On $p$-adic lattices and Grassmannians

Abstract

It is well-known that the coset spaces $G(k((z)))/G(k[[z]])$, for a reductive group $G$ over a field $k$, carry the geometric structure of an inductive limit of projective $k$-schemes. This $k$-ind-scheme is known as the affine Grassmannian for $G$. From the point of view of number theory it would be interesting to obtain an analogous geometric interpretation of quotients of the form $\mathcal {G}(\mathbf {W}(k)[1/p])/\mathcal {G}(\mathbf {W}(k))$, where $p$ is a rational prime, $\mathbf {W}$ denotes the ring scheme of $p$-typical Witt vectors, $k$ is a perfect field of characteristic $p$ and $\mathcal {G}$ is a reductive group scheme over $\mathbf {W}(k)$. The present paper is an attempt to describe which constructions carry over from the function field case to the $p$-adic case, more precisely to the situation of the $p$-adic affine Grassmannian for the special linear group $\mathcal {G}=\mathbf {SL}_{n}$. We start with a description of the $R$-valued points of the $p$-adic affine Grassmannian for $\mathbf {SL}_{n}$ in terms of lattices over $\mathbf {W}(R)$, where $R$ is a perfect $k$-algebra. In order to obtain a link with geometry we further construct projective $k$-subvarieties of the multigraded Hilbert scheme which map equivariantly to the $p$-adic affine Grassmannian. The images of these morphisms play the role of Schubert varieties in the $p$-adic setting. Further, for any reduced $k$-algebra $R$ these morphisms induce bijective maps between the sets of $R$-valued points of the respective open orbits in the multigraded Hilbert scheme and the corresponding Schubert cells of the $p$-adic affine Grassmannian for $\mathbf {SL}_{n}$.

Appendix: fpqc-sheaves

In this “Appendix” we collect some general results on fpqc-sheaves which are used throughout the preceding sections. In particular, we discuss in detail the existence of sheafifications over the fpqc-site in situations relevant for the present paper.

Let $\mathcal {C}$ be the category of schemes. By a presheaf on $\mathcal {C}$ we mean simply a functor on the category of schemes to the category $\text {(Set)}$ of sets.

Lemma 48

Let $\mathcal {D}\subset \mathcal {C}$ be an inclusion of sites, such that fiber products in $\mathcal {D}$ are mapped to fiber products in $\mathcal {C}$. Assume that for every covering $\mathcal {U} = \lbrace U_{i}\rightarrow X \rbrace$ in $\mathcal {C}$ of an object $X\in \mathcal {D}$ there exists a refinement $\mathcal {V}=\lbrace V_{i}\rightarrow X\rbrace$ of $\mathcal {U}$ with $V_{i}\in \mathcal {D}$ such that $\mathcal {V}$ is also a covering of $X$ in $\mathcal {D}$. Then restriction of presheaves from $\mathcal {C}$ to $\mathcal {D}$ commutes with sheafification. In other words, if a presheaf $F:\mathcal {C}\rightarrow \mathbf (sets)$ has a sheafifcation $F^{a}$, then $F^{a}|_{\mathcal {D}}$ is a (the) sheafification of $F|_{\mathcal {D}}$.

Proof

We check that the canonical morphism $F|_{\mathcal {D}} \rightarrow (F^{a})|_{\mathcal {D}}$ is a sheafification on $\mathcal {D}$. Let $X\in \mathcal {D}$ and let $\xi ,\eta \in F(X)$ be such that their images in $F^{a}(X)$ coincide. By definition of sheafification there exists a covering (in $\mathcal {C}$) of $X$ on which $\xi$ and $\eta$ coincide. But by assumption this covering can be refined in order to obtain a covering of $X$ in $\mathcal {D}$. Of course, $\xi$ and $\eta$ still coincide on this refinement. On the other hand, every element $\xi \in F^{a}(X)$ can be represented locally (on a covering in $\mathcal {C}$) by sections of $F$. Refining this covering, we see that $\xi$ can be represented on a covering in $\mathcal {D}$ by sections of $F$. Thus $F|_{\mathcal {D}} \rightarrow (F^{a})|_{\mathcal {D}}$ is indeed a sheafification and induces an isomorphism $(F|_{\mathcal {D}})^{a} \rightarrow (F^{a})|_{\mathcal {D}}$. $\square$

Theorem 49

([11]) Let $F$ be a presheaf on $\mathcal {C}$. Assume that $F$ is a sheaf for the Zariski topology. Then $F$ is an fpqc-sheaf on $\mathcal {C}$ if and only if for every faithfully flat homomorphism of affine schemes $Y\rightarrow X$ the sequence

$$\begin{aligned} F(X) \rightarrow F(Y) \rightrightarrows F(Y\times _{X}Y) \end{aligned}$$

(15)

is an equalizer.

Proposition 50

Let $F$ be a presheaf on $\mathcal {C}$. Assume that $F$ satisfies the following two conditions:

• (1) for every faithfully flat morphism of affine schemes $Y\rightarrow X$ the sequence

  $$\begin{aligned} F(X) \rightarrow F(Y) \rightrightarrows F(Y\times _{X}Y) \end{aligned}$$

  is an equalizer, and

• (2) for every finite collection of affine schemes $Y_{1},\ldots ,Y_{n}$ we have

  $$\begin{aligned} F(Y_{1}\coprod \cdots \coprod Y_{n}) = F(Y_{1})\times \cdots \times F(Y_{n}). \end{aligned}$$

Then the Zariski-sheafification $F^{a}$ of $F$ is an fpqc-sheaf. In particular, $F^{a}$ is an fpqc-sheafification of $F$. Moreover, the natural transformation $F\rightarrow F^{a}$ restricts to an isomorphism on the category of affine schemes.

Proof

In view of Theorem 49 we only have to prove that the condition in (1) of the present proposition remains valid after Zariski-sheafification. Thus it will suffice to prove the last assertion, namely that the natural map $F(X)\rightarrow F^{a}(X)$ is indeed an isomorphism for every affine $X$. To this end, for an arbitrary scheme $X$ and any Zariski-covering $\mathcal {U}$ of $X$ let $K(\mathcal {U})$ be the equalizer of $F(\mathcal {U})\rightrightarrows F(\mathcal {U}\times _{X}\mathcal {U})$. If we set $F'(X) = \underrightarrow{\lim }_{\mathcal {U}}K(\mathcal {U})$, where the colimit is taken over all Zariski-coverings of $X$, then $F'$ will be a separated presheaf. Applying this procedure twice, i.e. forming $F''$, will yield a sheaf, and indeed $F''$ is equal to the Zariski-sheafification $F^{a}$ of $F$. Now observe the following: if $X$ is affine, there is a cofinal subsystem of all Zariski coverings of $X$ given by those coverings which consist of only finitely man! y affines. Thus, using assumption (2),

$$\begin{aligned} F'(X) = \underrightarrow{\lim }_{Y\rightarrow X} \ker (F(Y)\rightrightarrows F(Y\times _{X}Y)), \end{aligned}$$

where now the limit is taken over a certain family of faithfully flat morphisms $Y\rightarrow X$ of affine schemes. But by assumption (1) for every such $Y\rightarrow X$ we have $F(X) = \ker (F(Y)\rightrightarrows F(Y\times _{X}Y))$, whence $F'(X)=F(X)$. This implies $F^{a}(X)=F(X)$, as desired. $\square$

Corollary 51

Let $F$ be as in Proposition 50. Then the restriction of $F$ to the site of affine schemes (with arbitrary covering families consisting of affine schemes) is a sheaf for the fpqc-topology.

The preceding discussion shows that the category of fpqc-sheaves on the category of $k$-schemes is equivalent to the category of functors on affine $k$-schemes which satisfy the conditions (1) and (2) of Proposition 50. Mutually inverse equivalences are given by restriction and respectively by passing to the associated Zariski-sheaf. In [1], the authors indeed define a $k$-space to be a functor on the category of affine $k$-schemes which satisfies condition (1). On the other hand, they do not require condition (2), which, however, does not seem to be automatic.

The following proposition shows that indeed every directed system of $k$-schemes gives rise to a $k$-ind-scheme (i.e. the colimit in the category of $k$-spaces exists).

Proposition 52

A functor which is defined as an inductive limit of schemes admits an fpqc-sheafification. More precisely, its Zariski-sheafification is already an fpqc-sheaf(ification). Further, the restriction of this sheafification to the category of affine schemes coincides with the original presheaf defined by the inductive system of schemes.

Proof

We have to check that such a functor satisfies the assumptions (1) and (2) of Proposition 50.

To this end, let $(X_{i})$ be a direct system of schemes and let $\underrightarrow{\lim }X_{i}$ be its colimit in the category of presheaves. Let $T_{1},\ldots ,T_{n}$ be affine schemes. Then we have

$$\begin{aligned}&(\underrightarrow{\lim }X_{i})(T_{1}\coprod \cdots \coprod T_{n}) = \underrightarrow{\lim }(X_{i}(T_{1}\coprod \cdots \coprod T_{n})) \\&\quad = \underrightarrow{\lim }(X_{i}(T_{1})\times \cdots \times X_{i}(T_{n})) = (\underrightarrow{\lim }X_{i})(T_{1})\times \cdots \times (\underrightarrow{\lim }X_{i})(T_{n}), \end{aligned}$$

which is condition (2). It remains to check exactness of the sequence

$$\begin{aligned} (\underrightarrow{\lim }X_{i})(R) \rightarrow (\underrightarrow{\lim }X_{i})(S) \rightrightarrows (\underrightarrow{\lim }X_{i})(S\otimes _{R}S), \end{aligned}$$

where $R\rightarrow S$ is a faithfully flat homomorphism of rings. Thus let $P\in (\underrightarrow{\lim }X_{i})(S)$ be such that both images of $P$ in $(\underrightarrow{\lim }X_{i})(S\otimes _{R}S)$ coincide. Assume that $P$ is represented by an element $P'\in X_{i}(S)$. By definition of the inductive limit, there exists some $i\le j\in I$ such that that the induced objects in $X_{j}(S\otimes _{R}S)$ coincide. Now we can use the exactness of the sequence

$$\begin{aligned} X_{j}(R) \rightarrow X_{j}(S) \rightrightarrows X_{j}(S\otimes _{R}S) \end{aligned}$$

to obtain an $R$-valued point of $X_{j}$, and hence an $R$-valued point of $\underrightarrow{\lim }X_{i}$ which induces $P$. Injectivity of the map $(\underrightarrow{\lim }X_{i})(R) \rightarrow (\underrightarrow{\lim }X_{i})(S)$ is proved likewise, which shows that condition (1) holds as well. $\square$

Proposition 52 says that if we restrict the functor direct-limit $\underrightarrow{\lim }X_{i}$ to the category of affine schemes (or more generally: quasi-compact schemes), then it is already a sheaf for the fpqc-topology. This is Beauville and Laszlo’s point of view.

Once again let $k$ denote a field of positive characteristic $p$. Contrary to what Vistoli claims ([11], Thm. 2.64) arbitrary functors on the category of $k$-schemes do not in general admit an fpqc-sheafification. An example of such a functor is described by [12]. As Waterhouse explains, the general problem with constructing an fpqc-sheafification of an arbitrary functor is that one is forced to consider direct limits over “all” fpqc-coverings of a given scheme. However, the entirety of “all” fpqc-coverings will not be a set, but a proper class. One way out of this problem would be to restrict to a fixed universe, which has the drawback that sheafifications depend on the particular choice of the universe. On the other hand, Waterhouse proves that for basically bounded functors it suffices to look at direct limits over certain sets of fpqc-coverings, which resolves the above described set-theoretical problems.

Let $m$ be a cardinal number not less than the cardinality of $k$, fix a set $S$ of cardinality $m$, and let $(k\text {-Alg(m)})$ be the category of $k$-algebras whose underlying set is contained in $S$. Let $(k\text {-Alg})$ denote the category of $k$-algebras, and let $j:(k\text {-Alg(m)}) \hookrightarrow (k\text {-Alg})$ be the inclusion. For any set-valued functor on the category of $k$-algebras, let $j^{*}$ denote the restriction to $(k\text {-Alg(m)})$. Right-adjoint to $j^{*}$ is the Kan extension $j_{*}$ along $(k\text {-Alg(m)}) \hookrightarrow (k\text {-Alg})$.

Definition 53

A functor $F$ on the category of $k$-algebras is $m$-based if it has the form $j_{*}G$ for some functor $G$ on $(k\text {-Alg(m)})$. A functor is basically bounded if there exists a cardinal $m$ such that it is $m$-based.

Theorem 54

([12], Cor. 5.2) If a functor $F$ on the category of $k$-algebras is $m$-based, then it has an fpqc-sheafification. More precisely, if $j^{*}F\rightarrow G$ is a sheafification for the fpqc-topology on $(k\text {-Alg(m)})$, then $F=j_{*}j^{*}F \rightarrow j_{*}G$ is an fpqc-sheafification on $(k\text {-Alg})$.

We use the following two observations made by Waterhouse: (a) A functor which is represented by an affine scheme whose underlying ring has cardinality $\le m$ is $m$-based, and (b) the Kan extension $j_{*}$ preserves colimits, and in particular, the colimit over a system of basically bounded functors is again basically bounded.

Corollary 55

Let $H$ and $G$ be functors on the category of $k$-algebras with values in groups, which are represented by $k$-ind schemes. Let $H\rightarrow G$ be a functorial group homomorphism. Then the presheaf-quotient $G/H$ is basically bounded, and hence has a well-defined fpqc-sheafification.

Proof

By (a) the ind-schemes $H$ and $G$ are colimits of basically bounded functors. Thus they are themselves basically bounded by (b) above. Further, since $G/H$ is the colimit of a direct system of the form

$$\begin{aligned} G \times H \rightrightarrows G, \end{aligned}$$

this presheaf-quotient is basically bounded, again by (b). By Waterhouse’s theorem, it thus has a well-defined fpqc-sheafification on the category of $k$-algebras. $\square$