All Questions
6,053 questions
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Is the semigroup M(n, Z) finitely presented? If so, where can I find a presentation of it?
I am new to semigroup research, so I apologize if this is an easy question.
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3
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1
answer
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Decomposition of modules using computer packages
I am interested in computing direct sum decomposition of modules over some quotients of polynomial rings over a field (do not care much about the field at this point). Does any one know a package ...
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1
answer
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First-order UFD (factorial ring) condition / pre-Schreier rings
All rings in this post are commutative and with $1$.
Everyone knows the definition of a factorial ring, a. k. a. unique factorization domain (UFD). I have been wondering about some variations ...
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The inverse limit of locally free module
A is an I-adic complete Noetherian ring. M is a finitely generated A module. For any n>0, $M/I^nM$ is a finitely generated locally free A/I^n-module. Is M necessarily a locally free A-module?
CommunityBot
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2
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Normality of an affine semigroup
An affine monoid is a finitely generated commutative submonoid of $\mathbb Z^k$ for some positive integer k. Let S be an affine monoid and let G(S) be the group generated by S. We say the monoid S is ...
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2
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When do primes lift uniquely (provided they lift at all)?
Given a ring $R$, a prime ideal $\mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $R\to S$ is injective), are there any nontrivial sufficient conditions for the induced map $Spec(S) \...
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3
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When is the radical of the extension of a prime ideal prime?
(All rings assumed to be commutative and unital)
Given a ring $R$, a prime ideal $\mathfrak{p}$ of $R$, and an extension ring $S$ (the algebra map $R\to S$ is injective), are there any nontrivial ...
- 156k
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Computing the structure of the group completion of an abelian monoid, how hard can it be?
Cherry Kearton, Bayer-Fluckiger and others have results that say the monoid of isotopy classes of smooth oriented embeddings of $S^n$ in $S^{n+2}$ is not a free commutative monoid provided $n \geq 3$. ...
CommunityBot
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1
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Adjunction for underlying reduced subschemes
Let $k$ be a perfect field (so reduced = geometrically reduced) and $f:X\rightarrow \mathrm{Spec}(k)$ a Cohen-Macaulay morphism. Denote by $i:X_{red}\rightarrow X$ the underlying reduced subscheme ...
- 57.6k
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Wants: Polynomial Time Algorithm for Decomposing a Multiset of Rationals into Two Additive Subsets.
First, allow me to say that this problem was posed to me by a professor in the department. It is related to his research in a way that I do not know. However, since I couldn't come up with anything ...
- 4,842
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Is ΩΣ in {simplicial commutative monoids} group completion?
Let C be the model category of simplicial commutative monoids (with underlying weak equivalences and fibrations), or equivalently the (∞,1)-category PΣ(Top), where T is the Lawvere theory ...
CommunityBot
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What does primary mean geometrically?
Given a primary ideal I in a ring A, we can consider the subscheme V(I) of Spec(A).
It is a nilpotentification (?) of the integral subscheme V(rad(I)) given by the radical rad(I) of I.
My question is ...
- 57.6k
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2
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Class groups of normal domains over finite fields
Let R be a local, normal domain of dimension 2. Suppose that R contains a finite field. I am interested in knowing when the class group of R is torsion. In characteristic 0, this is known to be ...
- 65.4k
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ring-valued points of locally ringed spaces
of course, one should expect that the concept of ring-valued points is not well-behaved for locally ringed spaces (LRS). I want to see examples for this.
so consider $LRS \to Set^{Ring}, X \mapsto X(-...
- 63.1k
4
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412
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F_q-structures on schemes
Let $k|\mathbb{F}_q$ be a field extension. An $\mathbb{F}_q$-structure on a $k$-algebra $A$ is an $\mathbb{F}_q$-subalgebra $A _0$ of $A$ such that $A _0 \otimes _{\mathbb{F}_q} k \cong A$ via the ...
- 65.4k
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is there a good computer package for working with bicomplexes?
I'm interested in working with bicomplexes of modules over polynomial rings, specifically tensoring them together, and the operation of taking cohomology in one direction, and then the other. Is ...
- 4,394
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4
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8k
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Serre intersection formula and derived algebraic geometry?
Let $X$ be a regular scheme (all local rings are regular). Let $Y,Z$ be two closed subschemes defined by ideals sheaves $\mathcal I,\mathcal J$. Serre gave a beautiful formula to count the ...
- 4,909
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Isolated hypersurface singularities, Chow groups and D-branes
Say a ring $R$ is an isolated hypersurface singularity if $R = k[x_1, \ldots, x_n]_{(x_1, \ldots, x_n)}/(W)$, where $k$ is a field and $W \in k[x_1, \ldots, x_n]$ is such that the ideal $(\partial_1 W,...
CommunityBot
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An "Elementary" Math Question Generalized (Ring Theory Perhaps)
The following question is posed in the book "The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics"
"Prove that if integers a_1, ..., a_n are all distinct, then the ...
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2
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611
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Computing Integral Closures
I'm wondering if there's an algorithm, or a program I can use, to compute integral closures. Specifically, what I have in mind are variants of questions of the sort: what is the integral closure of &#...
- 156k
4
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Parametric polynomial solution of a single polynomial equation
Let $P$ be a polynomial in $n$ variables with rational coefficients,
$P \in {\mathbb Q}[Z_1,Z_2, \ldots ,Z_n]$, and consider the algebraic
set
$Z=\lbrace (z_1,z_2,z_3, \ldots ,z_n) \in {\mathbb Q}^n |...
- 30.5k
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Do DG-algebras have any sensible notion of integral closure?
Suppose R → S is a map of commutative differential graded algebras over a field of characteristic zero. Under what conditions can we say that there is a factorization R → R' → S ...
- 4,909
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Examples of algebraic closures of finite index
So there are easy examples for algebraic closures that have index two and infinite index: $\mathbb{C}$ over $\mathbb{R}$ and the algebraic numbers over $\mathbb{Q}$. What about the other indices?
...
- 65.4k
6
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3
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797
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Tensor product is to flat as Hom is to ?
Sorry if I'm missing something here, but what do we call $M$ if the functor $H_M:N\mapsto Hom(M,N)$ is exact? Is this in fact equivalent to being flat through some adjointness properties?
- 57.6k
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Does homology detect chain homotopy equivalence?
Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.
- 15.6k
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Classifying Algebra Extensions over a fixed extension?
There are lots of "Ext groups" in homological algebra which measure extensions of various things. I'm sure there must be a homological algebra machine for computing the following, and I'm hoping that ...
- 3,396
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Field structure for R^n
Hi!
Is it possible to define a product on R^n for n>2 such that R^n can be made into a field?
R is a field in its own right with the standard operations and R^2 can be made into a field by ...
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What is the localization of Q[x]/(x) at 0
Q is a rational field. Q[x] is polynomial ring over Q 。(x) is maximal ideal of Q[x].
Take Q[x]/(x) as a module over Q[x]. Then what is Q[x]-module Q[x]/(x) localize at 0??
I think the result is
Q[x]/...
- 1,033
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604
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Homomorphism between exterior powers of a free module of finite rank
I´m looking for homomorphisms between exterior powers of a free module M of rank m
ΛmR M → Λm-1R M
Exactly, I´m looking for an explicit isomorphism
M → Hom R (ΛmR M , Λm-1R M)
I compare the ranks ...
- 47.3k
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Simple example of a ring which is normal but not CM
I try to keep a list of standard ring examples in my head to test commutative algebra conjectures against. I would therefore like to have an example of a ring which is normal but not Cohen-Macaulay. I'...
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409
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Is tensoring with a module representable iff it is locally free of finite rank?
Motivation:
It's nice when you can think of the elements of an $A$-module $M$ as sections some $A$-scheme $Y\to Spec(A)$. That is, maps $Spec(A)\to Y$ such that $Spec(A)\to Y \to Spec(A)$ is the ...
CommunityBot
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Splitting matrix of rank one
Let R a normal domain, that is an integrally closed noetherian domain, like Dedekind domains, UFD, etc
Let A=(a i j ) a matrix with elements in R and dimension n x m.
Suppose
rank A=1 ↔ all 2 x ...
- 156k
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Examples of finite local rings of length 2 or 3
What is an example of a finite local rings, that has length 2 or 3?
I want something different from $F_{q}[x] / x^{i}$ for $i=2, 3$; I'm looking for something more interesting. If you can give me ...
CommunityBot
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3
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Factorization of elements vs. of ideals, and is being a UFD equivalent to any property which can be stated entirely without reference to ring elements?
Why exactly is the unique factorization of elements into irreducibles a natural thing to look for? Of course, it's true in $\mathbb{Z}$ and we'd like to see where else it is true; also, regardless of ...
- 65.4k
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When can one localize Ext?
Let $R\to S$ be a ring map such that $S$ is projective over $R$ (I am willing to assume $S=R[X_1,...,X_n]$). Let $M,N$ be finite $S$-modules. Let $P\in Spec R$ such that $M_P$ is $R_P$-flat. Under ...
- 1,609
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Is (relatively) algebraically closed stable under finite field extensions?
Let $F\subset F'$ be a field extension such that $F$ is algebraically closed inside $F'$, i.e. if $x\in F'$ is algebraic over $F$ then $x$ belongs to $F$ itself.
Let now $F\subset L$ be a finite field ...
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500
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Are any finitely generated reflexive module a 2nd syzygy?
Are any finitely generated reflexive module a second syzygy?
(I´m thinking especially in normal noetherian domains)
More general...
Are any divisorial lattice a second syzygy?
(I´m thinking ...
- 30.5k
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When is a commutative ring the limit of its local rings?
Let $A$ be a commutative ring. Then we get local rings $A_p$ by localizing at each prime ideal $p$. Moreover, we get $A_p \rightarrow A_q$ when $p$ contains $q$. So we get a big diagram indexed by the ...
CommunityBot
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Can different modules have the same symmetric algebra? (answered: no)
Algebraic geometry allows one to think of an $A$-module $M$ geometrically as a module of functions on the $A$-scheme $\mathrm{Spec}(\mathrm{Sym}(M))$.
I'm wondering if anything is lost in just ...
- 11.3k
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Intuition for Nagata's altitude formula?
This is theorem 14.C on p.84 of Matsumura's commutative algebra.
Let $A$ be a noetherian domain, and let $B$ be a finitely generated overdomain of $A$. Let $P \in Spec(B)$ and $p = P \cap A$. Then ...
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Solvable subgroups of groups of polynomial automorphisms
Does every finitely generated free solvable group embed into the group of polynomial automorphisms of some C^n?
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R2 and S3 for rings.
For a noetherian ring R, Serre's criterion for normality states that R is normal if and only if R satisfies conditions R1 and S2, where R1 is regularity in codimension one, and S2 is Serre's condition ...
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Dense section of sheaves of modules
Here is something that isn't yet very clear to me. Say, I've got a commutative ring A. I consider the affine scheme from A, so it's a sheaf of rings over Spec A.
EDIT: And additionally let's say ...
- 41.4k
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Spectra of rings that are projective module over a subring
This question is motivated from my last question here. I wonder if one has a ring A and an over-ring of this ring say B, and if we know that B is a projective A-module can we have a particular idea of ...
CommunityBot
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Atiyah-MacDonald: exercise 5.29 - "local ring of a valuation ring"
The exercise is the following:
Let $A$ be a valuation ring, $K$ its field of fractions. Show that every subring of $K$ which contains $A$ is a local ring of $A$.
Does anyone know what is meant by "...
- 47.5k
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6
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Solving polynomial equations when you know in which number field the solutions live
Suppose I have a bunch of polynomial equations with coefficients in a number field, and suppose further that I'm guaranteed a priori that they have a solution in that number field. Can I leverage ...
CommunityBot
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What representative examples of modules should I keep in mind?
So here's my problem: I have no intuition for how a "generic" module over a commutative ring should behave. (I think I should never have been told "modules are like vector spaces.") The only ...
- 2,992
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Recovering a monoidal category from its category of monoids
What kind of additional properties and/or structures one needs to impose on the category
of (commutative or noncommutative) monoids of some monoidal category
so that one can recover the original ...
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What is the homology of the real coordinate ring of SO(n,R)? Other compact matrix groups?
As someone whose knowledge of cohomology is patchy and picked up on a need-to-know basis, and whose algebraic geometry is even worse, I wondered if someone could help with this question. (I ran into ...
- 25.8k
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Which commutative rigs arise from a distributive category?
A rig is an algebraic object with multiplication and addition, such that multiplication distributes over addition and addition is commutative. However, instead of requiring that the set forms an ...
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