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2 votes
0 answers
129 views

Is the deformation of a $C^{\infty}$-manifold over Artin local algebra trivial?

$\DeclareMathOperator\Spec{Spec}$Let $X$ be a compact $C^{\infty}$-manifold without boundary. Let $(A,m)$ be a Artin local $\mathbb{C}$-algebra such that $A/m\cong \mathbb{C}$. Intuitively, a ...
Zhaoting Wei's user avatar
  • 9,019
2 votes
1 answer
363 views

Is the completed tensor product (over a complete dvr) of two reduced complete Noetherian local rings again reduced?

To be more specific, Let $\mathcal{O}$ be a finite extension of $\mathbb{Z}_{p}$. Let $A=\mathcal{O}[[X_{1},\ldots, X_{n}]]/\left( f_{1},\ldots,f_{r}\right) $ and $B=\mathcal{O}[[Y_{1},\ldots, Y_{m}]]/...
user521295's user avatar
1 vote
0 answers
155 views

Homogeneous deformation of isolated singularities

Let $f\in \mathbb{C}[x_1, \dots,x_n]$ be a homogeneous polynomial of degree $p$ and let $F \in \mathbb{C}[x_1, \dots,x_n,t]$ be a polynomial such that $F(x_1, \dots, x_n,0)=f$, for every $t_0$ we have ...
Serge the Toaster's user avatar
2 votes
1 answer
205 views

Deformation of isolated singularities and non zero divisors

Consider $f \in \mathbb{C}\{x_1,\dots,x_n\}$ such that $(V(f),0)$ has an isolated singularity. Let $F \in \mathbb{C}\{x_1,\dots,x_n,t\}$ be a deformation of $f$ such that there exists some integer $m$...
Serge the Toaster's user avatar
4 votes
0 answers
248 views

Artin's "Versal Deformations and Algebraic stacks": Question concerning proof of Theorem 3.3

I have been reading Artin's paper titled "Versal deformations and algebraic stacks" and am a bit confused about a statement he makes in the proof of Theorem 3.3, in the first few lines of pg....
user's user avatar
  • 749
0 votes
0 answers
72 views

Explicit representative for an extension class

Let $A$ be a regular local ring and $I\subset A$ a complete intersection ideal. We have the natural map $\delta:Hom_A(I,A/I)\rightarrow Ext_A^1(A/I,A/I)$. For a given $\alpha\in Hom_A(I,A/I)$ is there ...
pi_1's user avatar
  • 1,463
3 votes
0 answers
214 views

formal smoothness and cotangent complex

If $k$ is a field and $A$ is a formally smooth $k$-algebra, then we know that $\Omega^{1}_{A/k}$ is projective. What about its cotangent complex $L_{A/k}$? When is it quasi-isomorphic to $\Omega^{1}_{...
prochet's user avatar
  • 3,472
1 vote
0 answers
105 views

formal smoothness and McQuillan formal schemes

Let $k$ be an algebraically closed field, $A\rightarrow B$ be a continuous map of weakly admissible topological $k$-local algebras. We assume that it is formally smooth and topologically of finite ...
prochet's user avatar
  • 3,472
1 vote
0 answers
88 views

Infinitesimal neighbourhoods and simultaneous normalization

Let $B$ be a local, complete, integral $\mathbb{C}$-algebra of Krull dimension $1$ and $n:B \to \mathbb{C}[[t]]$ the normalization map. Given any local artinian $\mathbb{C}$-algebra $A$, we say that ...
Ron's user avatar
  • 2,126
0 votes
1 answer
149 views

Power series rings and the formal generic fibre

Let $S = K[[S_1,\ldots,S_n]]$ and consider $d$ elements \begin{equation*} f_1,\ldots,f_d \in S[[X_1,\ldots,X_d]] \end{equation*} and the prime ideal ${\frak P} \colon\!= (f_1,\ldots,f_d)$ generated ...
Pierre's user avatar
  • 563
1 vote
0 answers
138 views

Power series ring $R[[X_1,\ldots,X_d]]$ over a domain $R$

Let $R$ be a domain and \begin{align*} T \,\colon= R[[X_1,\ldots,X_d]]. \end{align*} Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...
Pierre's user avatar
  • 563
4 votes
0 answers
363 views

Does the local Bertini theorem in mixed characteristic imply the global Bertini theorem

Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Assume that the characteristic of $K$ is $0$ and of $k$ is $p>0$. Let $\pi:X \to \mbox{Spec}(R)$ be a flat, ...
Jana's user avatar
  • 2,032
2 votes
1 answer
609 views

Generators of an ideal of $K[[X_1,X_2,X_3]]$

Let us consider an irreducible polynomial $$f \colon= \alpha_e + \alpha_{e-1}X_1 + ... + \alpha_1X_1^{e-1} + X_1^e \in K[[X_2,X_3]][X_1]$$ and set $$\iota_1 \colon K[[X_2,X_3]] \hookrightarrow K[[X_1,...
Pierre's user avatar
  • 563
5 votes
1 answer
367 views

Do singularities of plane curves deform independently?

If $C\subset\mathbb{P}^2$ is an integral curve of degree $d$, do its singularities deform independently as we vary $C$ over degree $d$ curves? If not, what about in the case $C$ is a nodal curve or $C$...
DCT's user avatar
  • 1,537
4 votes
1 answer
512 views

Being Cohen-Macaulay open in Hilbert scheme?

Let $H$ be the Hilbert scheme of closed subschemes of $\mathbb{P}^n$ with a given Hilbert polynomial. I would like to have a reference (preferably from a published paper or book, not stacks project) ...
Hans's user avatar
  • 3,031
2 votes
0 answers
132 views

Local diffeomorphism (étale maps) in terms of infinitesimal tubular neighborhood?

In this MO question I asked for some help with several definitions of formally etale maps. The definitions I'm asking about are 'isomorphism of tangent spaces', i.e the square below is a pullback $$\...
Arrow's user avatar
  • 10.5k
1 vote
0 answers
357 views

Base change and geometrically generic reduced fiber

Let $k$ be an algebraically closed field of characteristic $p>0$ and $f:X \to Y$ be a quasi-projective morphism between noetherian $k$-schemes. Assume that $Y$ is regular and the geometric generic ...
Ron's user avatar
  • 2,126
2 votes
0 answers
136 views

quasi-ordinary singularities on a versal deformation?

Let $V$ be a variety over $\mathbb{C}$ and suppose $O$ is a singular point of $V$. Are there conditions on $(V,O)$ such that a versal deformation $W$ of $(V,O)$ has only quasi-ordinary singularities. ...
Andrew Stout's user avatar
6 votes
0 answers
712 views

Spectral sequences and Koszul complexes in Deformation Theory

I'm reading this paper of A. Vistoli and I have some questions about the discussion in page 5. This is the context (If you don't want to download the paper): Let $A'$ be a noetherian local ring with ...
Pedro Montero's user avatar
4 votes
1 answer
1k views

Formal criterion of flatness

Let $k$ be a field, $S$ and $R$ be local $k$-algebras with residue field $k$ and $\phi:S\to R$ be a local homomorphism. Then $\phi$ induces (obviously) a natural transformation of "functors of points" ...
Piotr Achinger's user avatar
18 votes
1 answer
4k views

Deformations of the punctured affine plane

Let $k$ be some field, algebraically closed and of characteristic $0$, if you like. Let $U= \mathbb{A}^2_k \setminus \{ (0,0) \}$ be the punctured affine plane over $k$. Write $U$ as the union of $...
Peter Scholze's user avatar
3 votes
0 answers
197 views

which deformation of a matrix lead to flat deformations of determinantal varieties (fitting ideals)?

Let $(R,m)$ be a complete local ring over a field (of char=0). Consider a (not necessarily square) matrix $A$ over $R$. Consider its fitting ideal, $I_j(A)$. In general, a deformation of the matrix, $...
Dmitry Kerner's user avatar
4 votes
2 answers
354 views

"un-nil-ifying" ideals via deformation

This is perhaps a naïve question; given an irreducible scheme $X$, is there a general procedure to find a flat family $Y \to T$ such that over some point $t_0$ we have $Y \times_T {t_0} \cong X$ but ...
anon's user avatar
  • 467
3 votes
0 answers
325 views

Obstructions for reduced embedded deformation of Artinian rings

Let $A$ be an Artinian local $k$-algebra. $A$ is said to have a reduced embedded deformation if there is another local ring $S$ such that $A = S/(f_1,\cdots, f_n)$ with $S$ reduced and $(\underline f)$...
Hailong Dao's user avatar
  • 30.5k