## $sl_{2,k}$ and the automorphism group of the trivial first order inf. deformation of $\mathbb{P}_k^1$

Let $$X=\mathbb{P}_k^1$$ and let $$X_{k[\epsilon]} = X \times_k \operatorname{Spec} k[\epsilon]/(\epsilon)^2$$ denote the trivial first order deformation of $$X$$ over $$k[\epsilon]/(\epsilon)^2$$.

Let $$G= PGL(2,k)$$ so that we have the following string of isomorphisms $$sl_2 \cong \operatorname{Lie}(G) \cong (\mathcal{T}G)_{e} \cong G(k[\epsilon])_{(\epsilon) \to e} = \operatorname{Aut}_{k[\epsilon]}(X_{k[\epsilon]}).$$

If $$\phi \in \operatorname{Aut}_{k[\epsilon]}(X_{k[\epsilon]})$$, then $$\phi$$ is locally of the form, $$z \mapsto z + \epsilon(a_0 + a_1 z + a_2 z^2)$$ where $$z$$ is a choice of homogeneous coordinate on $$X$$.

I am trying to recover $$sl_2$$ from this local description. If I let $$a_0=b, a_1= a-d$$, and $$a_2= -c$$, then $$\frac{(1+a\epsilon)z+b\epsilon}{c\epsilon z + (1+d\epsilon)}=z+\epsilon\big( b+(a-d)z-cz^2 \big),$$ And, letting $$z=v/u$$, we can find that $$u \mapsto u + \epsilon(d u + cv)$$ $$v \mapsto v + \epsilon(bu + av)$$

But I don’t see how this relates to $$sl_2$$. Is it even possible for one to recover $$sl_2$$ from this description?

## Automorphisms of a First Order Inf. Deformation which pull back to the Identity Morphism

Let $$X$$ be a scheme over an algebraically closed field $$k$$ and let $$X’$$ be a deformation of $$X$$ over the dual numbers $$k[\epsilon]/(\epsilon)^2$$. Here we are assuming that $$X$$ has non-trivial first order inf. deformations.

I am interested in classifying automorphisms of $$X’$$ which pullback to the identity on $$X$$. Let $$U_i$$ be an open affine cover of $$X$$ so that $$U_i’ \cong X’ \vert_{U_i}$$ is an open cover of $$X’$$ by trivial first order inf. deformations of $$U_i$$.

Now let $$\phi$$ be an automorphism of $$X’$$ which pulls back to the identity morphism on $$X$$. Then $$\phi \vert_{U_i’}$$ is an automorphism of $$U_i’$$. Since $$U_i’$$ is trivial we know that its automorphism group is isomorphic to $$\Gamma(U_i, \mathcal{T}X \vert_{U_i})$$.

Is it possible to glue together these local automorphism (i.e global sections of the associated tangent sheaf) to form a global automorphism?

Based on the description given above, I think that the group of automorphisms of $$X’$$ which pullback to the identity on $$X$$ is isomorphic to $$\Gamma(X, \mathcal{T}_{X’/k[\epsilon]})$$.

Is this correct?

## What should be OpenSSL .cnf file equivalent of certreq .inf for S/MIME?

Currently I’m using certreq to prepare CSRs for S/MIME certificates. I want to move away from it and start using OpenSSL for key/CSR generation.

My .inf file looks like this:

[Version] Signature="$Windows NT$  "  [NewRequest] RequestType=PKCS10 Subject="CN=$name,O=$  org,L=$loc,C=$  cc,E=$email" KeyLength=2048 MachineKeySet=FALSE UseExistingKeySet=FALSE Exportable=TRUE ProviderName="Microsoft Enhanced Cryptographic Provider v1.0" ProviderType=1 KeySpec=1 KeyUsage=0xe0 [Extensions] 2.5.29.17 = "{text}" _continue_ = "email=$  email&" 

I’d like to prepare equivalent OpenSSL .cnf file (so it results in CSR as similar as possible), but I’m kind of lost in myriad config options. Can someone more experienced with OpenSSL help?

The CSR will be used to obtain commercial S/MIME certificate.

## What does Arg {Inf I(d)} means?

I am currently studying the phase-field method for fracture modeling. In an article by Miehe -“Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementation”, I came across on an identity that I don’t understand basically it says:

enter image description here

Can anyone tell me what does that mean? I added image because I don’t know how to add this identity in another way.