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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Grothendieck group of a variety and Grothendieck group of its derived category

Let $X$ be a smooth projective variety over a field $K$. Let $K_0(X)$ denote the Grothendieck group of the abelian category of coherent sheaves on $X$. Let $\newcommand{\Pf}{\mathbf{Pf}}K_0(\Pf(X))$ ...
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Closed sets of $Spec(R)$.

I have to dounbs: Let $R$ be the ring $\mathbb{C}[x]_{(x)}$. So we have that $Spec(R)=\{(0),(x)\}$. I have to descrive closed sets of $Spec(R)$. I think that they are $\{\emptyset, Spec(R), (x) \}$ ...
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Relationship between maps and maps of rings

I take this exercise from Ravi Vakil's book. Consider the map of complex manifolds sending $\mathbb{C} \rightarrow \mathbb{C}$ via $x \mapsto y=x^2$. We interpret $\mathbb{C}$ as the $x$-line and ...
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$Spec(k[x]_{(x)})$ as a shred of curve.

Let $k$ be a field and consider the ring $k[x]$. x is a prime ideal so we can take the localization $k[x]_{(x)}$ in which there are all inverses except for $x$. So we have that ...
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How can I calculate $\mathrm{Spec}(\mathbb{Z}_{(3)})$? And $\mathrm{Spec}(\mathbb{Z}_3)$?

Let $\mathbb{Z}_{(3)}$ be the localization (in $\mathbb{Z}$) of the ideal generated by $3$. So I have to put in $\mathbb{Z}$ all the inverses of the complement of $(3)$. How can I calculate ...
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Examples of affine schemes

I have to introduce affine schemes as topological spaces in a small seminar. Could you suggest me three examples of affine schemes in order to put in evidence the correlation among traditional ...
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114 views

Why do the Zariski distinguished open subsets form a base?

Let $R$ be commutative ring with unit. I have to prove that the distinguished open sets form a base for the Zariski topology i.e. any non-empty open set is a union of distinguished ones. We have that ...
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Generic point of a curve in affine plane

Consider the affine plane over $k$, i.e. Spec $k[x,y]$. There are three kinds of prime ideals: $(0)$, $(x-a,y-b)$, and $(f(x,y))$, for $f$ irreducible. Let the ideal $(f(x,y))$ correspond to the point ...
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158 views

What is Galois theory for schemes?

I have heard about "Galois theory for schemes" in this note. I haven't read it yet and I know Galois theory and a litle bit about schemes (as what it is, some properties like seperated, irreducible, ...
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144 views

Composition of Cartesian product is Cartesian

Let $X_1,X_2, X_3$ and $Y$ be schemes over $\mathbb{C}$. Let $f_1:X_1 \to Y, f_2:X_2 \to Y$ and $f_3:X_3 \to X_2$ be three morphism of schemes. Under the composition map $f_2 \circ f_3$ from $X_3$ to ...
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Tensoring a flasque resolution by a line bundle give s a flasque resolution

I had an argument explained to me the other day and I didn't quite understand one of the steps. Here's my best reconstruction: Let $\mathscr{F}$ be a quasi-coherent sheaf, and $\mathscr{L}$ a ...
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Concentrated schemes are closed under finite gluings

Let $X$ be a scheme. Assume that $X = \cup_i X_i$ is finite open covering, such that the $X_i$ as well as their intersections $X_i \cap X_j$ are concentrated, i.e. quasi-compact and quasi-separated ...
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Can I check smoothness after a base-change

Let $X\to S$ be a flat morphism of noetherian schemes. I know that I can check smoothness on the geometric fibers to see whether $X\to S$ is smooth. Let $T\to S$ be a surjective morphism. Under what ...
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Orbits of affine group scheme actions

Motivation Let $F$ and $K$ be distinct algebraically closed fields. Then $GL_n(F)$ acts on $M_n(F)$ by conjugation, and the nilpotent orbits of this action can be characterized by the Jordan normal ...
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Why is every Noetherian zero-dimensional scheme finite discrete?

In the book The geometry of schemes by Eisenbud and Harris, at page 27 we find the exercise asserting that Exercise I.XXXVI. The underlying space of a zero-dimensional scheme is discrete; if ...
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understanding schemes finite over Spec $K$

I am following Vakil's FOAG, exercise 7.3.H: Let $X\to $Spec $K$ be a finite morphism, prove that $X$ is a finite union of points with the discrete topology. I am following the guidance there. If we ...
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Is this the smallest non-affine scheme?

Exercise I.XXV. of the book Geometry of Schemes by Eisenbud and Harris claims that the smallest non-affine scheme has three elements with a constructed topology and sheaf. But I am wondering if this ...
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morphism of schemes factors through diagonal

Suppose that $X$ is a scheme over $Y$. Let $f:U \to X \times_{Y}X$ be a morphism such that $p_1 \circ f = p_2 \circ f$ where $p_1, p_2$ are projections maps from $X \times_{Y} X \to X$. Show that $f$ ...
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Are the hom sets in the category of varieties abelian groups?

This is supposedly (though I know of the proof bud haven't read it) for the Hom sets of noetherian schemes. Since every variety can be thought of as a noetherian scheme then it seems right... when ...
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Fiber product in scheme category

Let $X'$, $X$, and $Y$ be $S$-schemes. Let $f:X'\to X$ be an $S$-morphism such that $f_{x'}^{\#}$ is an isomorphism for all $x'\in X'$. Denote $g:=f\times_S \text{id}_Y:X'\times_SY\to X\times_SY$. ...
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Sheaf of meromorphic functions on an integral scheme

It is a theorem that the sheaf of meromorphic functions on an integral scheme is equal to the constant sheaf where each open set is assigned the function field of the scheme. See, for example, the ...
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Where can I find this result?

A Noetherian scheme $X$ over an algebraically closed field $k$ the set of derivations $\mathcal{O}_{X,x} \to \kappa(x)=k$, is isomorphic to the Zariski tangent space $(\mathfrak{m}/\mathfrak{m}^2)^*$ ...
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Is a prevariety the same as a notherian scheme?

By a pre-variety I mean a quasi-compact locally ringed space which can be covered by (a finite # of) affine varieties. I was wondering, this seems to be the same in scheme language as a Noetherian ...
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How does Liu intend his readers to compute $H^0(X,\Omega^1_{X/k})$ of this scheme?

In the chapter on duality theory in Liu's Algebraic Geometry and Arithmetic Curves, there is the following exercise. I'm having trouble seeing how one can apply the duality theory developed in Liu to ...
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When are complete intersections also local complete intersections?

First, let us recall some definitions. Let $P=\mathbb P^d_k$ be a projective space over a field $k$. Let $X$ be a closed subvariety of $P$ of dimension $r$. We say that $X$ is a complete ...
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On an openess property

Let $S=Spec(A)$ where $A$ is a noetherian integral domain. Let $f:X\rightarrow S$ be a flat, proper morphism of schemes. Let $U\subset X$ be an open and $V=f(U)$ (in particular $V$ is open by ...
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Computing Kähler differentials on $\mathbb P^1_A$ (What did Liu intend?)

Let $A$ be a ring and $X=\mathbb P^1_A$. On $D_+(T_0)\cap D_+(T_1)$, we have $$T^2_0d(T_1/T_0)= -T^2_1d(T_0/T_1).$$ How does one formally compute this transition? Of course this is ...
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133 views

What is the stalk at a point of the quotient of a scheme?

This came up when I was talking to @Benjalim on the chat. Consider the space $X=\operatorname{Spec} \mathbb C[x]$. With the usual structure sheaf, this is a scheme. Let $Y$ be $X$ with the points ...
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Why do these two constructions of a sheaf associated to a module on a projective scheme agree?

Let $B$ be a graded ring an $M$ be a $\mathbb Z$-graded $B$-module. We can associate to $M$ a sheaf of modules on $\operatorname{Proj} B$ by defining the sheaf on the principal open sets $D_+(f)$ to ...
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Some questions on the basics of invertible sheaves

Let $X$ be a scheme. A $\mathcal O_X$-module $\mathcal L$ is called invertible if, for every point $x\in X$, there is an open neighborhood $U$ of $x$ and an isomorphism of $U$-modules $\mathcal O_X|_U ...
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Elementary questions on ample invertible sheaves

Let $X$ be a quasi-compact scheme. Let $\mathcal L$ be an invertible sheaf on $X$. We say $\mathcal L$ is ample if for any finitely generated quasi-coherent sheaf $\mathcal F$ on $X$, there exists ...
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$f(Y)$ is closed iff stable under specialization

Let $f\colon Y \to X$ be a quasicompact morphism of schemes and suppose that $f(Y)$ is stable under specialization. Then $f(Y)$ is closed. I'm trying to follow the proof given here ...
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A canonical homomorphism of sheaves of modules

Let $\mathcal F$ be a sheaf of $\mathcal O_X$ modules on a scheme $X$. Fix an affine open subset $U$. If $M$ is a module over the coordinate ring of $U$, we let $\tilde{M}$ denote the associated sheaf ...
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Problem 3.1.2 in Liu — Omission in problem statement?

Exercise 3.1.2 in Liu's Algebraic Geometry and Arithmetic Curves is as follows. Let $f:X\rightarrow Y$ be a morphism of schemes. For any scheme $T$, let $f(T):X(T)\rightarrow Y(T)$ denote the ...
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Finite fiber of scheme morphism is zero-dimensional?

Let $X$ and $Y$ be locally Noetherian schemes and $f:X\rightarrow Y$ be an étale morphism of finite type. Let $x\in X$ and $y=f(x)$. I would like to know why the fiber $X_y$ is a zero-dimensional ...
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Does this morphism necessarily give rise to a finite extension of residue fields?

Let $f:X\rightarrow Y$ be a morphism of finite type of locally Notherian schemes. Let $x\in X$ and $y=f(x)$. Recall that $f$ is said to be unramified if the map of stalks $g:\mathcal O_{Y,y} ...
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Flat closed immersion into a Noetherian scheme is open

Let $X$ be an irreducible Noetherian scheme. Consider some flat closed immersion into it. I want to show that it is also open, so that the morphism is surjective. I have a few thoughts, but I can't ...
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Stalk map on fiber products of schemes

Let $X$ and $Y$ be schemes and $x$ a point on $X$. Let $f:X\rightarrow Y$ be a morphism. Recall this induces a map $f_x:\mathcal O_{Y,f(x)}\rightarrow \mathcal O_{X,x}$ on the level of sheaves. ...
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Inverse of open affine subscheme is affine

This seems ridiculously simple, but it's eluding me. Suppose $f:X\rightarrow Y$ is a morphism of affine schemes. Let $V$ be an open affine subscheme of $Y$. Why is $f^{-1}(V)$ affine? I noted that ...
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Why is this composition of scheme morphisms proper?

I am learning about proper morphisms from Liu's book. I have a question about the proof of the Lemma 3.17 on page 104. Let $A$ and $B$ be rings and suppose $\operatorname{Spec} B$ is proper over ...
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Why is the category of coherent sheaves not grothendieck?

Let $(X, \mathscr{O}_X)$ be a ringed space. It is well-known that the category $\mathbf{Mod}(\mathscr{O}_X)$ of $\mathscr{O}_X$-modules is a grothendieck abelian category (see e.g. Grothendieck's ...
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Is the product of non-separated schemes non-separated?

My question is the title, but let me be more specific: for schemes $X$ and $Y$ over $S$, with at least one non-separated over $S$, is it true that the fibered product $X\times_S Y$ is also not ...
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Why are projective schemes $\mathbb P_A^n$ over a ring not affine for $n>1$?

I recently posted a very similar question, but I hid the question I really wanted answered in it. I'm posting this to make that question explicit. Let $A$ be a nonzero commutative ring with unit. ...
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When is the image of a closed point closed under a morphism between schemes?

Let $f: X \rightarrow Y$ be a morphism between schemes. When is the image of a closed point closed? In another question , some remarks were already made. For example if $X$ and $Y$ are of finite type ...
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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 ...