I tried to give a proof that fppf (faithfully flat) descent implies Galois descent purely at the level of modules and I stumble to obtain the Galois cocycle condition. I’m interested to consider some questions of twisted sheaves with a Galois cohomological description and understanding how to obtain the former would be useful to me.

I obtained the following the following conditions: Given a finite Galois extension $ L/K$ of Galois group $ G$ and $ M$ an $ L$ -vector space $ M$ , we have for each $ \sigma \in G$ an isomorphism of $ L$ -vector spaces $ \psi_\sigma : M \to M^\sigma$ satisfying $ \psi_\sigma(am) = \sigma(a) \psi_\sigma(m)$ , where $ a \in L$ and $ m \in M$ and such that for every pair $ (\sigma, \tau) \in G \times G$ we have

$ $ \psi_{ ( \sigma, \tau), (\sigma, \tau), \sigma } \circ \psi_{ ( \sigma, \tau), (\sigma, \tau), \tau } = \psi_{ ( \sigma, \tau), (\sigma, \tau), \sigma \tau } $ $

as isomorphisms of $ L$ -modules. The $ L$ -module structure of $ M^\sigma$ is twisted by $ \sigma$ , i.e given by $ a \cdot m:= \sigma(a)m$ .

My issue is that because of what one obtains for sheaves of modules (https://stacks.math.columbia.edu/tag/0CDQ) I would expect the cocycle condition for modules to also have a twisting in the formula.

Translating what is done in stacks project into modules is certainly possible but I wasn’t able to do so. Instead I will present a (somehow long) sketch of what I did. I can provide more details upon request and I apologize if there are too many. My difficulty is on **Step 5**.

**The context**: Let $ L/K$ be a finite Galois extension and let $ M$ be an $ L$ -module together we an isomorphism of $ L \otimes_K L$ -modules $ \phi: M \otimes_K L \to L \otimes_K M$ satisfying the cocyle condition $ p_{13}^* \phi = p_{23}^* \phi \circ p_{12}^* \phi$ as isomorphisms of $ L \otimes_K L \otimes_K L$ -modules.

**What I did**:

**Step 1: Describing some isomorphisms** We have an isomorphism of $ K$ -algebras $ L \otimes_K L \to \prod_{\sigma \in G} L$ given by $ a \otimes 1 \mapsto ( a )_{\sigma \in G}$ and $ 1 \otimes a \mapsto ( \sigma(a) )_{\sigma \in G}$

and another one $ L \otimes_K L \otimes_K L \to \prod_{\sigma \in G} \Big( \prod_{\tau \in G} L \Big)$ given by

$ $ a \otimes 1 \otimes 1 \mapsto \Big( (a)_{\tau \in G} \Big)_{\sigma \in G}, $ $

$ $ 1 \otimes a \otimes 1 \mapsto \Big( (\tau(a)_{\tau \in G} \Big)_{\sigma \in G}, $ $

$ $ 1 \otimes 1 \otimes a \mapsto \Big( \tau\sigma(a)_{\tau \in G} \Big)_{\sigma \in G}. $ $

We can then describe the above isomorphisms as

$ $ (1_L \coprod \sigma) : L \otimes_K L \to \prod_{\sigma \in G} L $ $

and

$ $ (1_L \coprod \tau \coprod \tau \sigma): L \otimes_K L \otimes_K L \to \prod_{\sigma \in G} \Big( \prod_{\tau \in G} L \Big). $ $

**Step 2: Obtain some $ \prod_{\sigma \in G} L$ -module structures**

We then have a commutative diagram of modules.

$ $ \begin{array}{ccccc} M \otimes_K L & \xrightarrow{} & \prod_{\sigma \in G} \Big( M \otimes_K L \Big) & \xrightarrow{} & \prod_{\sigma \in G} M\ \downarrow & & \downarrow & & \downarrow \ L \otimes_K M & \xrightarrow{} & \prod_{\sigma \in G} \Big( L\otimes_K M \Big) & \xrightarrow{} & \prod_{\sigma \in G} M^\sigma \end{array} $ $

where the left most vertical arrow is $ \phi$ and we denote by $ \psi$ the induced the right most vertical arrow.

Using **Step 1** we have ring morphisms $ L \xrightarrow{1_L} L$ and $ L \xrightarrow{\sigma} L$ for each $ \sigma \in G$ . Tensoring $ M$ with these morphisms give $ L$ -module structures for $ M \otimes_K L$ by $ a \cdot ( m \otimes c ) = m \otimes ac$ and for $ L \otimes_{L,\sigma} M$ by $ a \cdot (c \otimes m) = \sigma(a)c \otimes m$ . Now the isomorphism of $ L$ -modules $ \mu: M \otimes_L L \to M: m \otimes c \mapsto cm$ then gives to $ M$ the $ L$ -module structure $ a \otimes m = am$ . We also have the composite diagram

$ $ L \otimes_{L,\sigma} M \xrightarrow{ \sigma^{-1} \otimes 1_M } L \otimes_L M \xrightarrow{\mu’} M : c \otimes m \mapsto \sigma^{-1}(c) \otimes m \mapsto \sigma^{-1}(c)m. $ $

Then this $ M$ has $ L$ -module structure given by $ a \cdot m := \sigma(a)m$ and we denote it by $ M^\sigma$ (We can relabel to get $ M^\sigma$ instead of $ M^{\sigma^{-1}}$ .).

Now a structure of $ \prod_{\sigma \in G} L$ -module on $ \prod_{\sigma \in G} M$ (resp. on $ \prod_{\sigma \in G} M^\sigma$ ) is determined by an $ L$ -module structure on $ M$ (resp. on $ M^\sigma$ ) for each $ \sigma \in G$ . Therefore, if $ (a_\sigma)_{\sigma \in G} \in \prod_{\sigma \in G}$ and $ (m_\sigma)_{\sigma \in G} \in \prod_{\sigma \in G} M$ (resp. in $ \prod_{\sigma \in G} M^\sigma$ ), then

$ $ (a_\sigma)_{\sigma \in G} \bullet (m_\sigma)_{\sigma \in G} = (~a_\sigma m_\sigma)_{\sigma \in G} ( \text{ resp. } (a_\sigma)_{\sigma \in G} \circ (m_\sigma)_{\sigma \in G} = ( \sigma(a)_\sigma m_\sigma)_{\sigma \in G}~). $ $

**Step 3: Determine for each $ \sigma \in G$ the isomorphisms of $ L$ -modules $ \psi_\sigma$ .**

The isomorphism $ \psi$ induced by $ \phi$ must then satisfy

$ $ \psi \big( a_\sigma m_\sigma)_{\sigma \in G} ) = (a_\sigma)_{\sigma \in G} \circ \psi( ( m_\sigma)_{\sigma \in G} ). $ $

For each $ \sigma \in G$ we have an isomorphism of $ L$ -modules

$ $ M \xrightarrow{\iota_\sigma} \prod_{\sigma \in G} M \xrightarrow{\psi} \prod_{\sigma \in G} M^\sigma \xrightarrow{\pi_\sigma} M^\sigma $ $

given by

$ $ am \mapsto (0, \cdots, 0, am, 0, \cdots, 0) \mapsto \big( \sigma(a) \pi_\sigma \Big( \psi( \iota_\sigma(m) \Big) \big)_{\sigma \in G} \mapsto \sigma(a)\pi_\sigma \Big( \psi(m) \Big). $ $

So for each $ \sigma \in G$ we have an isomorphism $ \psi_\sigma : M \to M^\sigma$ defined by $ \psi_\sigma(m):=\pi_\sigma( \psi(m) )$ and such that $ \psi_\sigma(am) = \sigma(a)\psi_\sigma(m)$ .

**Step 4: Determine some $ \prod_{\sigma \in G} \prod_{\tau \in G} L$ -module structures**

I will skip some details, which I can provide upon request. I use the cocycle condition to determine three $ \prod_{\sigma \in G} \prod_{\tau \in G} L$ -module structures.

Consider $ p_{12}^* p_1^* M$ (or equivalently $ p_{13}^*p_1^*M$ ).

The $ \prod_{(\sigma,\tau) \in G \times G } L$ -module $ \prod_{(\sigma,\tau) \in G \times G } M$ is

$ $ (a_{g,h}) \cdot ( m_{g,h} ) = ( a_{g,h} m_{g,h} ). $ $

Consider $ p_{12}^* p_2^* M$ (or equivalently $ p_{23}^*p_1^*M$ ).

The $ \prod_{(\sigma,\tau) \in G \times G } L$ -module $ \prod_{(\sigma,\tau) \in G \times G } M^\tau$ is

$ $ (a_{\sigma,\tau}) \cdot ( m_{\sigma,\tau} ) = ( \tau(a_{\sigma,\tau}) m_{\sigma,\tau} ). $ $

Consider $ p_{13}^* p_2^* M$ (or equivalently $ p_{23}^*p_2^*M$ ).

The $ \prod_{(\sigma,\tau) \in G \times G } L$ -module $ \prod_{(\sigma,\tau) \in G \times G } M^{\sigma \tau}$ is

$ $ (a_{\sigma,\tau}) \cdot ( m_{\sigma,\tau} ) = ( (\sigma \circ \tau)(a_{\sigma,\tau}) m_{\sigma,\tau} ). $ $

**Step 5: Determine the cocycle condition**

Finally, for each pair $ (\sigma, \tau) \in G \times G$ we have three composite maps given as follows:

$ $ M_{(\sigma, \tau)} \xrightarrow{ \iota_{(\sigma,\tau)}} \prod_{(\sigma,\tau) \in G \times G} M \xrightarrow{ p_{12}^* \psi } \prod_{(\sigma,\tau) \in G \times G} M^\tau \xrightarrow{ \pi_{\sigma,\tau}} M_{(\sigma, \tau)}^\tau $ $

defining an $ L$ -module isomorphism $ \psi_{ ( \sigma, \tau), (\sigma, \tau), \tau } : M_{(\sigma,\tau)} \to M_{(\sigma,\tau)}^\tau$ satisfying

$ $ \psi_{ ( \sigma, \tau), (\sigma, \tau), \tau }(am) = \tau(a)\psi_{ ( \sigma, \tau), (\sigma, \tau), \tau }(m) $ $

$ $ M_{(\sigma, \tau)} \xrightarrow{ \iota_{(\sigma,\tau)}} \prod_{(\sigma,\tau) \in G \times G} M \xrightarrow{ p_{13}^* \psi } \prod_{(\sigma,\tau) \in G \times G} M^{\sigma \tau} \xrightarrow{ \pi_{\sigma,\tau}} M_{(\sigma, \tau)}^{\sigma\tau} $ $

defining an $ L$ -module isomorphism $ \psi_{ ( \sigma, \tau), (\sigma, \tau), \sigma \tau } : M_{(\sigma,\tau)} \to M_{(\sigma,\tau)}^{\sigma \tau}$ satisfying

$ $ \psi_{ ( \sigma, \tau), (\sigma, \tau), \tau }(am) = (\sigma \circ \tau)(a)\psi_{ ( \sigma, \tau), (\sigma, \tau), \tau }(m) $ $

and

$ $ \psi_{ ( \sigma, \tau), (\sigma, \tau), \tau }(am) = \tau(a)\psi_{ ( \sigma, \tau), (\sigma, \tau), \tau }(m) $ $

$ $ M_{(\sigma, \tau)}^\tau \xrightarrow{ \iota_{(\sigma,\tau)}} \prod_{(\sigma,\tau) \in G \times G} M^\tau \xrightarrow{ p_{23}^* \psi } \prod_{(\sigma,\tau) \in G \times G} M^{\sigma \tau} \xrightarrow{ \pi_{\sigma,\tau}} M_{(\sigma, \tau)}^{\sigma \tau} $ $

defining an $ L$ -module isomorphism $ \psi_{ ( \sigma, \tau), (\sigma, \tau), \sigma } : M_{(\sigma,\tau)}^\tau \to M_{(\sigma,\tau)}^{\sigma \tau}$ satisfying

$ $ \psi_{ ( \sigma, \tau), (\sigma, \tau), \sigma }(am) = \sigma(a)\psi_{ ( \sigma, \tau), (\sigma, \tau), \tau }(m). $ $

Indeed, $ \psi_{ ( \sigma, \tau), (\sigma, \tau), \sigma }$ sends $ \tau(a)m$ to $ (\sigma \circ \tau)(a)m$ and so sends $ am = \tau(\tau^{-1}(a))m$ to $ \sigma( am )$ .

Since $ p_{23}^* \phi \circ p_{12}^*\phi = p_{13}^* \phi$ , we have $ p_{23}^* \psi \circ p_{12}^*\psi = p_{13}^* \psi$ and therefore for each pair $ (\sigma, \tau) \in G \times G$ we have

$ $ \psi_{ ( \sigma, \tau), (\sigma, \tau), \sigma } \circ \psi_{ ( \sigma, \tau), (\sigma, \tau), \tau } = \psi_{ ( \sigma, \tau), (\sigma, \tau), \sigma \tau } $ $

as isomorphisms of $ L$ -modules.

**Remarks:** The map $ \psi_{ ( \sigma, \tau), (\sigma, \tau), \sigma }$ is from $ M^\tau$ to $ M^{\sigma \tau}$ and it is not clear to me how to re-express it as starting from $ M$ and twisting it by $ \tau$ . Another problem is that the maps I found on **Step 3** are not in an obvious way related to those of **Step 5** and there might be a need to twick something here as well.