Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,493 questions
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
944
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{...
- 7,328
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
879
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 ...
71
votes
11
answers
9k
views
How to introduce notions of flat, projective and free modules?
In the coming spring semester I will be teaching for the first time an introductory (graduate) course in Commutative Algebra. As many people know, I have been plugging away for a while at this ...
- 65.4k
2
votes
2
answers
669
views
Maximal Cohen Macaulay modules over regular factor rings.
Hi,
my question is simple. Let (R,m) be a commutative regular local noetherian ring. Is it true that for every prime p \in Spec(R), the factor ring R/p has maximal cohen-macaulay R/p-module?
Best ...
- 203
5
votes
0
answers
517
views
Monomial-type ideals in polynomial rings
Let $R=k[x_1,x_2,...,x_n]$ be the polynomial ring in $n$ indeterminates over a field $k$. A monomial in $R$ is an element which is product (with repetitions allowed) of the indeterminates. Monomial ...
- 805
6
votes
1
answer
641
views
The Jacobian ideal generates the socle of a complete intersection
This is with reference to theorem 5.20 in Vasconcelos book linked (google books) here:
http://tinyurl.com/2967eov
I shall restate the theorem here for easy reference: "If $A=k[[x_1,x_2,...,x_n]]/I$ ...
- 805
0
votes
1
answer
262
views
Subtleties in the construction of base change morphisms
Given a flat and projective morphism of noetherian schemes, $f: X \rightarrow Y$ and $F$, $G$ two coherent $O_X$-modules, flat over $Y$. Furthermore given a morphism $u: Y' \rightarrow Y$ of ...
- 1,391
4
votes
2
answers
1k
views
Kaplansky's theorem for graded local rings
Hello!
This is a very short question:
Given a local graded Noetherian ring $R_{\bullet}$, is it true that any graded projective module over $R_{\bullet}$ is free?
In the ungraded case, this is true,...
- 2,756
5
votes
2
answers
1k
views
Structure theorem of f.g. modules over a (non) PID
I am looking for an example of a commutative ring with $1$, in which every ideal is generated by a single element, for which the conclusion of the structure theorem for finitely generated modules is ...
- 11.1k
7
votes
3
answers
969
views
Basepoints in the canonical system of algebraic surfaces
Let $X$ be a smooth projective variety defined over $\mathbb{C}$. In the context of the minimal model program it is often important to understand the geometry of the maps defined by the complete ...
- 2,087
3
votes
2
answers
787
views
Rees algebra for non-radical ideals
Today in my introductory algebraic geometry class we defined the so-called Rees algebra associated with an ideal $I$ of a ring $R$ (with strong conditions on $R$, if you like: I don't mind restricting ...
- 3,605
3
votes
1
answer
447
views
About a corollary of the Briançon-Skoda theorem
The following is a corollary of the Briançon-Skoda theorem:
If $R$ is a regular Noetherian ring of Krull dimension $d$ and $f_1,f_2,...,f_{d+1}\in R$. Then, $f_1^df_2^d...f_{d+1}^d \in (f_1^{d+1},f_2^...
- 805
10
votes
1
answer
2k
views
Is Illusie's generalization of the cotangent complex to arbitrary ringed toposes necessary in algebraic geometry?
André and Quillen both gave constructions of the relative cotangent complex for commutative rings, so pretty immediately that gives us that we understand the cotangent complex for affine schemes. ...
- 19.6k
9
votes
3
answers
2k
views
(Krull) dimension of any associated graded ring of a ring R equals the dimension of R
I am not sure if this is appropriate for MO. If not, I shall be happy to take it to SE.
For a local ring $(R,m)$, given any proper ideal $I$, the (Krull) dimension (from here on dimension means Krull ...
- 805
11
votes
3
answers
1k
views
Minimum of Milnor number for the curve singularities of fixed multiplicity
An element $F\in \mathbb{C}[[x,y]]$ defines a germ of plane curve.
We assume $F(0,0)=0$.
The multiplicity $mult$ of the germ is defined to be a minimal number $i$
such that $F\in m^i$ where $m=(x,y)$ ...
- 333
1
vote
2
answers
722
views
Do subsets of generators of a toric ideal generate a toric ideal?
Given a toric ideal, say $J$, in a polynomial ring $k[x_1,...,x_n]$ we can find a finite
generating set for $J$. Is it possible, perhaps under additional assumptions on the structure of $J$, to give ...
- 805
1
vote
1
answer
257
views
Are pullbacks from a factor of a product scheme flat over the other factor?
Given two smooth projective surfaces $X$ and $Y$ over some algebraically closed field.
Given a torsion free coherent sheaf $M$ on $X$. One has the projections $\pi_X$ and $\pi_Y$ from the product $X\...
- 1,391
10
votes
3
answers
1k
views
Strong Nullstellensatz
Let $I\subseteq{\mathbb C}[X_1,\dotsc,X_n]$ be an ideal, and
let $V\subseteq{\mathbb C}^n$ be the corresponding algebraic set
($V$ consists of those $x$ at which all $f\in I$ vanish).
Is it true ...
- 809
9
votes
1
answer
2k
views
Formally smooth morphisms, the cotangent complex, André-Quillen cohomology, and representability of nilpotent extensions as trivial extensions over a cofibrant replacement
Recall that an $R$-algebra $R\to S$ is called formally smooth (resp. formally unramified resp. formally étale) if given any lifting problem of the form
$$\begin{matrix}
R&\to &T\\
\downarrow&...
- 19.6k
4
votes
1
answer
555
views
Base change and relative Ext over noncommutative rings
Given two smooth projective schemes $X$ and $Y$ over some algebraically closed field $k$, we have $X\times Y$ with the projections $p$ to $X$ and $q$ to $Y$. Furthermore we have a "nice" sheaf of ...
- 1,391
16
votes
4
answers
1k
views
Algebraic analogue of the Moebius bundle over the circle
Let $R$ be the ring $R[X,Y]/(X^2+Y^2−1)$. The space of $\mathbb{R}$-rational points of the affine scheme associated to $R$ is the topological circle $S^1$.
An algebraic vector bundle over $R$ is an $...
- 2,782
29
votes
5
answers
5k
views
Why does the (S2) property of a ring correspond to the Hartogs phenomenon?
Hartogs Theorem says every function whose undefined locus is of codim 2 can be extend to the whole domain. I saw people saying this corresponds to the (S2) property of a ring. But I can't see why this ...
- 5,052
32
votes
3
answers
4k
views
What facts in commutative algebra fail miserably for simplicial commutative rings, even up to homotopy?
Simplicial commutative rings are very easy to describe. They're just commutative monoids in the monoidal category of simplicial abelian groups. However, I just realized that a priori, it's not clear ...
- 19.6k
1
vote
0
answers
276
views
Generalizations of divided-power algebras over finite fields
In Andrews, Askey, and Roy's Special Functions, the authors state that Gauß sums are finite field analogs of the $\Gamma$-function as Jacobi sums are to B-function. The $\Gamma$-function is well-...
- 1,049
6
votes
1
answer
952
views
Is it possible to recover the degree of a field extension from a list of elements and the ground field?
I'm interested to know if there is anything known about recovering the degree of a field extension, $E/k$, given $E=k(\alpha_1,\ldots, \alpha_n)$ (here I'm assuming that the extension is of finite ...
- 1,049
9
votes
0
answers
513
views
E(n) Deformations of the infinity category Qcoh(X) with it's E(n)-tensor product
Let $X$ be a smooth scheme, then an infinity enchancement of $QCoh(X)$ has an $E_\infty$ structure and in particular an $E_n$ structure for any $n$. In this paper, http://arxiv.org/abs/0805.0157 Ben-...
- 3,758