# Questions tagged [affine-schemes]

The spectrum of a commutative ring with unit is the set of prime ideals endowed with the Zariski topology. One can define a sheaf of rings on this space : to each Zariski-open set is assigned a commutative ring, thought of as the ring of "polynomial functions" defined on that set. This topological space endowed with this sheaf is called the spectrum of the ring. Every locally ringed space isomorphic to such a spectrum is called an affine scheme.

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### Open set of the cuspidal curve is not principal

I'm struggling with this problem: Let $k$ a field and $X=\operatorname{Spec} k[x,y]\big/(x^3-y^2)$ and $U=X\setminus\{(x-1,y-1)\}$. Show that $U$ is not a principal open set in $X$. I tried to show ...
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### Why is $H^0(X_n, \mathcal{O}_{X_n})$ a local artin ring.

Let $V$ be regular, proper and of dimension 2 over $S =$ spec $R$, for $R$ a complete discrete valuation ring with uniformizer $t$, maximal ideal $\mathfrak{m}$, and algebraically closed residue ...
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### Where is the error in this proof that all morphisms of schemes are quasi-compact?

Lemma 1. All preimages of affine open subschemes are affine (Inverse of open affine subscheme is affine). Lemma 2. All affine schemes are quasi-compact (Proof that an affine scheme is quasi compact). ...
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### Defining morphism of sheaves [duplicate]

Suppose we have sheaves $\mathcal{F},\mathcal{G}$ on a topological space $X$ where $\mathcal{U}$ is a base of $X$. Then to define a morphism $\varphi:\mathcal{F}\rightarrow \mathcal{G}$, is it enough ...
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### Easy question regarding open subschemes of closed subschemes

I am trying to follow the proof of the theorem 3.42 of Wedhorn's and Görtz book on algebraic geometry. The step I can't follow is like this: suppose we have $Z \subseteq X = \operatorname{Spec}A$ a ...
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### Question about a proof that all étale morphisms are locally standard étale

In chapter 1 of Milne's Étale cohomology book from 1980, Theorem 3.14 states that : If $f: Y\longrightarrow X$ is étale in some open neighbourhood of a point $y\in Y$, then there are affine open ...
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Is there a field k such that $Spec(k)\times \mathbb P^1$ is affine? Probably I have no technical background to claim any good idea on this. However, I think the answer should be negative since $Spec(k)... 0 votes 1 answer 52 views ### Show that$Z_{red}=(Z,\mathcal{O}_{Z_{red}})$is a scheme and it satisfies the universal property I am trying to show that, given a scheme$(X,\mathcal{O}_X)$and a closed subset$Z\subset X$, with the subsheaf $$\mathcal{I}^Z_X(U)=\{f\in\mathcal{O}_X\mid f(z)=0 \forall z\in Z\}.$$ Then define a ... 0 votes 0 answers 11 views ### Computing residue fields of affine schemes I have taken a course on schemes, so I am familiar with the basic definitions, but I'm very rusty and I've forgotten how to do this (if I ever knew). Basically, I want to compute the residue fields of ... 2 votes 1 answer 91 views ### Generic zeros of a polynomial according to Lang's IGA The concept of generic zero of a prime ideal is defined in Lang's Introduction to Algebraic Geometry at page 27 (1972 ed.). Because the setting is quite weird in my opinion (I have posted another ... 1 vote 1 answer 117 views ### Picard Group of the spectrum of a Noetherian UFD. Let$A$be a Noetherian UFD, how can I compute$\text{Pic}(A):=\text{Pic}(\text{Spec}(A))?$The Picard group of a scheme$X$is defined as the group of isomorphism classes of invertible$\mathcal{O}_X$... 0 votes 0 answers 56 views ### Is a blowing up always surjective? Consider$A$a commutative ring, and$I\leq A$an ideal. Let$X=\mathrm{Spec}(A)$and$Z=\mathrm{Spec}(A/I)$. Consider the blowing-up algebra \begin{equation} \mathrm{Bl}_I(A):=\bigoplus_{i=0}^\infty ... 7 votes 0 answers 84 views ### What schemes correspond to varieties in the sense of Weil? Out of (perhaps morbid) curiosity I am trying to learn the basics of Weil's foundations of algebraic geometry. I tried to ask a question earlier but it turned out I had misunderstood some more basic ... 0 votes 2 answers 95 views ### Prove that if a ring homomorphism is surjective, the associated spectral map is injective I recently had a bunch of questions on a problem set that I could not solve and the instructor is not providing solutions. Let$f: A \to B$be a surjective ring homomorphism. Prove that the induced ... 1 vote 0 answers 77 views ### Is the "vanishing locus" of a section of a quasi-coherent sheaf closed? Suppose$A$is a (possibly noetherian) ring and M a (possibly finitely generated)$A$-module. Consider for$m \in M$the set $$\{m = 0\} := \{\mathfrak p \subset A \text{ prime ideal} \,|\, m_{\... 1 vote 0 answers 39 views ### When is the closed subscheme of \Bbb P^n_{\Bbb C} cut out by a homogeneous polynomial f integral? I am new to algebraic geometry. I am trying to understand whether closed subschemes of \mathbb P^n_{\mathbb C} is an integral scheme. Let X = \operatorname{Proj} (A/(f)) where f is an ... 1 vote 2 answers 71 views ### If f:\operatorname{Spec} A\to\operatorname{Spec} B is a map of schemes, how can I show V(\varphi^{-1}(I)) = \overline{f(V(I))} for I\subset B? If X = \operatorname {Spec}(A) and Y = \operatorname{Spec}(B) are affine schemes and I have a morphism f : X \to Y, then I am trying to show$$V(\varphi^{-1}(I)) = \overline{f(V(I))}$$for any ... 0 votes 0 answers 28 views ### How to prove Neron model is not affine scheme? Let K be a local field and E/K is an elliptic curve over it. Neron model ε of E/K is a smooth group scheme over SpecR which satisfies Neron mapping property. Neron model is known to be ... 0 votes 1 answer 91 views ### Example of scheme which is not an affine scheme In Hartshorne, On Page 75 Example 2.3.6. I understood that gluing between two schemes is a scheme but how to prove that it is not an affine scheme without using the notions of next Chapters like ... 0 votes 1 answer 47 views ### Is quotient of group scheme is again group scheme? Is quotient of group scheme is again group scheme ? The following is the case I'm interested in. Let K be a local field, and R be ring of integers of K, then,let ε be a group scheme over SpecR ... 0 votes 1 answer 65 views ### On closed points in quasi-compact schemes I'm working on this exercise: Show that if X is a quasicompact scheme, then every point has a closed point in its closure. Show that every nonempty closed subset of X contains a closed point of X. I ... 3 votes 1 answer 90 views ### Structure sheaf of \operatorname{Spec} A with A an integral domain Let A be an integral domain, K = \operatorname{Quot}(A) the field of fractions of A and X=\operatorname{Spec}(A) = \{p\subset A \mid \ p \ \text{prime ideal of }A \} the spectrum of A with ... 4 votes 1 answer 122 views ### Proving that the disjoint union of two affine schemes is isomorphic to an affine scheme An "Easy Exercise" from Vakil's text: Show that the disjoint union of a finite number of affine schemes is also an affine scheme. To me it's not so easy. Let's do it for two affine schemes ... 1 vote 1 answer 93 views ### Quasi-coherent sheaves of algebras over affine scheme Hi there I am study algebraic geometry for the first time and I have a question. If we consider an affine scheme \operatorname{Spec}(R), is the category of quasi-coherent sheaves of \mathcal{O}_{R}... 0 votes 0 answers 25 views ### Equality of two Zariski closures Let X be an affine scheme, R=\Gamma(X,\mathcal O_X), R_0 a subring of R, Y=\mbox{Spec}(R_0), and \phi:X\rightarrow Y the morphism induced by the inclusion of R_0 in R. Let \{I_i\} be ... 1 vote 1 answer 64 views ### Open subsets of affine schemes (refined question) We work over the complex numbers, \mathbb{C}. Let X be a smooth affine variety. Let U be an open subvariety of X. Then we have a natural map$$ i:\operatorname{Spec}\mathbb{C}[U]\to X $$My ... 1 vote 0 answers 68 views ### Exercise 8.1.H from Vakil's FOAG: criterion for ideals in affine open sets defining closed subscheme I'm trying to solve the following exercise, namely Exercise 8.1.H, from Vakil's FOAG: As hints, there are three approaches provided, of which I'm following roughly the first. The way I want to do ... 1 vote 1 answer 77 views ### Example of a non-affine scheme whose reduction is affine In Hartshorne Exercise III.3.1, he asks the reader to prove that a noetherian scheme X is affine iff X_{\bf{red}} is affine. I am not sure if Noetherian is needed here (of course I only know the ... 1 vote 1 answer 94 views ### Induced morphism between affine schemes is injective \DeclareMathOperator{\Spec}{Spec} Problem Let X = \Spec A be an affine scheme and \Spec B = Y \subset X an affine open subset of X. Let \rho be the restriction map of \mathcal{O}_X. We ... 3 votes 2 answers 79 views ### Understanding the morphism of schemes involving nilpotents induced by: \mathbb C[x] \to \mathbb C[x]/(x^2) where x \mapsto ax \pmod {x^2} From Vakil's FOAG: The following imprecise exercise will give you some sense of how to visualize maps of schemes when nilpotents are involved. Suppose a \in \mathbb C. Consider the map of rings \... 0 votes 1 answer 140 views ### Proving a surjective k-algebra homomorphism to be an isomorphism I am trying to prove the following statement: Let f:A\to B be a surjective k-algebra homomorphism. Let A and B be local Noetherian rings with same finite Krull dimension. Then f is an ... 0 votes 1 answer 30 views ### Inclusion of open affines implies injection of coordinate rings? Let U, V be two affine open subsets of a scheme X (Noetherian, over an algebraically closed field of characteristic 0) and assume that U \cap V is also affine. Is it true that the map \... 0 votes 1 answer 80 views ### How to prove the Frobenius induces a homeomorphism? Suppose A and B are two \mathbb F_p-algebras, and denote the Frobenius on A by \sigma. Then \sigma \otimes id is an endomorphism on A\otimes_ {\mathbb F_p} B. How to prove it induces a ... 1 vote 1 answer 72 views ### Can we define \operatorname{Spec}(H^0(X, \mathcal O_X)) for a scheme X? If X=\operatorname{Spec} A, then H^0(X,\mathcal O_X)=\mathcal O_X(X)=A is a ring. (1) For a general scheme X, does the space H^0(X,\mathcal O_X) of global sections of structure sheaf still ... 1 vote 1 answer 61 views ### Definition of the structure sheaf on \text{Spec} A In his book Algebraic Geometry, Hartshorne defines the structure sheaf of \text{Spec} A to be the set of functions s:U\to\coprod_{p\in U}A_p such that s(p)\in A_p and s is locally a quotient ... 1 vote 0 answers 59 views ### Is a pure coherent sheaf locally pure? Let X be a Noetherian scheme and F be a pure coherent sheaf of O_X-modules, that is any non-zero subsheaf E\subset F satisfies dim(E)=dim(F) (by dimension of a sheaf, I mean the dimension of ... 0 votes 0 answers 42 views ### How to get such a scheme? Here is an exercise from the Wedhorn's book. Exercise 3.2. Prove that there exists a scheme which admits a covering by countably many closed subschemes each of which is isomorphic to the affine line \... 1 vote 1 answer 46 views ### How to find the lifting in the spectra of tensor product? Let A,B be two R-algebra. (i.e. we have two homomorphisms: f:R\rightarrow A and g:R\rightarrow B) We denote two canonical homomorphism A\rightarrow A\otimes_R B and B\rightarrow A\otimes_R ... 1 vote 1 answer 114 views ### How to show compact, Hausdorff and totally disconnected space is homeomorphic to the spectra of a boolean ring? Let X be a compact, Hausdorff and totally disconnected space. By giving the field of two elements \mathbb F_2 the discrete topology, let A denote the set of continuous maps from X to \mathbb ... 1 vote 1 answer 122 views ### Tensor product of quasi-coherent sheaves on an affine scheme Given an affine scheme X = \text{Spec}(A), then from an A-module M we can form the associated sheaf \tilde{M}, where$$ \tilde{M}(D_f) = M_f = M \otimes_A A_f. $$Now, given A-modules M ... 1 vote 1 answer 58 views ### Alternative definition of sheaf associated to a module? Given an affine scheme X = \text{Spec}(A), then from an A-module M we can form the associated sheaf \tilde{M}, where$$ \tilde{M}(D_f) = M_f = M \otimes_A A_f$$This agrees with the presheaf$...
As the title says, could you provide a local ring which is nonreduced and its $Spec$ is reducible? It's one of the exercise in the Wedhorn's book. To get a local ring I only need to do localization at ...