All Questions
6,057 questions
11
votes
1
answer
693
views
The word problem in the ring of polynomials
This question must be well known but I cannot find it in the literature.
Question: What is the computational complexity of the word problem in a subring of the ring of polynomials in $n\ge 1$ ...
user6976
5
votes
0
answers
538
views
Picard Group of a singular surface with a non-rational singularity
I've been spending some time looking at the surface $X=\mathcal{Z}(z^3-y(y^2-x^2)(x-1))$ in $\mathbb{A}^3$ (over an algebraically closed field of characteristic different from 3). The surface has four ...
- 319
2
votes
0
answers
244
views
How many generators for rings of partially symmetric polynomials?
Let $k$ be a field, $n$ a positive integer. The group $S_n$ acts on $R_n=k[x_1,\dots,x_n]$ by permuting indices, and $\mathcal{S}_n=R_n^{S_n}=k[s_1,\dots,s_n]$ where the $s_i$'s are the usual ...
- 13.1k
0
votes
1
answer
267
views
Embedding commutative associative rings in non associative rings [closed]
Let $R$ be a commutative and associative ring with unit. Can $R$ be embedded in a ring $\hat{R}$ wich is both non commutative and non associative ?
Thanks guys !
- 1
3
votes
1
answer
564
views
Alternative module-theoretic characterization of flatness
Let $A \to B$ be a homomorphism of commutative rings. I would like to find a criterion for the flatness of $A \to B$ which does not involve the notion of kernels; it should rather involve cokernels. ...
- 63.1k
32
votes
3
answers
5k
views
Krull dimension less or equal than transcendence degree?
Let $k$ be a field, and $A$ a $k$-domain, so that the fraction field of $A$ has transcendence degree $n$ over $k$.
If $A$ is finitely-generated over $k$, then $A$ has Krull dimension $n$ (Theorem A ...
- 13k
14
votes
3
answers
1k
views
How Gr(2,7) and Gr(3,6) are related?
Consider the two types of Grassmannians Gr(2,7) and Gr(3,6) having their plucker embeddings in $\mathbb P^{20}$ and $\mathbb P^{19}$ respectivley. The first one is 10-dimensional and latter is 9-...
- 143
4
votes
1
answer
244
views
Unique matrix satisfying a system of equations
Assume I have a $n\times n$ positive semidefinite matrix $G$ of rank $p$ satisfying a set of $np - p(p-1)/2$ equations $v^T_jGv_j = 1$, $j = 1 \ldots np - p(p-1)/2$ for some given vectors $v_j$. It is ...
- 149
3
votes
1
answer
271
views
A particular Isomorphism of graded algebras over a regular local ring
In Hartshorne's "Algebraic Geometry", the following statement is a weaker form of Theorem 8.21A (e), which he quotes from Matsumuura's book on commutative algebra:
Proposition. Let $R$ be a regular ...
- 4,902
2
votes
1
answer
797
views
UFD property descends?
Hi,
Let $k$ be a field and $A$ a local noetherian $k$-algebra. If its completion is a UFD,
is it true that $A$ is a UFD? Proof?
Thanks
- 2,842
11
votes
2
answers
860
views
Incarnations of a theorem of Eilenberg
Let $R$ be any ring, let $\text{Mod}_R$ be the category of right $R$-modules and let $\text{Ab}$ be the category of abelian groups. There is a classical theorem of Eilenberg (I think) which says that ...
- 5,163
14
votes
2
answers
1k
views
Constructing the surreal numbers as iterated Hahn series
A theorem due to N. Alling (Foundations of Analysis over Surreal Number Fields, §6.55) states that the surreal numbers are isomorphic, as an ordered and valued field, to the field of Hahn series with ...
- 32.5k
3
votes
2
answers
929
views
Modules of finite support
I'm reading Dwyer and Fried's paper "Homology of free abelian covers, I". In it, they make the following claim, which I'm having trouble verifying.
Let $F$ be a field and $A = F[x_1^{\pm 1},\ldots,...
- 33
3
votes
1
answer
544
views
Which monomial subalgebras are direct summands of polynomial rings
Let $S=k[x_1,\dots,x_n]$ be a polynomial ring, and $A:=k[x^{u^{(1)}}, \dots x^{u^{(l)}}]$ a monomial subalgebra, generated by monomials $x^{u^{(i)}} = \prod_{j=1}^n x_j^{u^{(i)}_{j}}$ with $u^{(i)} \...
- 1,961
2
votes
0
answers
329
views
check the equality of two ideals in Singular http://www.singular.uni-kl.de [closed]
How one can check in Singlar that two ideals are equal? For example in macaulay 2, it is enough to write I==J, and then we will get True or False...
5
votes
1
answer
436
views
Are plethories a theory of basis-free polynomials?
This question is a follow-up to a question about the theory of polynomials.
It should be quite clear by now that matrix theory and linear algebra are quite different topics. As the various answers ...
- 11.8k
3
votes
1
answer
200
views
Does a regular pair of elements in a noetherian domain remain regular if their order is switched?
Recall that in a commutative ring $A$ an ordered pair of elements (a,b) is said to form a regular sequence if the ideal $\langle a,b\rangle $ is strictly included in $A$ ,if $a$ is not a zero-divisor ...
- 47.5k
1
vote
1
answer
392
views
If all localizations of an algebra at primes are of finite type over a field
Let $k$ be a field, and $R$ a $k$-algebra. Suppose that $R_{\mathfrak{p}}$ is a finitely generated $k$-algebra for all prime ideals $\mathfrak{p}$ of $R$. Does this imply that $R$ is also a finitely ...
- 13
11
votes
0
answers
324
views
Subrings of invariants in divided power algebras
I am wondering to what extent the functors "ring of invariants under a group action $G$"
and "divided power envelope with respect to a $G$-stable ideal" commute.
To be precise, let $R$ be a ...
- 1,609
1
vote
3
answers
519
views
Example of a commutative algebra object in a braded monoidal category C
Hi,
I am looking for an example of a commutative algebra object in a braided monoidal category C which it can also be turned into a commutative Frobenius algebra. If you have any examples could you ...
- 13
10
votes
2
answers
1k
views
When the determinant of a 2x2 polynomial matrix is a square?
Consider a 2x2 matrix $A$ with entries from $\mathbb{C}[x,y]$. Assume that $\mathrm{det} A$ is a square. Is it true that then $A$ can be represented as a noncomuting product $A=A_1 A_2 … A_{2n}$, in ...
- 1,488
0
votes
0
answers
212
views
Homomophism from Koszul complex to the original ring
In an article, I encounter an isomorphism relation as follows:
Let S be a comm. ring, x an element in S. K[x,S] be corresponding Koszul complex. The article says "this is a classical isomorphism":
$...
- 61
5
votes
4
answers
1k
views
What is the "strongest" non-local property of a ring/module that is true of all localizations at maximal ideals?
Given a commutative ring $A$ we say that a property P is local if
$A$ has P $\leftrightarrow$ $A_{p}$ has P for all prime ideals $p$ of $A$
It is usually the case that this requirement is ...
- 497
2
votes
1
answer
652
views
Searching for polynomials with squarefree discriminant
In connection with the Galois theoretic results surrounding the irreducibility of $f(x)= x^{N}-x-1$ over $\mathbb{Q}$, I've been trying to prove for a while that the discriminant of $f$ is actually ...
- 625
7
votes
1
answer
2k
views
Spectral sequence of symmetric or exterior algebras?
This question is inspired by Hartshorne's exercise II.5.7 (c-d): the problem reads:
Let $0\rightarrow \mathcal{F}'\rightarrow\mathcal{F}\rightarrow\mathcal{F}''\rightarrow0$ be a short exact sequence ...
- 757
3
votes
1
answer
216
views
Simple reference for valuative criterion of integrality?
I'd like to see a complete proof of the simplest version of the following rough statement: "If $f/g$ is a rational function on a reduced scheme ($g$ not a zero divisor), and $f/g$ doesn't have poles ...
- 27.9k
2
votes
1
answer
336
views
Extension of radical ideal after adjunction of roots
I want to apologize in advance if this is blatantly trivial, but I already posted on math.stackexchange.com and got no answer at all.
Let $A$ be a Noetherian domain containing an algebraically ...
- 4,902
4
votes
1
answer
303
views
A fast way to decide satisfiability of a set of simple fewnomial inequalities?
Background
Considering a set of points $(x_i, y_i)$ in $\mathbb R^2$ and constraints between some triples of them, which state, whether the three points of the triple are oriented clockwise (R), ...
- 191
30
votes
2
answers
2k
views
When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?
Let $R$ be a commutative ring with unit and let $q$ be an ideal of $R$. There is thus a natural map $SL(n,R) \rightarrow SL(n,R/q)$ for all $n$. This map is surjective if $SL(n,R/q)$ is generated by ...
- 418
0
votes
0
answers
775
views
Discrete valuation rings.
Given an algebraically closed field $\mathbb F$ of characteristic $p$, let $\mathbb A$ be a discrete valuation ring of characteristic zero having $\mathbb F$ as its residue field ( it does exist, but ...
- 1
3
votes
1
answer
571
views
Reference for submultiplicativity of length of tensor product
I am looking for a reference, in the form of a textbook, that contains proofs of following statements.
NOTE: I am NOT looking for the proofs, I am looking for a reference! Proofs of these statements ...
- 3,101
4
votes
4
answers
630
views
A question about the additive group of a finitely generated integral domain
Let $R$ be an integral domain of characteristic 0 finitely generated as a ring over $\mathbb{Z}$. Can the quotient group $(R,+)/(\mathbb{Z},+)$ contain a divisible element? By a "divisible element" I ...
- 6,199
3
votes
1
answer
529
views
Study of free monoids of the recursive S. Eilenberg.
Compared to the usual treatises on recursion (eg, Rogers H. "Computability and Undecidability." McGraw-Hill, New York) the book of Samuel Eilenberg & Calvin C. Elgot "Recursiveness" treats such ...
- 4,165
11
votes
2
answers
3k
views
Is there an alternative formula for solving cubic equations?
It is known that Cardano's formula for solving cubic equations is not good in the case of positive discriminants. In this case it expresses the solution through cubic roots of complex numbers. ...
- 111
4
votes
2
answers
789
views
Criteria for system of parameters in polynomial rings
Let $k$ be a field and let $R=k[x_1,...,x_n]$ be a polynomial ring over $k$. A subset $\lbrace y_1,...,y_n\rbrace$ of $R$ is called a homogenous system of parameters (hsop) of $R$, if
the $y_i$ are ...
- 16.2k
3
votes
1
answer
367
views
submonoid of a matrix monoid with a common eigenvector
Hello,
I am considering two real invertible $3\times 3$ matrices $A$ and $B$ and a nonzero vector $v\in\mathbb{R}^3$ and i am wondering if the submonoid $E$ of the monoid $(A,B)$ genererated by $A$ ...
- 69
9
votes
1
answer
689
views
When and where did the term "module" enter commutative algebra?
Bruns/Herzog "Cohen-Macaulay-Rings" has a note in the notes for Chapter 1, saying roughly that after the influx of homological algebra into commutative ring theory, modules became popular objects (...
- 1,961
20
votes
2
answers
4k
views
Ideals of the ring of smooth functions
The ring $C^\infty(M)$ of smooth functions on a smooth manifold $M$ is a topological ring with respect to the Whitney topology and the usual ring operations. Is it possible to describe, maybe under ...
- 101
8
votes
3
answers
3k
views
Quotient of flat module is flat - a property in Mumford's Red book
Hi,
In Mumford's "Red Book of Varieties and Schemes", Chapter III, Paragraph 10 (entitled "Flat and smooth morphisms"), the following property is stated:
Let $M$ be a $B$-module, and $B$ an algebra ...
- 5,562
10
votes
2
answers
356
views
Is the ideal of a closure of a Bruhat cell generated by generalized minors?
Let $G$ be your favorite complex semi-simple algebraic group, and let $B\supset T$ be your favorite Borus. For any $w\in W$, we have the Bruhat cell $BwB$, and its closure $\overline{BwB}$.
Now, ...
- 44.7k
2
votes
0
answers
134
views
Infinitesimal lifting for hensel schemes?
I have local hensel ring $A$ and a finite flat $A$-algebra $B$ (which is therefore a direct product of local henselien $A$-algebras) and I would like a section of the canonical map $B \to B / N$ where ...
- 1,347
0
votes
1
answer
475
views
How to use Nakayama [closed]
Hi there,
Let R be a local commutative ring. If M and N are two R-modules with the condition that their direct sum is equal to R^n. How do I use Nakayama to show that M and N are in fact free R-...
- 1
4
votes
1
answer
323
views
Filtrant (not necessarily totally ordered) projective system commuting with direct sums
Hello,
Let $R$ be a commutative (not necessarily Noetherian) ring.
Let $I$ be a small filtrant (not necessarily totally ordered) category.
Let $(M_i)_{i\in I}$ be a projective system of $R$-modules ...
- 41
5
votes
2
answers
966
views
Krull-Schmidt Analogue for Complete / Graded Rings
Over the ring $\mathbb{Z}$, all finitely-generated modules decompose uniquely as a direct sum of indecomposable submodules; that's the Krull-Schmidt theorem.
I'm given to understand that if a (...
- 146
15
votes
0
answers
1k
views
Is the category of smooth manifolds equivalent to the opposite category of the category of commutative monoids of some additive symmetric monoidal category?
This is a followup to my previous question, which asked whether
the category of commutative or noncommutative C*-algebras or von Neumann algebras
is equivalent to the category of commutative or ...
- 37.8k
11
votes
2
answers
3k
views
Is the sum of two prime ideals in different polynomial rings, K[X_i] and K[Y_i] a prime ideal in K[X_i Y_i]?
Let $P$ be a prime ideal in the polynomial ring $K\left[x_1,...,x_m\right]$ and $Q$ be a prime ideal in the polynomial ring $K\left[y_1,...,y_n\right]$.
Is $P+Q$ a prime ideal in $K\left[x_1,...,x_m,...
- 113
46
votes
0
answers
1k
views
Enriched Categories: Ideals/Submodules and algebraic geometry
While working through Atiyah/MacDonald for my final exams I realized the following:
The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...
- 3,249
2
votes
2
answers
326
views
Where did the multigraded Segre product appear in the literature?
Let $k$ be a field and $A\subset \mathbb{N}^d$ a vector configuration. Let $R,S$ be commutative $k$-algebras, both graded by the affine semigroup $\mathbb{N}A$. Is the 'multidgraded Segre product' $R ...
- 1,961
1
vote
2
answers
388
views
quotient of integral polynomials not being integral
So let $R$ be an integral domain and $K$ its fraction field. Let $f,g,h\in K[x]$
be 3 monic polynomials such that $f=gh$. So I would like to have a simple example
of a ring $R$ for which one has that $...
- 7,609
0
votes
1
answer
1k
views
Conjugate Matrix
Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix $...
- 301