Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

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3
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127 views

polynomial relations between modular functions

$\newcommand{\Qbar}{\overline{\mathbb{Q}}}$ We define a modular function to be a meromorphic modular form of weight 0 for some subgroup (not necessarily congruence) $\Gamma\le\text{SL}_2(\mathbb{Z})$ ...
7
votes
1answer
275 views

What are some examples of total derived functors that can't be computed from a functorial replacement?

(Or more generally, what are some examples of Kan extensions which are not pointwise?) By a total derived functor of a functor $F: C \to D$, I simply mean a (left or right) Kan extension of $F$ along ...
2
votes
0answers
66 views

quasi-ordinary singularities on a versal deformation?

Let $V$ be a variety over $\mathbb{C}$ and suppose $O$ is a singular point of $V$. Are there conditions on $(V,O)$ such that a versal deformation $W$ of $(V,O)$ has only quasi-ordinary singularities. ...
6
votes
1answer
234 views

Vector bundles on open (affine) curves

It is well-known by Grothendieck (or earlier by Dedekind-Weber) that every vector bundle on $\mathbb{P}^1_k$ for $k$ a field decomposes into a sum of the line bundles $\mathcal{O}(k)$. As ...
2
votes
1answer
119 views

On conflicting descriptions for tor of a local cohomology group

Let $X$ be a smooth projective surface and $C$ a Cartier divisor on $X$. Denote by $\mathcal{H}^1_C(\mathcal{O}_X)$ the sheaf associated to the presheaf $U \mapsto H^1_{C \cap U}(\mathcal{O}_X|_U)$. ...
5
votes
1answer
171 views

In a noetherian commutative ring with only one associated prime, is the nilradical locally free?

The title says it all. I suspect that the answer in general is no, although my intuition tells me that a jump in the dimension of the fibre of the nilradical at some point of Spec(A) can occur only ...
3
votes
0answers
104 views

Which sheaves on a projective bundle are flat over the base scheme?

Assume $X$ is a noetherian scheme over $\mathbb{C}$ and $E$ a locally free sheaf of finite rank on $X$. Denote the the associated projective bundle by $f: \mathbb{P}(E)\rightarrow X$. Are there any ...
2
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0answers
86 views

When does $R [x]/I $ have infinitely many idempotents in special case?

At < When does $R [x]/I $ have infinitely many idempotents? >, Er_Ro asked the following question. Let $R $ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking ...
13
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0answers
252 views

What are retracts of polynomial rings?

Is there a known example of a ring endomorphism $f: \mathbb{Z}[x_1, \ldots, x_n] \to \mathbb{Z}[x_1, \ldots, x_n]$ such that $f \circ f = f$ but whose image is not isomorphic to a polynomial ring? ...
1
vote
1answer
289 views

Bertini-type theorem in positive characteristic [closed]

Let $f:X \to Y$ be a morphism of finite type of irreducible schemes over an algebraically closed field of characteristic $0$. Assume that $Y$ is non-singular. Let $x \in X$ be a closed point and ...
3
votes
1answer
185 views

A basic question on local cohomology

I had posted this question on stackexchange but did not get any response, hence putting it up on mathoverflow. Let $X$ be a smooth, projective variety, $i:X \hookrightarrow \mathbb{P}^n$ a closed ...
2
votes
1answer
386 views

When does $R [x]/I $ has infinitely many idempotents?

Let $R$ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking for a ring with finitely many idempotents and an unextended ideal $I$ in $R[x]$ such that $R[x]/I$ has ...
5
votes
1answer
123 views

Can relative flatness of a sheaf be tested using (faithfully) flat morphisms?

Given a $\mathbb{C}$-scheme $S$, two $S$-schemes $X$ and $Y$ that are flat over $S$ and a coherent sheaf of $O_Y$-modules $F$. Assume we have a (faithfully) flat $S$-morphism $\pi: X \rightarrow Y$ ...
2
votes
0answers
67 views

Torsion ideal in symmetric algebra

Let D be a a commutative domain, M be a D-module without torsion and S(M) its symmetric algebra. Is the D-torsion ideal of S(M) the prime ideal of S(M) ?
4
votes
1answer
255 views

Completion of a finite field extension is also finite?

Let $(L,w)/(K,v)$ be a finite extension of valuation fields, and let $L_w$, $K_v$ be the respective completions of $(L,w)$, $(K,v)$. Is the field extension $L_w/K_v$ finite? For nonarchimedean ...
5
votes
1answer
211 views

Basic questions about simplicial commutative rings

I am trying to learn about simplicial commutative rings, and would be grateful if one can help with some basic facts about them. Basically, I would like to understand how to do homological algebra ...
5
votes
0answers
208 views

A roadmap to learn about finite-dimensional commutative associative real or complex unital algebras

I've always been secretly fascinated with the rich structure and applications of finite-dimensional associative unital algebras over complete fields. In particular, I am very much interested in the ...
2
votes
0answers
73 views

conditions for a subfield of a rational function field to be algebraically closed

Let's start with the following general question. Let $k$ be the ground field. Let $K=k(x_1,\cdots, x_n)$ be a rational function field and let $L$ be a subfield of $K$. Is there a condition to ...
1
vote
0answers
136 views

Does the functor of taking invariants commute with tensor products? [closed]

Suppose that $G$ is a group acting on a commutative ring $R$, inducing an action on each $R$-module. For any $R$-module $M$, let $M^G$ denote the collection of elements of $M$ invariant under the ...
3
votes
0answers
84 views

The Geometric Intuition behind Minimal Primary Decompositions versus an arbitrary Primary Decomposition

Given an ideal in a Noetherian domain, it has a primary decomposition. This decompositions may not be minimal. I am interested in the relationship of primary ideals that are not in a minimal primary ...
6
votes
0answers
84 views

Constructing the normal sheaf for the plucker embedding in MAGMA (or a similar programming language)

How would one construct the normal sheaf $N_{G(2,6)/\mathbb P^{14}}$ to the plucker embedding of the grassmannian $G(2,6) \rightarrow \mathbb P^{14}$ as a sheaf in MAGMA (or another programming ...
3
votes
1answer
263 views

Connections to physics, geometry, geometric probability theory of Euler's beta integral (function)

Euler"s integral for the beta function $B(s,\alpha) = $ (with $x = 1$) $$ \frac{(s-1)!(\alpha-1)!}{(s+\alpha-1)!} x^{s+\alpha-1} = \int_0^\infty t^{s-1}\; H(x-t) \; (x-t)^{\alpha-1} dt = \int_0^x ...
13
votes
2answers
518 views

Every finitely generated flat module over a ring with finitely many minimal primes is projective

Over a commutative ring $R$, a finite type locally free (weak sense) module for which the rank function is locally constant is projective. If we notice that for each minimal prime $p$ of the ring, ...
8
votes
1answer
293 views

q-Integer-valued polynomials

For $n \in \mathbb{Z}_{\geq 0}$, let $[n]_q := (1-q^n)/(1-q) = (1+q+...+q^{n-1})$ as is customary, with $[0]_q=0$. Let $R$ be the subring of $\mathbb{Q}(q)[x]$ consisting of all $f$ such that ...
1
vote
1answer
137 views

A characterization for the ideals of $A+XB[X]$ and $A+XB[[X]]$

Let $A \subseteq B$ be an extension of commutative rings with identity. Then $A+XB[X]$ and $A+XB[[X]]$ are the polynomial and power series rings over $B$ whose constant terms are in $A$. Is there any ...
1
vote
1answer
98 views

Castelnuovo- Mumford regularity properties

Let $R=k[x_1,\ldots,x_n]$ be a graded ring and $S,T,U$ be monomials ideals. $reg(S)=max\{j-i \backslash \beta_{i,j}(S) \neq 0\}$. Assume $S+T=U$ prove \disprove : $reg(S+T^2) \leq reg(U^2)$. We can ...
7
votes
1answer
346 views

Local cohomology groups and linearity

I am reading local cohomology and am confused on a silly point. Let $U$ be an affine, non-singular variety and $Z \subset U$ a hypersurface section on $U$ (i.e., complete intersection in $U$ of ...
26
votes
2answers
952 views

Why is there a duality between spaces and commutative algebras?

1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated algebras over $\mathbb{C}$. The equivalence associates to an affine variety its ...
5
votes
2answers
154 views

The finiteness of the associated primes of $Ext^i_R(R/I, M)$

Let $(R,m)$ be a Noetherian local ring and $M$ an $R$-module. If $M$ is a finite $R$-module, then we know that $Ext^i_R(R/I,M)$ is a finite $R$-module for all $i\geq 0$; Now suppose $supp(M)\subseteq ...
1
vote
2answers
221 views

Smoothness of a subalgebra of a smooth algebra

Given commutative rings $A \subseteq B \subseteq C$ with $C$ a smooth $A$-algebra, I am interested to know if there are some "mild" conditions that make $B$ a smooth $A$-algebra. For example: Assume ...
7
votes
1answer
156 views

Littlewood-Richardson-Type Rule for Restriction from $S_{2n}$ to $S_{2(n-t)} \times (S_2 \wr S_t)$

It is well-known that the Littlewood-Richardson coefficient $c^{\nu}_{\lambda \mu}$ is the number of times the irreducible representation $V_\lambda \bigotimes V_\mu$ of the product of symmetric ...
1
vote
1answer
137 views

Question regarding arbitrary chains of prime ideals

All rings considered are commutative with $1$. In my study of arbitrary chains of prime ideals of a ring $R$, I have noticed two possiblities for the cardinality of a chain when $\dim(R)=\infty$: ...
1
vote
0answers
165 views

Smoothness of $A \to A[T]/(h)$

Let $A$ be a commutative noetherian domain of characteristic zero, $T$ an indeterminate, $h \in A[T]$, $B= A[T]/(h)$ and assume $B$ is also a domain. When $B$ is (formally) smooth over $A$?; namely, ...
10
votes
0answers
225 views

comparison of completion and Henselization in class field theory

Given a ring $R$ with maximal ideal $\mathfrak{m}$, we can form the localization $R_\mathfrak{m}$, the completion $\hat{R}_\mathfrak{m}$ or the Henselization $\hat{R}^h_\mathfrak{m}$ of $R$ with ...
8
votes
1answer
229 views

Integral domains equal to intersection of their height one localizations

Which integral domains have the property that $R = \bigcap R_P$, the intersection being taken over all height one prime ideals of $R$? It is a standard fact that Krull domains, and thus noetherian ...
2
votes
1answer
99 views

Projective dimension of a quotient ring

Assume $A$ and $B$ are commutative algebras with $1$, $B = A[z] = A[Z]/(h(Z))$, $Z$ an indeterminate. The first comment in this question says that, if $A$ is noetherian, then $pd_{B\otimes_A B}(B) ...
4
votes
0answers
79 views

Unibranch partial normalization

In a paper I recently read something about the "unibranch partial normalization" of a curve. Say, $R$ is a local integral domain with maximal ideal $\mathfrak{m}$ and fraction field $K$. Is it ...
0
votes
2answers
142 views

Making idempotent element by a relation [closed]

Let $R$ be a commutative ring with identity and let $a, b \in R$ such that $a=ab$. How can we make a non zero idempotent element of $R$ by this relation?
10
votes
1answer
510 views

Is an irreducible ideal in $R$ also irreducible in $R[x]$?

Let $R$ be a commutative Noetherian ring and $I\subset R$ an ideal that is irreducible in the sense that if $I = J_1 \cap J_2$, then $I=J_1$ or $I=J_2$. Is (the ideal generated by) $I$ irreducible in ...
3
votes
0answers
133 views

Annihilators of elements in symmetric algebras

Let $M$ be a module over a commutative ring, and $S(M)$ its symmetric algebra. What elements of $S(M)$ annihilate a given element $m\in M$ ? ($M$ is considered as a submodule of $S(M)$.)
0
votes
1answer
165 views

Can you detect homological dimensions from homology?

Suppose you are given a bounded chain complex $M$ over a commutative ring $R$. Is there a clear relation between homological dimensions of $M$ and homological dimensions of its cohomologies? For ...
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votes
0answers
48 views

trying to understand particular modules, to contruct some nice free resolutions

Let $R$ be a local or graded ring, fix some morphism $\phi\in Hom_R (R^{\oplus n},R^{\oplus m})=:Mat(m,n;R)$. In various applications one meets modules like $\frac{Mat(m,n;R)}{Span(U ...
2
votes
2answers
214 views

Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theorem: $(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...
0
votes
1answer
132 views

perfect modules over polynomial algebra

This may be obvious. My question is short: $R$ is the polynomial algebra $\mathbb{k}[X_{1},\dots , X_{n}]$. Is the $R$-module $\mathbb{k}$ perfect in the sense that $\mathbb{k}$ is a compact object ...
7
votes
0answers
170 views

When is a polynomial ring free over a graded subalgebra?

Keep the setting of my previous question and let $I := k[x_1, \dots, x_n] \cdot A_{>0}$ be an ideal of the algebra $k[x_1, \dots, x_n]$ generated by the set $A_{>0}$. It is clear that $I$ is a ...
0
votes
0answers
92 views

A property of the semi-local ring of the normalization of a singular curve

I have two following questions. 1) Let $R$ be a local ring in an algebraic function field of one variable over an algebraic closed field $k$. Let $\bar{R}$ and $m$ be its integral closure and maximal ...
3
votes
2answers
189 views

Examples of (non-normal) unibranched rings?

For a local integral domain $R$ the following are equivalent: a) The integral closure of $R$ in its fraction field (i.e., the normalization of $R$) is again local. b) The henselization of $R$ is ...
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0answers
65 views

Concerning the $SBI$-sequence for dihedral homology (Loday, Cyclic Homology, 5.2)

I was wondering about the signs of the $SBI$-sequence ("Connes' periodicity exact sequence") in equations $(5.2.7.2)$ and $(5.2.7.3)$ of 'Cyclic Homology' by Jean-Louis Loday. Why is the sequence ...
0
votes
1answer
148 views

Is there a complete local analogue of the Artin-Tate lemma?

The Artin-Tate lemma states that if $A \subseteq B \subseteq C$ are commutative rings where $A$ and $C$ are Noetherian, $C$ is finitely generated as an $A$-algebra, and $C$ is finitely generated as a ...
2
votes
0answers
82 views

Centers of Noetherian Algebras and K-theory

I'll start off a little vauge: Let $E$ be a noncommutative ring which is finitely generated over its noetherian center $Z$. Denote by $\textbf{mod}\hspace{.1 cm} E$ the category of finitely ...