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Presumably the algebraic analogue of something that uses the Implicit Function Theorem involves saying the word "etale" a lot. Thus:

Let $x_1,...,x_m$ be a lift of a basis of $m_p/m_P^2$ to the ring regular functions of some open affine neighborhood $U$ of $P$. Then the map $U \to \textrm{Spec} k[x_1,..,x_m]$ has image some closed subscheme $\textrm{Spec} k[x_1,..,x_m]/I$. This map is etale at $P$ since it induces an isomorphism on profinite completions of local rings, so it is etale in some neighborhood of $P$, so it is given by a set of $n$ equations in $n$ variables whose Jacobian is a unit. Lift those equations arbitrarily from $k[x_1,...,x_m]/I$ to $k[x_1,...,x_m]$. Because everything in $I$ is degree two or higher at zero/$P$, the Jacobian is still invertible at zero/$P$, thus invertible on an open neighborhood.

Lifting the equations makes a neighborhood of $P$ into a closed subscheme of the cover of $k[x_1,...,x_m]$, cut out by the equations $I$. Since the cover is etale in an open neighborhood of the origin, it is smooth and dimension $m$ in an open neighborhood of the origin. Some neighborhood of $P$ embeds as a close subscheme in this smooth, dimension $m$ scheme.

Presumably the algebraic analogue of something that uses the Implicit Function Theorem involves saying the word "etale" a lot. Thus:

Let $x_1,...,x_m$ be a lift of a basis of $m_p/m_P^2$ to the ring regular functions of some open affine neighborhood $U$ of $P$. Then the map $U \to \textrm{Spec} k[x_1,..,x_m]$ has image some closed subscheme $\textrm{Spec} k[x_1,..,x_m]/I$. This map is etale at $P$ since it induces an isomorphism on formal completions of local rings, so it is etale in some neighborhood of $P$, so it is locally the vanishing set of $n$ equations in $n$ variables over $k[x_1,...,x_m]/I$ whose Jacobian is a unit. Lift those equations arbitrarily from $k[x_1,...,x_m]/I$ to $k[x_1,...,x_m]$. Because everything in $I$ is degree two or higher at zero/$P$, the Jacobian is still invertible at zero/$P$, thus invertible on an open neighborhood.

The vanishing set of the lift of those equations is a cover of $\textrm{Spec}k[x_1,..,x_m]$, etale in a neighborhood of the origin, thus smooth and dimension $m$ in a neighborhood of the origin. The closed subscheme cut out by the ideal $I$ is the same as the vanishing set of the original equations over $k[x_1,...,x_m]/I$, which was locally isomorphic to $U$. Thus $U$ is locally a closed subscheme of a variety smooth of dimension $m$.

Presumably the algebraic analogue of something that uses the Implicit Function Theorem involves saying the word "etale" a lot. Thus:

Let $x_1,...,x_m$ be a lift of a basis of $m_p/m_P^2$ to the ring regular functions of some open affine neighborhood $U$ of $P$. Then the map $U \to \textrm{Spec} k[x_1,..,x_m]$ has image some closed subscheme $\textrm{Spec} k[x_1,..,x_m]/I$. This map is etale at $P$ since it induces an isomorphism on profinite completions of local rings, so it is etale in some neighborhood of $P$, so it is given by a set of $n$ equations in $n$ variables whose Jacobian is a unit. Lift those equations arbitrarily from $k[x_1,...,x_m]/I$ to $k[x_1,...,x_m]$. Because everything in $I$ is degree two or higher at zero/$P$, the Jacobian is still invertible at zero/P, thus invertible on an open neighborhood.

Lifting the equations makes a neighborhood of $P$ into a closed subscheme of the cover of $k[x_1,...,x_m]$, cut out by the equations $I$. Since the cover is etale in an open neighborhood of the origin, it is smooth and dimension $m$ in an open neighborhood of the origin. Some neighborhood of $P$ embeds as a close subscheme in this smooth, dimension $m$ scheme.

Presumably the algebraic analogue of something that uses the Implicit Function Theorem involves saying the word "etale" a lot. Thus:

Let $x_1,...,x_m$ be a lift of a basis of $m_p/m_P^2$ to the ring regular functions of some open affine neighborhood $U$ of $P$. Then the map $U \to \textrm{Spec} k[x_1,..,x_m]$ has image some closed subscheme $\textrm{Spec} k[x_1,..,x_m]/I$. This map is etale at $P$ since it induces an isomorphism on profinite completions of local rings, so it is etale in some neighborhood of $P$, so it is given by a set of $n$ equations in $n$ variables whose Jacobian is a unit. Lift those equations arbitrarily from $k[x_1,...,x_m]/I$ to $k[x_1,...,x_m]$. Because everything in $I$ is degree two or higher at zero/$P$, the Jacobian is still invertible at zero/$P$, thus invertible on an open neighborhood.

Lifting the equations makes a neighborhood of $P$ into a closed subscheme of the cover of $k[x_1,...,x_m]$, cut out by the equations $I$. Since the cover is etale in an open neighborhood of the origin, it is smooth and dimension $m$ in an open neighborhood of the origin. Some neighborhood of $P$ embeds as a close subscheme in this smooth, dimension $m$ scheme.

Presumably the algebraic analogue of something that uses the Implicit Function Theorem involves saying the word "etale" a lot. Thus:

Let $x_1,...,x_m$ be a lift of a basis of $m_p/m_P^2$ to the ring regular functions of some open affine neighborhood $U$ of $P$. Then the map $U \to \textrm{Spec} k[x_1,..,x_m]$ has image some closed subscheme $\textrm{Spec} k[x_1,..,x_m]/I$. This map is etale at $P$ since it induces an isomorphism on profinite completions of local rings, so it is etale in some neighborhood of $P$, so it is given by a set of $n$ equations in $n$ variables whose Jacobian is a unit. Lif those equations arbitrarily from $k[x_1,...,x_m]/I$ to $k[x_1,...,x_m]$. Because everything in $I$ is degree two or higher at zero/$P$, the Jacobian is still invertible at zero/P, thus invertible on an open neighborhood.

Lifting the equations makes a neighborhood of $P$ into a closed subscheme of the cover of $k[x_1,...,x_m]$, cut out by the equations $I$. Since the cover is etale in an open neighborhood of the origin, it is smooth and dimension $m$ in an open neighborhood of the origin. Some neighborhood of $P$ embeds as a close subscheme in this smooth, dimension $m$ scheme.

Presumably the algebraic analogue of something that uses the Implicit Function Theorem involves saying the word "etale" a lot. Thus:

Let $x_1,...,x_m$ be a lift of a basis of $m_p/m_P^2$ to the ring regular functions of some open affine neighborhood $U$ of $P$. Then the map $U \to \textrm{Spec} k[x_1,..,x_m]$ has image some closed subscheme $\textrm{Spec} k[x_1,..,x_m]/I$. This map is etale at $P$ since it induces an isomorphism on profinite completions of local rings, so it is etale in some neighborhood of $P$, so it is given by a set of $n$ equations in $n$ variables whose Jacobian is a unit. Lift those equations arbitrarily from $k[x_1,...,x_m]/I$ to $k[x_1,...,x_m]$. Because everything in $I$ is degree two or higher at zero/$P$, the Jacobian is still invertible at zero/P, thus invertible on an open neighborhood.

Lifting the equations makes a neighborhood of $P$ into a closed subscheme of the cover of $k[x_1,...,x_m]$, cut out by the equations $I$. Since the cover is etale in an open neighborhood of the origin, it is smooth and dimension $m$ in an open neighborhood of the origin. Some neighborhood of $P$ embeds as a close subscheme in this smooth, dimension $m$ scheme.