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

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4
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71 views

Effective Nullstellensatz and bounds on the nilpotency index of reduced ideal together with linear forms

Let $K$ be an algebraically closed field of characteristic $p>0$ and let $I\subset K[x_{1},\dots,x_{n}]$ be an ideal generated by (homogeneous) polynomials of degree $d$. Assume that $I$ is ...
0
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0answers
55 views

An additive question on polynomials

Consider $S\cup T=\{0,1\}^n$ where $S\cap T=\emptyset$. Consider real multilinear (only monomials of form $x_ix_jx_k$) polynomials $P,Q$ such that: $$Q(S)=0\quad Q(T)\neq0\quad P(S)\neq0\quad ...
6
votes
0answers
240 views

Can every commutative ring of characteristic $p\in\mathbb P$ be written as the form $R/(p)$ with $R$ being a ring of characteristic $0$?

All rings here are associative, commutative and unital. By a ring of characteristic zero (resp. of characteristic $p$, for prime $p$) I mean a ring $A$ such that the canonical homomorphism $\mathbb ...
1
vote
1answer
88 views

Conditions for the consistency of a system of affine polynomials

Let $f_1, f_2,\ldots,f_N$ be some affine polynomials. We consider the question if these polynomials have a common (affine) root. By homogenizing these polynomials, we can associate a projective ...
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83 views

Flatness of a simple ring extension

Assume $A \subseteq B=A[b]$ are integral domains, $b \in B$ is algebraic over $A$ (but not necessarily integral over $A$), and $A$ and $B$ have the same field of fractions. (Notice that $b=u/v$ for ...
1
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0answers
89 views

Are finitely presented modules finitely presentable? [closed]

Over a ring $R$ we have a notion of finitely presented module, namely: Definition 1 A module $F$ is finitely presented if there are $m$, $n$ positive integers such that $R^m\to R^n\to F\to 0$ is ...
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0answers
70 views

Finding a set of generators of an ideal with certain property in $k[x_1, …, x_n]$

I am interested in the following problem, and I would appreciate any comments, inputs, answers, references! Let $k$ be a field. For each $1 \leq j \leq n$, let $$ I_j = (x_j, u_{2}^{(j)}, ..., ...
5
votes
1answer
133 views

discrete valuation ring and ring of witt vectors

Given a perfect field $F$ of prime characteristic the ring of Witt Vectors $W(F)$ is a discrete valuation ring. For example, $W(\mathbb{F}_p)$ is the ring of $p$-adic integers. Is it possible to embed ...
1
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0answers
93 views

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 ...
3
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0answers
157 views

Kähler differentials, intuition behind $\text{div}(\omega)$, canonical divisor on algebraic curves?

See my two previous questions here: Intuition for thinking about R-module of Kähler differentials, universal receptacles, derivations? and Kähler differentials, define valuation? for background. If ...
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102 views

Free resolutions of affine (non-projective!) varieties

Say, you have an ideal $I$ of a polynomial ring $R = K\lbrack X_1,\ldots,X_n \rbrack$ over an algebraically closed field $K$ (you can assume $K = \mathbb{C}$). What does a minimal free resolution of ...
0
votes
1answer
122 views

Change of grading used in the paper “The diagonal subring and the Cohen-Macaulay property of a multigraded ring” by Eero Hyry

I am currently reading the paper "The diagonal subring and the Cohen-Macaulay property of a multigraded ring" by Eero Hyry. I do not understand the following part in Lemma 1.1. here. Let ...
5
votes
1answer
193 views

Generalization of Krull dimension for commutative rings

In the paper How to construct huge chains of prime ideals in power series rings by B. Kang and P. Toan the Krull dimension of a commutative ring with $1$ is defined as follows: Let $R$ be a ...
5
votes
1answer
181 views

Intuition for thinking about $R$-module of Kähler differentials, universal receptacles, derivations?

Suppose $k$ is a field of characteristic zero, and $R$ is a $k$-algebra. The $R$-module of Kähler differentials $\Omega_{R/k}$ of $R$ over $k$ with generators $\{dr\}_{r \in R}$ is the module subject ...
0
votes
1answer
125 views

Chow groups of rational varieties

Let $X$ be a smooth projective rational variety over a field $k$. Let $CH^i(X)$ denote the Chow group of codimension $i$ algebraic cycles on $X$ modulo rational equivalence. What can one say about ...
0
votes
1answer
110 views

polynomial expression for counting number of integral points of a set

Let $v_i=a_ie_i\in\mathbb R^d$ and $w_i=b_ie_i\in\mathbb R^d$ for $i=1,\dots,d$ where $e_i$'s are unit vetcors and $a_i,b_i$ are positive integers. Let $$S=conv\{0,rv_i+sw_i:i=1,\dots,d\}.$$ Can we ...
4
votes
0answers
57 views

Minimal rank of a permutation resolution of a $G$-lattice

Let $G$ be a finite group. By a $G$-lattice I mean a finitely generated free abelian group $L$ with an action of $G$. One says that $L$ is a permutation lattice if $L$ has a $\mathbb{Z}$-basis ...
2
votes
0answers
63 views

Question related to $h$-invariant of a form

Let $k$ be a field. Given a form $f \in k[x_1, ..., x_n]$ of degree at least $2$, we define the Schmidt rank, also known as the $h$-invariant, $h_k(f)$ to be the least positive integer $h$ such that ...
2
votes
1answer
161 views

A question about Complete Intersections

Suppose that $(A,\mathfrak{m})$ is a complete intersection and $\mathbf{x}$ is a minimal basis for $\mathfrak{m}$. Consider the Koszul homologies $H_\bullet(\mathbf{x},A)$. It is well-known that ...
9
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1answer
328 views

Excellent rings

If A is an excellent commutative ring and G is a finite group of automorphisms of A, is the invariant subring A^G still excellent ? I think this is false -- because if not it would probably be written ...
1
vote
1answer
151 views

finite generation of a certain type of subring

Let $k$ be a field, and let $R$ be a finitely generated $k$-algebra. (If it helps, you may assume $R$ is an integral domain.) Let $I$ be an ideal of finite colength. Note that $A:=k+I$ is a subring ...
3
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0answers
112 views

Does there exist a continuous surjection? [closed]

Let $C$ be an irreducible projective cubic in $\mathbb{P}_2$ with a singular point $p$. So consider $f: \mathbb{P}_1 \to C$ defined as follows. Identify $\mathbb{P}_1$ with the set of lines in ...
3
votes
1answer
263 views

When a smooth algebra is regular?

Let $A \subseteq B$ be noetherian integral domains, $A$ regular (=every localization at maximal ideal is a regular local ring) and $B$ is a smooth $A$-algebra. For the definition of a smooth algebra, ...
2
votes
1answer
118 views

Finitely generated ordered monoids and noetherian subsets

(This question was asked a long time ago on MSE but got no answer so far...) Let $E$ be an additively written cancellable commutative monoid with no non-trivial units. We furnish $E$ with the order ...
0
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0answers
72 views

Lifting points of étale group scheme

Consider the following setting: let $(R,\mathfrak{m})$ be a Noetherian complete discrete valuation ring (with maximal ideal $\mathfrak{m}$) and let $K$ be its field of fractions. Now take $L$ be the ...
3
votes
2answers
193 views

Do discrete valuation rings correspond to local rings of points in fibre?

Given projective curves $C$ and $C'$ and a surjective morphism $\varphi\colon C\to C'$, such that $Q\in C'$ is a smooth point and its fibre $\varphi^{-1}(Q)$ consists of smooth points. Then ...
3
votes
1answer
237 views

Castelnuovo-Mumford regularity in multigraded case

Let $R=\oplus_{n\geq 0}R_n$ be a standard Noetherian commuative graded ring over a local ring $(A,m)$ where $R_0=A.$ Put $R_+=\oplus_{n\geq 1}R_n.$ Let $M$ be a finitely generated $\mathbb Z$-graded ...
2
votes
0answers
63 views

Newton polyhedron and product of ideals

Let $I$ be an ideal generated by monomials $\underline{x}^{\underline{a}_1},\ldots ,\underline{x}^{\underline{a}_s}$ and $J$ be the ideal generated by $\underline{x}^{\underline{b}_1},\ldots ...
2
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0answers
60 views

Geometric/algebraic interpretation of quadratic points of rank r

In the paper of Eckl/Puhklikov (http://arxiv.org/abs/1210.3715) the following terminology is introduced: " Let $X \subset Y$ be a subvariety of codimension 1 in a smooth quasiprojective complex ...
4
votes
1answer
177 views

Macaulay's example of prime ideals in $\mathbb C[X_1,X_2,X_3]$ having large number of generators

There is a famous example of Macaulay which shows that there are prime ideals of height two in $\mathbb C[X_1,X_2,X_3]$ having at least $l$ generators for any $l\ge 3$. In Macaulay's words, the ...
2
votes
1answer
97 views

Can height one maximal ideals in the normalization contract to non-height one primes in the base?

Let $R$ be a local (Noetherian) integral domain of dimension greater than one. Can the integral closure (i.e. normalization) of $R$ have a maximal ideal of height one?
2
votes
1answer
108 views

Classification of local and semi-local rings in function fields

Let $C$ be a non-singular algebraic curve over an algebraically closed field $k$, and $F$ a function field of this curve. It is well-known that non-trivial discrete valuation rings of $F$ correspond ...
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0answers
94 views

How to show integrally closed implies topologically unibranch

On p.52 of Mumford's book Algebraic Geometry: Complex projective varieties, he states that $$\mathcal{O}_{x.X} \text{is integrally closed} \ \Rightarrow X \ \text{is topologically unibranch at } \ ...
1
vote
2answers
149 views

Degree of sum of integral elements over a UFD

Is it possible to generalize Degree of sum of algebraic numbers (especially Pete L. Clark's answer, based on Keith Conrad's answer) in the following way: Let $D$ be a (noetherian) UFD of zero ...
4
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0answers
187 views

Algebraic Closure of the field of rational functions

Using the theorem of Puiseux, one concludes that the algebraic closure of $\mathbb C(X)$ is the set of algebraic elements (over $\mathbb C(X)$) of the algebraic closure of $\mathbb C((X))$, which is ...
1
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1answer
216 views

Composition of rational functions

Given a rational function $R\in\Bbb R(x_1,\dots,x_n)$ with multilinear numerator and denominator, is there always a rational function $G\in\Bbb R(x)$ such that $G\circ R\in\Bbb R[x_1,\dots,x_n]$, ...
0
votes
2answers
154 views

Regularity of a tensor product

Let $A \subseteq B$ and $A \subseteq C$ be commutative noetherian domains. Assume that $A$ and $C$ are regular rings (=every localization at a maximal ideal is a regular local ring). Assume that $B$ ...
0
votes
1answer
141 views

When every module is a scalar extension?

Let $A \subseteq B$ be commutative noetherian domains. Of course, if $M$ is an $A$-module, then $M \otimes_A B$ is a $B$-module. I am curious to know if there exist additional conditions on $A$ and ...
2
votes
1answer
99 views

formally etale deformations of algebras

Let $A$ be a local artinian ring with residue field $k$, $S$ a $k$-algebra. Suppose there is a formally etale deformation $B$ of $S$ over $A$, i.e. a flat $A$-algebra $B$ such that $S\cong ...
3
votes
1answer
227 views

Are essentially smooth schemes noetherian?

Let $k$ be a field. I am unable to find a precise definition of essentially smooth $k$ schemes, but I will stick to this definition below, since this is exactly what I need: Definition: A $k$-scheme ...
1
vote
1answer
168 views

a problem about ideals of polynomial rings

Let $\{f_n\}_{n=1}^\infty\in \mathbb{C}[x,y]$ be a sequence of polynomials given by the following expressions $$ f_n(x,y)=\sum_{i=0}^{[\dfrac{n}{2}]}(-1)^{n-i}{{n-i}\choose i}x^{n-2i}y^i. $$ Let ...
5
votes
2answers
447 views

The number of ideals in a ring

Here is a question that I first asked in math.stackexchange, but I think the question must be proposed here. Let $R$ be a finite commutative ring with identity. Under what conditions the number ...
2
votes
1answer
277 views

A generalization of miracle flatness theorem

I wonder if the miracle flatness theorem Generalizing miracle flatness (Matsumura 23.1) via finite Tor-dimension still works if the rings involved are not local (and the dimension condition is ...
1
vote
1answer
168 views

Dimension of Ext modules [closed]

Let $(R,m)$ be a noetherian local ring, and $M$ and $N$ be two finitely generated $R$-module. Then is it true that $\dim \text{Ext}^k(M,N)\leq \dim M-k$? If not does the reversed inequality hold?
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1answer
424 views

What is an excellent algebraic space?

What does it mean to say that an algebraic space $S$ is excellent? One knows that excellence of a Noetherian ring is not a property that is etale local (that is, excellence cannot be checked over an ...
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votes
0answers
71 views

Degree of function field extension in several variables (degree of an endomorphism over an AV)

I just want to know which is the best way to calculate the degree of a function field extension like this $[\mathbb{F}_q(a,b,c):\mathbb{F}_q(x,y,z)]$ where $x\mapsto f(a,b,c)$ $y\mapsto g(a,b,c)$ ...
0
votes
0answers
111 views

Local-cohomology and Hom

Let $f:R\to S$ be a flat homomorphism of commutative Noetherian rings. "Flat Base Change Theorem", compares the local cohomology modules $H^i_a(M) \otimes_R S$ and $H^i_{aS} (M\otimes_R S)$ for $i ∈ ...
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0answers
221 views

Complete Intersection

Let $I$ be an ideal of the polynomial ring $P=K[x_{1},...,x_{n}]$ that is generated by degree two polynomials ${f_1,...,f_k}$. The zero set $\mathcal{Z}(I)$ is isomorphic to an affine space of ...
0
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0answers
73 views

Rees rings and a formula

Could someone help me to solve this question? Let $(R,\frak m)$ be a commutative, Noetherian, local, and complete domain. and let $R(I)=\bigoplus _{n ‎‎\geqslant 0} I^n t^n$ be be Rees ring of $R$ ...
2
votes
1answer
176 views

Automorphisms of complete local rings

Let $k$ be a field and $(A,m)$ be the completion of the local ring of a smooth point of a $k$-variety. Let $x_1,x_2\in m\backslash m^2$ be regular elements. I am interested in knowing if one can find ...