# Sheafification of loop scheme/group

Let $$X$$ be a scheme over $$K = k((t))$$, where $$k$$ is a field. We define the loop scheme $$LX$$ to be the functor from the category of $$k$$-algebras to sets by $$R \mapsto LX(R) := X(Spec (R((t))))$$.

Do we need sheafification in order to make $$LX$$ a sheaf? A particular interest is when $$X$$ is a connected linear reductive group over $$K$$ and fpqc site is chosen.

• I think not. It’s easy to check for affine schemes, including reductive groups, by just using coordinates. – Will Sawin Oct 19 '19 at 16:53

Despite Will Sawin's answer in comment, it was not so obvious for me at first glance, so I give a detailed (at least for $$X$$ affine) proof here.
Lemma. Let $$A$$ be a $$K$$-algebra and let $$A\to B$$ be fpqc. Then the following sequence is exact $$A((t))\to B((t))\rightrightarrows (B\otimes_{A}B)((t)).$$
Proof. Since $$A\to B$$ is fpqc, Grothendieck's fpqc-descent shows that the sequence $$A\to B\rightrightarrows B\otimes_A B$$ is exact. Consider the sequence $$A[[t]]\to B[[t]]\rightrightarrows (B\otimes_{A}B)[[t]].$$ We claim that this sequence is also exact.
The first arrow $$A[[t]]\to B[[t]]$$ is injective since $$A\to B$$ is injective and since $$A[[t]] = \prod_{n\in \mathbf{N}}A$$ and $$B[[t]] = \prod_{n\in \mathbf{N}}B$$ as $$A$$-modules. It remains to show that for $$f(t)\in B[[t]]$$, $$f(t)\otimes 1 = 1\otimes f(t)\in (B\otimes_{A}B)[[t]] \quad\Longrightarrow \quad f(t)\in A[[t]].$$ Suppose that $$f(t) = \sum_{n\in \mathbf{N}} b_n t^n\in B[[t]]$$ satisfies the condition $$f(t)\otimes 1 = 1\otimes f(t)$$. It is enough to show that $$f(t) \mod{t^N} \in A[t]/t^N$$ for each $$N\ge 1$$. Indeed, if we tensorise the first exact sequence by $$\otimes_{A}A[t]/t^N$$, we get $$A[t]/t^N\to B[t]/t^N\rightrightarrows (B\otimes_{A} B)[t]/t^N,$$ which implies that $$b_n\in A$$ for $$n < N$$. Hence $$f(t)\in A[[t]]$$.
Inverting $$t$$, we deduce the exactness of $$A((t))\to B((t))\rightrightarrows (B\otimes_{A}B)((t)).$$ End of Proof
Now if $$X = \mathrm{Spec}\, C$$ is an affine $$K$$-scheme and A,B are as above, then the exactness of the following sequence follows from the lemma $$\mathrm{Hom}(C, A((t)))\to \mathrm{Hom}(C, B((t)))\rightrightarrows \mathrm{Hom}(C, (B\otimes_{A} B)((t))).$$ Hence, if $$X$$ is affine, $$LX$$ is automatically an fpqc sheaf. The general case follows from the standard procedure of glueing.