The Stacks project

Lemma 92.7.3. In Example 92.7.1 let $P$ be a graded $k$-algebra. Then

\[ T\mathcal{D}\! \mathit{ef}_ P \quad \text{and}\quad \text{Inf}(\mathcal{D}\! \mathit{ef}_ P) = \text{Der}_ k(P, P) \]

are finite dimensional if $P$ is finitely generated over $k$.

Proof. We first deal with the infinitesimal automorphisms. Let $Q = P \otimes _ k k[\epsilon ]$. Then an element of $\text{Inf}(\mathcal{D}\! \mathit{ef}_ P)$ is given by an automorphism $\gamma = \text{id} + \epsilon \delta : Q \to Q$ as above where now $\delta : P \to P$. The fact that $\gamma $ is graded implies that $\delta $ is homogeneous of degree $0$. The fact that $\gamma $ is $k$-linear implies that $\delta $ is $k$-linear. The fact that $\gamma $ is multiplicative implies that $\delta $ is a $k$-derivation. Conversely, given a $k$-derivation $\delta : P \to P$ homogeneous of degree $0$, we obtain an automorphism $\gamma = \text{id} + \epsilon \delta $ as above. Thus we see that

\[ \text{Inf}(\mathcal{D}\! \mathit{ef}_ P) = \text{Der}_ k(P, P) \]

as predicted in the lemma. Clearly, if $P$ is generated in degrees $P_ i$, $0 \leq i \leq N$, then $\delta $ is determined by the linear maps $\delta _ i : P_ i \to P_ i$ for $0 \leq i \leq N$ and we see that

\[ \dim _ k \text{Der}_ k(P, P) < \infty \]

as desired.

To finish the proof of the lemma we show that there is a finite dimensional deformation space. To do this we choose a presentation

\[ k[X_1, \ldots , X_ n]/(F_1, \ldots , F_ m) \longrightarrow P \]

of graded $k$-algebras where $\deg (X_ i) = d_ i$ and $F_ j$ is homogeneous of degree $e_ j$. Let $Q$ be any graded $k[\epsilon ]$-algebra finite free in each degree which comes with an isomorphsm $\alpha : Q/\epsilon Q \to P$ so that $(Q, \alpha )$ defines an element of $T\mathcal{D}\! \mathit{ef}_ P$. Choose a homogeneous element $q_ i \in Q$ of degree $d_ i$ mapping to the image of $X_ i$ in $P$. Then we obtain

\[ k[\epsilon ][X_1, \ldots , X_ n] \longrightarrow Q,\quad X_ i \longmapsto q_ i \]

and since $P = Q/\epsilon Q$ this map is surjective by Nakayama's lemma. A small diagram chase shows we can choose homogeneous elements $F_{\epsilon , j} \in k[\epsilon ][X_1, \ldots , X_ n]$ of degree $e_ j$ mapping to zero in $Q$ and mapping to $F_ j$ in $k[X_1, \ldots , X_ n]$. Then

\[ k[\epsilon ][X_1, \ldots , X_ n]/(F_{\epsilon , 1}, \ldots , F_{\epsilon , m}) \longrightarrow Q \]

is a presentation of $Q$ by flatness of $Q$ over $k[\epsilon ]$. Write

\[ F_{\epsilon , j} = F_ j + \epsilon G_ j \]

There is some ambiguity in the vector $(G_1, \ldots , G_ m)$. First, using different choices of $F_{\epsilon , j}$ we can modify $G_ j$ by an arbitrary element of degree $e_ j$ in the kernel of $k[X_1, \ldots , X_ n] \to P$. Hence, instead of $(G_1, \ldots , G_ m)$, we remember the element

\[ (g_1, \ldots , g_ m) \in P_{e_1} \oplus \ldots \oplus P_{e_ m} \]

where $g_ j$ is the image of $G_ j$ in $P_{e_ j}$. Moreover, if we change our choice of $q_ i$ into $q_ i + \epsilon p_ i$ with $p_ i$ of degree $d_ i$ then a computation (omitted) shows that $g_ j$ changes into

\[ g_ j^{new} = g_ j - \sum \nolimits _{i = 1}^ n p_ i \partial F_ j / \partial X_ i \]

We conclude that the isomorphism class of $Q$ is determined by the image of the vector $(G_1, \ldots , G_ m)$ in the $k$-vector space

\[ W = \mathop{\mathrm{Coker}}(P_{d_1} \oplus \ldots \oplus P_{d_ n} \xrightarrow {(\frac{\partial F_ j}{\partial X_ i})} P_{e_1} \oplus \ldots \oplus P_{e_ m}) \]

In this way we see that we obtain an injection

\[ T\mathcal{D}\! \mathit{ef}_ P \longrightarrow W \]

Since $W$ visibly has finite dimension, we conclude that the lemma is true. $\square$


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