**3**

votes

**0**answers

90 views

### Does there exist a continuous surjection? [on hold]

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

168 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

83 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

61 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

178 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 ...

**0**

votes

**0**answers

38 views

### Is a toric blow-up in codimension 2 a real toric blow-up?

Let $X, Y$ be toric projective algebraic varieties over $\mathbb{C}$. Suppose that $X$ and $Y$ are $\mathbb{Q}$-factorial and smooth in codimension two (e.g. they have terminal singularities).
Let ...

**1**

vote

**0**answers

120 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

56 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

57 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

166 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

91 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

94 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

88 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**

votes

**0**answers

30 views

### Regularity of simple ring extensions, subrings and quotients [migrated]

Let $R$ be a regular UFD of zero characteristic, $I=(p)$ a prime ideal of $R$ and $Q(R)$ the field of fractions of $R$.
Assume $R[a]$ is integral and flat over $R$, for some $a \notin Q(R)$. Is it ...

**1**

vote

**2**answers

135 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

155 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

196 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

143 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

123 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

96 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

218 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

159 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 ...

**1**

vote

**0**answers

87 views

### Chinese remainder theorem for discrete valuation rings in a field

Let $v_1,\dots,v_n:K\to \mathbb Z$ be different nontrivial additive discrete valuations on a field $K$ and $(R_i,m_i)$ the corresponding DVR for each $v_i$.
1- Does always there exist an element ...

**5**

votes

**2**answers

414 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

243 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**

votes

**0**answers

48 views

### Coefficients of the pull-backs of divisors by resolving morphism

Let $\varphi : X \dashrightarrow X$ be a rational map. By a theorem of Hironaka we can find a resolution of singularities $(\tilde{X}_\varphi,\pi)$ of $\varphi$, where $\tilde{X}_\varphi$ is a ...

**1**

vote

**1**answer

159 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

399 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

58 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

103 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

188 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

65 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

169 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 ...

**1**

vote

**0**answers

34 views

### Antisymmetrization of the Hochschild cocycle

Let $A$ be a commutative (unital, complex) algebra and let $\varphi$ be a $n+1$-linear functional on $A$ (we will call it cochain). Define ...

**0**

votes

**0**answers

65 views

### 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 ...

**1**

vote

**0**answers

80 views

### Condition for a finite group scheme to be étale [closed]

My question comes from the reading of Tate's paper $p$-divisible groups. In the last few pages there is an argument which gives as trivial the following fact. If we take a $p$-divisible group over a ...

**0**

votes

**0**answers

71 views

### compute standard basis in local rings

Let$>'$ be the order in $k[t,x_1,\cdots,x_n]$ as follows:
Each semigroup order > on monomial in the $x_i$ extends to a semigroup order >' on monomial in $t,x_1,\cdots,x_n$ in the following way. We ...

**2**

votes

**3**answers

185 views

### ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$.
Let $f,g\in \mathbb{C}[x,y]$.
Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$.
...

**4**

votes

**0**answers

156 views

### On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes

Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...

**0**

votes

**0**answers

107 views

### Length of quotients and relations between $\ell(\mathrm{coker}\varphi),\ell(R/\det\varphi)$ [migrated]

Let $R$ be domain(not necessary local) with maximal ideal $\mathfrak{p}$ and $d \in R, d \neq 0$. $(R/(d))_{\mathfrak{p}} = 0 \iff (d) \not\subset \mathfrak{p}$(?). And if $ (d) \subset \mathfrak{p}$ ...

**2**

votes

**0**answers

180 views

### Algebraic closedness in field of fractions

If $A\subseteq B$ are affine domains over an algebraically closed field $k$ of characteristic zero, such that $Q(A)$ is algebraically closed in $Q(B)$, how can one show that $Q(A)$ is also ...

**0**

votes

**1**answer

163 views

### irreducibility of general fiber

I would like to get a reference of the following fact.
Let $A\subseteq B$ be affine domains over an algebraically closed field of characteristic zero. If $Q(A)$ is algebraically closed in $Q(B)$, ...

**1**

vote

**0**answers

79 views

### Syzygies in integral domains

Let $f$ be a homogeneous irreducible element in a graded commutative Noetherian ring.
What is the possible set of elements $f_1$ and $f_2$ such that $f=f_1g_1+ f_2g_2$?
Even in very particular cases ...

**-1**

votes

**0**answers

38 views

### Cohen-Macaulay rings and Normal rings [migrated]

is there an example that R is Cohen-Macaulay but not normal ring?
what about the converse example?

**0**

votes

**1**answer

129 views

### $\inf\{i\in \mathbb N \cup \{0\}\cup\infty\mid Ext^i_R(R/I,R)\neq 0\}=0 ?$

Let $R := k[x_1, \cdots, x_n, \cdots]/(x_1^1, \cdots , x^n_n, \cdots),$ where $k$ is a field. Set $I:=(x_1, \cdots, x_n, \cdots)$. the questions are:
Is $\inf\{i\in \mathbb N \cup \{0\}\cup ...

**3**

votes

**1**answer

243 views

### Can states on commutative Banach algebras be understood as probability measures?

Suppose $\mathcal{A}$ is a commutative Banach algebra (over $\mathbb{R}$) or commutative Banach *-algebra (over $\mathbb{C}$). Is there always a measurable space $(\Omega,\mathcal{F})$ such that there ...

**2**

votes

**1**answer

174 views

### Uncountable cardinals and Prufer $p$-groups

Let $A$ be an elementary Abelian uncountable $p$-group. Is it known if there is an action of a Prufer $q$-group (here $q$ is a prime not necessarily distinct from $p$) $C_{q^{\infty}}$ onto $A$ such ...

**10**

votes

**1**answer

746 views

### What is the mathematical structure called if we replace commutative group by commutative monoid in the definition of linear space?

Could anyone tell me what the mathematical structure is called if we replace commutative group by commutative monoid in the definition of linear space?
Also, are there any names for "commutative ...

**1**

vote

**2**answers

190 views

### Surjectivity of trace map

Let $R$ be a closed integral domain with its fraction field $F$. Let $K$ be a finite separable extension field of $F$, and let $A$ be the integral closure of $R$ in $K$.
It is well known that the ...

**0**

votes

**0**answers

79 views

### $K_1(R)$ and splitting

Let $R$ be a commutative ring with unit. Under what conditions does the following exact sequence split?
$1\to E(R)\to Gl(R)\to K_1(R)\to 1$.