All Questions
1,159 questions
5
votes
1
answer
344
views
Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
- 28.8k
7
votes
0
answers
277
views
Is every normalization a blowup?
I asked this at math.stackexchange, but received no reply.
Is the normalization of a variety always a blowup along some coherent ideal sheaf? If not, I would like to see a concrete counter-example.
...
5
votes
0
answers
166
views
Galois ascent - When is a variety a Weil restriction?
Let $L|K$ be a finite Galois extension of degree $d$ and $X$ be a variety over $K$.
Is there a simple criterion, similar to Galois descent, allowing to determine whether $X$ is the Weil restriction (...
- 413
5
votes
1
answer
350
views
Dévissage of stratified structures in Grothendieck's "Esquisse d’un programme"
I have a question about the intuition behind Grothendieck's proposed notion of so called "Tame topology" in his Esquisse d’un programme. Grothendieck insisted that theory should admit “...
- 28.8k
3
votes
0
answers
230
views
Action of an algebraic group $G$ on a scheme $X$ with fixed rational point
Let $X$ be an irreducible locally noetherian $k$-scheme ($k$ any field) and $G$ an algebraic $k$-group acting on $X$.
Proposition 3.1.6 in these notes by M. Brion claims
Let $a : G \times X \to X$ be ...
- 5,996
0
votes
0
answers
118
views
Induced action on infinitesimal thickenings by an algebraic group
Let $X$ be an irreducible locally noetherian $k$-scheme (for $k$ any field), $G$ an algebraic group acting on $X$ via $a:G \times X \to X$ and $x \in X$ a closed point, which is by Zariski's lemma ...
- 5,996
1
vote
1
answer
366
views
Proj construction and nilpotent homogenous elements in graded ring
Let $A= \bigoplus_{n \ge 0} A_n$ be a commutative Noetherian graded ring and $f \in A_d$ a nonzero homogeneous element of degree $d>0$. The natural ring map $q:A \to A/(f)$ induces a well defined ...
- 5,996
25
votes
4
answers
6k
views
When is an irreducible scheme quasi-compact?
The standard examples of schemes that are not quasi-compact are either non-noetherian or have an infinite number of irreducible components. It is also easy to find non-separated irreducible examples. ...
- 11.1k
4
votes
0
answers
200
views
Questions about the fixed point functor $X^G$ of a $G$-scheme
Let $X$ a (locally Noetherian; but not sure if that's really matter) $k$-scheme, $G$ a $k$-group scheme acting on $X$ via morphism $a:X \times G \to X$.
The fixed point functor of $X$ (where $X$ is ...
- 5,996
1
vote
1
answer
110
views
Torsor of finite presentation and surjectivity of map of $\overline{k}$-valued points
I have a question about the content of remark 2.6.6. (i) (p 18) from M. Brion's notes on structure of algebraic groups.
Let $G$ be a group scheme over certain fixed base field $k$ (as all other ...
- 12.9k
3
votes
1
answer
517
views
Classical $k$-prevarieties vs reduced $k$-schemes of finite type. What happens when $k$ is not algebraically closed?
$\def\cpvar{\mathsf{CPVar}}
\def\spvar{\mathsf{SPVar}}
\def\Spec{\operatorname{Spec}}
\def\class{\mathrm{class}}
\def\sO{\mathcal{O}}
\def\Hom{\operatorname{Hom}}$Let $k$ be a field. By classical $k$-...
CommunityBot
- 1
4
votes
0
answers
135
views
Specialization map Chow groups preserves algebraic equivalence
Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$.
Let $\pi\colon X\rightarrow \text{Spec}(R)$ be a smooth projective morphism with geometrically integral fibers.
In ...
- 984
5
votes
1
answer
288
views
Origin of the name Trace resp Integral symbol for the trace map of Dualizing Sheaf
Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) ...
- 5,996
2
votes
1
answer
176
views
Isomorphisms $S^d(S^m(V)^*) \cong \Lambda^d(S^{m+d-1}(V)^*)$
$\DeclareMathOperator\SL{SL}$Let $K$ be an algebraically closed field. Let $V$ be a 2-dimensional vector space over $K$. The group $G=\SL_2(K)$ acts naturally on $V$ by left-multiplication on column ...
- 12.9k
39
votes
6
answers
9k
views
What is the inverse image sheaf necessary for in algebraic geometry?
Given a continuous map $f \colon X \to Y$ of topological spaces, and a sheaf $\mathcal{F}$ on $Y$, the inverse image sheaf $f^{-1}\mathcal{F}$ on $X$ is the sheafification of the presheaf
$$U \mapsto \...
- 18.4k
2
votes
0
answers
233
views
Representability of moduli problem of elliptic curves with complex multiplication
I'd like to know whether the moduli problem for elliptic curves with complex multiplication by a fixed imaginary quadratic number field $K$ (and with suitable level structure to be picked) is ...
- 91
2
votes
1
answer
179
views
Blowup formula for a morphism
Let $f: X\to S$ be a smooth projective morphism between smooth schemes over $\mathbb C$, $i: Z \to X$ a closed subscheme of codimension $c$, also smooth over $S$, and let $g: Y\to S$ be the blowup ...
24
votes
2
answers
2k
views
Does Zariski's Main Theorem come with a canonical factorization?
Zariski's Main Theorem (EGA IV, Thm 8.12.6): Suppose $Y$ is a quasi-compact and quasi-separated scheme, and $f:X\to Y$ is quasi-finite, separated, and finitely presented. Then $f$ factors as $X\...
- 4,713
2
votes
1
answer
210
views
Surjective étale map from simply connected curve over $\mathbb{C}$
Let $X$ be a simply connected algebraic curve over $\mathbb{C}$ and $f:X\rightarrow \mathbb{A}^1_{\mathbb{C}}$ is a surjective étale map.
Then is it true $f$ is finite?
All the domains of non finite ...
- 149k
6
votes
1
answer
269
views
Criteria for when Gauss-Manin sheaves are vector bundles
Let $(X,D_X)$ and $(S, D_S)$ be smooth normal crossings pairs over $\mathbb C$; i.e. smooth schemes of finite type over $\mathbb C$ with a normal crossings divisor. If $f:X \to S$ is a proper, flat ...
- 16.2k
5
votes
1
answer
773
views
Compact quasi-coherent sheaves
Let $X$ be a scheme. What are the compact objects in the category of quasi-coherent $\mathcal{O}_X$-modules? All references seem to discuss the derived category but I need the abelian category.
- 615
46
votes
0
answers
2k
views
Mikhalkin's tropical schemes versus Durov's tropical schemes
In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of ...
- 20.7k
2
votes
1
answer
200
views
Find stratification to decompose constructible sheaf to constant parts (example from Wikipedia)
I have a question about techniques used in determining the stratification over which a constructible sheaf falls into even constant pieces demonstrated on this example from Wikipedia.
Let $f:X = \text{...
- 149k
1
vote
0
answers
52
views
Symmetric 0-dimensional schemes with generic Hilbert function and Grassmannians
I've came across this problem while thinking about some properties of fat schemes.
Let me give you an explicit (motivating) example:
We have $S=\mathbb{C}[x,y,z]$, the coordinate ring of $\mathbb{P}^2$...
- 63.9k
2
votes
1
answer
186
views
Finite étale cover of factorial ring
Let $A$ be a regular factorial ring.
Consider $B=A[X]/(P)$ such that $B$ is finite étale over $A$. When do we have that $B$ is also factorial?
- 6,262
2
votes
0
answers
139
views
What are the categories of IND and PRO schemes?
below is a mathexchange question with no answers so I drop it here.
I have some difficulties to figure out what the category of IND-schemes and PRO-schemes are, in particualer the relations with ...
- 1,270
5
votes
1
answer
267
views
Some questions on derived pull-back and push-forward functors of proper birational morphism of Noetherian quasi-separated schemes
Let $f: X \to Y$ be a proper birational morphism of Noetherian quasi-separated schemes. We have the derived pull-back $Lf^*: D(QCoh(Y))\to D(QCoh(X))$ (https://stacks.math.columbia.edu/tag/06YI) and ...
- 9,229
2
votes
0
answers
113
views
Singularities of curves over DVRs with non-reduced special fibre
Let $R$ be a complete DVR of mixed characteristic with fraction field $K$ of characteristic $0$ and residue field $k$ of characteristic $p>0$. Suppose that $\mathcal{X}$ is a normal $R$-curve such ...
- 305
4
votes
0
answers
108
views
Shafarevich conjecture for Abelian varieties over global function fields
Let $S$ be a finite set of places of a global function field $K$. Are there finitely many Abelian varieties over $K$ with good reduction outside $S$? What if we exclude isotrivial families?
- 679
2
votes
1
answer
106
views
Does a polynomial $P(X,Y)$ that specializes to a polynomial $P(x_0,Y)$ with distinct roots in $\overline{k}$ have distincts roots in $\overline{k(X)}$
Let $k$ be a field, $P\in k[X][Y]$ be a monic polynomial of degree $n$ in $Y$.
I am looking for a simple proof of the following fact.
"If there exists $x_0\in k$ such that $P(x_0,Y) \in k[Y]$ has ...
- 12.9k
1
vote
0
answers
218
views
Interpretation of model theory in algebraic geometry
I found a paper Some applications of a model theoretic fact to (semi-) algebraic geometry by Lou van den Dries. In this paper, the author uses model theoretical methods to prove the completeness of ...
- 5,031
4
votes
0
answers
135
views
Nilpotent orbits in characteristic $0$ vs. positive characteristics
Let $G_\mathbb{C}$ be a connected reductive group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}_{\mathbb{C}}$. For any algebraically closed field $k$, let $G_k$ denote the connected reductive group ...
- 2,751
2
votes
0
answers
101
views
formal smoothness for henselian thickening
Assume that $A,I$ is an henselian pair over $R$ and $X$ is a smooth $R$ scheme. can we say that $X(A)\to X(A/I)$ is surjective? I know that this is true if $X$ is affine(or even quasi-projective) or ...
- 127
3
votes
0
answers
152
views
Locus where a family of cycles is rationally trivial is countable union of closed subvarieties?
Following up on this question which received a negative answer, I wonder if something weaker is true.
We work in the same set-up as the previous question. Let $B$ be a smooth quasi-projective variety ...
- 984
1
vote
0
answers
137
views
Locus where a family of cycles is rationally trivial is closed?
Let $B$ be a smooth quasi-projective variety over a field of characteristic zero.
Let $\pi\colon \mathcal{X} \rightarrow B$ be a smooth and projective morphism with geometrically integral fibres. Let $...
- 984
4
votes
0
answers
183
views
Characters of finite fields - Reference request
Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of ...
- 63.9k
0
votes
0
answers
109
views
Affine scheme over ring of meromorphic functions with finite poles on unit circle
I am looking into the set $S$ of meromorphic functions with a finite number of poles on the unit circle (i.e., rational functions with poles on the unit circle). I assume that any $h\in S$ has the ...
- 91
10
votes
0
answers
371
views
How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?
$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
- 1,335
1
vote
0
answers
169
views
Étaleness of Isom scheme $\operatorname{Isom}_S(X,Y)$
Let $S$ be a quasi-projective scheme over base field $k$ and $X, Y$ two finite étale schemes over $S$ and assume we are in situation we know that the isom space $\operatorname{Isom}_S(X,Y)$ exists as ...
- 12.9k
3
votes
2
answers
374
views
Find a non-quasi-compact scheme s.t. all finitely generated + globally generated quasi-coherent modules are finitely globally generated
Closely related to this question in MSE, but the difference is that we will set $X$ to be scheme and $\mathcal{F}$ to be quasi-coherent.
Let $X$ be a locally ringed space. We say an $\mathcal{O}_X$-...
- 452
0
votes
0
answers
331
views
Smooth morphisms under base change, Qing Liu's proposition 4.3.38
I have a concern about the first assertion in the proof of proposition 4.3.38 of Qing liu's "Algebraic Geometry and Arithmetic Curves". Referring to smooth morphisms, he says "The ...
- 6,377
2
votes
0
answers
200
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Proposition 4.3.8 Qing Liu about flat morphisms of schemes
I have a problem with a detail of Qing Liu's proof of Proposition 4.3.8 (pag. 137 of "Algebraic Geometry and Arithmetic Curves").
The statement is:
Let $Y$ be a scheme having only a finite ...
- 149
4
votes
0
answers
396
views
Non-Noetherian (classical) algebraic geometry
My starting point for this question is that, in a very classical sense, algebraic geometry is the study of solution spaces of systems of polynomial equations over an algebraically closed field. It is ...
- 365
4
votes
1
answer
254
views
Frobenius and regular scheme
Let $X$ be a noetherian regular scheme over $\mathbb{F}_{p}$. Then by Kunz's theorem, the absolute Frobenius $F: X\rightarrow X$ is flat and integral. Can it be written as a projective limit of finite ...
- 28.8k
1
vote
0
answers
98
views
Adjunction correspondence for Blow up of double point
Let $C$ a curve over an algebr closed field $k$ with a singular double point singularity at $x$ and $\pi: C' \to C$ the blowup in $x$ and let $x_1,x_2 \in C'$ be the two points over $x$.
Why holds for ...
- 5,996
2
votes
0
answers
127
views
Classification of restricted Lie algebras of reductive groups
$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
- 461
6
votes
1
answer
268
views
Hochschild cohomology of an Azumaya algebra
Let $k$ be a field. Given a commutative $k$-algebra $Z$ and an associative algebra $A$ that is Azumaya over $Z$, do we have an isomorphism of Hochschild cohomologies: $HH^*(A) \cong HH^*(Z)$?
This is ...
- 63.9k
1
vote
0
answers
50
views
Do parabolic/Levi pairs admit dynamic descriptions over disconnected base?
In Gille, Thm. 7.3.1, it is proven that given a reductive group scheme $G \to S$ over a connected base $S$, every parabolic-Levi pair $(P, L)$ over $S$ admits a dynamic description, i.e. is of the ...
- 605
4
votes
0
answers
286
views
Dévissage for a stratification in Grothendieck's Esquisse d’un programme: What is it?
I have a question about the the intuition of what Grothendieck proposed as tame topology in his "Esquisse d’un programme" as a "better suited" geometric structure in order to have ...
- 12.9k
3
votes
0
answers
251
views
Grothendieck's vs Gruson and Raynaud's dévissages
In which sense is Gruson and Raynaud's relative dévissage an "extension/ generalization" of Grothendieck's "classical" dévissage concept except that the first one works in "...
- 5,996