All Questions
5,985 questions
16
votes
5
answers
5k
views
An advanced exposition of Galois theory
My knowledge of Galois theory is woefully inadequate. Thus, I'd be interested in an exposition that assumes little knowledge of Galois theory, but is advanced in other respects. For instance, it would ...
CommunityBot
- 1
4
votes
4
answers
2k
views
Why are principal ideal domains and Dedekind domains prominent, but I always seem to see Noetherian rings rather than Noetherian domains?
It seems to me that these 3 algebraic systems are closely related, but it always seems to be Noetherian rings rather than Noetherian domains appearing, and conversely I rarely seem to see principal ...
CommunityBot
- 1
16
votes
3
answers
3k
views
Is being torsion a local property of module elements?
Say $R$ is a ring, not necessarily a domain, and $M$ is an $R$-module. All rings are commutative with 1. An element $m\in M$ is called torsion if $r.m=0$ for some regular element (non-zerodivisor) $r\...
CommunityBot
- 1
11
votes
0
answers
1k
views
Reverse mathematics strength of identically zero polynomials are the zero polynomial
According to wikipedia, the statement "every polynomial over a countable field that is not the zero polynomial has only finitely many roots" is equivalent to RCA0 over RCA0* (which is called ERCA-0 in ...
CommunityBot
- 1
3
votes
0
answers
240
views
Flatness of a homogeneous quasi-coherent sheaf on the formal plane from flatness on lines through the origin?
(This is a follow-up to a previous question; as often happens, I didn't include enough hypotheses, so I'm asking a new, more-likely-to-be-true question).
Consider the formal plane $\operatorname{Spec}...
CommunityBot
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11
votes
1
answer
1k
views
Can ⨁_I A be isomorphic to ∏_I A for infinite I?
Suppose $A$ is a non-zero ring (say commutative unital) and $I$ is an infinite set. Can it happen that there is an isomorphism of $A$-modules $\bigoplus_{i\in I}A\cong \prod_{i\in I}A$?
The obvious ...
- 20.8k
1
vote
1
answer
268
views
Flatness on the formal plane from flatness on lines through the origin?
Consider the formal plane $\operatorname{Spec}\mathbb C[[t,h]]$, and let $\mathcal F$ be a quasi-coherent sheaf. Now assume that $\mathcal F$ is flat over infinitely many lines through the origin of ...
CommunityBot
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3
votes
1
answer
928
views
How exotic can DVRs be in the ring of rational functions over a local field?
Suppose that $R$ is a complete DVR with field of fractions $K$, uniformiser $\pi$ and residue field $k$.
Let $B$ be a subring of the ring $K(t)$ of rational functions over $K$. Moreover assume that $...
- 11.1k
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 ...
- 30.5k
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 ...
CommunityBot
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6
votes
1
answer
761
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 ...
- 39.9k
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
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~ \...
- 5,846
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
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 ...
- 66.3k
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 ...
CommunityBot
- 1
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
7
votes
2
answers
566
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
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 $g_i$ ...
- 2,537
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'...
- 22.6k
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 ...
CommunityBot
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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{...
- 23k
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?
- 20.5k
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
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
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 ...
- 27.8k
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 ...
- 20.5k
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 $\...
- 42.9k
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?
- 30.5k
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 ...
- 12.4k
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
4
votes
2
answers
2k
views
What are non-trivial examples of non-singular blow-ups of a non-singular variety?
This question arose from the responses to this question. The references to the comments of Karl Schwede and VA are to comments made there.
The blow-up of the variety $X=\mathbb{A}^2$ along the ...
CommunityBot
- 1
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 ...
- 3,093
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 ...
- 30.5k
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,...
- 20.5k
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 ...
- 39.3k
18
votes
2
answers
2k
views
What does primary decomposition of (sub) modules mean geometrically?
I want to know how I should visualize modules in algebraic geometry. The way we visualize rings, via their spectra, automatically (or by the beauty of its design) depicts primary decomposition of ...
- 12.4k
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 ...
- 20.5k
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 ...
CommunityBot
- 1
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 $m=(x,y)$ ...
- 66.3k
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&...
- 27.2k
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. ...
- 8,004
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\...
- 39.3k
40
votes
1
answer
3k
views
Is every connected scheme path connected?
Every (?) algebraic geometer knows that concepts like homotopy groups or singular homology groups are irrelevant for schemes in their Zariski topology. Yet, I am curious about the following.
Let's ...
CommunityBot
- 1
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 $...
CommunityBot
- 1
5
votes
3
answers
2k
views
The correspondence between affine vector bundles and f.g. projective modules
The definition of a (geometric) vector bundle over a scheme $X$ can be rewritten as follows in terms of 'not-so-geometrical algebra' if $X=Spec R$ is affine and if I am not missing something.
A ...
- 13k
6
votes
1
answer
951
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 ...
- 13k
37
votes
2
answers
3k
views
How can I define the product of two ideals categorically?
Given a commutative ring $R$, there is a category whose objects are epimorphisms surjective ring homomorphisms $R \to S$ and whose morphisms are commutative triangles making two such epimorphisms ...
- 63.1k
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