**4**

votes

**1**answer

254 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

283 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

144 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

150 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

262 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

87 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

144 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

96 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

128 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

161 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

103 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

273 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

64 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

38 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

221 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

334 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

258 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

47 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

261 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

42 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

100 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

333 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

258 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

149 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?

**1**

vote

**0**answers

106 views

### Automorphism on F_2[[X,S]]

Let us define the automorphism $\sigma$ on ${\Bbb F}_2[[X,S]]$ such that
$\sigma \colon S \mapsto S + S^2 + S^3$
$\sigma \colon X \mapsto X + S$.
It is easy to see that the ideal $(S)$ is stable ...

**0**

votes

**1**answer

163 views

### Projective Modules/Algebras: decomposition of linear functions, and the rank formula

Let $A$ be a ring, $B$ a finite projective $A$-algebra, and $C$ a finite projective $B$-algebra. We can show that $C$ is also finite and projective when regarded as an $A$-algebra (by, for instance, ...

**5**

votes

**1**answer

214 views

### Inverse limit of Gorenstein local rings is again Gorenstein?

If we have the system of surjective ring homomorphisms
$f_{i,i+1}: R_{i+1} \twoheadrightarrow R_i$
for an arbitrary $i \geq 0$ such that all $R_i$ are Gorenstein local ring. Let us put
...

**3**

votes

**1**answer

293 views

### Automorphisms of complete discrete valuation ring

Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb ...

**7**

votes

**2**answers

321 views

### convergence in Z-hat; modulo prime power

The following problem appears in Lenstra's Galois Theory for Schemes (p 14, Ex 1.16).
Let $b\in\mathbb Z_{\ge0}$. Define the sequence $(a_n)_{n=0}^\infty$
by $a_0=b, a_{n+1}=2^{a_n}$. Prove that ...

**0**

votes

**0**answers

84 views

### Super-Gorenstein ideal of ${\Bbb F}_p[[X_1,\ldots,X_n]]$

Let $A \colon= {\Bbb F}_p[[X_1,\ldots,X_n]]$ be a $n$-variable power series ring over a finite field ${\Bbb F}_p$. We put ${\frak m}_A \colon= (X_1,\ldots,X_n)$.
Definition(Super-Gorenstein ideal): ...

**4**

votes

**2**answers

393 views

### Characterizing $\mathbb{Q}[X]$ via a property of its tensor powers

Let $\varphi: \mathbb{Q}[X] \longrightarrow R$ an inclusion of commutative rings. Suppose that the map
$$- \circ \varphi: \operatorname{Hom}_{\mathbb{Q}\operatorname{-alg}}(R, R^{\otimes_{\mathbb{Q}} ...

**9**

votes

**2**answers

170 views

### Direct sum of Hopf algebras

I realise that this question might be rather basic but however I was unable to find the answer in any textbook nor manage to figure out the answer. The question is the following: given two Hopf ...

**3**

votes

**1**answer

133 views

### Is the ideal membership problem solvable for differential ideals? Is there a good notion of a Gröbner basis?

Let $K$ be a field of characteristic zero. Let $\Omega = K[x_1, \dots, x_n, dx_1, \dots, dx_n]$ be the differential ring of algebraic differential forms over $K[X_1, \dots, X_n]$.
Is there an ...

**3**

votes

**1**answer

250 views

### What is the geometric meaning of content or intersection flatness?

The polynomial extension $R \rightarrow R[X]$ ($X$ an indeterminate) has many nice properties beyond faithful flatness. The one I'm most interested in at the moment is the following. Say that a flat ...

**0**

votes

**0**answers

129 views

### Weighted projective space, its varieties, the universal hyperplane section description and the Segre embedding

We will view here the weighted projective space as an orbifold.
Let $S=k[x_0,...,x_n]$ be a positively graded ring (I don't assume $S$ to be finitely generated in degree 1). To the grading ...

**2**

votes

**1**answer

159 views

### Localisation of two rings which is an integral extension, then integral extension still holds?

Question seems simple, but I just can't find the solution.
Let A/B be an integral ring extension and let P be a prime ideal of B. By going-up theorem, there is Q, a prime ideal of A, lying over P. ...

**1**

vote

**0**answers

103 views

### Local Profinite Ring

I haven't received any substantial responses to a similar question on math.stackexchange, so let me try here.
Let $R$ be a profinite ring (that is a projective limit of finite rings). Assume
...

**0**

votes

**1**answer

158 views

### When is a power of an indeterminate in an ideal with 2 generators?

If I have an ideal ${\frak a} \colon= (f(X,S), g(X,S))$ of height $2$ in ${\Bbb C}[X, S]$, is it easy to know what power of $S$ is contained in $\frak a$? For example, what is the minimal number $m$? ...

**2**

votes

**0**answers

102 views

### Global sections of exceptional divisor in normalized blow-up

Let $(R, \mathfrak{m})$ be a Noetherian normal local domain and $I$ an $R$-ideal. Write $X$ for the normalization of $\mathrm{Proj}(R[It])$ and $E$ for the effective Cartier divisor defined by the ...

**2**

votes

**1**answer

178 views

### Finite-index free subgroups in lattices and matrix rings

It is a theorem of Selberg that a lattice $\Gamma$ in a linear group has a torsion-free subgroup of finite index. Page 64 in 'Introduction to Arithmetic Groups' by Dave Morris asserts these can be ...

**0**

votes

**0**answers

117 views

### Action on C[[X,Y]]/f(X,Y) giving complete intersection quotients

Let $R \colon\!= {\Bbb C}[[X,Y]]/(f(X,Y))$ be a complete local ring of Krull-dimension $1$. Assume that we have an action of $\Bbb Z$ on $R$ such that fixed elements by $\Bbb Z$ in $R$ are only ...

**3**

votes

**2**answers

281 views

### Does regular field extension preserve regularity?

Let $k$ be an arbitrary field and suppose that $K/k$ is a regular field extension. Let $V$ be regular scheme of finite type over $\text{Spec }k$ (not necessarily smooth). Is it true that $\text{Spec ...

**0**

votes

**1**answer

231 views

### Iwasawa theory for Mazur's deformation ring R

The ideal class group $\mathrm{Cl}({\cal O}_K)$ and Mazur's deformation ring $R(\overline{\rho})$ for a number field $K$ are said to be similar to each other.
Let ${\Bbb Q}_{\infty}$ be the unique ...

**7**

votes

**4**answers

357 views

### On the fixed point of automorphism of F_3[[T]]

Consider the automorphism $\sigma$ on ${\Bbb F}_3[[T]]$ such that $T \mapsto c_1T + f(T)$ with $c_1 = 1$ or $-1$, and $f(T) \not=0$ and the non-zero leading term $c_mT^m$ of $f(T)$ satisfies $m \geq ...

**7**

votes

**0**answers

103 views

### Constructively correct notion of unique factorization domain

Recall the well-known proof that a unique factorization domain is a GCD domain:
Let $x, y \in R \setminus \{ 0 \}$. Factor $x$ and $y$ into pairwise non-associated irreducible elements: ...