Questions tagged [schemes]

The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1). It was finalized by Alexandre Grothendieck, during the 1950s and the 1960s.

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121 views

Epi-mono factorisations in schemes via scheme-theoretic image

Suppose that $f : X \rightarrow Y$ is a morphism of schemes. Let $Z \hookrightarrow Y$ the scheme-thereotic image of $f$. Under what conditions is the morphism $X \rightarrow Z$ an epimorphism? If ...
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161 views

Are there nonaffine schemes over which every exact sequence of vector bundles is split?

Is there an example of a non-affine scheme $X$ such that every short exact sequence of vector bundles over $X$ splits? If there are such examples then what if we ask it to be true of all (not ...
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61 views

The weight of a weighted filtration is given (for large $m$) by a polynomial

Let $I$ be an homogeneous ideal of $k[x_0, \dots, x_n]$. Suppose to give integral weights $\lambda_0, \dots, \lambda_n$ to $x_0, \dots, x_n$. We assign a weight to every homogeneous polynomial of ...
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Do rationally contractible presheaves have rationally contractible injective resolution

Given a presheaf $\mathcal{F}: Sm/k\rightarrow Ab$ we define a new presheaf $C\mathcal{F}= \varinjlim\limits_{X\times \{0,1\}\subset U \subset X\times \mathbb{A}^1}\mathcal{F}(U)$. The presheaf $\...
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82 views

Closed algebraic subset dominating a curve

In the book "C. Voisin. Hodge Theory and Complex Algebraic Geometry. Volume II. Cambridge studies in advanced mathematics 76 (2002)" page 228 says: Let $X$ be a smooth projective variety. If ...
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100 views

integral subschems of geometrically integeral schemes

Let $X$ be a geometrically integral surface over a field $k$ and let $C$ be an integral curve on $X$. I want to know whether $C$ is geometrically irreducible over $k$ or not.
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Help to use Statistics and algebra books for community [closed]

My father has 2000 statistics and higher algebra books (schaum series etc). Need to use these for community since he passed away (India) kindly guide me I just need to know if we can donate these ...
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365 views

Extending functors between K-algebras to schemes

Assume we have $K$ and $L$ (comm.) rings, and we have a functor $F$ from the category of $K$-Algebras to the category of $L$-Algebras (I work only with commutative rings). What conditions need to ...
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278 views

Intuition behind formal neighborhood and local ring and formal power series

In The Geometry of Schemes by David Eisenbud and Joe Harris, on page 57, there is an explanation on "node" of a plane curve. The book says that, a curve $X\subseteq \mathbb A_{\mathbb C}^2$ ...
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187 views

Flatness of affine cone due to semicontinuity theorem

I would like to clarify an important aspect from the discussion in this question. The OP discussed an obstacle to solve part (c) from Exercise 9.5 from Hartshorne's Algebraic Geometry Chap. III page ...
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108 views

Combinatorics of projective planes over commutative rings

An axiomatic projective plane is a point-line incidence structure with the following axioms: any two distinct points are collinear (via a unique line); any two distinct lines meet in a unique point; ...
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144 views

Relative affine schemes

I was reading these notes by D. Gaitsgory, and I don't understand a claim he makes about relative affine schemes. Namely, on page 3 he says that if $f: Y \rightarrow X$ is an affine scheme over $X$, ...
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Is every variety an image of a smooth variety?

Let $X$ be a finite type scheme over a field $k$. Is it true that there exists a surjective morphism $f : Y \rightarrow X$, where $Y$ is smooth over $k$? In other words, is every such scheme a ...
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Ring of global sections of a functor $\mathbf{CRing} \to \mathbf{Set}$

Let $U : \mathbf{CRing} \to \mathbf{Set}$ be the forgetful functor. For any functor $F : \mathbf{CRing} \to \mathbf{Set}$ consider the class of natural transformations $$\mathcal{O}(F) := \mathrm{Hom}(...
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Is the restriction of an injective sheaf on a closed subscheme still injective?

Let $X$ be a Noetherian scheme, and let $i:Z\to X$ be the inclusion of a closed subscheme $Z$. Let $\mathcal{I}$ be an injective sheaf of modules on $X$. Question. Is $i^*\mathcal{I}$ still an ...
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Looking for the exact and the precise statement of Ogus conjecture

I have been looking for several weeks for the exact and the precise statement of Ogus conjecture, but, I cannot find it. The only book which made me discover the statement of this conjecture is that ...
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Interesting property of a divisor contained in special fiber

Let $(A, \mathfrak{m}, \kappa=A/\mathfrak{m})$ be a local ring and $f:X \to \operatorname{Spec} (A)$ a scheme. Let $D \subset X$ a divisor on $X$ contained in special fiber $D \subset f^{-1}(\sigma_{\...
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589 views

Motivation for Henselian rings in algebraic geometry

In Andrew Kobin's script on Algebraic Geometry I found on page 355 a comment I would like better understand. It states Another way to view formal smoothness is as an abstraction of Hensel's Lemma. ...
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163 views

Generalizations of Artin–Verdier duality?

Constructible étale abelian sheafs on $Spec\ O_\mathbb K$, for number fields $\mathbb K$, satisfy Artin-Verdier duality. Are there known any algebraic schemes or algebraic stacks, other than $Spec\ O_\...
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100 views

Support of a coherent sheaf over a fiber product scheme

I'm trying to prove the following fact which I don't know if it is true since I am not able to find a counterexample: Let $X,S$ be two $K$-scheme of finite type with $K$ an algebraically closed field....
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202 views

$H^1(X, GL(n, \mathcal{O}_X))$ and Vector Bundles

Let $X$ be a nice space: a manifold, a variety over a field, or so on. We know that line bundles on $X$ up to isomorphisms are given by $H^1(X,\mathcal{O}_X^*)$. Can it be generalized to higher rankal ...
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Moduli stacks and representability of diagonal by schemes

The answer to my question might very well be standard, but I have had trouble finding the right keywords to search for it, so I apologize if this is something well-known to the experts. I am learning ...
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1answer
453 views

Does Grothendieck's algebraization imply existence of colimits of schemes?

I am a little bit confused about two lemmas regarding Grothendieck's algebraization. Assume all algebras are defined over some field. Here is the short version of my question: Does Tag 09ZT ("...
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138 views

Is quotient of projective scheme over arbitrary base by a finite group also projective

This question probably follows from standard geometric invariant theory. If true I'd to know a reference for it. Given a projective scheme $X\rightarrow S$ over the base $S$. Let's assume a finite ...
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237 views

On the definition of a principal ideal sheaf

In his book Algebraic Geometry and Arithmetic Curves Qing Liu claims in Exercise 3.4, page 56, the following for a scheme $X$ and a global function $f\in \mathcal O_X(X)$: "The map $U\mapsto f\...
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136 views

Colimits of the infinitesimal neighborhoods of symmetric product in the category of schemes

This problem is highly related to this one and in fact it is the same question applied to a very specific situation. Given a smooth projective curve $C$, let $\text{Sym}^i(C)$ be the $i$-th symmetric ...
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60 views

Foundational question: to nonunitial commutative rings correspond to schemes?

Affine schemes correspond to unitial commutative rings, of course. Further, let us draw upon the Gelfand correspondence, where commutative $C^*$-algebras are dual to compact hausdorff spaces. If we ...
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106 views

Equidimensional Morphism

I am reading the paper "Relative Cycles and Chow Sheaves" due to Suslin and Voevodsky. Here we have the following definition: Definition 2.1.2. A morphism of schemes $p:X\rightarrow S$ is ...
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138 views

How can we generalize the finite type property so that global sections still have the same property?

Motivation: When I was young(er), I was once shocked to learn that, for $X$ a scheme of finite type, $\Gamma(X,\mathscr{O}_X)$ can fail to be of finite type. Now that I am no longer so young and ...
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66 views

Is the blowup of a toric variety corresponding to a subdivision normal?

Toroidal Embeddings 1 by KKMS say subdividing the fan of a toric variety yields the fan of a normalized blowup. How do I avoid normalization? Do I need to choose my subdivision carefully, or is it ...
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113 views

Transversally intersecting divisors $C$ and $D$ in a Hartshorne's AG lemma

Question about proof of lemma V.1.3 in Robin Hartshorne's Algebraic Geometry on page 358. Let $X$ be surface. That's for us a nonsingular projective surface over an algebraically closed field $k$ and ...
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83 views

Quotient of $\text{Proj}(A)$ by the action of a finite group

Let $X$ be $ \operatorname{Proj}(A)$ for some graded ring A, and let $G$ be a finite group acting on $A$ with morphisms of graded rings; consequently $G$ acts on $X$. I know I can write $X = \bigcup_{...
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221 views

Colimit of nilpotent thickenings in the category of schemes

This question is highly related to this and this one. Given a ring $A$ and an ideal $I$, the direct system of schemes $\text{Spec}(A/I)\rightarrow \text{Spec}(A/I^2)\rightarrow \ldots$ has a colimit ...
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275 views

Vector bundles on complete rings

Given a ring $A$ and an ideal $I$, consider the completion $\hat{A}$. What does usually mean by a vector bundle on $\hat{A}$? One way is to consider projective $\hat{A}$-modules. Another one is a ...
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109 views

Reduction theory of higher dimensional algebraic varieties

If $X$ is a nonsigular curve over a number field $K$, one can obtain several arithmetic models of $X$. Namely, we can construct an arithmetic surface $\mathcal X\to\operatorname{spec} O_K$, such that $...
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1answer
191 views

How to show analytification functor commutes with forgetful functor?

Let $k$ be a field complete with respect to a non-trivial non-archimedean absolute value (so that rigid $k$-space makes sense). Denote $K$ a finite field extension of $k$. Denote $X\rightsquigarrow X^{...
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134 views

Curves of genus 0 over a DVR determined by fibers?

Closely related is this question. Suppose $R$ is a DVR with fraction field $K$ and residue field $k$ (say finite), and $S = \mathrm{Spec}(R)$. I am interested in regular, proper, flat schemes $X \to S$...
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236 views

What is the category of coherent sheaves on a logarithmic scheme?

I try to learn basic things on logarithmic geometry, and in particular I don't find much on the category of coherent sheaves on a logarithmic scheme: is it a notion that makes sense or differ from ...
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82 views

A Subfunctor of Quot-functor compatible with pullbacks

Let $X$ be a smooth projective irreducible algebraic curve over field $k$. For $d,r,k,m >0$ the representable Quot scheme $\mathcal {Quot}_X^{r,d}(\mathcal{O}_X(m)^k)$ is given for any test scheme $...
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Rank of the top Chow group

Let $X$ be a regular integal scheme of finite type over $\mathbb Z$ and assume that $X$ has dimension $d$. In general it is not known if the Chow groups $CH^q(X)$ ($q$ is the codimension) are finitely ...
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Valuation diagram of scheme

Let $f: X\rightarrow Y$ be a morphism of schemes, then if $f$ is quasi-compact, then there exist a valuation ring $A$ and its fraction field $K$ satisfying the following commutative diagram. The proof ...
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164 views

Is an algebraic space having a monomorphism to an affine scheme a scheme?

Definition An algebraic space is a functor $X$ from the opposite of the category of commutative rings to the category of sets satisfying the following conditions: The functor $X$ is a (large) etale ...
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292 views

Finiteness of the integral closure of an integral domain in its field of fractions

I've just started reading John Milne's book ''Etale Cohomology". Prop. 1.1 of Sec 1 in Ch1 reads as follows: {\em If $X$ is a normal scheme and $X'\to X$ is its normalization in a certain ...
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Gluing categorical limit over subgraphs

Let $C$ be a category, and $\Gamma$ a graph in $C$. Under good conditions it makes sense to talk about the limit $\lim \Gamma$ of $\Gamma$ in $C$. Suppose $\Gamma$ is the union of two subgraphs $\...
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Gluing two affine schemes along a different intersection

Given a quasi-projective $X$ variety that is the union of two affines $\text{Spec(A)}$, $\text{Spec(B)}$ with intersection $\text{Spec}(C)$. Let $f\in C$. Then $\text{Spec}(C_f)$ is an open in both of ...
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Building algebraic geometry without prime ideals

$\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\ev{ev}$Teaching algebraic geometry, in particular schemes, I am struggling to provide intuitive proofs. In particular, I find it counter-intuitive ...
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92 views

etale locally infinitesimal lifting property

For a morphism $X\rightarrow Y$ of qcqs schemes, one has the usual notion of formal smoothness which says that for a pair $(R,I)$ with $I^2=0$, if there is a point $y\in Y(R)$ such that $y_{\vert R/I}$...
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100 views

Existence of integral extension of DVR satisfying some conditions

Let $X^{\prime}$ and $X$ be integral noether schemes over $\mathbb{C}$, and $p:X^{\prime}\rightarrow X$ be a surjective morphism. Let $R$ be any discrete valuation ring over $\mathbb{C}$ with its ...
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1answer
429 views

Fpqc-locally constant if and only if étale-locally constant?

Also in SE. Let $\mathcal{F}$ be sheave over $S_\mathrm{fpqc}$. We say $\mathcal{F}$ is a fpqc-locally constant sheaf (of finitely generated abelian groups) if there exists a fpqc covering $(S_i\to S)...
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608 views

Smooth morphism (algebraic geometry) vs. Submersion (differential geo) & Ehresman's Lemma

I have a general question about the motivation behind to definition the smooth morphisms as we know it from algebraic geometry. The most common definition of a smooth morphism $: X \to Y$ between two ...

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