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Let $k$ be an algebraically closed field and let $\varphi :k^n\to k^r$ be a map such that there exist $f_1,\cdots ,f_r\in k[x_1,\cdots ,x_n]$ such that for every $a\in k^n$, $\varphi (a)=(f_1(a),\cdots ,f_r(a))$.

For $a\in k^n$ such that $f_i(a)=0$ for every $i$, define $\text{J}\varphi \left ( a \right )\in k^{n\times r}$ as $\left [\displaystyle \frac{\partial f_j}{\partial x_i}\left (a \right )\right ]_{ij}$. Let $\mathfrak{m}_a\subset k[x_1,\cdots ,x_n]/\langle f_1, \dots, f_r \rangle $ be the maximal ideal $\langle x_1-a_1,\cdots ,x_n-a_n \rangle$.

Prove that $\frac{\mathfrak{m}_a}{\mathfrak{m}_a^2}\cong \text{Coker}\left (\text{J}\varphi \left ( a \right ) \right )$ as $k$-vector spaces.

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  • $\begingroup$ Have you tried reading the proof of I 5.1 in Hartshorne? $\endgroup$ – Kenny Wong Feb 27 '17 at 22:28
  • $\begingroup$ I also recommend reading the proof of Theorem 3.14 in Hulek. $\endgroup$ – Kenny Wong Feb 27 '17 at 22:29
  • $\begingroup$ In both books they use definitions we did not see in class. Also, I could not find the word $\text{Coker}$ in any of them. I would like to have a proof of the statement with the vocabulary I used in my post. Which function $\mathfrak{m}_a\to \text{Coker}\left (\text{J}\varphi \left ( a \right ) \right )$ should I define? $\endgroup$ – solomeo paredes Feb 27 '17 at 23:04
  • $\begingroup$ Coker means cokernel. If $f : V \to W$ is a linear map between vector spaces, then $Coker(f) / W / Ker(f)$. $\endgroup$ – Kenny Wong Feb 27 '17 at 23:56
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    $\begingroup$ My professor uses the notation $\mathfrak{m}R_{\mathfrak{m}}$ to refer the maximal ideal in $R_{\mathfrak{m}}$, so $\mathfrak{m}_p$ should refer to the maximal ideal in $R$. $\endgroup$ – solomeo paredes Mar 1 '17 at 1:14
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What is $m_a$ exactly? It's the ideal containing all regular functions on the variety $(f_1 = \dots = f_r = 0) \subset A^n$ that vanish at $a$. In other words, it contains all regular functions such that the constant term of their Taylor expansion around $a$ vanish.

And what is $m_a^2$? It's the ideal containing all the regular functions whose Taylor series around $a$ have vanishing constant AND linear terms.

So $m_a/m_a^2$ is the vector space of "linear truncations of regular functions on $(f_1 = \dots = f_r = 0) \in \subset A^n$ vanishing at $a$". In other words $m_a / m_a^2$ is spanned by $x_1 - a_1, \dots , x_n - a_n$ as a $k$-vector space.

Except...

$x_1 - a_1, \dots , x_n - a_n$ are not linearly independent in $m_a/m_a^2$! This is because we are quotienting by the ideal $(f_1, \dots , f_r)$. So really, $m_a / m_a^2$ is the vector space generated by $x_1 - a_1, \dots , x_n - a_n$, quotiented by the subspace generated by the linear terms of the Taylor expansions of $f_1, \dots , f_r$.

The Taylor expansion of $f_j(x)$ is $$ f_j(x) = \partial_1 f_j(a) (x_1 - a_1) + \dots + \partial_n f_j(a) (x_n - a_n)+ {\rm \ higher \ order \ }. $$ (Remember that $f_j(a) = 0$.)

So $m_a / m_a^2$ is the vector space generated by $x_1 - a_1, \dots , x_n - a_n$, quotiented by the subspace generated by linear combinations of $$\partial_1 f_1(a) (x_1 - a_1) + \dots + \partial_n f_1(a) (x_n - a_n), \\ \vdots \\ \partial_1 f_r(a) (x_1 - a_1) + \dots + \partial_n f_r(a) (x_n - a_n).$$

To finish off, consider the linear map $k^n \to m_a / m_a^2$, mapping $$ (c_1, \dots c_n) \mapsto c_1 (x_1 - a_1) + \dots + c_n (x_n - a_n).$$ It is surjective, and its kernel is spanned by the vectors $$(\partial_1 f_1(a) , \dots , \partial_n f_1(a) ), \ \dots \ , (\partial_1 f_r(a) , \dots , \partial_n f_r(a) ).$$ This kernel is clearly ${\rm Im \ }J (\varphi (a))$. So ${\rm Coker \ J(\varphi(a))} = m_a / m_a^2$.

Finally, as you pointed out in your comment, $m_a / m_a^2 \cong m_aA_a / (m_a A_a)^2 $. So we've shown ${\rm Coker \ J (\varphi(a))} = m_a A_a / (m_a A_a)^2$. Both of these vector spaces are representations of the cotangent space at $a$. The expression ${\rm Coker \ J (\varphi(a))}$ is closer to your intuitive notion of the cotangent space, but the expression $m_aA_a / (m_a A_a)^2$ has the advantage that it is intrinsic to the local ring at the point of the variety.

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  • $\begingroup$ Hi Kenny, isn't the maximal ideal the maximal ideal of the localization of the coordinate ring at the point in question? Explicitly, if we call $M=(x_1-a_1,\dots,x_n-a_n)$ and the coordinate ring is $A$: $$ m_a=MA_M=\left\{\frac{g}{h}:g\in A, h\in A-m\right\} $$ $\endgroup$ – user347489 Feb 28 '17 at 19:39
  • $\begingroup$ That's what I asked solomeo in a comment under the question. $\endgroup$ – Kenny Wong Feb 28 '17 at 19:46
  • $\begingroup$ The coordinate ring has a maximal ideal. The local ring also has a maximal ideal. It's not clear which we're talking about. $\endgroup$ – Kenny Wong Feb 28 '17 at 19:47
  • $\begingroup$ So I answered on the assumption that we're not localising. $\endgroup$ – Kenny Wong Feb 28 '17 at 19:48
  • $\begingroup$ Obviously this whole story is more geometrically useful if we do localise. Essentially the same argument works. $\endgroup$ – Kenny Wong Feb 28 '17 at 19:49

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