All Questions
15 questions with no upvoted or accepted answers
6
votes
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answers
712
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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 ...
- 987
4
votes
0
answers
248
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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....
- 749
4
votes
0
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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, ...
- 2,022
3
votes
0
answers
214
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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}_{...
- 3,472
3
votes
0
answers
197
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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, $...
- 2,232
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 $(\underline f)$...
- 30.5k
2
votes
0
answers
129
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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 ...
- 9,019
2
votes
0
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132
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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
$$\...
- 10.5k
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.
...
- 761
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 ...
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 ...
- 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 ...
- 2,126
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,\...
- 563
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 ...
- 2,126
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 ...
- 1,463