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 1950's and ...

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4
votes
1answer
523 views

What sort of ind-scheme is this?

It apparently follows from work of Velu (MathSciNet) that every isogeny between elliptic curves in (long) Weierstrass form over $k$ can be written in the form $$ \left(\frac{u(x)}{v(x)}, ...
4
votes
0answers
172 views

A suspected typo, and Deligne's image of the general fiber swallowing the special

In SGA 4.5 (Arcata V.1) Deligne writes: Let $X$ be a complex analytic variety and $f:X\rightarrow D$ map $X$ into the disk. Write $[0,t]$ for closed line segment with extremities 0 and $t$ in ...
20
votes
5answers
3k views

Reduced scheme and closed points

In The Geometry of Schemes by Eisenbud and Harris, Exercise I-32 asks one to show that a scheme $X$ is reduced if and only if every local ring $\mathcal{O}_{X,p}$ is reduced for closed points $p \in ...
0
votes
0answers
68 views

Do closed points have “locally maximal” codimensions?

Let $S$ be a Noetherian excellent irreducible scheme of finite Krull dimension; since the question is local it may also be assumed to be affine. Let $s$ be a closed point of $S$. My question is: does ...
7
votes
0answers
175 views

A Hartogs-type criterion for flatness

Let $U$ be a smooth affine connected variety over $\mathbb C$ and let $V\subset U$ be an open whose complement is of codimension at least two. Now, let $Y$ be a smooth quasi-affine connected variety ...
8
votes
0answers
278 views

Are all formal schemes *really* Ind-schemes?

I'm trying to understand whether there's a fully faithful functor $LRS \supset FormalSch \to IndSch$ and in what sense. Here's my progress so far: Let $\mathsf{A}$ be the category of adic rings. The ...
3
votes
0answers
95 views

Equivalent definitions of the Hasse invariant

As probably many others before me, I got stuck in verifying all the nice properties of the Hasse invariant. Let me start by recalling one definition: Let $E\to S$ be an elliptic curve in ...
5
votes
1answer
169 views

Spin structures on schemes

This is a very naive question, but I have been wondering about the role of spin geometry and spinor structures in the context of algebraic geometry. I know the definition of spin structures and ...
155
votes
14answers
11k views

What elementary problems can you solve with schemes?

I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary ...
22
votes
1answer
556 views

Useful, non-trivial general theorems about morphisms of schemes

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians. I'm trying to ...
1
vote
0answers
217 views

About complete residues on curves

Preliminaries: Let $X$ be a projective smooth curve (scheme of finite type, integral and of dimension $1$) over a perfect field $F$. Let $K=K(X)$ be the function field of $X$ and for a closed point ...
8
votes
1answer
224 views

Tube of a mod p point on a smooth Z_(p)-scheme

Let $R$ be a smooth, integral, finite-type $\mathbb{Z}_{(p)}$-algebra of relative dimension $n$ and $\overline{f} \colon R \to \mathbb{F}_p$. Then Hensel's lemma tells us that this lifts to a map $R ...
2
votes
0answers
101 views

Quotients of quasi affine varieties and extension of scalars

I have some questions about GIT quotients and extensions of scalars of categorical quotients: 1) Let $X$ be a complex algebraic quasi-affine variety, $G$ an algebraic reductive group over ...
2
votes
1answer
163 views

Some questions about ruled surfaces defined over a number field

definitions: A non-singular complex projective surface $S$ is a ruled surface if it is birationally equivalent to $C\times_{\text{Spec} \mathbb C}\mathbb P^1_{\mathbb C}$ where $C$ is a non-singular ...
0
votes
0answers
95 views

Action on algebraic variety and adjoint bundles

Let $X$ be a complex algebraic variety and let $G$ be a complex algebraic group; I mean that $X$ is a reduced, separated scheme of finite type on $\operatorname{Spec}\mathbb{C}$, and the underlying ...
2
votes
0answers
89 views

representability of some mapping stack

Let $S$ be an Artin stack of finite type. We assume that it contains a point as an open dense. Is it always true that the mapping stack: $Hom^{0}(\mathbb{P}^{1},S)$ which consists of sections ...
3
votes
2answers
376 views

When is the determinant of the push-forward of an ample line bundle ample

Let $f:X\to S$ be a "nice" morphism of "nice" schemes. Let $L$ be an ample line bundle on $X$. When is $\det f_\ast L$ also ample? A "nice" morphism could be anything from "finite type separated" to ...
1
vote
0answers
121 views

Verdier duality on excellent schemes

Let $f:X\rightarrow Y$ be a regular morphism between $k$-schemes which are noetherian and excellent with a funcion of dimension. In the book by Illusie-Laszlo-Orgogozo, there is a theorem (4.4.1 in ...
11
votes
1answer
361 views

Obstructed automorphisms of schemes

Let $X$ be a smooth projective scheme over a field $\mathbf{k}$ of characteristic zero such that $\mathrm{H}^0(X, \mathrm{T}X)$ vanishes, and let $f$ be an automorphism of $X$. I would like to have an ...
3
votes
0answers
86 views

extending local systems on a neighbourhood

Let $Y$ an affine finite type scheme over an algebraically closed field $k$. Let $S$ be a closed subscheme of $Y$ and $Y'$ the henselization of $Y$ along $S$. If we have a $\mathbb{Z}_{\ell}$ local ...
1
vote
1answer
153 views

l-adic local system. on hensel schemes

Let $k$ be a field, $\ell$ a prime different from the characteristic. If I take $S$ a closed subscheme of $Y$, which is a $k$-scheme of finite type, is it true that any $\mathbb{Z}_{\ell}$-local ...
6
votes
0answers
141 views

Comparison of sheaves of modular forms

Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$: $e^*\Omega^1_{E/X}$ and ...
3
votes
0answers
89 views

henselizations along closed subscheme

Where can I find some references about henselizations ablong a closed subscheme? For example if I take a map $Y\times\mathbb{A}^{1}\rightarrow Y$ and $Z$ a closed subscheme. Let $Y_{Z}^{h}$ the ...
4
votes
1answer
155 views

infinitesimal lifting criterion for non-noetherian schemes

We have the "standard criterion" which says that a morphism $f:X\rightarrow Y$ is smooth if: 1/ $Y$ is locally noetherian. 2/ $f$ is locally of finite type and satisfies lifting criterion for ...
5
votes
1answer
255 views

Mapping scheme from a proper variety

Let $X$ be a proper scheme over a field $k$. Let $T$ be a scheme over $k$. Is it true that morphisms $T \times X \to \mathbb{A}^1$ are in bijection with morphisms $T \to \Gamma (X, \mathcal{O}_X)$ ...
2
votes
0answers
147 views

Description of the equalizer of $\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$

This is a crosspost of this MSE question. I have asked several questions in an attmept to get a general version of the Chinese remainder theorem without conditions on the ideals which will ...
16
votes
0answers
372 views

History of the functor of points

Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar. However, in this note by Lawvere the author writes: "I myself had learned the ...
2
votes
0answers
109 views

Morphisms of locally ringed spaces into affine schemes

In Görtz and Wedhorn's Algebraic Geometry I, there's the following proposition: Proposition 3.4. Let $(X,\mathcal O_X)$ be a locally ringed space. If $Y$ is an affine scheme then the natural map ...
32
votes
1answer
3k views

Why “open immersion” rather than “open embedding”?

When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic ...
2
votes
2answers
249 views

$\{$Affine schemes over $S$$\}$ $\cong$ $\{$$\mathcal{O}_S$ - algebras$\}$?

I've asked something very similar before on MSE but unfortunatly it hasn't recieved a lot of attention. I decided to ask again here. Let $S$ be a fixed scheme. Is the following true? "Theorem": ...
3
votes
1answer
241 views

The SGA1 version of Riemann's Existence Theorem is about analytic spaces. How does one relate it to topological covering spaces?

In SGA1, Theoreme 5.1 (Riemann's Existence Theorem) states: Let $X$ be a $\mathbb{C}$-scheme locally of finite type, $X^{\operatorname{an}}$ the associated analytical space. Then the functor which ...
8
votes
0answers
198 views

Arithmetic zeta function and local zeta functions

For the arithmetic zeta function of (say) a nonsingular projective variety $X$, one has the following Euler product \begin{equation} \zeta_X(s) = \prod_{p\ \mbox{prime}}\zeta_{X\vert\mathbb{F}_p}(s), ...
8
votes
1answer
248 views

Base schemes and Bayesian priors

One of Grothendieck's dicta about algebraic geometry is to consider "the relative situation", where one doesn't consider the category of schemes but of schemes over a fixed base scheme. In Bayesian ...
8
votes
2answers
333 views

Are there varieties with non finitely generated Picard group and vanishing irregularity?

Let $X$ be a smooth projective variety over an algebraically closed field $k$. Can it happen that $q(X) := \dim H^1(X,\mathcal O_X) =0$ and $\textrm{Pic} \,X$ is not finitely generated? Certainly, ...
19
votes
1answer
589 views

Reference for de Rham cohomology in positive characteristic

It is known in characteristic $0$ that (algebraic) de Rham cohomology is a Weil cohomology theory. However, in characteristic $p > 0$ it isn't, if only because it has mod $p$ coefficients, whereas ...
3
votes
1answer
180 views

Chevalley devissage

Let $G$ be an algebraic group over a perfect field $k$. Then it is know that it can be written as an extension of an affine algebraic group and a proper algebraic group. Is there a similar result for ...
19
votes
1answer
517 views

Why would one “attempt” to define points of a motive as $\operatorname{Ext}^1(\mathbb{Q}(0),M)$?

I'm a novice when it comes to motives. (I've read multiple introductory texts.) I'm attempting to read Galois Theory and Diophantine geometry by Minhyong Kim. In it, he says that "One might attempt, ...
0
votes
1answer
488 views

Artin approximation of a diagram

Let consider $f:(X,x)\to (Z,z)$ and $g:(Y,y)\to (Z,z)$ morphisms of pointed $k$-schemes of finite type ($k$ is a field). Suppose that there exists a map on the level of formal neighborhoods ...
7
votes
1answer
226 views

Higher-dimensional Artin L-functions

I begin by clarifying that the "higher-dimensional" in my question refers to analogues of Artin L-functions over higher dimensional base schemes than $\mathrm{Spec}(\mathbb{Z})$. Now for the set-up. ...
14
votes
2answers
802 views

Homotopy types of schemes

Let $X$ be a scheme over $\mathbb{C}$. When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex? When does the topological ...
2
votes
0answers
170 views

Blow up along a section of a smooth morphism

Let $C$ be a ground locally notherian and quasiprojective scheme. Let $\pi:S\to B$ be a $C$-morphism of finite type $C$-schemes. We call $\pi$ a $C$-smooth morphism if the morphisms $S\to C$ and $B\to ...
8
votes
2answers
292 views

Parahoric Group Scheme

I am looking for the definition of a parahoric group scheme in the sense of Bruhat and Tits? I couldn't find a reference for that? at least a "clear" reference! thanks
7
votes
1answer
321 views

About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've asked this question http://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text on math.stackexchange , however I don't think I will receive ...
2
votes
1answer
101 views

Are the fibers of this morphism geometrically regular?

Let $A\rightarrow B$ be a local morphism of complete noetherian rings making $B$ a formally smooth $A$-algebra. Does the induced morphism $\textrm{Spec}(B)\to\textrm{Spec}(A)$ have geometrically ...
6
votes
1answer
293 views

Isotrivial families with non-zero Kodaira spencer map

Let $S$ be a smooth quasi-projective curve over the complex numbers. Let $P$ be a closed point in $S$. Let $f:\mathcal X \to S$ be a polarized family of smooth projective connected varieties. To this ...
1
vote
0answers
100 views

In how many ways can one extend the zero section of the affine line with a double origin

Let $X$ be the affine line with a double origin over Spec $\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over Spec $\mathbb Q$. Let $0$ be one of the origins of ...
6
votes
0answers
217 views

Is the stack of varieties with a big line bundle algebraic

In Starr's paper https://www.math.stonybrook.edu/~jstarr/papers/moduli4.pdf the folk result that the fibred category of pairs $(X\to S, L)$, where $S$ is an affine scheme, $X\to S$ is flat proper ...
0
votes
0answers
97 views

Irreducible component of a scheme over a dvr

Let $\mathcal M$ be a (reduced) quasi-projective scheme over a dvr (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...
-3
votes
1answer
244 views

The scheme $y^n = x^{2n}$ for $n$ a rational number [closed]

Let $n\geq 1$ be an integer. If $A$ is a ring, then the spectrum of $A[x,y]/(y^n - x^{2n})$ is a well-defined (affine) scheme, say $X_n$. This scheme describes the "variety" given by the equation ...
9
votes
1answer
241 views

Torsors trivializing over a fixed finite etale cover

Let $S$ be an integral regular scheme and let $T\to S$ be a finite etale morphism. Let $G$ be a smooth affine finite type group scheme over $S$. Is the set of $S$-isomorphism classes of $G$-torsors ...