All Questions
Tagged with commutative-algebra or ac.commutative-algebra
5,492 questions
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. ...
- 7,994
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
950
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
1
vote
0
answers
351
views
Regularity and limits of smooth rational curves.
Fix integers $2 < d \leq n$.
Suppose that $T$ is a smooth complex curve with marked point $0 \in T$, and $X$ is a closed subscheme of $\mathbb{P}^n_T$, flat over $T$ such that each fiber has ...
- 1,990
1
vote
0
answers
169
views
Sum of two free o-submodules in a vector space over a local field
Let $V$ be a countably infinite dimensional $K$-vector space over a local field $K$ (nontrivially discretely valued with finite residue field). Let $o$ be the ring of integers of $K$.
Given two free ...
- 107
12
votes
1
answer
480
views
Extending properties of commutative rings to schemes
I'm trying to pin down the various ways we can extend a property of commutative rings to a corresponding property for schemes. Let $P$ be a property of commutative rings. We could define a scheme $(X,\...
- 4,165
0
votes
2
answers
232
views
Commutation of $GL_{n}$ with projective limits
Let $A$ denote a unital commutative ring. Given a system of ideals $(I_p)_{p\in P}$ indexed by a partially ordered set $P$ such that if $p \leq q$, then $I_p$ is contained in $I_q$, when is
$$GL_n ...
- 12.3k
3
votes
2
answers
467
views
Chern character of Hom-sheaves
I'm reading the book about moduli spaces by Huybrechts and Lehn, and i'm stuck understanding a proof, it is Theorem 6.1.8.:
Given a K3-surface $X$ and a 2-dimensional space $M$, coherent and torsion ...
- 1,391
1
vote
1
answer
434
views
Equality of chern classes and isomorphism
Given two torsion free coherent sheaves $M$ and $N$ wit $rk(M)=rk(N)=r$ on an smooth projective surface $S$, by definition $det(M):=\Lambda^r(M)^{\*\*}$.
Is the following criterion correct?
$M\cong ...
- 2,181
6
votes
1
answer
301
views
Orbits in commutative groups.
Let A be finite commutative group say $(Z_m)^h$. I will say that $S \subset A$ is an orbit if exist group $H$
which acts on A such that $S$ is an orbit of $H$.
Can one give a simple characterization ...
- 5,654
0
votes
3
answers
891
views
local Artin algebras
Given a commutative Artin algebra $A$ over an algebraically closed field $k$ one has a decomposition $A=A_1\oplus\ldots\oplus A_n$ into local Artin subalgebras, see for example Atiyah-McDonald, ...
- 163
1
vote
1
answer
146
views
Is every nontrivial morphism already injective in this case?
I'm a little bit suprised at the moment, so i'll ask here if I see this wrong:
Given a sheaf of algebras $R$ ( e.g. maximal order or Azumaya) on a smooth projective scheme $X$ with generic point $p$. ...
- 42.9k
4
votes
2
answers
610
views
Are schematic fixed-points of a Cohen-Macaulay scheme Cohen-Macaulay?
I'm not sure how long this iterative questions can go on, but let me try again. Let's say $X$ is a Cohen-Macaulay scheme with an action of $\mathbb{G}_m$ (i.e. if $X$ is affine, a grading on the ...
CommunityBot
- 1
4
votes
1
answer
375
views
Regular sequence of elements of degree 1 for a homogeneous Cohen-Macaulay ring
Assume that a positively graded ring R is generated in degree 1. Is it true that, if R is Cohen-Macaulay, then there exists a regular sequence x of elements of degree 1 so that R/x is zero dimensional?...
- 5,846
4
votes
3
answers
621
views
Examples of DVRs of residue char p and ramification e
I am looking for concrete examples of a complete discrete valuation ring $R$ of characteristic 0, residue characteristic $p$ and ramification index $e$. By residue characteristic, I mean the ...
CommunityBot
- 1
2
votes
1
answer
504
views
A question arising from the Krull intersection theorem.
Let R be a local ring, I an ideal, M a finitely generated module and $N=\cap _nI^nM$. Then the Krull intersection theorem states that $N=IN$. Now if R is a local ring of characteristic $p>0$, for ...
- 19.6k
2
votes
2
answers
369
views
vectors with entries from a finite ring
I've been working recently with vectors over finite fields, but I was hoping to work in a more general setting and consider vectors over finite commutative rings. The question I had is as follows: if ...
- 56.6k
3
votes
2
answers
589
views
Comparing homomorphisms over different base rings
I am trying to compare some homomorphism groups over different base rings, so given a commutative local ring $(A,\mathfrak{m})$ and a finite dimensional Azumaya algebra $R$ over $A$.
If $M$ and $N$ ...
- 27k
16
votes
1
answer
2k
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Commuting Matrices and the Weak Nullstellensatz
In the Wikipedia article on Hilbert's Nullstensatz,
http://en.wikipedia.org/wiki/Hilbert%27s_Nullstellensatz
the following application of the Weak Nullstensatz is mentioned:
Commuting matrices
...
- 16.1k
1
vote
0
answers
198
views
Seek for good methods of computing the Krull dimension of a module?
Hi, everyone. Recently I am interested in computing the Krull dimensions of modules without using any software. However, it is not an easy job for me to do so only by its definition. Therefore, I ...
- 1,207
1
vote
1
answer
375
views
Any implemented algorithm to compute the closure of an affine variety in a product of projective spaces?
Let $I$ be an ideal of $k[x_1, \ldots, x_m, y_1, \ldots, y_n]$, $k$ being a field. Does any of the computer algebra systems implement any algorithm to calculate the generators of the 'bi-...
- 24.8k
5
votes
2
answers
495
views
Sub-Hopf algebras of group algebras
Let $k$ be a field and $G$ a finite group. Is every sub-Hopf algebra over $k$ of the group algebra $k[G]$ of the form $k[U]$ for a subgroup $U$ of $G$ ?
- 12.3k
0
votes
0
answers
183
views
Standard system of parameters and an example
Let $(R,m)$ be a local Noetherian ring. A system of parameters $\bf{x}$$:=x_{1}, \dots, x_{d}$ is a standard system of parameters if $(\bf{x})H^{i}_{m}(R/(x_{1}, \dots, x_{j}))=0$ holds for all non-...
- 113
5
votes
0
answers
817
views
morphism which is open but not universally open
In someone's note, I have seen such an example, but I can't show that it is not universally open. Here is the example:
Let $k$ be a field and $A = k[T]_{(T)}$, the discrete valuation ring obtained ...
- 1,109
17
votes
2
answers
2k
views
How badly can Krull's Hauptidealsatz fail for non-Noetherian rings?
Krull's Hauptidealsatz (principal ideal theorem) says that for a Noetherian ring $R$ and any $r\in R$ which is not a unit or zero-divisor, all primes minimal over $(r)$ are of height 1. How badly can ...
- 982
1
vote
1
answer
336
views
Need an example of finitely generated graded algebra such that each its graded subspace has infinite dimension.
More accurately, let $\displaystyle A=\sum_{i=0}^{\infty}A_i$ be a finitely generated graded algebra over say $\mathbb{Q}$ but $\dim A_i=\infty$ for each $i.$ Is it possible?
- 301
6
votes
2
answers
418
views
Does regularity of a prime ideal in the fibre imply regularity of the prime?
Recall that a prime $\mathfrak{p}$ is called nonsingular (regular) if the localization at that prime is a regular local ring. If all primes of a ring $R$ are nonsingular, $R$ is called regular. Let $...
- 30.5k
14
votes
2
answers
8k
views
Choosing the algebraic independent elements in Noether's normalization lemma
Given a field $k$ and a finitely generated $k$-algebra $R$ without zero divisors, one knows that there exist $x_1, \ldots, x_n$ algebraically independent such that $R$ is integral over $k[x_1, \ldots, ...
- 6,262
4
votes
0
answers
350
views
Artin approximation theorem for analytic functions over a field of zero characteristic
Artin's approximation theorem states: "if a system of locally analytic equations in several complex variables has a formal solution then it has a locally analytic solution".
(Artin 1968, "On the ...
- 2,232