All Questions
6,056 questions
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 ...
- 2,097
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
2
answers
752
views
Adding a formal inverse of an element to a free monoid
Let $FM_2=\langle a,b\rangle$ be the free monoid of rank 2. If we add a formal inverse to the word $aba$, we get the free group $F_2$ (because both $a$ and $b$ will have inverses).
Question: For ...
user6976
8
votes
1
answer
1k
views
Lattice-ordered commutative monoids
By a lattice-ordered monoid, I mean a structure $(A,0,{+},{\vee},{\wedge})$ such that $(A,0,{+})$ is a (not necessarily commutative) monoid, $(A,{\vee},{\wedge})$ is a lattice, and the two ...
- 44.4k
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
1
vote
1
answer
400
views
Transitive Semigroups of $2\times 2$ matrices
Suppose $G$ is a semigroup (i.e., closed under matrix multiplication) of invertible $2\times 2$ real matrices. Suppose also that $G$ is transitive i.e., for any two non-zero vectors $u$ and $v$ there ...
- 1,045
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
514
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
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
13
votes
2
answers
3k
views
Wikipedia's definition of 'locally free sheaf'
Let $R$ be a, say, noetherian ring and $M$ an $R$-module. The Wikipedia article on 'locally free sheaf' tells me that the following two statements are equivalent:
The module $M$ is locally free (Edit:...
- 2,782
5
votes
2
answers
537
views
If $k[S]$ is noetherian, is S finitely generated?
Let $S$ be a semigroup. If $S$ is abelian, then it follows that the semigroup algebra $k[S]$ is finitely generated if and only if $S$ is.
What if we relax the condition on $k[S]$, so that $k[S]$ is ...
- 11.6k
2
votes
0
answers
281
views
Is evaluating limits with dual numbers sound?
Let $D$ be the ring $\mathbb{C}[\epsilon]/\langle \epsilon^2\rangle$. Define the functions $dual : \mathbb{C} \to D$ and $stdPart : D \to \mathbb{C}$ by $dual(x) := x+0\cdot \epsilon$ and $stdPart(x+...
user5810
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
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 ...
- 2,782
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 ...
user5810
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
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 ...
- 91
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 ...
- 1,391
0
votes
3
answers
892
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, ...
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 $p$. ...
- 1,391
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?...
- 157
14
votes
1
answer
911
views
Can a single DVR witness all specializations on a variety?
If $X$ is a noetherian scheme with points $x$ and $\xi$ so that $x$ is in the closure of $\{\xi\}$, then there exists a discrete valuation ring $V$ and a map $Spec(V)\to X$ sending the generic point ...
- 24.1k
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 ...
- 44.7k
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 ...
- 1,207
4
votes
3
answers
622
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 ...
- 317
16
votes
1
answer
2k
views
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
...
- 839
4
votes
2
answers
607
views
Invertible elements in monoid rings of unital monoids without non-trivial invertible elements
This question is somewhat related to Tilmans notorious problem in this post. Let $(M,\cdot)$ be a monoid with unit $1$ and set
$$(M,\cdot)^{\times} := \lbrace x \in M \mid \exists y \in M : xy=yx=1 \...
- 25.5k
1
vote
0
answers
538
views
Functoriality of a standard integral domain construction.
The evident forgetful functor from fields to integral domains has a left adjoint, namely the construction of the quotient field for a given integral domain. Another standard construction is taking the ...
- 21
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 $N$ ...
- 1,391
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
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 $G$ ?
- 16.2k