# Tagged Questions

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

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $Y'$ ...
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### Wanted: A purely algebraic proof of the Frobenius theorem on distributions

Is there a purely algebraic proof of the Frobenius theorem? Here's a rough sketch of what i'm looking for: Let $Der(R)$ denote the $R$-module of ($R$-valued) derivations of the algebra $R$ endowed ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Ideals in $C[0,1]$ which are not finitely generated (From Atiyah- Macdonald )

I'm trying to solve the following problem from Atiyah-Macdonald: Is the ring of continuous function on $[0,1]$ is Noetherian ? Certainly not, here are two non terminating ascending chain of ...
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### 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 effective ...
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### Finding irreducible components of Spec$(R/I^n)$

Let $R= k[x,y,z]/(xy,yz,zx)$. Let $I=(x)$. What are the irreducible components of $\mathrm{Spec}(R/I^n)$ where $n \geq 2$ and $k$ is a field? For solving this problem I'm trying to use following ...
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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 $$(R[t]/I')\otimes_{\,\mathbb{C}[t]} \mathbb{C}[t]/\langle t-c \... 0answers 426 views ### Trivial intersection of algebraic sets? The question came up while reading a bit more into the Hilbert-Zariski theorem I asked about the other week. Suppose V is an algebraic variety over arbitrary field k. (For this situation, I'll ... 0answers 46 views ### Is torsion of a topological module closed? I was asking to myself the following question. Consider a p-adically complete and separated topological algebra R over \mathbb{Z}_{p}. As \mathbb{Z}_{p} is a domain, we know that the \mathbb{... 0answers 81 views ### Looking for a special class of ideals such that If every ascending chain of ideals from this class stabilizes, then R is a Noetherian ring. A commutative ring R is called Noetherian if any one of the following holds: 1. Every ascending chain of ideals in R stabilizes, that is,$$ I_1\subseteq I_2\subseteq I_3\subseteq\cdots $$... 0answers 173 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 ... 0answers 76 views ### \mathbb C[X_1, \ldots, X_n] is a free module over \mathbb C[X_1, \ldots, X_n]^G Let G be finite subgroup of GL_n( \mathbb C ). Let \mathbb C[X_1, \ldots, X_n]^G be the set of all G-invariant polynomials of \mathbb C[X_1, \ldots, X_n]. Is there any rule by which we can ... 0answers 86 views ### Geometric intuition for coherent rings, modules, and sheaves Throughout, all rings are commutative. Definition 1. A ring R is coherent if the solutions \mathbf x=(x_1,\dots,x_n) to a linear equation \mathbf{rx}=0 are a finitely generated R-submodule of ... 0answers 146 views ### Which of the algebra isomorphisms hold? Fix m, n \ge 1. Which of the algebra isomorphisms below hold? k\langle t_1, \dots, t_m\rangle \otimes_k k\langle s_1, \dots, s_n\rangle \cong k\langle t_1, \dots, t_m, s_1, \dots, s_n\rangle k[ ... 0answers 95 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. (... 0answers 130 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. ... 0answers 93 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 one-... 0answers 103 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 ... 0answers 164 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 ... 0answers 108 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 ... 0answers 197 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 ... 0answers 158 views ### Help requested to understand the abstract cotangent complex construction I am trying to thoroughly understand one way of constructing of the cotangent complex (I am using here the Lichtenbaum's way) The first question I have is about the definition of an extension of ... 0answers 473 views ### Regular Noetherian local rings are integral domains - questions about the proof I am reading a proof that if (A,\mathfrak m) is a regular local ring, then A is an integral domain. I put the major questions I'm worried about in bold, but there are a lot of little things I'm ... 0answers 214 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 ... 0answers 661 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 ... 0answers 138 views ### An example of prime ideal P in an integral domain such that \bigcap_{n=1}^{\infty}P^n is not prime I am looking for an example of prime ideal P in an integral domain such that the ideal \bigcap_{n=1}^{\infty}P^n is not a prime ideal. This is a followup to this question where the ring was not ... 0answers 70 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 = \... 0answers 120 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 ... 0answers 63 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 S-... 0answers 176 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 ... 0answers 104 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 ... 0answers 133 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 \... 0answers 94 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 ... 0answers 89 views ### What are some important examples of differential objects that aren't naturally graded? [By a "differential object" I mean an object A in some abelian category \mathcal{A} together with a morphism d : A \to A such that d \circ d = 0. By a "differential module" I mean a ... 0answers 160 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 ... 0answers 223 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$$\...
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 ...
### 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, ...