Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
1
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2answers
37 views
Ascending chain condition on radical ideals
There is a basic theorem in the geometry of schemes saying that the Spec of a Noetherian ring is a Noetherian topological space. It can be formulated as the ACC condition implies the ACCR condition ...
0
votes
0answers
26 views
Unimodular triangulation and affine toric variety
Let $\mathcal{K}$ be a $pointed$ rational cone in $\mathbb{R}^d$ with extremal rays generated by $r_1,r_2,\dots, r_m\in \mathbb{Z}^d$.
Here, pointed means that all $r_i$ lie strictly on one side of ...
4
votes
1answer
230 views
Is the ring of invariants Noetherian?
Let $R$ be a complete regular local ring whose residue field is perfect. Suppose that a finite group $G$ acts on $R$ by ring automorphisms in such a way that the induced action on the residue field is ...
5
votes
0answers
63 views
flatness and derived completion
Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion.
If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat.
If $A$ is not noetherian, ...
0
votes
1answer
104 views
Properties of Integral Closure [on hold]
Definition(Integral closure): Let $R$ be a ring and $I$ an ideal of $R$. An element $x$ is said to be integral over $I$ if $x$ satisfies a monic equation
$x^n + i_1x^{n−1} + ··· + i_n = 0$ such ...
6
votes
2answers
181 views
Homological criterion for $A(B \cap C) = AB \cap AC$?
Is there a homological criterion for the condition $A(B \cap C) = AB \cap AC$ for ideals in a ring $R$? I mean a statement such as "the given equation holds if and only if (some $\operatorname{Tor}$, ...
10
votes
1answer
216 views
When is $X \rightarrow \text{Spec}(C(X))$ a homeomorphism?
Let $X$ be compact Hausdorff topological space. Consider the ring $C(X)$ of continuous functions $X \rightarrow \mathbb C$ (we do not consider the C* algebra structure, just consider $C(X)$ as a ring) ...
1
vote
1answer
164 views
Relation between intersection and product of ideals
Let $C$ be a smooth projective (irreducible) curve in $\mathbb{P}^n$ for some $n$. Denote by $I_C$ the ideal of $C$. Let $g \in I_C\backslash I_{C}^2$, an irreducible element. Is it true that for any ...
1
vote
1answer
122 views
Noetherianess of a finite module over a noethrian ring without Axiom of Choice
All rings are assumed to be commutative with 1.
We say a module over a ring is strictly noetherian if every non-empty set of submodules has a maximal member. We say a ring is strictly noetherian if it ...
0
votes
1answer
90 views
if $R$ is Noetherian local with a finite module of finite injective dimension and if “?” , then $R$ is *Gorenstein*
I know that if $R$ is Noetherian local with a finite module of finite injective dimension, then $R$ is Cohen-Macaulay.
Can one add assumptions on $M$, so that $R$ be Gorenstein or Complete ...
1
vote
0answers
52 views
The kernel of the residue map before passing to Milnor's K-theory
Let $F$ be a field of zero characteristic. All groups are taken modulo torsion.
Consider a residue map from the exterior algebra of the multiplicative group of the function field of the projective ...
1
vote
0answers
64 views
Are all (graded) Artinian complete intersections like this?
I'm trying to prove some stuff (it's not important what) about (graded) Artinian complete intersections $R=\mathbb{C}[x_1,\ldots,x_n]/I$, where the $x_i$ have certain positive weights and where $I$ is ...
1
vote
0answers
67 views
“Exceptional components” of the exceptional divisor of a blow up
Assume we are blowing up an ideal $I$ on an affine variety $X$, let $E$ be the exceptional divisor, and $P$ be a (closed) point in $V$, the zero set of $I$. Is there any algorithm to check that $E$ ...
6
votes
1answer
254 views
Find a polynomial not in any ideal generated by polynomials of total degree $o(n)$
Is there an explicit nontrivial (= not a constant) polynomial $p \in \mathbb{C}[x_1, \ldots, x_n]$ such that, for any ideal $I \not= \mathbb{C}[x_1, \ldots, x_n]$ generated by $f_1, f_2, \ldots, f_m$ ...
0
votes
0answers
28 views
normality of truncated arc space
Let $X=Spec(A)$, with $A$ a normal $k$-algebra of finite type, $k$ is a field.
For any integer $n$, let $X(k[t]/(t^{n}))$ the $n$-th truncated arc space, is it also normal?
Same question for ...
7
votes
0answers
79 views
Weierstrass division theorem for henselian rings
Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...
3
votes
1answer
171 views
Generators vs minimal degree polynomials of ideals
Given an ideal $I$ of $\mathbb{R}[X_1,X_2,X_3,X_4,X_5]$ generated by two unknown polynomials. I know two homogenous polynomials $p_1 \in I$ and $p_2 \in I$ such that
$p_1$ is of degree 2 and up to a ...
3
votes
3answers
146 views
canonical module can be identified with an ideal. how can one reach that ideal?
Let $R[[X,Y,Z]]/(X,Y)\cap (Y,Z)\cap(X,Z)$. then $R$ is Cohen-Macaulay ring and has a canonical
module, $K$. By Proposition 3.3.18 of Bruns_Herzog, $K$ can be identified with an ideal in $R$.
So we ...
3
votes
1answer
255 views
Invertibility of a matrix whose entries are certain binomial coefficients
Let $l$ be a positive integer. Does the matrix
$$
M_l \ := \ \left( \binom{l-(2p+1)}{j} \right)_{0\leq p,j \leq[(l-1)/2]}
$$
have nonzero determinant?
5
votes
0answers
62 views
How far finiteness dimension can be from edges? Example for $f_m(S/I)\ge depth S/I+2$
Let $ (R,m) $ be a commutative unital noetherian local ring (with $m$ as its maximal ideal), $ I $ an ideal of $ R $, and $ M $ a finite $R$-module with $\dim M\gt 0$. $f_I(M) = \inf\ \{i : H_I^i(M)\ ...
1
vote
1answer
100 views
Graded version of Baer's Criterion
Baer's Criterion for injectiveness of modules says: "An $R$-module $E$ is injective iff for all ideals $I$ of $R$, every homomorphism $f\colon I \to E$ can be extended to $R$." I wonder if there is a ...
1
vote
0answers
90 views
A Characterization of Closed Ideals in $C^{\infty}(\mathbb{R}^n)$
The space $C^{\infty}(\mathbb{R}^n)$ can be turned into a topological ring using the Whitney topology. Whitney's Spectral Theorem says that the closure of an ideal in this ring is the ideal of all ...
0
votes
0answers
74 views
: References on complete intersections rings
As we know, for the local ring, we have
regular $\subset$ complete intersection $\subset$ Gorenstein $\subset$ Cohen-Macaulay
but it seems that the refenences of complete intersection are more ...
0
votes
1answer
61 views
Canonical module of rees algebra
[Example 4.27, Integral Closure, Rees Algebras, Multiplicities, Algorithms] by Vasconcelos, says that if $I=(f_1,\ldots,f_g)$ is an ideal generated by a regular sequence with $g\ge 2$ then the ...
2
votes
2answers
58 views
Does $fd(M)\lt \infty$ and $id(M)\lt \infty $ imply that $R$ is Gorenstein?
$(R,m)$ is a local Noetherian ring. $M$ is a nonzero finite $R$-module of finite injective dimension($id$). It is known that if $R$ is Gorenstein, then $M$ has finite flat dimension ($fd$). I wonder ...
-1
votes
0answers
36 views
Integrality of a field extension
Let $L\subset K$ be fields such that $K=L(a_0,a_1,...,a_m, b_0,b_1,...,b_n)$. Is the extension of fields $L(a_0,b_0,a_m,b_n,\{\sum_{i+j=r}a_ib_j)\}_{r=1}^{m+n-1})\subset K$ an algebraic extension?
2
votes
1answer
238 views
What is the explicit ideal (wrt the Lazard ring) generated by the associativity of formal group laws?
Quick Preliminaries:
A commutative formal group law is a formal power series $F(x,y)=\sum_{ij}c_{ij}x^iy^j$ that satisfies:
Commutativity: $F(x,y) = F(y,x)$
Identity: $F(x,0)=x=F(0,x)$
...
-2
votes
0answers
168 views
intuitive interpretation of the multiplicity and a reference [migrated]
Although logically i can understand and use multiplicity, yet, the concept of multiplicity of a module is not completely clear for me.
I wonder what idea is behind this definition and What is ...
2
votes
1answer
87 views
A criterion for complete intersection in terms of the Hilbert series?
Let $(R, \mathfrak{m})$ be a complete local ring (of dimension $2$ if that makes a difference). I would like to be able to decide whether or not $R$ is a complete intersection (meaning, a quotient of ...
0
votes
2answers
158 views
Algebraic characterization of commutative rings of Krull dimension 1,2, or 3
A commutative ring $R$ (with $1$) is $0$-dimensional if and only if $R/\sqrt 0$ is von Neumann regular. Besides this result, there is a wealth of information about the algebraic structure of ...
10
votes
2answers
323 views
Computing intersection of subrings
Let $R$ be a finitely generated commutative ring over a field, for concreteness.
If $S,T \leq R$ are two finitely generated subrings, is their intersection
also finitely generated?
(Certainly ...
1
vote
0answers
51 views
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. ...
0
votes
0answers
98 views
How can every divisor be reached by a sequence of blow-ups?
The following is a result of Zariski [cf. Lemma 2.45 of Birational Geometry of Algebraic Varieties].
$X$ : an algebraic variety over a field $k$.
$(R,m)$ : a DVR of the quotient field $K(X)$ ...
0
votes
1answer
76 views
Examples of fractional ideals whose inverse does not commute with the product
Let $R$ be an integral domain, $K$ its field of fractions, and $I,J$ fractional ideals.
If $R$ is a Krull domain, then $(R:_KIJ)=(R:_KI)(R:_KJ)$, or $(IJ)^{-1}=I^{-1}J^{-1}$.
But I can't see any ...
3
votes
1answer
77 views
Example of fractional ideal whose inverse does not commute with localization
Let $R$ be an integral domain, and $K$ its field of fractions. It is well known that for a finitely generated fractional ideal $I$ of $R$, and $S$ a multiplicative set we have ...
1
vote
1answer
143 views
specific question related to the extension of an integrally closed domain and the residual fields
I really need help !
In a previous thread, I have asked for the solution of a general question, without getting answers. Since this question was posted, I have reduced the problem to the following ...
0
votes
0answers
62 views
An embedding of modules by tensor product over a Noetherian domain
I have a problem on Ring theory. I would like to prove or disprove the following statement:
Let $R$ be a Noetherian domain. Then by the Goldie theorem $R$ have $Q$ as a full ring of quotients and $Q$ ...
2
votes
1answer
228 views
Automorphisms of ideals of $\mathbb{C}[t]$
Let $f(t)\in\mathbb{C}[t]$, and let $I_f$ be the ideal in $\mathbb{C}[t]$ generated by $f(t)$.
The ideal $I_f$ has a natural $\mathbb{C}$-algebra structure. My question is the following:
For ...
3
votes
0answers
108 views
Looking for the correct version of a wrong statement from Barvinok's book on convex polyhedra
The book I'm concerned with is "Integer Points in Polyhedra" by A. Barvinok, which, I must say, is turning out to be highly fascinating.
A real finite-dimensional vector space $V$ defines the ...
0
votes
0answers
41 views
Extension of canonical surjection to a place of fraction field
Let $R$ be an integrally closed domain, $K$ its field of fractions, and $m$ a maximal ideal of $R$.
By Chevalley theorem, the canonical surjection $R\to R/m$ extends in at least one way to a place ...
6
votes
1answer
294 views
Irreducibility of a polynomial
For $n\ge 1$, let $g(x_1,x_2,\ldots,x_n)$ be an irreducible homogeneous polynomial in $n$ variables over a field $k$ and $f(x)$ an irreducible polynomial of $k[x]$. Is $f(g(x_1,x_2,\ldots,x_n))$ ...
4
votes
0answers
144 views
Inversion, Koszul duality, combinatorics and geometry
According to this MO answer Koszul duality is related to operations on generating series;
1) multiplicative inversion for quadratic algebras,
2) compositional inversion for quadratic operads,
3) ...
3
votes
2answers
130 views
Divisibility of the degree of an extension by the degree of its residual field
Let $A$ be an integrally closed domain whose quotient field is $K$, $L$ be a finite Galois extension of $K$, and $B$ be the integral closure of $A$ in $L$. Let $M_A$ be a maximal ideal of $A$, and ...
0
votes
0answers
78 views
Moduli space of points - Gorenstein ideal
I've been working on algebraic covers, $\varphi\colon X\rightarrow Y$, ($\varphi_*\mathcal{O}_X$ is a locally free $\mathcal{O}_Y$-algebra of rank d).
I'm more interested in the algebraic point of ...
4
votes
0answers
72 views
Multiplicity of $Ext^{d-t}(M,\omega_R)$, ($d=\dim R, t=\dim M$)
Let $R=\bigoplus_{i \geq 0} R_i$ be a Cohen-Macaulay graded ring ($R_0$ is a field and $R$ is generated by $R_1$) of dimension $d$ with canonical module $\omega_R$, and $M$ a graded Cohen-Macaulay ...
1
vote
1answer
107 views
Reducedness of a ring with prime nilradical
Let $A$ be a regular ring and $\mathfrak q$ be an ideal, such that $\sqrt{\mathfrak q}$ is prime. Further assume that $\mathfrak q$ is locally principal (i.e. $\mathfrak q$ is an irreducible divisor ...
4
votes
2answers
242 views
Irreducible/prime/indivisible elements
in what follows all the rings are commutative, nontrivial, with unit.
Recall the following definitions:
1) $\pi\in A$ is prime if $(\pi)$ is a nonzero prime ideal
2) $\pi\in A$ is irreducible if ...
1
vote
0answers
59 views
ramification index generalized
I am trying to rewrite the theory of decomposition/inertia/ramification groups independently of the theory of Dedekind or valuation rings (I believe this has been done elsewhere, but I found only few ...
6
votes
2answers
550 views
Is being reduced a generic property of schemes?
(Naive formulation:) Let $X$ be an (irreducible) affine variety (over an algebraically closed field $k$) and $I$ be an ideal of the coordinate ring $R$ of $X$. Assume $Y = V(I)$ is equidimensional. ...
7
votes
0answers
170 views
What is a good introduction to cluster algebras from surfaces?
What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory?
In my view, that means it should start off with unpunctured surfaces (and in ...
