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

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190
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9k 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 ...
135
votes
0answers
4k views

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 ...
29
votes
0answers
805 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 ...
22
votes
0answers
421 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 $Y'$ ...
16
votes
0answers
181 views

Can any commutative ring of characteristic $p\in\mathbb P$ be written as the form $R/(p)$ with $R$ being a ring of characteristic $0$?

Let $S$ be a commutative ring with identity with $\operatorname{char}S=p$, where $p$ is a prime number. I wonder if we can always find a ring $R$ such that $\operatorname{char}R=0$ and $R/(p)\cong ...
14
votes
0answers
117 views

What is the importance of modules in algebraic geometry?

I have been trying to teach myself the basics of algebraic geometry. I understand the basic premise, how we define geometry spaces (algebraic sets and schemes) in terms of commutative rings. And I ...
12
votes
0answers
135 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 ...
12
votes
0answers
561 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 ...
12
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0answers
160 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 ...
10
votes
0answers
107 views

Intuitive/geometric way of thinking about effective divisors?

What is the motivation/intuition/geometric way of thinking about an effective divisor? I know that a divisor is effective if all its coefficients are non-negative. We write $D \ge 0$ for ...
10
votes
0answers
91 views

Study of rings of the form $R+I$

In my life I saw lots of ways of constructing rings: polynomial rings, quotient rings, localizations, endomorphism rings, rings of fractions, integral closure of a ring, center of a ring, etc... These ...
10
votes
0answers
230 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 ...
9
votes
0answers
93 views

Abelian category induced by commutative ring

If $R$ is any ring, then ${}_R \mathsf{Mod}$ is an abelian category. We cannot detect commutativity of $R$ from ${}_R \mathsf{Mod}$, since for example $R$ and the matrix ring $M_n(R)$ are always ...
9
votes
0answers
177 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 ...
9
votes
0answers
366 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 ...
7
votes
0answers
72 views

A Geometric Description of Injective Modules

I've found that when studying commutative algebra, thinking of things in terms of their algebro-geometric interpretation helps them stick as well as motivates otherwise odd and abstract concepts. ...
7
votes
0answers
76 views

What is the “projective limit” of a polynomial?

Bayer and Mumford, What can be computed in algebraic geometry, reads (in part): Let $S = k[x_0, \ldots, x_n]$ be the homogeneous coordinate ring of $\mathbb{P}^n$. [. . .] Choose a ...
7
votes
0answers
57 views

Does the inverse of a polynomial matrix have polynomial growth?

Let $M : \mathbb{R}^n \to \mathbb{R}^{n \times n}$ be a matrix-valued function whose entries $m_{ij}(x_1, \dots, x_n)$ are all multivariate polynomials with real coefficients. Suppose that ...
7
votes
0answers
122 views

Check whether a polynomial ideal is prime in the power series ring

I would like to know whether the ideal $I = \langle y^{2}(y^{2}-x^{2}) + w^{7}, y^{2}(y^{4}-x^{4}) + z^{7}\rangle$ is prime in $\mathbb{C}[[x,y,z,w]]$, the ring of formal power series in the ...
7
votes
0answers
153 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. ...
7
votes
0answers
169 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 ...
7
votes
0answers
185 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
166 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
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0answers
561 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
109 views

What is the group structure on the ring of power series around a point that makes it “the completion of an elliptic curve” along that point?

I've been struggling to understand the explicit details of the completion of an elliptic curve about the origin, and am desperately confused by the explicit details of the resulting group operation. ...
6
votes
0answers
55 views

Geometric statement of Prime Avoidance?

The Prime Avoidance Theorem is very clean to state in algebraic terms: Let $I \subset R$ be an ideal (with $R$ noetherian) and $I \subseteq \bigcup_{i=1}^r P_i$, where each $P_i$ is prime. Then $I ...
6
votes
0answers
58 views

Finding equations for projective curves, low genus, Riemann-Roch.

Let $C \subset \mathbb{CP}^n$ be a nonsingular projective curve, and let $L \subset \mathbb{CP}^n$ be a hyperplane. We have that $L \cdot C$ is a divisor $H$ on $C$ if $C \subset L$. Let $R = ...
6
votes
0answers
56 views

$E \to S$ surjective in degrees $\geq 1$ implies $\widetilde{E} \to \widetilde{S}$ surjective

In the proof of Theorem II.8.13 in Hartshorne (which is the Euler sequence), the author writes: Let $S = A[x_0, \ldots, x_n]$. [...] The exact sequence $$0 \to M \to E \to S$$ of graded ...
6
votes
0answers
81 views

Computing Hodge numbers of a complete intersection

The situation is this: I have a 5-dimensional irreducible projective variety $Y$ embedded in $\mathbb P^{13}$. This variety is singular, the singularities being a disjoint union of two curves. I have ...
6
votes
0answers
155 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
105 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
77 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
85 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 ...
6
votes
0answers
186 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
191 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
356 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
155 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
263 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
132 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
177 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
49 views

What is the obstruction to extending a linear map on tangent spaces of a variety to a regular map on neighborhood?

Suppose that $X$ and $Y$ are algebraic varieties of the same dimension $n$. If $p$ and $q$ are points in $X$ and $Y$ respectively, suppose that there is a linear map $i : T_p X \to T_q Y$. My vague ...
5
votes
0answers
110 views

What about $\mathrm{Spec}(\mathbf{Q})$?

I've heard a lot about $\mathrm{Spec}(\mathbf{Q})$ (see for example Minhyong Kim's answer here), but $\mathbf{Q}$ is a field. So isn't $\mathrm{Spec}(\mathbf{Q})$ trivial? What's the point of studying ...
5
votes
0answers
71 views

Bass' paper on Gorenstein rings

I am currently reading the paper On the ubiquity of Gorenstein rings by Hyman Bass. I found difficulty to understand the proof of Proposition (7.2). Under the the following setting: $A$: commutative ...
5
votes
0answers
64 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
600 views

Hilbert's Basis Theorem - Clever Proof?

So I am studying commutative algebra at the moment and I have come across the proof of the Hilbert Basis Theorem (the proof I have is the same as the one in Reid's "Undergraduate Commutative ...
5
votes
0answers
91 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 ...
5
votes
0answers
235 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
113 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
123 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
76 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 ...