**4**

votes

**0**answers

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**

votes

**0**answers

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

**0**answers

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

**1**answer

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 ...

**0**

votes

**0**answers

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**

vote

**0**answers

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 ...

**0**

votes

**0**answers

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

**1**answer

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**

vote

**0**answers

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**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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**

votes

**1**answer

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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**

votes

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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 ...

**0**

votes

**0**answers

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

**2**answers

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**

votes

**0**answers

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**

vote

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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?

**8**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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

**0**answers

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 ∈ ...

**0**

votes

**0**answers

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**

votes

**0**answers

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

**1**answer

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 ...