The concept of scheme is from algebraic geometry: A locally ringed space that is glued together (much like a manifold) from affine schemes, that is spectra of rings with Zariski topology and structure sheaf.

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Why are projective spaces over a ring of different dimensions non-isomorphic?

Let $A$ be a nonzero commutative ring with unit. Define $\mathbb P_A^n$ to be the scheme $\operatorname {Proj} A[T_0,\dots,T_n]$, where the grading on the polynomial ring is by degree. Why is it true ...
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'$R$-rational points,' where $R$ is an arbitrary ring

On page 49 of Liu's Algebraic Geometry and Arithmetic Curves, we find Example 3.32. In it, he shows that if $k$ is a field and $X=k[T_1,\dots,T_n]/I$ is an affine scheme over $k$, the sections $X(k)$ ...
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Quasicoherent ideal sheaves on open subschemes

Let $X$ be an open subscheme of an affine scheme $\operatorname{Spec} A$, let $f : X \to \operatorname{Spec} A$ be the inclusion, and let $\mathscr{I}$ be a quasicoherent ideal sheaf on $X$. Since ...
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A particular closed subscheme

Look at the following definition: Let $(X,\mathscr O_X)$ be a scheme. A closed subscheme of $(X, \mathscr O_X)$ is a scheme $(Z, \mathscr O_Z)$ such that: $Z$ is a closed subset of $X$ ...
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Toric Varieties: gluing of affine varieties (blow-up example)

Let $\Delta$ be a fan, consisting of cones $\sigma_0=conv(e_1,e_1+e_2)$ and $\sigma_2=conv(e_1+e_2,e_2)$ and $\tau=\sigma_0\cap\sigma_1=conv(e_1+e_2)$. The dual cones are $\sigma_0^\vee= ...
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Morphism of finite type between affine schemes is quasi-projective

I want to prove that given $A \to B$ a ring homomorphism of finite type, then the induced morphism of schemes $X \to Y$ is quasi-projective. A morphism is quasi-projective if it factors into an open ...
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Morphisms between locally ringed spaces and affine schemes

I need some hints to understand the conclusion of the proof of the following lemma from the Stacks Project: Lemma $\mathbf{6.1.}\,$ Let $X$ be a locally ringed space. Let $Y$ be an affine scheme. ...
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Points contained in the diagonal of the product of schemes

Let $X,Y$ schemes over $S$, and $f,g$ two $S$-morphisms of schemes, $h:X \to Y\times_{S} Y$ the morphism obtained from $f$ and $g$ and $\Delta:Y \to Y \times_{S} Y$ the diagonal morphism. I tried to ...
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von Neumann Stability help

Using the forward time centered space scheme, I transformed the equation: $u_t-2u_{xx}-u_{yy}=0$ to ...
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247 views

Dominant morphism, equal dimensions: always finite?

Let $f:X\to Y$ be a dominant morphism of varieties (integral separated schemes of finite type over an algebraically closed field) such that dim $X$ = dim $Y$. Question: must f be finite? It seems ...
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Finishing the proof that $\textrm{Spec}K[X,Y]\setminus\{(X,Y)\}$ is not an affine scheme

If $K$ is a field, $\mathbb A^2_K=\textrm{Spec}K[X,Y]$ and $U=\mathbb A^2_K\setminus\{(X,Y)\}$, I want to prove that $(U,\mathscr O_{\mathbb A^2_K|U})$ is not an affine scheme. I know that this topic ...
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Degree of a Cartier Divisor under pullback

This is question 7.2.3 in Liu's book Algebraic Geometry and Arithmetic Curves and I have been trying with this for some time now. Let $f:X \rightarrow Y$ be a morphism of Noetherian schemes, and ...
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Hartshorne Proposition II.6.5

The statement of part (a) of this proposition is as follows: Let $X$ be a noetherian integral separated scheme which is regular in codimension 1. Let $Z$ be a proper closed subset of $X$, and let ...
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Intersection of open affines can be covered by open sets distinguished in *both*affines

Suppose $X$ is an arbitrary scheme and $U \cong \operatorname{Spec} A$ and $V \cong \operatorname{Spec} B$ are affine upon subsets of $X$. It's not true in general that $U \cap V$ is affine, so if we ...
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Algebraic analogue of maximum modulus principle applied to Riemann surface.

Let $X$ be an abstract curve in the following sense: $X$ is a scheme, proper over $k$ which is noetherian, integral, dimension 1, and normal. The important thing to point out is that I am not assuming ...
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Definition of prime ideal sheaf

In the chapter I.3.3 of Eisenbud & Harris "The Geometry of Schemes" they give a definition of "prime ideal sheaf": Let $\mathcal{O}_S$ be the structure sheaf of a scheme $S$, and let ...
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criterion for geometrically integral scheme

I want to prove the remark 3.2.9 of the book Algebraic Geometry of Arithmetic Curves (of Quing Liu) that is: let $X$ be an integral scheme with function field $K(X)$, if $K(X)\otimes_k \overline{k}$ ...
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The image of the diagonal map in scheme

Let $X\rightarrow S$ be a separated morphism of schemes, that is, the diagonal map $\Delta:X\rightarrow X\times_S X$ is a closed inmersion. In general (see Closure of image of diagonal morphism of ...
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When are the sections of the structure sheaf just morphisms to affine space?

Let $X$ be a scheme over a field $K$ and $f\in\mathscr O_X(U)$ for some (say, affine) open $U\subseteq X$. For a $K$-rational point $P$, I can denote by $f(P)$ the image of $f$ under the map ...
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Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!) I have been working on giving ...
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1answer
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Does the direct image functor on sheaves reflect epimorphisms?

Let $f:X\to Y$ be a morphism of schemes, and let $f_*:\mathbf{Sh}(X)\to\mathbf{Sh}(Y)$ be the direct image functor. Does $f_*$ reflect epimorphisms? That is, suppose ...
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Irreducibility is preserved under base extension

I want to prove that if $A$ is a finitely generated $k$-algebra ($k$ is a field) with prime nilradical then for any field extension $k\rightarrow K$, the $K$-algebra $A\otimes_kK$ has also prime ...
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Glueing sheaves on Grothendieck sites

Let $X$ be a topological space and $\{V_i\}$ a cover of $X$. Let $F_i:\mathsf{Open}(V_i)^{\mathrm{op}}\to \mathsf{Sets}$ be a family of sheaves. One can glue this family to obtain a sheaf ...
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How to understand elements of the scheme $GL_n$ as linear isomorphisms?

Let as usual $GL_n$ be the scheme given by the equation $$det(\{x_{kl}\}_{1\leq k,l\leq n})y=1$$ in $\mathbb{A}^{{n^2}+1}$. I have seen that one considers elements of $GL_n$ likewise as linear ...
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Deducing that a locally ringed space is a scheme

I would like to know if the following is true: Let $\mathscr{F}$ and $\mathscr{G}$ be two sheaves on a topological space $X$ and let $\varphi:\mathscr{F}\rightarrow\mathscr{G}$ be a morphism such that ...
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A scheme from a quasi-compact morphism

Let $f: X \to Y$ be quasi-compact, set $\mathcal{J} = \ker f^{\#}$, and $Z = V(\ker f^{\#}) = \{ y \in Y \: | \: \mathcal{J}_y \neq \mathcal{O}_{Y,y} \}$. $Z$ is a locally ringed topological space ...
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Nice proof for étale of degree 1 implies isomorphism.

Let $f: X \rightarrow Y$ be a finite étale covering of degree 1 of varieties over some field $k$, so $$ [K(Y):K(X)] = 1. $$ If $k=\mathbb{C}$ and the varieties are smooth, one can apply complex ...
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Definition of degree of finite morphism plus context

Let $f: X \rightarrow Y$ be a finite morphism of schemes, defined here, http://en.wikipedia.org/wiki/Finite_morphism I always assumed that the degree of $f$ was the degree of the induced field ...
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Zariski Tangent Space and $k[\varepsilon]/\varepsilon^2$

Let $X$ be a scheme over a field $k$, and let $x\in X$ be a rational point, that is, we have $k(x):=\mathcal{O}_x/\mathfrak{m}_x\cong k$. Let $\alpha:\mathfrak{m}_x/\mathfrak{m}_x^2\rightarrow k$ be ...
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Showing that intersection multiplicity at a point is finite for prime divisors

My question has two parts two it: one vaguely more elementary, one perhaps less so. In Beauville (Complex Algebraic Surfaces), we define the multiplicity of intersection of two (irreducible, no ...
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Restricting a Weil divisor to a local scheme

Apologies if the title is garbage. I couldn't think up anything more pertinent, as it's really just a notational question. In Hartshorne, p.141, Proposition 6.11, it is shown that Weil divisors are ...
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A question on an answer on Math Overflow about the tangent bundle

I have a question on the accepted answer of this Math Overflow question. Let $K$ be a field and $X$ a $K$-scheme. Define the morphism of schemes $T=\operatorname{Spec}\operatorname{Sym}(\Omega_{X/K}) ...
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Morphism between varieties

Let $V$ and $W$ be two (irreducible) varieties over an algebraic closed field $k$. Then there is a definition of what a morphism of varieties $f:V\rightarrow W$ is. On the other hand we can see $V$ ...
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Comparison between etale and singular cohomology for a singular variety

In his Lectures on Etale Cohomology Milne proves in Theorem 21.1, that for all $r\geq 0$ $$ H^r_{\acute{e}t}(X,\Lambda)\cong H^r(X_{cx},\Lambda) $$ with $X$ a nonsingular $\mathbb{C}$-variety and ...
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Serre's Theorem for $\operatorname{Proj}$

Let $k$ be an algebraically closed field. If $S$ is a positively graded $k$-algebra which is finitely generated by $S_1$ over $S_0 = k$ then quasi-coherent sheaves on $\operatorname{Proj}S$ are ...
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Does the inclusion from affine schemes into schemes preserve pushouts?

Let $K$ be a field. What is an example of two $K$-algebra morphisms $R\to T$ and $S\to T$ such that $\operatorname{Spec}(R\times_T S)$ is not the pushout of the diagram $$ ...
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Scheme glued out of spectra of local rings

This is a follow up question to this question. Is every scheme over a field $K$ the colimit (over some arbitrary complicated diagram) of affine schemes $\operatorname{Spec}(R_\alpha)$ where each ...
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Fiber of morphism of integral schemes

Let $X$ and $Y$ be integral schemes and $f : X \to Y$ a morphism. Let $X_{\eta'}$ be the fiber of $f$ over the generic point $\eta'$ of $Y$, i.e. the base change $X_{(K)} = X \times_Y K$ where $K = ...
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Invertible rational functions

I am looking for references for the following facts. ...
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Connected reduced proper $k$-scheme

Any connected reduced proper $k$-scheme is irreducible? Here $k$ is a field.
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A proof that every projective morphism is proper?

I am currently working my way through Q. Liu's book "Algebraic Geometry and Arithmetic Curves". I'm puzzled by the proof that every projective morphism is proper, see below I understand that ...
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An intermediate result needed in proving the existence of quotients of a ringed topological space by a group

Setting up notation: Let $G$ be a group acting on a scheme $(X, \mathcal{O}_X)$, so that for all $\sigma \in G$ we have a sheaf morphism $\sigma^{\#}: \mathcal{O}_X \to \sigma_* \mathcal{O}_X$ such ...
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Scheme glued out of finitely many spectra of local rings

Let $X$ be a noetherian separated scheme over the spectrum of a field $K$. By definition, I can find an open covering of $X$ by finitely many affine schemes $Spec(R_i)$ where each $R_i$ is a ...
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Sheaves on $\mathbb{P}^n \times \mathbb{P}^m$, and a commutation relation for derived functors of global sections and tensor products on it.

I'll state my questions first and then provide some background. Question 3 is by far my most important one. We work over $k=\mathbb{C}$ whenever necessary. Is it true that $\text{Pic}(\mathbb{P}^n ...
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Graph of morphism , reduced scheme.

Let $f:X \rightarrow Y $ be a morphism of S-schemes. Let us suppose that X is reduced and endow the image of the graph morphism $\Gamma_f:X \rightarrow X \times_S Y $ , call it A, with a reduced ...
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Is there any equivalence between the category of schemes over $\mathbb R$ and the category real manifolds

The equivalence of the category of smooth projective curves over $\mathbb C$ and the category of compact Riemann surfaces is, I believe, well documented. For example, it is mentioned on the wiki page: ...
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About normalization of scheme

I have the following definition for normalization of scheme: Let $X$ a integral scheme and $L\supseteq K(X)$ an algebraic extension. So $\pi:X'\to X$ is a normalization of $X$ in $L$ if $X'$ is ...
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An exercise in Liu regarding a sheaf of ideals (Chapter II 3.4)

I'm fairly certain there is both a typo and an omission in this exercise. It reads "Let $X$ be a scheme and $f \in \mathcal{O}_X(X)$. Show that $U \mapsto f|_U \mathcal{O}_X(U)$ for every affine open ...
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To what extent is a scheme morphism determined by its topological map?

I am just beginning to learn scheme theory. This question is aimed at getting a feel for something so apologies in advance for the lack of precision. I am struck by the following difference from the ...
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Property kept under base change and composition is preserved by products

The following is true? Why? Let $P$ be a property of morphisms preserved under base change and composition. Let $X\to Y$ and $X'\to Y'$ be morphisms of $S$-schemes with property $P$. Then the unique ...