L$. Denote by $I$ the kernel of the morphism $m$. Is $I$ ...
2
votes
0
answers
307
views
Endomorphism algebra of a coherent sheaf is locally free
What is an example of a Noetherian ring $A$ and a finitely generated $A$-module $M$ such that the endomorphism algebra $\mathrm{Hom}_{A}(M,M)$ is flat as an $A$-module but $M$ is not flat?
2
votes
1
answer
54
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A question on isomorphism between factor modules over commutative semi-simple ring
Let $R$ be a commutative semi-simple ring with unity (i.e. all modules over $R$ is semi-simple) , let $P \le N \le M$ be a chain of $R$ modules such that $M \cong M/N$ ; then is it true that $M \cong ...
1
vote
0
answers
234
views
Quotient of polynomial ring over a Dedekind domain
Let $A$ be a Dedekind domain and let $B:=A[X]/(f(X))$, where $f(X)\in A[X]$ is some monic polynomial such that $B$ is a domain. If I take the canonical map $A\longrightarrow B$, then it nduces a ...
9
votes
0
answers
165
views
When is the rank 2 free metabelian group of exponent $n$ center free?
Let $M_n$ be the rank 2 free metabelian group of exponent $n$. For which $n$ is $M_n$ center-free?
The abelianization $M_n^{ab}\cong C_n\times C_n$, so the commutator subgroup $M_n'$ is a cyclic $(\...
4
votes
0
answers
104
views
A Euclidean domain in which radix expansion is not possible
It is well known that given two nonconstant polynomials $f,g\in F[x]$ where $F$ is a field, there are unique polynomials $r_0,\dots ,r_n$ such that
$$f=r_n g^n +\dots+r_1 g +r_0,$$
where $\deg r_i &...
3
votes
1
answer
198
views
Torsion submodules of non-noetherian modules
Let $R$ be a commutative ring, let $\mathfrak{a}\subseteq R$ be an ideal, and let $M$ be an $R$-module. The $\mathfrak{a}$-torsion submodule of $M$ is defined as $$\Gamma_{\mathfrak{a}}(M)=\{x\in M\...
1
vote
1
answer
774
views
On the set of non-zero elements in an integral domain whose generating principal ideal is of a special kind
Let $R$ be an integral domain. Consider the set $$S := \big\{a \in R\smallsetminus \{0\}
: Ra+Rx \text{ is a principal ideal } \forall x \in R \big \}.$$ Is $S$ a saturated multiplicative closed ...
1
vote
1
answer
100
views
Functions on rings and polynomials with coefficients in a certain kind of localisation
Let $R$ be a commutative ring with unity and let $S$ be a multiplicatively closed subset of $R$ such that $S$ contains no zero divisor . So the canonical map $f : R \to S^{-1}R$ is invective , hence w....
1
vote
0
answers
182
views
generators for a graded algebra
I have a graded commutative algebra over a field $K$ and I also know that it is not generated by the degree 1 elements. Somehow I could guess a set of generators, is there a criterion to check that ...
7
votes
0
answers
211
views
Invariant theory over rings
Apologies if this is a silly question, but I have had cause to briefly introduce myself to invariant theory. I have noticed that authors primarily work over (algebraically closed) fields. I was ...
2
votes
1
answer
626
views
Gluing Schemes along subschemes
Let X be the union of two planes in $\mathbb{A}^4$ touching at origin. Blow up X at the origin. Call it $\overline{X}$. It has two disjoint copies of $\mathbb{A}^2$ blown up at the origins. Their ...
2
votes
0
answers
97
views
If $H$ is an atomic, unit-cancellative monoid such that the set of atoms of $H$ is finite up to associates, then $H$ is BF
In a previous version of this post, $H$ was an atomic commutative monoid such that the quotient $H/H^\times$ is finitely generated, and I was asking if such conditions were enough for $H$ to be BF. ...
2
votes
0
answers
60
views
Different term's contribution in zonal polynomial
I am very interested in the contribution of different terms in a zonal polynomial. Let's focus on the simplest case. In (20) in "Distributions of matrix variates and latent roots derived from normal ...
2
votes
1
answer
194
views
Why is $C^\infty(M)$-module homomorphism $P\mapsto\Gamma(P)$ surjective?
$\DeclareMathOperator{\Id}{Id}
\require{cancel}$
Jet Nestruev's "Smooth Manifolds and Observables" contains following exercise:
Exercise. Show that $P$ is geometric if and only if the two modules $...
5
votes
0
answers
762
views
Picard group of non reduced scheme
Let X be a non reduced scheme. $X_{red}$ be the reduced scheme. Is it true that Picard group of $X_{red}$=Picard group of X?
Is this map surjective $Pic (X)\rightarrow Pic(X_{red})$?
Is there a ...
1
vote
0
answers
272
views
Is it true that the functor of completion of a module over a local ring is injective on isomorphism classes?
Let $A$ be a commutative Noetherian local ring and $\hat A$ be its completion. Then we have the functor of completion from the category of finitely generated $A$-modules to the category of finitely ...
1
vote
2
answers
327
views
Finding all submodules
Given a finite dimensional local commutative algebra over a finite field $K$ and a finite dimensional module $M$. What is the fastest/best way to obtain all submodule from $M$ using a Computer algebra ...
1
vote
0
answers
514
views
An infinitely-many-variable formal power series ring ${\Bbb F}_p[[X_1,\ldots]]$
We shall define a infinitely-many-variable formal power series ring ${\Bbb F}_p[[X_1,\ldots]]$ as follows$\colon$
${\Bbb F}_p[[X_1,\ldots]]\colon= \underset{n \geq 1}{\varprojlim}\, {\Bbb F}_p[[X_1,\...
15
votes
1
answer
1k
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Applications of cluster algebras
Why are so many algebraists nowadays interested in cluster algebras?
(This is a rewording of one half of the closed question Cluster algebras and teichmuller theory.)
4
votes
1
answer
323
views
An analogue of rational functions for Hahn series
For any field $k$, we have both the field $k(t)$ of rational functions (formal quotients of polynomials, i.e. the field of fractions of $k[t]$) and the field $k((t))$ of formal Laurent series (which ...
10
votes
0
answers
836
views
Scholze's infinite to finite type ring theory reductions?
In following essay "The Perfectoid Concept: Test Case for an Absent Theory" by Michael Harris, there is the following sentence I found to be quite striking.
The most virtuosic pages in Scholze's ...
10
votes
2
answers
1k
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Is it possible to describe the image of the $p$-adic logarithm on $1+\mathfrak{m}$, where $\mathfrak{m}$ is the maximal ideal of a $p$-adic field?
Let $R$ be the ring of integers of some finite extension of $\mathbb{Q}_p$. In particular, I'm interested in the case $R = \mathbb{Z}_p[\zeta_{p^k}]$ (the totally ramified extension of $\mathbb{Z}_p$ ...
7
votes
1
answer
274
views
Uniquely ordered commutative rings
I am wondering whether there are reasonable necessary and/or sufficient conditions to dedice whether a commutative ring can be uniquely ordered (like for instance $\mathbb{Z}$) or not. In the field ...
1
vote
0
answers
36
views
Quadratic suborders of an imprimitive quartic order
Let $Q$ be an irreducible quartic order; that is, $Q$ is a subring of the ring of integers $\mathcal{O}_K$ in a quartic extension $K$ over $\mathbb{Q}$ such that the fraction field of $Q$ is equal to $...
4
votes
1
answer
308
views
Understanding full set of sections as in Katz-Mazur
I was reading this question, specifically Brian's answer. In particular I am having a bit of trouble digesting the following sentence:
Being a "full set of sections" of $Z/S$ is something which is ...
1
vote
1
answer
242
views
minimal number of generators of a k-algebra over commutative rings
I have that $R$ is the $k$-algebra ($k$ is a field) finitely generated by $S=\{f_1, ..., f_m \}\subset k[x_1, \cdots, x_n]$ and this set of polynomials is minimal with respect to inclusion (i.e., e ...
28
votes
4
answers
4k
views
What are traces?
Let $A$ be a Noetherian commutative ring and Let $A\rightarrow B$ be a finite flat homomorphism of rings. We can thus form the so called "trace" $\mathrm{Tr_{B/A}}:B\rightarrow A$, which is a ...
1
vote
1
answer
233
views
Localization of a maximal Cohen-Macaulay module
Let $(R,m)$ be a Cohen-Macaulay local ring of dimension $d\geq 2$ and $M$ an module with depth$M=d.$
Is there any example of $M$ such that
$(1)$ $M_p$ is not free for some $p\in Ass(R)$ and
$(2)$...
9
votes
2
answers
210
views
Which elements of $1+(x_1,x_2)\subset\mathbb{Z}_p[[x_1,x_2]]^\times$ are in $\langle 1+x_1,1+x_2\rangle$?
It's a classical fact that the commutative power series ring $\mathbb{Z}_p[[x_1,x_2]]$ is isomorphic to the completed group algebra $\mathbb{Z}_p[[\mathbb{Z}_p\times\mathbb{Z}_p]]$, the isomorphism ...
-1
votes
1
answer
434
views
tangent bundle of $\mathbb{P}^1$
Let V be a two dimensional vector space over k. Consider the set of isomorphism classes of all rank 1 $k[x]/x^2$-submodules of $V\otimes k[x]/x^2$. Does this set has a variety structure? Is it ...