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Tag 06IU

Chapter 80: Formal Deformation Theory > Section 80.13: Versal formal objects

Lemma 80.13.2. Let $\mathcal{F}$ be a predeformation category satisfying (S1) and (S2). Let $\xi$ be a formal object of $\mathcal{F}$ corresponding to $\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$, see Remark 80.7.12. Then $\xi$ is versal if and only if the following two conditions hold:

  1. the map $d\underline{\xi} : T\underline{R}|_{\mathcal{C}_\Lambda} \to T\mathcal{F}$ on tangent spaces is surjective, and
  2. given a diagram in $\widehat{\mathcal{F}}$ $$ \vcenter{ \xymatrix{ & y \ar[d] \\ \xi \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & B \ar[d]^{f} \\ R \ar[r] & A } } $$ in $\widehat{\mathcal{C}}_\Lambda$ with $B \to A$ a small extension of Artinian rings, then there exists a ring map $R \to B$ such that $$ \xymatrix{ & B \ar[d]^{f} \\ R \ar[ur] \ar[r] & A } $$ commutes.

Proof. If $\xi$ is versal then (1) holds by Lemma 80.8.8 and (2) holds by Remark 80.8.10. Assume (1) and (2) hold. By Remark 80.8.10 we must show that given a diagram in $\widehat{\mathcal{F}}$ as in (2), there exists $\xi \to y$ such that $$ \xymatrix{ & y \ar[d] \\ \xi \ar[ur] \ar[r] & x } $$ commutes. Let $b : R \to B$ be the map guaranteed by (2). Denote $y' = b_*\xi$ and choose a factorization $\xi \to y' \to x$ lying over $R \to B \to A$ of the given morphism $\xi \to x$. By (S1) we obtain a commutative diagram $$ \vcenter{ \xymatrix{ z \ar[r] \ar[d] & y \ar[d] \\ y' \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ B \times_A B \ar[d] \ar[r] & B \ar[d]^{f} \\ B \ar[r]^{f} & A . } } $$ Set $I = \mathop{\rm Ker}(f)$. Let $\overline{g} : B \times_A B \to k[I]$ be the ring map $(u, v) \mapsto \overline{u} \oplus (v - u)$, cf. Lemma 80.10.8. By (1) there exists a morphism $\xi \to \overline{g}_*z$ which lies over a ring map $i : R \to k[\epsilon]$. Choose an Artinian quotient $b_1 : R \to B_1$ such that both $b : R \to B$ and $i : R \to k[\epsilon]$ factor through $R \to B_1$, i.e., giving $h : B_1 \to B$ and $i' : B_1 \to k[\epsilon]$. Choose a pushforward $y_1 = b_{1, *}\xi$, a factorization $\xi \to y_1 \to y'$ lying over $R \to B_1 \to B$ of $\xi \to y'$, and a factorization $\xi \to y_1 \to \overline{g}_*z$ lying over $R \to B_1 \to k[\epsilon]$ of $\xi \to \overline{g}_*z$. Applying (S1) once more we obtain $$ \vcenter{ \xymatrix{ z_1 \ar[r] \ar[d] & z \ar[r] \ar[d] & y \ar[d] \\ y_1 \ar[r] & y' \ar[r] & x } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ B_1 \times_A B \ar[d] \ar[r] & B \times_A B \ar[r] \ar[d] & B \ar[d]^{f} \\ B_1 \ar[r] & B \ar[r] & A . } } $$ Note that the map $g : B_1 \times_A B \to k[I]$ of Lemma 80.10.8 (defined using $h$) is the composition of $B_1 \times_A B \to B \times_A B$ and the map $\overline{g}$ above. By construction there exists a morphism $y_1 \to g_*z_1 \cong \overline{g}_*z$! Hence Lemma 80.10.8 applies (to the outer rectangles in the diagrams above) to give a morphism $y_1 \to y$ and precomposing with $\xi \to y_1$ gives the desired morphism $\xi \to y$. $\square$

    The code snippet corresponding to this tag is a part of the file formal-defos.tex and is located in lines 3715–3752 (see updates for more information).

    \begin{lemma}
    \label{lemma-versal-criterion}
    Let $\mathcal{F}$ be a predeformation category satisfying (S1) and
    (S2). Let $\xi$ be a formal object of $\mathcal{F}$ corresponding to
    $\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}$, see
    Remark \ref{remark-formal-objects-yoneda}.
    Then $\xi$ is versal if and only if the following two conditions hold:
    \begin{enumerate}
    \item the map
    $d\underline{\xi} : T\underline{R}|_{\mathcal{C}_\Lambda} \to T\mathcal{F}$
    on tangent spaces is surjective, and
    \item given a diagram in $\widehat{\mathcal{F}}$
    $$
    \vcenter{
    \xymatrix{
                &  y \ar[d] \\
    \xi \ar[r]  &  x
    }
    }
    \quad\text{lying over}\quad
    \vcenter{
    \xymatrix{
             &   B  \ar[d]^{f} \\
    R \ar[r] &   A
    }
    }
    $$
    in $\widehat{\mathcal{C}}_\Lambda$ with $B \to A$ a small extension of
    Artinian rings, then there exists a ring map $R \to B$ such that
    $$
    \xymatrix{
             &   B  \ar[d]^{f} \\
    R \ar[ur] \ar[r] &   A
    }
    $$
    commutes.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    If $\xi$ is versal then (1) holds by
    Lemma \ref{lemma-smooth-morphism-essentially-surjective}
    and (2) holds by
    Remark \ref{remark-versal-object}.
    Assume (1) and (2) hold. By
    Remark \ref{remark-versal-object}
    we must show that given a diagram in $\widehat{\mathcal{F}}$ as in (2),
    there exists $\xi \to y$ such that
    $$
    \xymatrix{
                &  y \ar[d] \\
    \xi \ar[ur] \ar[r]  &  x
    }
    $$
    commutes. Let $b : R \to B$ be the map guaranteed by (2). Denote
    $y' = b_*\xi$ and choose a factorization $\xi \to y' \to x$
    lying over $R \to B \to A$ of the given morphism $\xi \to x$.
    By (S1) we obtain a commutative diagram
    $$
    \vcenter{
    \xymatrix{
    z  \ar[r] \ar[d]          &  y \ar[d] \\
    y' \ar[r]  &  x
    }
    }
    \quad\text{lying over}\quad
    \vcenter{
    \xymatrix{
    B \times_A B \ar[d] \ar[r] &   B  \ar[d]^{f} \\
    B \ar[r]^{f} &   A .
    }
    }
    $$
    Set $I = \Ker(f)$. Let $\overline{g} : B \times_A B \to k[I]$
    be the ring map $(u, v) \mapsto \overline{u} \oplus (v - u)$,
    cf.\ Lemma \ref{lemma-lifting-along-small-extension}.
    By (1) there exists a morphism $\xi \to \overline{g}_*z$ which lies over a ring
    map $i : R \to k[\epsilon]$. Choose an Artinian quotient
    $b_1 : R \to B_1$ such that both $b : R \to B$ and $i : R \to k[\epsilon]$
    factor through $R \to B_1$, i.e., giving
    $h : B_1 \to B$ and $i' : B_1 \to k[\epsilon]$.
    Choose a pushforward $y_1 = b_{1, *}\xi$, a factorization
    $\xi \to y_1 \to y'$ lying over $R \to B_1 \to B$ of $\xi \to y'$, and a
    factorization $\xi \to y_1 \to \overline{g}_*z$ lying over
    $R \to B_1 \to k[\epsilon]$ of $\xi \to \overline{g}_*z$.
    Applying (S1) once more we obtain
    $$
    \vcenter{
    \xymatrix{
    z_1  \ar[r] \ar[d] & z \ar[r] \ar[d] & y \ar[d] \\
    y_1 \ar[r] & y' \ar[r] &  x
    }
    }
    \quad\text{lying over}\quad
    \vcenter{
    \xymatrix{
    B_1 \times_A B \ar[d] \ar[r] & B \times_A B \ar[r] \ar[d] & B \ar[d]^{f} \\
    B_1 \ar[r]  & B \ar[r] &  A .
    }
    }
    $$
    Note that the map $g : B_1 \times_A B \to k[I]$ of
    Lemma \ref{lemma-lifting-along-small-extension}
    (defined using $h$)
    is the composition of $B_1 \times_A B \to B \times_A B$ and the map
    $\overline{g}$ above. By construction there exists a morphism
    $y_1 \to g_*z_1 \cong \overline{g}_*z$! Hence
    Lemma \ref{lemma-lifting-along-small-extension}
    applies (to the outer rectangles in the diagrams above)
    to give a morphism $y_1 \to y$ and precomposing
    with $\xi \to y_1$ gives the desired morphism $\xi \to y$.
    \end{proof}

    Comments (2)

    Comment #2639 by Xiaowen Hu on July 10, 2017 a 10:27 am UTC

    Set $I=\ker(k)$ should be $I=\ker(f)$.I=Ker(k)

    Comment #2662 by Johan (site) on July 28, 2017 a 5:08 pm UTC

    Thanks, fixed here.

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