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

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105
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0answers
4k views

A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
86
votes
0answers
3k views
+50

The Ring Game on $K[x,y,z]$

I recently read about the Ring Game on Mathoverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ...
27
votes
0answers
660 views

What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?

I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ...
19
votes
0answers
363 views

Classification of local Artin (commutative) rings which are finite over an algebraically closed field.

A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ ...
12
votes
0answers
395 views

Class group of $k[x,y,z,w]/(xy-zw)$

I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
11
votes
0answers
107 views

How to name these “ideals”?

Background. Let $\mathcal{C}$ be a symmetric monoidal category with unit $\mathbf{1}$. A subobject of $\mathbf{1}$ is just a monomorphism $I \to \mathbf{1}$. We may also call this an ideal of ...
10
votes
0answers
205 views

Non-reflexive module isomorphic to its double dual

Could you give me an example of a non-reflexive module isomorphic to its double dual? I found an example here but I cannot understand it, do you have any simpler examples? By this question we ...
10
votes
0answers
156 views

checking that an element of a module is zero, point-wise

Let $M$ be a module over a commutative ring $R$. Let $s \in M$ be an element such that for any $x \in \mathrm{Spec}\,R$, the image of $s$ in $M \otimes \kappa(x)$ is 0 (where $\kappa(x)$ is the ...
9
votes
0answers
342 views

Trivial intersection of algebraic sets?

The question came up while reading a bit more into the Hilbert-Zariski theorem I asked about the other week. Suppose $V$ is an algebraic variety over arbitrary field $k$. (For this situation, I'll ...
8
votes
0answers
165 views

Tensoring is thought as both restricting and extending?

I hope these questions are not too trivial. Let $I$ be an ideal in $R$. Write $I'\subseteq R[t]$. Then the notion of tensoring $$ (R[t]/I')\otimes_{\,\mathbb{C}[t]} \mathbb{C}[t]/\langle t-c ...
7
votes
0answers
157 views

Divisor class group of an affine surface

In this topic the OP considers the following surface: $X=\mathcal{Z}(z^3-y(y^2-x^2)(x-1))$. (The field it's not explicitely mentioned, but for geometric reasons this can be algebraically closed.) He ...
7
votes
0answers
150 views

Question about definition of $\mathrm{Ext}$

One can define $\mathrm{Ext}^n(M,N)$ (where $M,N$ are $R$-modules) in two ways, either by taking an injective resolution of $N$ and applying $\mathrm{Hom}(M,-)$or by taking a projective resolution and ...
7
votes
0answers
475 views

A problem about the twisted cubic

I have some difficulty with the following problem: Let $f : k → k^3$ be the map which associates $(t, t^2, t^3)$ to $t$ and let $C$ be the image of $f$ (the twisted cubic). Show that $C$ is an ...
6
votes
0answers
132 views

Andre-Quillen Homology of the cuspidal curve $k[x,y]/(x^2 - y^3)$

I was wondering if I am in the right track here. Let $A := k[x,y]/(x^2 - y^3)$, the cuspidal curve. Obviously this isn't etale or smooth over $k$ so its cotangent complex is not contractible. Now, I ...
6
votes
0answers
121 views

Construct a minimal free resolution, Bruns and Herzog, Exercise 2.3.18(a)

Here is a question form Bruns-Herzog, Cohen-Macaulay Rings, exercise 2.3.18(a). Let $S$ be a regular local ring of dimension $4$, and $y_1$, $y_2$, $y_3$, $y_4$ a regular system of parameters. ...
6
votes
0answers
111 views

Module of Kähler differentials for a formal power series ring

Let $A$ be a ring and $A[[T]]$ the formal power series over A. Then, one can show that $\Omega^1_{A[[T]]/A}$ is not finitely generated over $A[[t]]$. Now, in $\Omega^1_{A[[T]]/A}$ I am trying to ...
6
votes
0answers
80 views

The automorphism group of a toric variety

Let $X$ be a projective toric variety (assume nonsingular, if it helps). Is there a nice description of its automorphism group $\operatorname{Aut}(X)$? I can see that for $\mathbb P^n$ it is ...
6
votes
0answers
67 views

For this morphism of integral schemes $X\to Y$, if $X$ is geometrically reduced (irreducible) then is $Y$ also geometrically reduced (irreducible)?

Let $k$ be an arbitrary field. Suppose that $C/k$ is an integral curve which is birationally equivalent to a projective line. Is it true that $C$ is geometrically reduced and irreducible? This is of ...
6
votes
0answers
165 views

Description of $\mathrm{Ext}^1(R/I,R/J)$

Let $R$ be a commutative ring with unit and $I$ and $J$ are nonzero ideals of $R$. Do we have a nice description for $\mathrm{Ext}^1_R(R/I,R/J)$? What do I mean by a nice description? For example ...
6
votes
0answers
178 views

Punctured plane algebraic over a finite field?

I've been asked to prove, by my algebraic geometry teacher, that the punctured affine plane $\mathbb{A}_k^2 \backslash \{0,0\}$ is not an algebraic set, i.e. is not the zero set of any set of ...
6
votes
0answers
299 views

Rational quartic curve in $\mathbb P^3$

By using similar arguments to the ones from my answer to this question, I can prove that the homogeneous coordinate ring of the rational quartic curve in $\mathbb P^3$, that is, $$R = K[x_1, x_2, x_3, ...
6
votes
0answers
117 views

The geometric interpretation of Gorenstein local ring

Many local rings have geometric interpretations. Cohen–Macaulay rings correspond to equi-dimensionality, and regular local rings correspond to non-singularity, but what is a geometric interpretation ...
6
votes
0answers
230 views

Showing an ideal is prime in polynomial ring

Let $k=\mathbb{C}$ and let $J$ the ideal $(xw-yz,y^{3}-x^{2}z,z^{3}-yw^{2},y^{2}w-xz^{2})$. I want to see why $J$ is a prime ideal in $k[x,y,z,w]$. I know that $Z(J)$ (the zero set of $J$) is ...
6
votes
0answers
123 views

On the order of $\mathbb{Z}[X]/(f,g)$ and the resultant $R(f,g)$.

I suspect that $\#\mathbb{Z}[X]/(f,g)=|R(f,g)|$ holds for any two non-constant polynomials $f,g\in\mathbb{Z}[X]$, where $R(f,g)$ is the resultant of $f$ and $g$. I am however unable to prove it. I'd ...
6
votes
0answers
163 views

When does base change preserves Homs

Let $A \to B$ be a ring homomorphism and $M,N$ be two $A$-modules. Consider the natural map $\alpha_{M,N} : \mathrm{Hom}_A(M,N) \otimes_A B \to \mathrm{Hom}_B(M \otimes_A B,N \otimes_A B)$ Consider ...
5
votes
0answers
60 views

Basic question: Condition for a map associated to a linear series to be an immersion

I am reading this set of lectures of a class by Prof. Harris. There is a theorem. Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without ...
5
votes
0answers
209 views

Irreducible homogeneous ideals

I have the following question: Let $I$ be a homogeneous ideal. Is it true that $I$ is irreducible if and only if it can't be written as the intersection of two homogeneous ideals? So, is it ...
5
votes
0answers
78 views

“Graded free” is stronger than “graded and free”

This topic suggested me the following question: If $R$ is a commutative graded ring and $F$ a graded $R$-module which is free, then can we conclude that $F$ has a homogeneous basis (that is, a ...
5
votes
0answers
95 views

Counterexamples to the avoidance lemma for arbitrary ideals

Let $A$ be a commutative ring with $1$. Let $I$ and $J_k$, $k=1,\dots,n$ be ideals of $A$ with $I\subseteq \cup _{k=1}^n J_k$. Then I have obtained the following: (1) If $J_k$, $k=1,\dots,n$, are ...
5
votes
0answers
105 views

When is $K_0(i)$ an injection?

Suppose that $\mathcal A$ and $\mathcal B$ are two abelian categories such that $\mathcal A$ is a full subcategory of $\mathcal B$. If $i: \mathcal A\rightarrow\mathcal B$ is the inclusion functor, ...
5
votes
0answers
67 views

regular map of Noetherian rings

We say a homomorphism of Noetherian rings $\varphi:A\rightarrow B$ is regular if $\varphi$ is flat and for every prime ideal $p$ of $A$, the fiber ring $B\otimes_Ak(p)$ is geometrically regular over ...
5
votes
0answers
169 views

generic regularity of affine varieties

Suppose that $V\subset {\mathbb C}^n$ is an affine subvariety of codimension $p$. How does one prove that $V$ is regular (i.e., is a smooth manifold) at its generic points? In view of the Jacobian ...
5
votes
0answers
82 views

approximating a variety locally by a vector space

Suppose we have $m$ homogeneous equations with integer coefficients in $n$ variables and that $m >> n$. Let $x_0 \in \mathbb{C}^n$. Question 1: is there a way to approximate the variety ...
5
votes
0answers
135 views

Question about the nullstellensatz for projective schemes

Assume that $ G $ is a graded ring. Assume that $A$ is a relevant homogeneous ideal (that is, it does not contain the irrelevant ideal $ \oplus_{n > 0}G_n$). I am having trouble proving the ...
5
votes
0answers
65 views

Flatness over Jacobson ring

I need either a reference or a counter-example to the following statement. Let $A$ be a noetherian Jacobson ring (i.e. a noetherian ring where every prime ideal $\mathfrak{p} \subset A$ is an ...
5
votes
0answers
295 views

Ideals generated by regular sequences are generated by regular sequences in any order/Eisenbud, Exercise 17.6

I have a question regarding an exercise I found in Eisenbud's Commutative Algebra with a view towards Algebraic Geometry: Exercise 17.6: Any ideal of a Noetherian ring generated by a regular ...
5
votes
0answers
84 views

Injective map on power series ring

Suppose $k$ is a field and let $n > m$. Does there exist injective homomorphisms of the form: $$ k [[x_1, x_2, \ldots, x_n]] \rightarrow k[[x_1, x_2, \ldots, x_m]]$$
4
votes
0answers
50 views

Flatness and Cohen-Macaulay rings

Let $A$ be a local Artin ring, $R$ a local Noetherian ring, $f:A \to R$ a flat morphism and $R$ is cohen-Macaulay. Let $I$ be an ideal in $R$ such that $R/I$ is also Cohen-Macaulay. Under what ...
4
votes
0answers
86 views

Flatness of homomorphisms of graded-commutative rings

Algebraic geometry offers some properties and criteria for homomorphism of commutative rings to be flat. What about homomorphisms of graded-commutative rings? You can define flatness as usual: $R \to ...
4
votes
0answers
48 views

a subtle detail in the proof of Theorem 3.3.7 of Bruns and Herzog

Let $\phi: (R,m,k) \rightarrow (S,n,l)$ be a local homomorphism of Artinian rings, with $k,l$ being the corresponding residue fields. Let $E_R(k)$ be the injective hull of $k$ over $R$ and $E_S(l)$ ...
4
votes
0answers
79 views

Functorial approach to Ideals and Quotients, Multiplicative Sets and Localizations

I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with ...
4
votes
0answers
64 views

Condition on a field that makes every subring an integrally closed domain

I want to know what condition would need to be additionally imposed on a field to make every subring of the field an integrally closed domain.
4
votes
0answers
72 views

Application of Zariski's Main Theorem

Suppose $f: A \to B$ is a local homomorphism, $B$ is isomorphic to a localization of an $A$-algebra of finite type. Let $L$ be the field of fractions of $B$, and suppose that $B$ contains the normal ...
4
votes
0answers
106 views

Direct image of an ideal sheaf along a blow-up

Suppose that $I\subseteq\mathbb{C}[x_0,\ldots,x_n]$ is a saturated homogeneous ideal. Let $\mathcal{I}\subseteq\mathcal{O}_{\mathbb{P}^n}$ denote the corresponding coherent ideal sheaf, and then let ...
4
votes
0answers
102 views

Existence of finite projective resolution

The situation I'm considering is quite involved. All rings are noetherian commutative with $1$. All modules are finitely generated. First of all we fix a non reduced local ring $A$ where all zero ...
4
votes
0answers
84 views

How to compute Ext over an exterior algebra

I found this question in several places (even on mathoverflow and mathstackexchange), but I never found a satisfying answer. Let $k$ be a field and $V$ a finite dimensional $k$-vectorspace, I would ...
4
votes
0answers
219 views

Computing toric ideals via saturation

I have recently got interested in toric varieties and I have a question concerning their ideals. Let $A \in \mathbb{Z}^{m \times n}$ and $\ker A = \{ u \in \mathbb{Z}^n \; | \; Au = 0 \}$. For any $u ...
4
votes
0answers
81 views

Criterion of nonsingular varieties

It's well-known fact, that if $X$ is non-singular algebraic variety over algebraically closed field $k$ and $Y \subset X$ is its irreducible closed subscheme defined by sheaf of ideals $J$, then $Y$ ...
4
votes
0answers
44 views

Extension of morphism of Coherent sheaves over the projective space

Let $\mathcal{F}_1, \mathcal{F}_2$ be coherent sheaves over $\mathbb{P}^n_{\mathbb{C}}$ for $n \ge 3$. Denote by $U_i$ the fundamental affine schemes defined by the non-vanishing of the coordinates ...
4
votes
0answers
59 views

Endomorphism rings of MCM Modules

Let $k$ be a field (algebraically closed of characteristic not equal to two, if you like) and let $R = k[[t^2, t^{2n+1}]]$. It is well known $R$ has finite type and the MCM (maximal Cohen-Macaulay) ...