6 fixed a typo

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) asociates associates to a ring k $k$ 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!

5 added 60 characters in body

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) asociates 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!

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) asociates 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 $RHom_{A \otimes_{\mathbb{F}_p} A}(A, m[\tau])$

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 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!

3 added 4 characters in body
2 added 7 characters in body
1