**1**

vote

**1**answer

171 views

### Does $\mathbb P^1 \times \mathbb P^1$ admit an Ulrich bundle?

In an answer to a MathOverflow question on the following link
Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$, it is mentioned that $\mathbb P^1 \times \mathbb P^1$ has an Ulrich sheaf. However, ...

**1**

vote

**0**answers

114 views

### Classification of rings between a PID and its field of fractions?

Let $D$ be a PID and let $\mathrm{Frac}(D)$ be its field of fractions. I want to classify the intermediate rings $D\subseteq R\subseteq \mathrm{Frac}(D)$.
Theorem: Every such ring $R$ is a ...

**5**

votes

**1**answer

160 views

### Kernel of the differential in de Rham complex in positive characteristic

Roughly, I'd like to ask how does the first terms in de Rham complex behaves for singular varieties.
Let $Y$ be a potentially singular integral scheme over a perfect field $k$ of characteristic $p$ ...

**1**

vote

**0**answers

77 views

### The structure of symmetric powers of finite-dimensional local rings

Fix an algebraically closed field $k$ of arbitrary characteristic $p$ and let $R$ be a finite-dimensional local $k$-algebra (so in particular $R$ is Artinian and Noetherian). Let $S_n$ be the ...

**1**

vote

**1**answer

154 views

### Does the normalization morphism induce isomorphism on residue fields?

The question is basically coming from the following situation:
Let $C$ be an integral curve over a field $k$ (EDIT and assume that $k$ is not algebraically closed) and let $\phi\colon C^N\to C$ be the ...

**0**

votes

**1**answer

136 views

### Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial

The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ...

**3**

votes

**2**answers

254 views

### Counterexample to Openness of Flat Locus

Let $A$ be a commutative Noetherian ring and $B$ a finitely generated $A$-algebra. Then the set $$U\colon=\{P\in\operatorname{Spec}B\mid B_P\ \mathrm{is\ flat\ over}\ A\}$$ is open in ...

**3**

votes

**1**answer

184 views

### Is the induced ring homomorphism surjective for a finite injective morphism between affine varieties?

Let $X$ and $Y$ be affine varieties over $\mathbb C$, and consider a
morphism $f:X\to Y$ and the induced homomorhism
$$ \varphi=f^*:B=\mathbb C[Y]\to A=\mathbb C[X]. $$
It is very easy to see that ...

**1**

vote

**1**answer

98 views

### Morphisms between a globally generated sheaf and a coherent sheaf(Edited)

Let $X$ be a quasi-projective irreducible scheme, $\mathcal{F}_1$ a globally generated $\mathcal{O}_X$-module and $\mathcal{F}_2$ a coherent sheaf over $X$. Suppose that $\mathcal{F}_1$ is globally ...

**1**

vote

**0**answers

80 views

### When is a power series of two variables formally a rational function?

If we have a (formal) power series of two variables with positive coefficients.
Is there any necessary and sufficient condition for this to be (formally) a rational function?

**1**

vote

**1**answer

246 views

### Cohomology after completion

I've been scouring google and asking friend about something I was certain must be absolutely the easiest thing to people who do homological algebra, and none seem to know the answer to this, so if ...

**6**

votes

**1**answer

203 views

### A question on non noetherian ring

Let $R$ be a reduced commutative non noetherian ring of dimension $d$ and $a$ a non zero divisor. Can I say that Krull dimension of $R/(a)$ is at most $d - 1$?

**8**

votes

**1**answer

402 views

### Commutative algebras whose bidual is commutative

Let $k$ be a commutative ring and $A$ a commutative $k$-algebra. Call $D(A) := \mathrm{Hom}_k(A,k)$ the dual of $A$ as a $k$-module, and $DD(A) := \mathrm{Hom}_k(D(A),k)$ the dual of the latter. Let ...

**2**

votes

**0**answers

149 views

### Weak assassins and essential morphisms

Let $R$ be a commutative ring and let $M\rightarrow N$ be an essential morphism of $R$-modules. Then, $M$ and $N$ have the same associated primes.
Over non-noetherian rings the notion of associated ...

**5**

votes

**3**answers

410 views

### Finite index free subgroups of $\mathrm{SL}(3,\mathbb{Z})$

Does $\mathrm{SL}(n,\mathbb{Z})$ have a free subgroup of finite index for some $n \geq 3$? I know that $\mathrm{SL}(3,\mathbb{Z})$ has many free subgroups and that in the case of ...

**3**

votes

**1**answer

126 views

### Toric ideal of slice of a polytope?

Given a collection $A:=\{a_1, \ldots ,a_n \}$ of different integer points in $\mathbb{N}^d$, which span an affine hyperplane when viewed in $\mathbb{R}^d$, one can define a toric ideal $I_A$ from a ...

**0**

votes

**0**answers

145 views

### Is the universal enveloping algebra of the free Poisson algebra generated by finite set (left)-noetherian?

Let $P$ be the free Poisson algebra over $k$ (a field) generated by a finite set $x_1,\dots,x_n$. Let's consider the universal enveloping algebra $P^e$ of the free Poisson algebra $P$. A Poisson ...

**2**

votes

**1**answer

232 views

### Localisation of $\mathbb{Z}_p[[X]]$ at ideal $(p)$

Let $R=\mathbb{Z}_p[[X]]$ where $\mathbb{Z}_p$ denotes the $p$-adic integers and $p$ is a prime. Then what is $R_{(p)}$ $(R$ localised at the ideal $pR)$ $?$

**0**

votes

**2**answers

237 views

### Tensor powers of an algebra all isomorphic

Let $k$ be a commutative ring with total quotient ring $K$, and let $A$ be a commutative $k$-algebra such that the multiplication map $A \otimes_k A \longrightarrow A$ is an isomorphism.
EDIT: Assume ...

**2**

votes

**2**answers

212 views

### Condition for a local ring whose maximal ideal is principal to be Noetherian

The statement "a local ring whose maximal ideal is principal is Noetherian" is (I think) false. The ring of germs about $0$ of $C^\infty$ functions on the real line seems to be a counterexample since ...

**4**

votes

**1**answer

406 views

### Is there a relative version of Artin's approximation theorem?

I've been thinking about the following situation. I have schemes $X$ and $Y$, smooth of dimension $n$ over a base scheme $S$, together with sections of the structure maps, which are closed embeddings ...

**4**

votes

**1**answer

248 views

### Torsors and the fpqc topology

Fix a scheme $S$, a group scheme $G/S$ (let us say smooth, maybe even affine with some finiteness conditions if you like), and suppose I have some other $S$-scheme $P$ with a right $G$-action. We want ...

**8**

votes

**1**answer

282 views

### Presenting $\mathbb{Q}[[t]]$ as an explicit colimit of smooth $\mathbb{Q}$-algebras: an explicit example for the Popescu's theorem

By the seminal Popescu's theorem, $R=\mathbb{Q}[[t]]$ is a filtered colimit of smooth $\mathbb{Q}$-algebras. Could you give me a hint: which $\mathbb{Q}$-algebras can yield such a colimit? My problem ...

**3**

votes

**0**answers

204 views

### Is there fppf descent of locally free modules

Being locally free is a property of quasi-coherent modules which
does not descend in the fpqc topology (see Remark Tag 05VF). But what happens for fppf coverings? More precisely we ask:
Suppose $A ...

**4**

votes

**0**answers

138 views

### Is pushforward along a closed immersion in the fppf topology exact?

Let $i : Z \to X$ be a closed immersion of schemes. Is $i_* : Ab((Sch/Z)_{fppf}) \to Ab((Sch/X)_{fppf})$ an exact functor?
The answer is yes in the \'etale or syntomic topology. It seems likely the ...

**5**

votes

**0**answers

149 views

### Are monomorphisms between algebraic spaces representable?

The question in the title can be reformulated as follows. Let $f : Y \to X$ be a monomorphism of algebraic spaces where $X$ is a scheme. Is it true that $Y$ is a scheme?
If $f$ is locally of finite ...

**3**

votes

**1**answer

260 views

### Splitting as $\mathbb{F}_p[[X]]$-modules

Let $A$ be a finitely generated torsion $\mathbb{Z}_p[[X]]$-module, $B$ = { $x \in A$ such that $px=0$ } and $C=A/B$ where $\mathbb{Z}_p$ denotes the $p$-adic integers. Given $ 0 \rightarrow B/pB ...

**0**

votes

**0**answers

37 views

### What is maximum number of m-complex solutions to a order n polynomial (say with real coefficients)?

I know the answer is n^2 for bicomplex numbers. Does anyone know if a general answer has been found for m-complex numbers (
http://en.wikipedia.org/wiki/Multicomplex_number)?

**0**

votes

**1**answer

86 views

### Normality and fiber product

Let $A$ and $B$ be noetherian normal rings and let $f:A\rightarrow B$ be a finite but non-flat ring homomorphism. We can also assume $Spec(A)$ connected if necessary. We put on $B$ the structure of ...

**0**

votes

**0**answers

120 views

### Invariants of the Determinant Form

Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial
$$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n}
$$
After the linear change of ...

**3**

votes

**2**answers

140 views

### Is every commutative group structure underlying at least one (unitary, commutative) ring structure

From the theorem of classification of finitely generated abelian groups, we can see that every finitely generated commutative group can be considered as the additive structure underlying (at least) ...

**1**

vote

**1**answer

95 views

### Module structure of the abelianization of the commutator subgroup

Let $G$ be a (non-abelian) group, and let $G_2$ denote its commutator subgroup. Then the abelianization $G_2^{ab} = H_1(G_2,\mathbb{Z})$ is a module over the group ring $\mathbb{Z}[G^{ab}]$. The ...

**2**

votes

**1**answer

125 views

### Which algebras can be presented as filtered colimits of f.g. regular ones with smooth connecting morphisms?

Let $R$ be a regular (commutative associative unitial) algebra over a prime field $F$ (i.e. $F=F_p$ or $F=\mathbb{Q}$); assume that it is noetherian excellent (and even of Krull dimension $1$). What ...

**3**

votes

**0**answers

160 views

### Computing the Abelianization of an Automorphism Group

Setup: We are working in a Henselian local ring $(R, \mathfrak m, k)$ that way may assume is Cohen-Macaulay, admits a canonical module and is of finite type (so is an isolated singularity). Let ...

**2**

votes

**1**answer

143 views

### Tangent cone and embedded components

Is it possible for a reduced, equidimensional germ of complex analytic singularity to have a tangent cone with embedded components but without multiple irreducible components?
If it is, how can you ...

**0**

votes

**1**answer

102 views

### The Existence of Pure Resolutions, Given a Degree Sequence?

I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important ...

**6**

votes

**0**answers

265 views

### Can we prove that the ring of formal power series over a noetherian ring is noetherian without axiom of choice?

Let $A$ be a commutative ring with an identity.
Suppose that every non-empty set of ideals of $A$ has a maximal element.
Let $A[[x]]$ be the formal power series ring over $A$.
Can we prove that every ...

**0**

votes

**0**answers

62 views

### Graded Betti Numbers of a Graded Ideal with Linear Quotients

Exercise 8.8 in Monomial Ideals by Herzog and Hibi:
Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators ...

**0**

votes

**0**answers

37 views

### Graded Betti Numbers of a Stable Monomial Ideal

Exercise 8.8 in Monomial Ideals by Herzog and Hibi:
Let $I\subset S=K[x_{1},...,x_{n}]$ be a stable monomial ideal with $G(I)=\{u_{1},...,u_{m}\}$ and such that for $i<j$, either ...

**2**

votes

**1**answer

217 views

### Explicit basis for the space of global sections of a twisted arithmetic ideal sheaf

Assume $x\in X=\mathbb{P}^1_{\mathbb{Z}}$ is a closed point with $f(x)=p\in Y$ where $f:X\rightarrow Y$, here $Y=Spec(\mathbb{Z})$. Assume $k(x)=\mathbb{F}_p$ and denote by $I_x$ the ideal sheaf of ...

**1**

vote

**1**answer

332 views

### Iwasawa algebra [closed]

Let $\mathbb{Z}_p$ denotes the $p$-adic integers for a prime $p$. Suppose $M$ is a finitely generated torsion $\mathbb{Z}_p[[T]]$-module such that $\mu(M)=0$. Then $M/pM$ and $M[p]$($p$-torsion ...

**8**

votes

**1**answer

252 views

### Irreducibility of a class of polynomials

This question is directly inspired by this question. Consider polynomials of the form
$$p(x) = \prod_{i=1}^n(x-i)^2 - d.$$ For which values of $n$ and $d$ is $p(x)$ irreducible? There is a theorem of ...

**2**

votes

**0**answers

44 views

### Nullspace of a matrix modulo an ideal

Suppose $R$ is a multivariate polynomial ring and $I$ is an ideal in $R$.
Let $M$ be a $n\times n$ square matrix with entries in $R$, and suppose that det($M$) lies in $I$.
Thus, $M$ has a ...

**6**

votes

**1**answer

258 views

### Discrepancies in different definitions of a rank of a module?

I have seen different definitions of a rank of a module $M$ over a commutative ring $R$.
1- In nlab (here http://ncatlab.org/nlab/show/rank), for quite general modules, the rank is defined locally at ...

**0**

votes

**0**answers

40 views

### About Linear Quotients of Square of an Ideal with Linear Quotients

Let $I$ be a monomial ideal generated by quadratic monomials $u_{1},...,u_{s}$ and suppose that $I$ has linear quotients with respect to this given ordering. Is it true or false that $I^{2}$ has ...

**2**

votes

**0**answers

68 views

### Determining Hilbert polynomial from some values of Hilbert function

For simplicity, let $(R,m)$ be a Noetherian local ring and $I$ an $m$-primary ideal. The Hilbert function of $I$ is defined as
$$
H_I(n): \mathbb{Z}_{\ge 0} \to \operatorname{length}_{R/m} I^n / ...

**0**

votes

**1**answer

96 views

### The closure of an effective Cartier divisor in a special situation

I am studying first order deformations and a natural question arises.
Situation: Let $X_1$ be a scheme. $\pi: X_1 \to {\rm Spec}~ k[t]/(t^2)$ is a flat morphism of finite type, where $k$ is an ...

**7**

votes

**2**answers

330 views

### Smoothness and Kähler differentials

Let $X$ be a complex variety. It is well-known that $X$ is smooth if and only if the sheaf of Kähler differentials $\Omega_X^1$ is locally free (Hartshorne p. 177).
Question: What happens for forms ...

**2**

votes

**2**answers

255 views

### Generic methods to check irreducibility of polynomials in $K[[X,Y]]$

I usually find it difficult to check irreducibility of polynomials in $K[[X,Y]]$ ($K$ algebraically closed). Does anyone know about generic methods that can be used ? And especially of ones that can ...

**2**

votes

**2**answers

145 views

### Domains $D$ for which for any prime $P$, $D_P$ is a PID

Is there any name or alternative characterization for the class of integral domains $D$ such that for any prime ideal $P$, $D_P$ is a principal ideal domain?