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# 数学代写|交换代数代写Commutative Algebra代考|The Smooth Complete Intersection Case

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## 数学代写|交换代数代写Commutative Algebra代考|The Smooth Complete Intersection Case

We treat the case of using two equations to define a smooth complete intersection. The generalization to an arbitrary number of equations is straightforward.

Let $\mathbf{R}$ be a commutative ring, and $f(\underline{X}), g(\underline{X}) \in \mathbf{R}\left[X_1, \ldots, X_n\right]$. Consider the R-algebra
$$\mathbf{A}=\mathbf{R}\left[X_1, \ldots, X_n\right] /\langle f, g\rangle=\mathbf{R}\left[x_1, \ldots, x_n\right]=\mathbf{R}[\underline{x}]$$
The Jacobian matrix of the system of equations $(f, g)$ is
$$J(\underline{X})=\left[\begin{array}{lll} \frac{\partial f}{\partial X_1}(\underline{X}) & \cdots & \frac{\partial f}{\partial X_n}(\underline{X}) \ \frac{\partial g}{\partial X_1}(\underline{X}) & \cdots & \frac{\partial g}{\partial X_n}(\underline{X}) \end{array}\right] .$$
We say that the algebraic manifold $S$ defined by $f=g=0$ is smooth and of codimension 2 if, for every field $\mathbf{K}$ “extension of $\mathbf{R}$ ” and for every point $(\underline{\xi})=$ $\left(\xi_1, \ldots, \xi_n\right) \in \mathbf{K}^n$ satisfying $f(\underline{\xi})=g(\underline{\xi})=0$, then one of the $2 \times 2$ minors of the Jacobian matrix $J_{k, \ell}(\underline{\xi})$, where
$$J_{k, \ell}(\underline{X})=\left|\begin{array}{ll} \frac{\partial f}{\partial X_k}(\underline{X}) & \frac{\partial f}{\partial X_{\ell}}(\underline{X}) \ \frac{\partial g}{\partial X_k}(\underline{X}) & \frac{\partial g}{\partial X_{\ell}}(\underline{X}) \end{array}\right|$$
is nonzero.
By the formal Nullstellensatz, this is equivalent to the existence of polynomials $F, G$ and $\left(B_{k, \ell}\right){1 \leqslant k<\ell \leqslant n}$ in $\mathbf{R}[\underline{X}]$ which satisfy $$F f+G g+\sum{1 \leqslant k<\ell \leqslant n} B_{k, \ell}(\underline{X}) J_{k, \ell}(\underline{X})=1$$
Let $b_{k, \ell}=B_{k, \ell}(\underline{x})$ be the image of $B_{k, \ell}$ in $\mathbf{A}$ and $j_{k, \ell}=J_{k, \ell}(\underline{x})$. We therefore have in $\mathbf{A}$
$$\sum_{1 \leqslant k<\ell \leqslant n} b_{k, \ell} j_{k, \ell}=1$$

## 数学代写|交换代数代写Commutative Algebra代考|The General Case

We treat the case of using $m$ equations to define a smooth manifold of codimension $r$.
Let $\mathbf{R}$ be a commutative ring, and $f_i(\underline{X}) \in \mathbf{R}\left[X_1, \ldots, X_n\right], i=1, \ldots, m$. Consider the $\mathbf{R}$-algebra
$$\mathbf{A}=\mathbf{R}\left[X_1, \ldots, X_n\right] /\left\langle f_1, \ldots, f_m\right\rangle=\mathbf{R}\left[x_1, \ldots, x_n\right]=\mathbf{R}[\underline{x}]$$
The Jacobian matrix of the system of equations $\left(f_1, \ldots, f_m\right)$ is
$$J(\underline{X})=\left[\begin{array}{ccc} \frac{\partial f_1}{\partial X_1}(\underline{X}) & \cdots & \frac{\partial f_1}{\partial X_n}(\underline{X}) \ \vdots & & \vdots \ \frac{\partial f_m}{\partial X_1}(\underline{X}) & \cdots & \frac{\partial f_m}{\partial X_n}(\underline{X}) \end{array}\right] .$$
We say that the algebraic manifold $S$ defined by $f_1=\cdots=f_m=0$ is smooth and of codimension $r$ if the Jacobian matrix taken in $\mathbf{A}$ is “of rank $r$,” i.e.
every minor of order $r+1$ is zero, and the minors of order $r$ are comaximal
This implies that for every field $\mathbf{K}$ “extension of $\mathbf{R}$ ” and at every point $(\xi) \in \mathbf{K}^n$ of the manifold of the zeros of the $f_i$ ‘s in $\mathbf{K}^n$, the tangent space is of codimension $r$.

If the ring $\mathbf{A}$ is reduced, this “geometric” condition is in fact sufficient (in classical mathematics).

Let $J_{k_1, \ldots, k_r}^{i_1, \ldots, i_r}(\underline{X})$ be the $r \times r$ minor extracted from the rows $i_1, \ldots, i_r$ and from the columns $k_1, \ldots, k_r$ of $J(\underline{X})$, and taken in $\mathbf{A}: j_{k_1, \ldots, k_r}^{i_1, \ldots, i_r}=J_{k_1, \ldots, k_r}^{i_1, \ldots, i_r}(\underline{x})$.

The condition on $r \times r$ minors indicates the existence of elements $b_{k_1, \ldots, k_r}^{i_1, \ldots, i_r}$ of $\mathbf{A}$ such that
$$\sum_{1 \leqslant k_1<\cdots<k_r \leqslant n, 1 \leqslant i_1<\cdots<i_r \leqslant m} b_{k_1, \ldots, k_r}^{i_1, \ldots, i_r} j_{k_1, \ldots, k_r}^{i_1, \ldots, i_r}=1 .$$
The $\mathbf{A}$-module of differential forms with polynomial coefficients on $S$ is
$$\Omega_{\mathbf{A} / \mathbf{R}}=\left(\mathbf{A} \mathrm{d} x_1 \oplus \cdots \oplus \mathbf{A} \mathrm{d} x_n\right) /\left\langle\mathrm{d} f_1, \ldots, \mathrm{d} f_m\right\rangle \simeq \mathbf{A}^n / \operatorname{Im}^{\mathrm{t}} J$$
where ${ }^{\mathrm{t}} J={ }^{\mathrm{t}} J(x)$ is the Jacobian matrix transpose (seen in $\left.\mathbf{A}\right)$.

## 数学代写|交换代数代写Commutative Algebra代考|The Smooth Complete Intersection Case

$$\mathbf{A}=\mathbf{R}\left[X_1, \ldots, X_n\right] /\langle f, g\rangle=\mathbf{R}\left[x_1, \ldots, x_n\right]=\mathbf{R}[\underline{x}]$$

$$J(\underline{X})=\left[\begin{array}{lll} \frac{\partial f}{\partial X_1}(\underline{X}) & \cdots & \frac{\partial f}{\partial X_n}(\underline{X}) \ \frac{\partial g}{\partial X_1}(\underline{X}) & \cdots & \frac{\partial g}{\partial X_n}(\underline{X}) \end{array}\right] .$$

## MATLAB代写

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