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I am going through the proof of Thm 1.5.1 of Laumon, Cohomology of Drinfeld modular varieties, which says that a certain map of stacks is smooth. To prove this, Laumon considers the tangent space of a moduli stack:

Very roughly, a Drinfeld module over $k$ is a certain ring homomorphism $A \to k[\tau]$, the moduli stack of Drinfeld modules (of a fixed rank, but never mind) associates to a ring $k$ the groupoid of such ring homomorphisms.

The tangent space of this stack at a $k$-rational point corresponding to a Drinfeld module $A \to k[\tau]$ parametrizes deformations of this, i.e. Drinfeld modules $A \to (k[\epsilon])[\tau]$ ($k[\epsilon]$ the ring of dual numbers), who become the given module after modding out $\epsilon$.

So deformations of a Drinfeld module are lifts of this homomorphism $A \to k[\tau]$ to $A \to (k[\epsilon])[\tau]$ and these are controlled by Hochschild cohomology: Let $m[\tau]:=\epsilon \cdot k[\tau]$ denote the ideal of $(k[\epsilon])[\tau]$.

The existence of lifts has an obstruction in $HH^2(A,m[\tau])$ and this cohomology group vanishes, so lifts always exist. The existing lifts are then parametrized by $HH^1(A,m[\tau])$ and so that is the tangent space of the moduli stack of Drinfeld modules of rank $d$ at the $k$-rational point given by the Drinfeld module $A \to k[\tau]$.

Hochschild cohomology is given by the following Ext-groups $HH^n(A,m[\tau]) \cong Ext^n_{A \otimes_{\mathbb{F}_p} A}(A,m[\tau])$ and the Ext-groups are the cohomology groups of the complex $RHom_{A \otimes_{\mathbb{F}_p} A}(A, m[\tau])$. One knows that these $Ext^n$-groups vanish for $n \geq 2$.

This is all fine but then Laumon asserts the following quasi-isomorphism:

$$ RHom_{A \otimes_{\mathbb{F}_p} A}(A, m[\tau]) \cong (T_{A/ \mathbb{F}_p} \otimes^L_{A \otimes_{\mathbb{F}_p} A} m[\tau])[-1] $$

where $T_{A/\mathbb{F}_p} := Hom_A(\Omega^1 _{A/\mathbb{F}_p}, A)$ and where this is said to be considered as an ${A \otimes_{\mathbb{F}_p} A}$-module "via the augmentation map" ${A \otimes_{\mathbb{F}_p} A} \to A$.

This is where I am stuck. I would be grateful for an explanation of the above quasi-isomorphism or a reference which provides one. Thanks!

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  • $\begingroup$ Dear Peter Arndt, I have tried to edit your question without altering its content. Is it ok for you? $\endgroup$
    – agt
    Jan 19, 2013 at 16:28
  • $\begingroup$ Sure, thanks. I was struggling with the Latex-displaying... $\endgroup$
    – Peter Arndt
    Jan 19, 2013 at 16:35
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    $\begingroup$ It is well-known that for a commutative $k$-algebra $R$ and and $R$-module $M$, $HH^1(R, M)\cong Der(R, M) \cong \hom_R(\Omega_{R/k}, M)$. Could it be just this? I am not sure if the dimension shift signifies something important or is just a matter of grading... $\endgroup$
    – Gregory Arone
    Jan 19, 2013 at 20:36
  • $\begingroup$ Ah, thanks! This might actually be enough to continue the proof. Do you have a reference for this fact? $\endgroup$
    – Peter Arndt
    Jan 20, 2013 at 0:46
  • $\begingroup$ Googling "Hochschild cohomology and derivations" will bring up a few references. It also is easy to figure it out from scratch by considering the Hochschild complex $M\stackrel{d^0}{\to}\hom(R, M) \stackrel{d_1}{\to} \hom(R\otimes R, M)\to \cdots$. $d_0$ sends $m$ to the inner derivation $f(r)=rm-mr$. So if $R$ is commutative, $d^0=0$. $d^1$ sends a homomorphism $f$ to the homomorphism $f(r_1\otimes r_2)-r_1f(r_2)-f(r_1)r_2$. So the kernel of $d^1$ consists exactly of derivations from $R$ to $M$. $\endgroup$
    – Gregory Arone
    Jan 20, 2013 at 7:24

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