# Computations in condensed mathematics, page 32-34

I started reading the Lectures on Condensed Mathematics. I am looking at the material at page 32-34. I have three fundamental computation questions:

1. At the last line of pg 32 - it seems to imply that for finite sets $$S$$, $$\Bbb Z[S] \simeq \underline{Hom}(C(S, \Bbb Z ), \Bbb Z)$$?

1. In pg 33 line 4 how is $$Hom(C(S,\Bbb Z), \Bbb Z)$$ identified with measure on $$S$$?

1. In proof of Proposition 5.7, there was the following equivalence,

$$RHom(\prod_J \Bbb R, \Bbb Z) \simeq RHom_{\Bbb R} (\prod_J \Bbb R, R\underline{Hom}(\Bbb R, \Bbb Z) )=0$$

why is this true and what exactly does $$RHom_{\Bbb R}$$ mean? Is there some adjunction happening here from $$\Bbb Z$$ modules of $$Cond(Set)$$ to $$\Bbb R$$-modules of $$Cond(Set)$$?

I would appreciate if there are some related references for the general set up in 2 and 3.

For 1, I would like to compute the $$T$$ points for $$T$$ extremely. disconnected. What i don't see is an easy expression for lhs.
$$\Bbb Z[S] (T)= \bigoplus_{C(T,S)}\Bbb Z$$ Conversely for rhs we have

$$\underline{Hom}(C(S,\Bbb Z), \Bbb Z)(T)=Hom(\Bbb Z[T] \otimes C(S,\Bbb Z), \Bbb Z)$$ This doesn't seem an easy expression to handle too.

1. Correct, as both sides are the $$S$$-indexed direct sum of copies of $$\mathbb{Z}$$. For the LHS this holds by the universal property of $$\mathbb{Z}[S]$$, and for the RHS note that $$C(S,\mathbb{Z}) = \prod_S \mathbb{Z} = \oplus_S \mathbb{Z}$$ which lets you calculate.
2. By definition, one could say. It's reasonable to define a $$\mathbb{Z}$$-valued measure on a profinite set to be an element of $$Hom(C(S,\mathbb{Z}),\mathbb{Z})$$. You could also define a measure to be a finitely additive assignment of an integer to each clopen subset of S. You can consider the indicator functions of clopen subsets to see the equivalence.
3. $$RHom_{\mathbb{R}}(M,N)$$ means a complex calculating Ext's from M to N in the category of $$\mathbb{R}$$-modules in condensed abelian groups. The equality holds because both are the same as $$RHom_{\mathbb{R}}(\mathbb{R} \otimes_\mathbb{Z}^L \prod_J \mathbb{R},\mathbb{Z})$$ by some adjunctions.
There's also another way of looking at 3. Abstractly, the claim is that if $$A$$ is a condensed algebra and $$N$$ a condensed abelian group with $$\underline{RHom}(A,N)=0$$, then $$RHom(M,N)=0$$ whenever $$M$$ has an $$A$$-module structure. By adjunction, the hypothesis is equivalent to saying that $$RHom(A\otimes_{\mathbb{Z}}^L M,N)=0$$ for all condensed abelian groups $$M$$. If $$M$$ has an $$A$$-module structure, then $$M$$ is a retract of $$A\otimes^L M$$ by the action map $$A\otimes^LM\rightarrow M$$ in one direction and the unit $$M=\mathbb{Z}\otimes^L M\rightarrow A\otimes^LM$$. Thus we deduce $$RHom(M,N)=0$$ as claimed.