All Questions
6,056 questions
12
votes
1
answer
5k
views
intersection of ideals in a commutative ring vs their product
This question was inspired by this one. Given two ideals $A,B$ in a finitely generated commutative ring $R$. Is it possible to decide whether $A\cap B=AB$? Here $R$ is given by generators and ...
user6976
65
votes
4
answers
22k
views
When is the product of two ideals equal to their intersection?
Consider a ring $A$ and an affine scheme $X=\operatorname{Spec}A$ . Given two ideals $I$ and $J$ and their associated subschemes $V(I)$ and $V(J)$, we know that the intersection $I\cap J$ corresponds ...
- 914
8
votes
1
answer
1k
views
Give an example of monoid with property $m^2 = m^3$
Give an example of finitely generated, infinite monoid $M$ with property that for all $m \in M$ we've got $m^2 = m^3$.
This question comes from the problem I was given during algebraic languages ...
15
votes
4
answers
1k
views
What formal properties should resolution of singularities have?
If I were going to propose a new construction as a "replacement for resolution of singularities", what properties would my replacement have to have? [I am going to do no such thing -- this is purely ...
- 5,846
0
votes
1
answer
183
views
Projectively splitting module
Is there a name for such class of modules $M$ such that $M\rightarrow N\rightarrow 0$ splits for every $N$?
- 2,857
5
votes
2
answers
3k
views
Zariski topology and compact \paracompact space?
Does the Zariski topology on a ring (not commutative in common) form a compact or paracompact space and why?
- 53
4
votes
2
answers
493
views
Explicit Bézout cofactors
$\DeclareMathOperator\lcm{lcm}$This is a rather severe revision of a question I asked recently. We know over the integers that $\gcd(a^2,b^2)=\gcd(a,b)^2$. We might prove this via unique factorization....
- 30.1k
6
votes
1
answer
762
views
Nontrivial criteria for polynomials to have no common zeros?
When we work in $C[x_1,x_2,...,x_n]$,here $C$ denotes the complex field, we know that when polynomials $f_1,f_2,...,f_k$ have no common zeros, then there exists polynomials $g_1,...,g_k$ , such that ...
- 1,528
4
votes
0
answers
1k
views
Commutative ring Notes by M. Artin
In 1966, Professor Michael Artin gave a course for first-year graduate students at MIT on commutative algebra. In that course he covered many classical topics, (the Spectrum of a commutative ring, ...
- 165
5
votes
2
answers
2k
views
Iterated calculation of determinants
Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...
- 11.1k
13
votes
4
answers
4k
views
Reference for tensor products of fields
Does anybody know a reference for basic properties of tensor products of (finite) algebraic extensions of fields?
Ideally, I would like a description of $L \otimes_k K$ for arbitrary finite ...
- 467
0
votes
1
answer
251
views
What is a certain cartesian product of algebras?
Suppose $F$ is a field and $A$ the $F$-algebra $F[X]\times_{(F\times F)} F$ given by the missing corner of a cartesian square in $F$-algebras
\begin{equation}
F~\xrightarrow{\Delta} ~F\times F~ \...
- 1
7
votes
3
answers
1k
views
Can the I-fold direct product be free?
Let $A$ be a non-zero commutative ring with unit, $I$ a infinite set.
Can $\prod_{i\in I}A$ be free as an $A$-module?
I found when $A$ is a field or is isomophic to $\mathbb{Z}/m\mathbb{Z}$, then ...
- 268
3
votes
2
answers
810
views
What is the divisibility theory for Bezout Domains?
There are many facts about integer gcds which can be proved by appealing to unique prime factorization (up to sign). for example $\gcd(a^2,b^2)=\gcd(a,b)^2$. One way to get the machinery (if that is ...
- 30.1k
29
votes
3
answers
7k
views
Non finitely-generated subalgebra of a finitely-generated algebra
Ok, I feel a little bit ashamed by my question.
This afternoon in the train, I looked for a counter-example:
— $k$ a field
— $A$ a finitely generated $k$-algebra
— $B$ a $k$-subalgebra of $A$ that ...
3
votes
2
answers
4k
views
How to calculate Tor(R/I, R/J) ??
How can I prove that $\text{Tor}_1(R/I,R/J) = (I \cap J)/IJ$, where $R$ is a ring and $I, J$ ideals.
Moreover, if we suppose $R=I+J$, how do I prove that $\text{Tor}_1(R/I,R/J)=0$?
Ps: No, this is ...
- 33
11
votes
2
answers
1k
views
Geometric motivation for the Stanley-Reisner correspondence
The Stanley-Reisner ring of an abstract simplicial complex $\Delta$ on the vertex set $\{1,...,n\}$ is the $k$-algebra
$$
k[X_1,...,X_n]/I_\Delta
$$
where $I_\Delta$ is the ideal generated by the $X_{...
- 2,782
12
votes
2
answers
799
views
Normal Macaulayfications
Given any reduced excellent scheme $X$, there exists a Macaulayfication $\pi : Y \to X$. In other words, there exists a proper birational map $\pi$ from a Cohen-Macaulay scheme $Y$ to $X$. These ...
- 20.5k
8
votes
2
answers
2k
views
Can we deduce that two rings $R_1$ and $R_2$ are isomorphic if their polynomial ring are isomorphic?
Given two rings $R_1$ and $R_2$ (with or without identity: it's not specified). If $R_1[x]$ is isomorphic to $R_2[y]$ (No such requirement that the isomorphism sends the constant terms to constant ...
- 1,079
5
votes
3
answers
688
views
examples of finitely generated semigroups that are not residually finite
Does anybody know of any finitely generated semigroups that are not residually finite and whose group of units (if there is an identity) is trivial? Basically, I'm looking for finitely generated ...
- 549
2
votes
0
answers
261
views
On a characterization of the symbolic square of prime ideals in polynomial rings
If $R=k[x_1,...,x_n]$ is a polynomial ring in $n$ indeterminates over a field $k$ of characteristic $0$, there is a characterization of the symbolic square of a prime ideal $P$ (the $n$-th symbolic ...
- 805
15
votes
1
answer
649
views
Primes that must occur in every composition series for a given module
Let $M$ be a finitely generated module over the commutative noetherian ring $R$. Let ${\cal C}(M)$ be the set of all primes $P$ in $R$ such that $R/P$ appears as a quotient in every composition ...
- 23k
4
votes
0
answers
338
views
What to call the following variant of tame ramification
Suppose that $R \subseteq S$ is a generically separable extension of 1-dimensional normal domains (you can assume that $R$ is local if you'd like) of equal-characteristic $p > 0$ (for simplicity, ...
- 20.5k
4
votes
2
answers
354
views
"un-nil-ifying" ideals via deformation
This is perhaps a naïve question; given an irreducible scheme $X$, is there a general procedure to find a flat family $Y \to T$ such that over some point $t_0$ we have $Y \times_T {t_0} \cong X$ but ...
- 467
7
votes
0
answers
249
views
Does there exist a commutative ring R such that SL_3(R) and SL_2(R) have the same finite subgroups?
This question is inspired, of course, by this question, and I don't know enough commutative algebra to know whether it's answered by silence dogood's answer to this follow-up question. If the answer ...
- 118k
7
votes
2
answers
567
views
Rational powers of ideals in Noetherian rings
Let $R$ be a Noetherian ring, and let $I$ be an ideal of $R$. Fix a rational number $a=\frac{p}{q}$ with $p, q\in \mathbb{Z_\geq 0}$ $q\neq 0$. We define $I_a = \{x \in R: x^q\in \overline{I^p}\}$, ...
- 805
3
votes
4
answers
945
views
What conditions are needed for $-\otimes_A B$ to be faithful?
For $A$ a (commutative) ring, $f:A\to B$ an $A$-algebra, what conditions do we need on $A$ and $B$ (and $f$) for the functor $-\otimes_A B:A-mod\to B-mod\quad$ to be faithful (i.e. injective on $Hom$-...
- 1,488
12
votes
1
answer
2k
views
Is every ring the direct limit of Noetherian rings?
Are there any examples of commutative rings that do not occur as direct limits of Noetherian rings?
- 441
0
votes
1
answer
329
views
What is correct name of the following construction?
Consider an ideal $I=\langle f_1,f_2,\ldots,f_s\rangle$ in the polynomial ring $\mathbb{Q}[x_1,x_2,\ldots,x_n].$ Build the following set
$$
\{ g_1 f_1+g_1 f_2+\cdots+g_n f_n \},
$$
where $g_i$ ...
- 301
6
votes
1
answer
806
views
Radicals of binomial ideals
Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. An ideal (that can be) generated by monomials is called a monomial ideal. For the monomial ideal $M=(m_1,m_2,.....
- 805
6
votes
1
answer
1k
views
Explicit injective resolutions of (Laurent) polynomial rings
Hi,
Despite being nothing but the dual notion of projective resolution, injective resolutions seem to be harder to grasp. For example, the general form of Poincaré-Lefschetz duality given in Iversen'...
- 558
4
votes
2
answers
670
views
term for a "faithful" module
Is there a term for an $A$-module $M$ such that $M \otimes_A -$ takes nonzero modules to nonzero modules?
Motivation: It is a standard theorem that if $B$ is faithfully flat over $A$, then $\hbox{...
- 7,338
5
votes
1
answer
959
views
Multiplicative Structures On Free Resolutions
Hello,
this question is related to Differential graded structures on free resolution?.
Given a regular local ring $S$ and $f\in{\mathfrak m}_S\setminus\{0\}$, I am interested in studying $R$-modules ...
- 2,756
2
votes
0
answers
1k
views
Decomposition group vs Galois group of completed extension for height > 1 primes
Assume
Let $R$ be a Noetherian normal excellent domain, $F$ its field of fractions.
Let $S$ be a finite $R$-algebra, $L$ its field of fractions.
$L/F$ a (finite) Galois extension
$S$ normal in $L$
...
- 21
4
votes
1
answer
2k
views
Arithmetically Cohen-Macaulay varieties
What do we mean by a variety being arithmetically Cohen-Macaulay? Is every such variety also Gorenstein?
- 61
10
votes
2
answers
706
views
CM for radical ideal
Let $R$ the polynomial ring in $n$ variables with complex coefficients and $I$ an ideal of $R$. Is it true that if $R/I$ is CM also $R/J$ is CM (where $J$ is the radical of $I$)?
Is there a relations ...
- 671
44
votes
7
answers
5k
views
Does $SL_3(R)$ embed in $SL_2(R)$?
Is there any non-trivial ring such that $SL_{3}(R)$ is isomorphic to a subgroup of $SL_{2}(R)$?
$SL_{3}(\mathbb{Z})$ is not an amalgam, and has the wrong number of order $2$ elements to be a subgroup ...
- 1,012
15
votes
1
answer
637
views
When is a local Artin C-algebra a subring of C[t]/t^n
Let $A$ be a local ring over $\mathbb{C}$, which moreover is a finite dimensional $\mathbb{C}$-vector space.
When is $A$ a subring of $\mathbb{C}[t]/t^n$?
What does the minimal ...
- 8,723
5
votes
1
answer
327
views
When is the projective line the seminaive projective line?
Excuse the possible naivete of this question. Since reading a nice survey article by Daniel Biss a few years ago, I'm always worried about what $P^1(R)$ is, for a ring $R$.
So that I stop worrying, ...
- 13.3k
1
vote
1
answer
518
views
Cardinality of a linear independent subset of a free module over a commutative ring which is not an integral domain
If R is a commutative ring with unity and not an integral domain and F is a free R-module with rank k,is there a linear independent set with cardinality > k?
I prooved that this is not true if R is an ...
- 345
15
votes
2
answers
881
views
injectivity of torsion submodules of injectives
Local cohomology with respect to an ideal $\mathfrak{a}$ is often studied over a Noetherian ring $R$. However, the proof of a lot of basic results does not rely on noetherianity of $R$, but rather on ...
- 6,700
11
votes
2
answers
916
views
Homologically nice commutative rings
Let $R$ be a commutative regular local ring. Is it true that for every $\mathfrak p \in \mathrm{Spec}(R)$ there is a finitely generated $R$-module $M$ such that $\mathrm{projdim}(M)=\mathrm{ht}(\...
- 203
5
votes
1
answer
2k
views
Intersections of irreducible components
Let $V$ be an algebraic variety (not irreducible) over $\mathbb{C}$, defined by an ideal $I = \{f_1,f_2,\dots, f_n\}$. $V$ is not necessarily pure dimensional. Suppose $V = R_1\cup R_2\cup\dots\cup ...
- 1,510
1
vote
1
answer
343
views
primary regular sequences
Let $R$ be commutative regular local ring. Is it true, that for every $\mathfrak p \in \mathrm{Spec}(R)$, there is a $\mathfrak p$-primary $R$-regular sequence? (I.e. an $R$-regular sequence $\bf x$ ...
- 203
1
vote
1
answer
769
views
Application of the base change theorem
Given a flat and projective morphism $f:X\rightarrow Y$ of noetherian schemes over some algebraically closed $k$ and $F$, $G$ coherent $O_X$-modules, flat over $Y$.
Then the base change theroem for ...
- 1,391
1
vote
1
answer
573
views
Generalization of the Structure theorem for artinian rings?
Let $A$ be a commutative ring with identity. If $A$ is a ring with only a finite set of prime ideals $p_1...p_n$ and moreover $\prod_{i=1}^n p_i^{k_i}=0$ for some k_i. Is $A$ then isomorphic to $\...
- 43
3
votes
0
answers
2k
views
Cohomology and tensor product
Let $G$ be a profinite group, $A$ a free $\mathbb{Z}_p$-module of finite rank with a continuous action of $G$ and $B$ any $\mathbb{Z}_p$-module (I am not supposing it to be free), with the trivial ...
- 657
2
votes
0
answers
384
views
What do you call an algebraic element with the property that the generated field extension is normal?
Let $L/K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Is there an established terminology for the property of $\alpha$ that $K(\alpha)/K$ is a normal field extension? Would you ...
- 2,399
2
votes
1
answer
330
views
CM module is height-unmixed?
$A$ a Cohen-Macaulay ring (not necessarily local), $M$ a Cohen-Macaulay $A$-module. Then does it necessarily follow that $\mbox{ann}(M)$ is height-unmixed?
- 2,857
7
votes
2
answers
1k
views
Upper bound to the number of generators
When defining noetherian ring/module there's no condition on the number of generators of ideals/submodules (apart from being finite).
However, in some cases we can do better:
-A noetherian module ...
