Tagged Questions

Questions about commutative rings, their ideals, and their modules.

96 views

Injective homomorphism of finite free modules and Nakayama's lemma [duplicate]

Let $f:E→F$ be a homomorphism of modules, finite over a local ring $A$. Assume that $E, F$ are free. Let $\mathfrak m$ be the maximal ideal, $f_{\mathfrak m}: E/\mathfrak mE→F/\mathfrak mF$ be the ...
25 views

Monomial order with all weights equal?

Consider a set of $n$ polynomials $P_i$ with $i=1,2,...,n$ in variables $z_1,z_2,...,z_n\in\mathbb{C[\textbf{z}]}$. Furthermore, let all polynomials $P_i$ be completely homogeneous in all variables ...
158 views

Is every ideal of $R$ a sum of a nilpotent ideal and an idempotent ideal?

I have a question concerning the following local ring: $$R=K[X_1,...X_n,...]/(X_1,X_2^2-X_1,...,X^2_{n+1}-X_n,...).$$ Is every ideal of $R$ a sum of a nilpotent ideal and an idempotent ideal? ...
20 views

Polynomial rings: if $A \otimes_B A$ is free over $A$, is it a complete intersection?

Let $A = k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$ of characteristic zero and $\{y_j\}_{1 \leq j \leq \ell}$ a family of homogeneous polynomials. Write $B$ for the subring ...
35 views

$\mathcal{T}^i(X/Y,\mathcal{F})$ forms a sheaf

In Hartshorne's Deformation Theory, given an $A$-algebra $B$ and a $B$-module $M$, he defines these functors $T^i$ for $i=0,1,2$ that outputs $B$-modules $T^i(B/A,M)$. In Exercise 3.5, he asks the ...
69 views

Ideal of $(u^3,u^2v,uv^2,v^3)$

Let $k$ be a algebraically closed field, consider $f: \mathbb A_k^2\rightarrow \mathbb A_k^4$ given by $f(u,v)=(u^3,u^2v,uv^2,v^3)$. Let $X=f(\mathbb A_k^2)$. Then how to determine $I(X)$? If I let ...
90 views

An example from Lang's Algebra about primary ideal

On page 421 in Lang's Algebra, the author writes Let $R$ be a factorial ring with a prime element $t$. Let $A$ be the subring of polynomials $f(X)∈R[X]$ such that $$f(X)=a_0 + a_1X + \dotsb$$ ...
38 views

Prime ideal in $R[x]$ lying above $P$ [duplicate]

I am trying to understand this proof, I think "$Q$ is lying above $P$" means $Q \cap P = R$, but I don't know why we can assume "$P=0$" or "$R$ is a field", can someone explain it to me?
28 views

Valuation ring and integral closure

Let $A$ be a one-dimension local noetherian domain and suppose that we know that $K=\text{Frac}(A)$ is a complete discrete valuation field (valuations for me are surjective). Let's denote with ...
21 views

Local subring of a DVR and finite residue field extension

Let $\mathcal O$ be a complete DVR with fraction field $K$, maximal ideal $\mathfrak p$ and residue field $\widetilde K=\mathcal O/\mathfrak p$. Now consider a subring $A\subset \mathcal O$ with the ...
38 views

Geometric generic fibre

I have a question concerning the following exercise in Hartshorne: The inclusion of $k[s]$ into $k[s,t]/(s-t^2)$ induces a morphism of the corresponding affine schemes $X\to Y$. The exercise itself ...
68 views

Are polynomial rings finitely generated modules over the base ring?

Let $R$ be a commutative ring. Consider $R[X_1,...,X_n]$. Clearly it is a natural $R$-module. Is it true that $R[X_1,...,X_n]$ is a finitely generated $R$-module? If it is, then every its ideal is ...
47 views

Nilpotent or just nil idempotent ideal?

Let $R$ be a ring with zero Krull dimension and $I$ be an idempotent ideal contained in the Jacobson radical $J(R)$ of $R$. Could one infer just with these hypotheses that $I$ is a nilpotent ideal? I ...
109 views

29 views

Does a free $R$-module of countable rank ever arise as a (double) dual of some $R$-module?

The usual counterexample to all vector spaces being canonically isomorphic to their double duals goes something like this: if $F$ is a field, take $F^{\omega}$. Since contravariant hom takes colimits ...
25 views

generating radical of an ideal with small degree polynomials

Say $f_1,f_2,\ldots,f_n$ are a finite set of polynomials in $M$ variables $x_1,x_2,\ldots,x_M$ over a field $k$. Say each $f_i$ has total degree at most $D$. If $I$ the ideal generated by the ...
37 views

When is a polynomial contained in the ideal generated by its partial derivatives?

Let $R = k[x_1,\dots,x_n]$ be a multivariate polynomial ring over a field $k$ of characteristic zero, and let $f\in R$. Is there an easy-to-test necessary and sufficient condition on $f$ such that ...
42 views

The equivalent conditions for commutative rings with a unique prime ideal [closed]

In a commutative ring $R$ the followig conditions are equivalent. (1) $R$ has a unique prime ideal (2) every nonunit is nilpotent (3) $R$ has a minimal prime ideal which contains all zero divisors, ...
41 views

Localization of $\Bbb k[x]$

Let $R= \Bbb K [x]$ and $S=\{x^n: n \in \Bbb Z, n \geq 0 \}$. Let $D$ be the localization of $R$ in $S$, that is $D = S^{-1}R = \{ \frac{r}{s}: r \in R, s\in S \}$. By using the Universal Property ...
47 views

If $A$ is an integrally closed domain, then is $GrSym A$ (the graded symmetric algebra on $A$) also integrally closed?

Question: If $A$ is an integrally closed domain, which is f.g. k algebra over an algebraically closed field, then is $A'$ (defined in the edit below) also integrally closed? (I am thinking about this ...
30 views

Dimension of an affine scheme, 24.5.7 of Vakil's book

In the remark 24.5.7 of Vakil's book, it claims that the dimension of the scheme $\operatorname{Spec}k(x) \otimes k(y)$ is a $k(x)$-scheme with dimension one, where $k$ is a field and $x,y$ are two ...
65 views

Why are Unique Factorization Domains (UFD's) geometrically significant?

We know that for $A$ a UFD, it's class group is trivial. More generally, for a factorial (stalks are UFD's) scheme $X$ (that is also noetherian and normal), we have an isomorphism between it's Picard ...
51 views

$T^i$ functors in Hartshorne's Deformation Theory

In chapter 3 of Hartshorne's Deformation Theory, he defines functors $T^i$ for $i=0,1,2$ that take as input a ring homomorphism $A\rightarrow B$ and a $B$-module $M$ and outputs $T^i(B/A,M)$, a ...
50 views

Flat schemes over artinian local ring with isomorphic special fibers

I'm sure this is standard, but I don't know where to find it. If $A$ is a local artinian $k$-algebra, $X_1,X_2$ are finite type schemes flat over $A$, and $f:X_1\rightarrow X_2$ is a morphism over ...
27 views

Zero divisors in a finite dimensional Poincaré duality algebra

Let $A= \bigoplus_{i=0}^{n}A_i$ be a finite dimensional algebra over a field $\mathbb{k}$ such that $A_0 \cong \mathbb{k} \cong A_n$. Consider the bilinear form \varphi: A_i \times A_{n-i} \to ...
Suppose $(A,m)$ is a discrete valuation ring and $[m]$ is the closed point of $Spec\,A$ and $\eta$ is its generic point, which is also the only nontrivial open set. If we have a morphism $\pi:X ... 0answers 20 views Finitely graded ring and zero divisors Let$R= \bigoplus_{i=0}^{d} R_i$be a finitely graded ring such that$R_0$is a field and$R_d \cong R_0$. I'm trying to understand how zero divisors work in such a ring; when is it true that for$r ...
I have a question about an inequality that was stated in my class but never proven. We stated p is simple $I_p(F \cap G+H) \ge \min ( I_p (F \cap G) , I_p(f \cap H))$. Where we defined \$I_p(F \cap ...