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10
votes
0answers
275 views

(When) is isomorphism on differentials enough to guarantee that a map is étale?

I'm sorry if this is too easy for MO. Let $S$ be a locally noetherian scheme, flat over $\mathrm{Spec}\,\mathbb{Z}$, $X$ and $Y$ be flat $S$-schemes locally of finite presentation, and let $f:X\to Y$ ...
1
vote
1answer
199 views

Hochschild cohomology and formal smoothness

Hochschild cohomology can be used to characterise formal smoothness of unital associative algebras; in that such an algebra $A$ is formally smooth if and only if it is of Hochschild cohomological ...
3
votes
1answer
134 views

Does formal smoothness work compatibly across morphisms?

This question is about the formal smoothness property for schemes. A morphism $X\to S$ is formally smooth if for every affine $S$-scheme $Y$ and every subscheme $Y_0\subset Y$ cut out by a nilpotent ...
1
vote
1answer
153 views

Does flatness/smoothness over special fiber imply flatness/smoothness globally?

Let $f:X \to \mbox{Spec } R$ be a projective morphism between irreducible Noetherian schemes. Assume that $R$ is a discrete valuation ring and its residue field is algebraically closed. Suppose now ...
2
votes
1answer
86 views

on smoothness of morphisms on an artinian base

Let $A$, $B$ two smooth $R$-algebras of finite type for a artinian local ring $R$. Let $I$ an ideal such that $I^{2}=0$ and $\bar{R}=R/I$. We assume that the map $Spec B/IB\rightarrow Spec A/IA$ is ...
3
votes
2answers
166 views

Kolmogorov superposition for smooth functions

Kolmogorov superposition theorem states that a continuous function $f(x_1,\ldots,x_n)$ can be written as $$f(x_1,\ldots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^{n}\phi_{q,p}(x_p)\right)$$ for ...
1
vote
0answers
53 views

lift sections on a thickened curve

Let $X$ a curve over an algebraically closed field $k$ and $D$ a divisor on X. Fix an integer $N$ and a closed point $x$ on $X$, we assume that $\deg(D)$ is big enough such that we have a surjective ...
2
votes
0answers
110 views

scheme of sections over complete local ring

Let $f:X\rightarrow S= Spec(k[[\pi]])$ a finite type faithfully flat morphism. Let $U\subset X$ be an open subset such that $U$ is smooth and surjective on $S$. We consider the $k$-scheme ...
3
votes
1answer
179 views

Structure of fundamental groups arising from smooth projective morphisms

Let $f:X\to B$ be a smooth projective morphism of complex algebraic varieties. If $f$ is of relative dimension zero, i.e., $f$ is a finite etale cover, then the image of the topological fundamental ...
2
votes
0answers
117 views

Steps required to recognise a $z$-smooth number

I am currently reading section 5 of Pomerance's paper The Number Field Sieve and I have a few questions about smooth numbers. A number $x\in\mathbb Z_{\ge1}$ is called $z$-smooth if every prime ...
1
vote
0answers
101 views

smooth morphism from a finite type source

Let $f: X\rightarrow Y$ a smooth morphism over a field $k$. We assume that $X$ is locally of finite type, does it imply that $Y$ is also locally of finite type?
2
votes
0answers
137 views

fpqc, formal smoothness

Based on Possible formal smoothness mistake in EGA, let $X$ and $Y$ $k$-schemes ($k$ a field), let $f:X\rightarrow Y$ a fpqc morphism such that $f$ is formally smooth and $X$ formally smooth, do we ...
2
votes
0answers
114 views

descent for formally smooth maps

Let $f:X\rightarrow Y$ a morphism between schemes and $Y'\rightarrow Y$ a fpqc morphism such that the base change $f'$ of $f$ to $Y'$ is formally smooth, does it imply that $f$ is formally smooth?
1
vote
1answer
137 views

Importance of Denjoy-Carleman classes as a class.

Denjoy-Carleman classes of differentiable functions, say in Roumieu's form: Given a log-convex sequence $M_n$ of positive number denote by $C_M=C_M(\mathbb{R}^n,0)$ the ring of germs of ...
0
votes
1answer
262 views

Non Smooth K3 surface?

Hi, My question is related to Algebraic Surfaces. I have seen that we always consider K3 surfaces which are smooth, but I wonder how can we define a non-smooth K3 surface. The problem I see is on ...
4
votes
0answers
213 views

Smoothness of the convolution of a singular measure with itself

Let $\gamma:I\subset\mathbb{R}\rightarrow\mathbb{R}^2, s\mapsto \gamma(s)$, denote the arclength parametrization of a smooth, convex curve $\Gamma:=\gamma(I)\subset\mathbb{R}^2$. Equip $\Gamma$ with ...
8
votes
0answers
357 views

Do smooth ind schemes have Dualizing sheafs?

Say I have an ind scheme $X = \cup_i X_i$ over a field $k$. I have its tangent bundle $\hom_k(k[\epsilon], X)$ which I can think of as ind scheme via $\cup_i \hom_k(k[\epsilon],X_i)$. The problem is ...
4
votes
1answer
367 views

Is the space of smooth partitions of unity connected? Simply-connected?

One of the requirements for a smooth manifold $M$ is that it be paracompact, and one of the equivalent definitions of paracompactness for a smooth space is that for overy open cover of $M$, there ...
3
votes
1answer
311 views

Line bundles on Ind Schemes

I've learned when you have a integral smooth scheme line bundles are the same as Cartier divisors are the same as Weil divisors. My question is to what extent does this continue to hold (if at all) ...
2
votes
2answers
497 views

Smoothness of hypersurfaces in Grassmannians

I have a general question, and then the specific version of that question I need for research. All vector spaces over $\mathbb{C}$. Grassmanians of planes The $(2,n)$-Grassmannian, denoted ...
13
votes
1answer
1k views

Interpolating between piecewise linear functions, with a family of smooth functions

Let $[a,b)\subset\mathbb R$, and $F,G:[a,b)\to\mathbb R$ two decreasing piecewise linear functions so that $F(x)\leq G(x)$ for any $x\in[a,b)$. We assume that: there is a number $k\in\mathbb ...
10
votes
0answers
374 views

Exotic smoothness and Parallelizability

Regarding the parallelizability of the Milnor's seven dimensional exotic spheres: Parallelizability of the Milnor's exotic spheres in dimension 7 The following question naturally arises: Suppose ...
54
votes
4answers
2k views

Parallelizability of the Milnor's exotic spheres in dimension 7

Are the Milnor's seven dimensional exotic spheres parallelizable?
2
votes
2answers
445 views

smoothness of solution for second order elliptic problem

Hello all, could someone point me to a reference that ties the smoothness of the solution $u$ to the classical elliptic problem $\nabla \cdot ( q \nabla u ) = f \;,\; x \in \Omega$ $u = g \;,\; x ...
1
vote
1answer
221 views

Cubic spline smooting question

Hello, I came across this link when searching for an algorithm for spline smoothing. Though I understand basically what I have to do I need further clarifications on the formula chosen for curvature ...
2
votes
0answers
277 views

Computable distribution on [0,1] with C-infinity distribution function

Does anyone know of an easily-describable distribution on $[0,1]$ with a density $p$ (with respect to Lebesgue measure) that satisfies the following properties: $p$ is $C^\infty$ $p(0) = a$, $p(1) = ...
1
vote
1answer
1k views

smooth approximation of the hinge loss function

I came across this paper but the smooth approximation for the hinge loss function is wrong. Can someone guide me to the proper smooth approximation (using polynomials) of the function ...
2
votes
2answers
325 views

If $K$ and $L$ are compact convex sets with smooth boundary, does their union have piecewise-smooth boundary?

Clarification: by "piecewise", I mean a finite number of pieces. I'm sure this must be true, but my search for a citation was in vain (although I did learn the new term "polyconvex"). Thanks!
1
vote
2answers
253 views

What are sufficient conditions on $f, g : \mathbb{R}^n \to \mathbb{R}$ such that `$\{x : f(x) \geq t \} \cup \{ x : g(x) \geq u\}$` has piecewise-smooth boundary?

What are sufficient conditions on $f, g : \mathbb{R}^n \to \mathbb{R}$ such that $\{x : f(x) \geq t \} \cup \{ x : g(x) \geq u\}$ has piecewise-smooth boundary? Some remarks: I don't mind if the ...
7
votes
2answers
1k views

Slice knots and exotic $\mathbb R^4$

In the http://arxiv.org/abs/math/0606464v1 I read "If you want to prove existence of exotic smooth structure on $\mathbb R^4$ you can do this if you are in possession of a knot which is ...
3
votes
5answers
609 views

Does smooth target space and smooth fibers imply smooth total space?

Suppose that $f: X \rightarrow Y$ is a morphism between algebraic varieties. If $Y$ is smooth, and the fibers of $f$ over closed points of $Y$ are proper and nonsingular, does it follow that $X$ is ...
16
votes
4answers
1k views

Example of a smooth morphism where you can't lift a map from a nilpotent thickening?

Definition. A locally finitely presented morphism of schemes $f\colon X\to Y$ is smooth (resp. unramified, resp. ├ętale) if for any affine scheme $T$, any closed subscheme $T_0$ defined by a square ...
13
votes
3answers
835 views

Is there an example of a variety over the complex numbers with no embedding into a smooth variety?

Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
10
votes
2answers
1k views

Is there an example of a formally smooth morphism which is not smooth?

A morphism of schemes is formally smooth and locally of finite presentation iff it is smooth. What happens if we drop the finitely presented hypothesis? Of course, locally of finite presentation is ...
6
votes
1answer
388 views

Can one check formal smoothness using only one-variable Artin rings?

Let f:X -> Y be a morphism of schemes over a field k. Can one check that f is formally smooth using only Artin rings of the form k'[t]/t^n, where k' is also a field? Considering cuspidal curves one ...
6
votes
3answers
528 views

If \Omega_X/Y is locally free of rank dim(X)-dim(Y), is X->Y smooth?

Suppose I have a morphism f:X→Y such that the relative sheaf of differentials ΩX/Y is locally free. Does it follow that f is smooth? The answer is no, but for a silly reason. You could ...