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

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1answer
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, ...
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0answers
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
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1answer
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 $p$ ...
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0answers
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 ...
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1answer
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 ...
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1answer
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
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2answers
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
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1answer
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 ...
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1answer
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 ...
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0answers
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?
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1answer
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 ...
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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 $d - 1$?
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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 ...
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0answers
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 ...
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3answers
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 ...
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1answer
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 ...
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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 ...
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1answer
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 $pR)$ $?$
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2answers
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
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2answers
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
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1answer
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
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1answer
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
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1answer
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 ...
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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 ...
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0answers
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 ...
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0answers
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
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1answer
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 ...
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0answers
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)?
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1answer
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 ...
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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
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2answers
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) ...
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1answer
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 ...
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1answer
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
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0answers
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 ...
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1answer
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 ...
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1answer
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 ...
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0answers
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 ...
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0answers
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 ...
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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
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1answer
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 ...
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1answer
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 ...
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1answer
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 ...
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0answers
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
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1answer
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 ...
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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
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0answers
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 / ...
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1answer
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 ...
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2answers
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 ...
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2answers
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 ...
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2answers
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?