Questions tagged [ac.commutative-algebra]
Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
5,497 questions
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Is every Noetherian *connected* ring a quotient of a Noetherian domain?
This question is a strengthening of this question (answered negatively), and arose due to David Speyer's answer here.
Geometrically, this asks if every Noetherian connected affine scheme can be ...
- 706
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0
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164
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Minimal free resolution of sum of ideals
Let $S$ be the polynomial ring in $n$ variables, and let $I_1$ and $I_2$ be ideals in $S$. What can be said about the $\mathbb{Z}$-graded minimal free resolution of $I_1+ I_2$ in terms of the $\mathbb{...
- 179
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112
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A question about a specific inverse proposition of Combinatorial Nullstellensatz
From the Hilbert's Nullstellensatz, we have the following consequence which is usually called Combinatorial Nullstellensatz:
Let $F$ be an arbitrary field, and let $f = f(x_1,x_2,\cdots,x_n)$ be a ...
- 1,997
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115
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cohomlogy of Diagonal ring
Let $S=\bigoplus_{\underline n\in\mathbb N^r } S_{\underline n}$ be a standard multigraded ring over a local ring and M be a finitely generated $\mathbb N^r $-graded $S$-module. Let $M_{\Delta}=\...
- 1,713
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Two definitions of smoothness?
This is confusing, there appear to be possibly two definitions of smoothness in algebraic geometry for a morphism $f: X \rightarrow Y$ of schemes of finite type over an arbitrary field $k$.
...
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246
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Global dimension of a subalgebra with all units
(All rings here are always assumed to be unital and associative).
Setup
Let $R$ be a ring, and $A$ and $B$ be $R$-algebras, with $A$ a commutative subalgebra of $B$ satisfying:
If $u$ is a unit in $...
- 5,405
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2
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872
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Rational power series
If we let $R=\mathbb{Z}[x]$ and $D=\mathbb{Z}[[x]]$. We say that $z\in D$ is rational if there is $g\in R$, $g\ne 0$ such that $zg\in R$. Let $S$ be the set of all rational elements in $D$. Then $S$ ...
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2
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976
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Question on a theorem of Eisenbud's and Harris' "The geometry of schemes"
My problem is perhaps a general lack of understanding but it occurred in a special case of a theorem in Eisenbud's and Harris' "The geometry of schemes" (Theorem VI-29). Let $K$ be a field and $n\in\...
- 2,782
2
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226
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why do we care about the irreducibility of parameter ideals?
It is well known that a local commutative unital ring $R$ is Gorenstein if and only if every parameter ideal is irreducible. Why the irreducibility of parameter ideals in a Gorenstein local ring is ...
- 591
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354
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Weak assassins and essential morphisms
Let $R$ be a commutative ring and let $M\rightarrow N$ be an essential morphism of $R$-modules. Then, $M$ and $N$ have the same associated primes.
Over non-noetherian rings the notion of associated ...
- 6,700
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1
answer
496
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Regular sequence of power sum symmetric polynomials in polynomial ring.
Let $S=\mathbb{C}[x_1,\dots,x_n]$ be a polynomial ring and $p_a=x_1^a+\cdots+x_n^a$ be a power sum symmetric polynomial in $S$. Let $n \geq 3$.
Question: To show $p_m,p_{2m}, \dots,p_{nm}$ forms a ...
- 446
2
votes
1
answer
570
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Localization of a pure-injective module is pure-injective?
Hi,
is there some work on localization of pure-injective modules? Is a localization of a pure-injective module pure-injective?
By localization I mean the standard localization defined for any ...
- 41
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355
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Cubic field and the corresponding cubic binary form
I am currently reading about binary cubic forms and cubic number fields (mainly about using binary cubic forms with integer coefficients to parametrize orders in the cubic field) and I thought it ...
- 21
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125
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Is there an explicit way to glue a stable map in projective space by writing down the family of maps explicitly in terms of polynomials?
Let $v_1:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$
and $v_2:\mathbb{P}^1 \longrightarrow \mathbb{P}^2$ be two holomorphic maps
of degree $d_1$ and $d_2$ respectively. Suppose they agree at some ...
- 3,245
1
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2
answers
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 ...
2
votes
1
answer
266
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On Non F-pure ideal and Sharp F-Purity for a pair $(X, \Delta)$ where $K_X+\Delta$ is NOT $\mathbb{Q}$-Cartier
Suppose $(X,\Delta\ge 0)$ is a pair such that $(p^g-1)(K_X+\Delta)$ is an Integral Weil Divisor for some $g>0$ and $X$ is a normal variety. Define $\mathcal{L}_{e,\Delta} = \mathcal{O}_X( (1-p^g)(...
- 165
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1
answer
235
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Projective bundles
Fix $n$ and let
$0\leftarrow \mathcal{F}\leftarrow \bigoplus \mathcal{O}_{\mathbb{P}^n}(a_i)\leftarrow \bigoplus \mathcal{O}_{\mathbb{P}^n}(b_i)\leftarrow \cdots$
be an exact sequence.
Then we can ...
- 39
10
votes
2
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610
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When is tensoring with a module representable by a scheme?
Consider the following: Let $A$ be a commutative ring, let $M$ be an $A$-module. When is the functor from $A$-algebras to Sets given by $R \mapsto R \otimes M$ representable by an $A$-scheme?
Unless ...
- 5,548
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1
answer
2k
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Multiplicity of a singular point
Let $X$ be a smooth projective complex variety. Assume that $Z$ is a subvariety. Let $T$ be a generic complete intersection of codimension $\dim Z-1$. Assume that $p$ is a point in $Z_T:=T\cap Z$. Is ...
- 2,444
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593
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Flattening techniques of Raynaud and Gruson
Suppose $R$ is an adic valuation ring with a finitely generated ideal of definition. Let $A$ be an $R$-algebra of topologically finite type, i.e. $A$ is isomorphic to $R<\zeta_1,\zeta_2,...,\zeta_n&...
- 31
1
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1
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382
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An equalizer in commutative algebras
I'm feeling quite a bit embarrassed to ask such a basic thing, but I can't seem to figure it out. Let $R$ be a commutative ring and $A$ be a commutative $R$-algebra. Is the fork
$$
A \xrightarrow{i} A ...
- 6,162
5
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0
answers
420
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Ring of invariants of a finite subgroup of $GL_2(\mathbb{C})$
In the paper:
Kac, Victor; Watanabe, Keiichi, Finite linear groups whose ring of invariants is a complete intersection. Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 2, 221–223
it is said in Remark 2 ...
- 221
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1
answer
327
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Examples of complexes of modules for wich homomorphisms "homological" implies "homotopic"
Let $V$ and $V^\prime$ - complexes of modules over ring $A$, and $f, g$ - homomorphisms $V\rightarrow V^\prime$.
I am interested in various conditions on $A, V, V^\prime$: ($f$ and $g$ are ...
- 101
2
votes
1
answer
268
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Rings satisfying a certain property
In the course of reading a paper , I've encountered the following property of interest.
If $R$ is a ring, say it satisfies (*) if: For any smooth, irreducible $R$-algebra $B$ of finite type such ...
- 287
1
vote
1
answer
438
views
Filter-regular sequence and regularity
Let $A$ be a commutative Noetherian ring, $R$ be a standard graded algebra over $A$, $M$ be finitely generated graded $R$-module. Let $R_{+}$ be the irrelevant ideal. The Castelnuovo-Mumford ...
- 325
4
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362
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Comparing different Euclidean algorithms on a Euclidean domain
I have posted this question at stackexchange (502413), without responses until now.
In the papers by T. Motzkin: The Euclidean Algorithm, Bull. AMS 55, 1949, pp. 1142--1146 and P. Samuel: About ...
- 2,454
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3
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1k
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Is there a software package that does Schubert Calculus computations?
Is there a good software package for doing computations in the cohomology ring of Grassmannians? Things like, I can write down a polynomial in, in fact, special Schubert classes, but it's one where ...
- 16k
5
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157
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Link between abelian groups and endomorphisms
When teaching Algebra, I try to share my fascination about two apparently unrelated questions, which turn out to involve the same theory:
classifying the finitely generated abelian groups,
...
- 52.3k
1
vote
1
answer
464
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Does the global dimension gldim R equal the projective dimension of R as bimodule over its enveloping algebra?
I know that generally the answer is no, for example the weyl algebra。
But is this true for commutative algebra？ or we may restrict to affine commutative algebras。
Maybe ，it is a classical result. So,...
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1
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0
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141
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Normalization (integral closure) of $\mathbb Z_p[x]$ in function field of a curve to obtain Model of curve
I want to follow this construction of a normal model of a curve:
Let $p\neq 2,3$ and $Y\to \mathbb P¹$ be a smooth projective curve over $\mathbb Q_p$ with function field $L/\mathbb Q_p(x)$ e.g. $L=\...
- 171
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436
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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
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186
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Categorical characterization of closed imbeddings
Let $f\colon X\to Y$ be a morphism of schemes.
Let $F_X$ and $F_Y$ be the contravariant functors from the category $Sch$ of schemes to the category of sets defined via the Yoneda construction, i.e. $...
- 21.8k
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0
answers
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Is the radical of a homogeneous ideal homogeneous?
Let $S$ be an $M$-graded $R$-algebra, where $M$ is some monoid, and $I\subset S$ an homogeneous ideal. The original, naïve, question, was: is it true that $\sqrt{I}$ is homogeneous? In this generality,...
- 1,811
4
votes
1
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594
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Morley's Theorem and real algebraic geometry
Consider the following attempt at a ``thought-free'' proof of Morley's
Theorem.
Let $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$ denote three vertices of
a generic triangle.
Let $(a_1,b_1)$, $(a_2,b_2)$ ...
- 17.6k
4
votes
1
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543
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Generators of a certain ideal
In view of Mariano Suárez-Alvarez's answer I see how badly phrased my question was, and decided to rewrite it. The drawback is that some comments of Martin Brandenburg are now incomprehensible, but I ...
- 4,416
1
vote
1
answer
928
views
Hilbert-Samuel function and that of the irreducible components.
How to obtain a relation between the Hilbert-Samuel function of the local ring at a point of a reduced, but not necessarily irreducible variety, and the Hilbert-Samuel functions of the corresponding ...
- 807
3
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168
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Invariant Theory over finite adeles
Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$.
I am ...
- 2,384
13
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1
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908
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Computational Question about finite local rings:
Let $(A,\mathfrak{m})$ be a local Artinian ring with
finite residue field, which I'm happy to assume is $\mathbf{F}_3$.
(In particular, $A$ has finitely many elements.)
I would like to do some ...
user631
1
vote
1
answer
573
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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 $\...
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230
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A question about the unbounded derived category of the polynomial ring in infinitely many variables
In this moment I am trying to understand the derived category of the polynomial ring in infinitely many variables over a field $k$, $R=k[x_{1},x_{2},\dots]$ and I am wonder if it is true that $Hom_{D(...
- 11
5
votes
2
answers
495
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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
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223
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Semidirect products of semigroups [closed]
Let $S,T$ be two semigroups. A function $f:S\to T$ is called multiplicative if for any $x,y\in S$ we have $f(xy)=f(x)f(y)$. If $T=S$ then $f:S\to S$ is called automorphism on $S$.
A function $f:S\to\...
- 131
5
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1
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541
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Localizability of differential operators a la Grothendieck
Hello,
Maybe this question is trivial, so sorry
Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1).
Then we can define the module of differential operators $D^{\leq n} (A)$,...
- 5,562
2
votes
1
answer
268
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Question on localization technique
In the book "Local cohomology : An algebraic introduction with geometric application", page 289 there is a proof of the following theorem :
Assume that $R=\bigoplus_{n}R_{n}$ is positive graded ...
- 87
11
votes
1
answer
1k
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The associated prime ideals of $Ext^i_R(M,N)$
If R is a commutative noetherian ring, M and N are modules with M finite. It is well known in commutative algebra that $AssHom_R(M,N)=Supp(M)\cap Ass(N)$. But I want to know whether there is a ...
- 1,207
4
votes
0
answers
212
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A doubt from the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring"
I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I have a doubt about the first part of Theorem 2.5. In the proof $(1)\iff ...
- 1,713
3
votes
1
answer
636
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Left Adjoint to the Forgetful Functor on $\lambda$-rings?
The forgetful functor from the category of $\lambda$-rings to that of rings has a right adjoint in the form of the universal $\lambda$ functor $\Lambda$, which is isomorphic to the big Witt vectors ...
- 822
1
vote
1
answer
771
views
Prime ideals in coordinate rings
Is there a way to characterise prime ideals in affine coordinate rings (i.e. quotients of polynomial rings). To be more specific, how can I say if principal ideals in such rings are prime or not in an ...
- 11
2
votes
1
answer
235
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Which algebras can be presented as filtered colimits of f.g. regular ones with smooth connecting morphisms?
Let $R$ be a regular (commutative associative unitial) algebra over a prime field $F$ (i.e. $F=F_p$ or $F=\mathbb{Q}$); assume that it is noetherian excellent (and even of Krull dimension $1$). What ...
- 16.9k
0
votes
2
answers
617
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An element in the product of schemes
Let $ f : X \to S, g : Y \to S$ be the scheme morphisms, and $ X \times_S Y$ be the product of shemes. Let $ z \in X \times_S Y$ , and $ x=p(z) , y=q(z) ,$ where $ p: X \times_S Y \to X, q: X \times_S ...
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