Projectivity of the Witt vector affine Grassmannian

Abstract

We prove that the Witt vector affine Grassmannian, which parametrizes W(k)-lattices in $W(k)[\frac{1}{p}]^n$ for a perfect field k of characteristic p, is representable by an ind-(perfect scheme) over k. This improves on previous results of Zhu by constructing a natural ample line bundle. Along the way, we establish various foundational results on perfect schemes, notably h-descent results for vector bundles.

Appendix: Determinants

Let R be a commutative ring. Our goal in this section is to recall the definition of the natural map

$$\begin{aligned} {\mathrm {det}}: K(R)\rightarrow {\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R) \end{aligned}$$

from the K-theory spectrum of R to the Picard groupoid of graded line bundles ${\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R)$. Intuitively, this is just the map sending a finite projective R-module M to $({\mathrm {det}}M, {\mathrm {rk}}M)$.

Recall that a symmetric monoidal category is a category C equipped with a functor $\otimes : C\times C\rightarrow C$, a unit object $1\in C$, as well as a unitality constraint $\eta _X: 1\otimes X\cong X$, a commutativity constraint $c_{X,Y}: X\otimes Y\cong Y\otimes X$ and an associativity constraint $a_{X,Y,Z}: X\otimes (Y\otimes Z)\cong (X\otimes Y)\otimes Z$ functorially in $X,Y,Z\in C$, satisfying certain compatibility conditions. Moreover, there is a notion of symmetric monoidal functor between symmetric monoidal categories, which is a functor which commutes with $\otimes$ in the appropriate sense. As usual, there is also a notion of a natural transformation between symmetric monoidal functors. The definitions were originally given by MacLane [40].

Definition 12.1

A symmetric monoidal category C is called strict if $c_{X,X}: X\otimes X\cong X\otimes X$ is the identity for all $X\in C$.

In general, the axioms of a symmetric monoidal category say that $c_{X,Y}\circ c_{Y,X} = {\mathrm {id}}$ for all $X,Y\in C$, so in particular $c_{X,X}$ is always an involution.

Note that whenever C is a symmetric monoidal category, the subcategory $C^\simeq \subset C$ consisting of all objects, but only isomorphisms as morphisms, is another symmetric monoidal category (as all extra data concerns isomorphisms). Let us refer to symmetric monoidal categories all of whose morphisms are isomorphisms as symmetric monoidal groupoids.

Below, we will recall that any symmetric monoidal category C admits a K-theory spectrum K(C), and any symmetric monoidal functor $F: C\rightarrow D$ induces a map of spectra $K(F): K(C)\rightarrow K(D)$. Applying this to the inclusion $C^\simeq \subset C$ will give an equivalence $K(C^\simeq )\simeq K(C)$; actually, an equality $K(C^\simeq ) = K(C)$. For this reason, we largely restrict to symmetric monoidal groupoids in the following.

The following examples are our main interest.

Example 12.2

Fix a commutative ring R.

• (i) The groupoid ${\mathcal {V}}{\mathrm {ect}}(R)$ of finite projective R-modules is a symmetric monoidal category with respect to $\oplus$, unit 0, and the standard unitality, commutativity and associativity constraints. For example, $c_{X,Y}: X\oplus Y\cong Y\oplus X$ sends (x, y) to (y, x). Note that $c_{X,X}$ is not the identity map for $X\ne 0$.

  We remark that the groupoid of finite projective R-modules is also symmetric monoidal with respect to the tensor product $\otimes$; we will not use this symmetric monoidal structure.

• (ii) The groupoid ${\mathcal {P}}{\mathrm {ic}}(R)$ of line bundles over R (i.e., finite projective R-modules of rank 1) is a symmetric monoidal category with respect to $\otimes$, unit 1, and the standard unitality, commutativity and associativity constraints. For example, $c_{X,Y}: X\otimes Y\cong Y\otimes X$ sends $x\otimes y$ to $y\otimes x$. Note that $c_{X,X}$ is the identity map for all $X\in {\mathcal {P}}{\mathrm {ic}}(R)$.

• (iii) The groupoid ${\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R)$ of ${\mathbf {Z}}$-graded line bundles. Here, an object is given by a pair (L, f) where $L\in {\mathcal {P}}{\mathrm {ic}}(R)$, and $f:{\mathrm {Spec}}(R) \rightarrow {\mathbf {Z}}$ is a locally constant function; the set ${\mathrm {Isom}}( (L,f), (M,g))$ is empty if $f \ne g$, and given by ${\mathrm {Isom}}(L,M)$ otherwise. The unit is given by (1, 0). This groupoid is endowed with a symmetric monoidal structure $\otimes$ where $(L,f) \otimes (M,g) {:=} (L \otimes M, f + g)$, and the commutativity constraint

  $$\begin{aligned} (L \otimes M, f+g) {=:} (L,f) \otimes (M,g) \simeq (M,g) \otimes (L,f) {:=} (M \otimes L, g+f) \end{aligned}$$

  determined by the rule

  $$\begin{aligned} \ell \otimes m \mapsto (-1)^{f \cdot g} m \otimes \ell , \end{aligned}$$

  and the obvious associativity constraint. Note that $c_{(L,f),(L,f)}: (L,f)\otimes (L,f)\cong (L,f)\otimes (L,f)$ is given by multiplication with $(-1)^f$, and is thus not in general the identity.

The following proposition is central.

Proposition 12.3

There is a natural symmetric monoidal functor

$$\begin{aligned} {\mathrm {det}}: {\mathcal {V}}{\mathrm {ect}}(R)\rightarrow {\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R) \end{aligned}$$

sending $M\in {\mathcal {V}}{\mathrm {ect}}(R)$ to $({\mathrm {det}}M,{\mathrm {rk}}M)\in {\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R) = {\mathcal {P}}{\mathrm {ic}}(R) \times H^0({\mathrm {Spec}}(R),{\mathbf {Z}})$.

Note that this functor does not factor through ${\mathcal {P}}{\mathrm {ic}}(R)$.

Proof

All verifications are automatic, and the only critical observation is the following. If $M, N\in {\mathcal {V}}{\mathrm {ect}}(R)$, then the commutativity constraint $c_{M,N}: M\oplus N\cong N\oplus M$ swapping N and M induces multiplication by $(-1)^{{\mathrm {rk}}M {\mathrm {rk}}N}$ on

$$\begin{aligned} {\mathrm {det}}(M)\otimes {\mathrm {det}}(N)\cong {\mathrm {det}}(M\oplus N)\buildrel {{\mathrm {det}}c_{M,N}}\over \cong {\mathrm {det}}(N\oplus M)\cong {\mathrm {det}}(N)\otimes {\mathrm {det}}(M), \end{aligned}$$

if one identifies ${\mathrm {det}}(M)\otimes {\mathrm {det}}(N)$ and ${\mathrm {det}}(N)\otimes {\mathrm {det}}(M)$ using the usual commutativity constraint of ${\mathcal {P}}{\mathrm {ic}}(R)$. $\square$

Now we recall the construction of the K-theory spectrum. Let C be a symmetric monoidal category, with $C^\simeq \subset C$ the underlying symmetric monoidal groupoid. By definition, $K(C) = K(C^\simeq )$, so assume that C is a groupoid to start with. Recall that a groupoid is equivalent to a space whose only nonzero homotopy groups are $\pi _0$ and $\pi _1$. Concretely, this can be realized by the nerve construction, which associates to any category C the simplicial set N(C) whose n-simplices are chains of $n-1$ morphisms,

$$\begin{aligned} X_0\buildrel f_0\over \rightarrow X_1\buildrel f_1\over \rightarrow \cdots \buildrel {f_{n-1}}\over \rightarrow X_n. \end{aligned}$$

In particular, the 0-simplices $N(C)_0$ are the objects of C, and the 1-simplices $N(C)_1$ are the morphisms of C. Higher simplices encode the composition law, and degenerate simplices encode identity morphisms. If C is a groupoid, then N(C) is a Kan complex, whose geometric realization |N(C)| is a (compactly generated) topological space whose only nonzero homotopy groups are $\pi _0$ and $\pi _1$. In fact, |N(C)| is homotopy equivalent to the disjoint union

$$\begin{aligned} \bigsqcup _{X\in C/\simeq } B{\mathrm {Aut}}(X) \end{aligned}$$

of the classifying spaces of the automorphism group ${\mathrm {Aut}}(X)$, over all isomorphism classes of objects $X\in C$. In the following, we refer to Kan complexes as ‘spaces’. They are naturally organized into the $\infty$-category ${\mathcal {S}}$ of spaces. Here and in the following, we make use of the theory of $\infty$-categories, cf. [33], which were previously defined under the name of weak Kan complexes (by Boardman–Vogt) and quasicategories (by Joyal). Using this language, we can easily state precise definitions and theorems without having to go into the pain of detailed constructions. Giving detailed proofs is more elaborate, and we only try to convey the meaning of the statements below.

As C is symmetric monoidal, the space N(C) is equipped with an addition law $N(\otimes ): N(C)\times N(C)=N(C\times C)\rightarrow N(C)$. Moreover, this addition law is commutative and associative up to coherent isomorphisms, as expressed by the commutativity and associativity constraints. However, the addition law is not strictly commutative and associative. Such spaces equipped with a coherently commutative and associative (but not necessarily invertible) addition law are known as (special) $\Gamma$-spaces, as defined by Segal [48], or as ${\mathbf {E}}_\infty$-monoids, as defined by May [39]. Although our definition is essentially that of a special $\Gamma$-space, we prefer to call them ${\mathbf {E}}_\infty$-monoids.²⁷

Definition 12.4

Let ${\mathrm {Fin}}_*$ be the category of finite pointed sets.²⁸ For $n\ge 0$, let $[n] = \{0,1,\ldots ,n\}\in {\mathrm {Fin}}_*$ be the object pointed at 0. An ${\mathbf {E}}_\infty$-monoid is a functor

$$\begin{aligned} X: N({\mathrm {Fin}}_*)\rightarrow {\mathcal {S}}\end{aligned}$$

of $\infty$-categories to the $\infty$-category ${\mathcal {S}}$ of Kan complexes, such that for all $[n]\in {\mathrm {Fin}}_*$, the natural map

$$\begin{aligned} X([n])\rightarrow \prod _{i=1}^n X([1]) \end{aligned}$$

is a weak equivalence, where the map is induced by the n maps $[n]\rightarrow [1]$ contracting everything to 0 except i.

The ${\mathbf {E}}_\infty$-monoids are naturally organized into an $\infty$-category (a full subcategory of the $\infty$-category of functors from $N({\mathrm {Fin}}_*)$ to ${\mathcal {S}}$), which we call the $\infty$-category of ${\mathbf {E}}_\infty$-monoids ${\mathrm {Mon}}_{{\mathbf {E}}_\infty }$.

For an ${\mathbf {E}}_\infty$-monoid X, the 0-th space X([0]) is weakly contractible, as it is weakly equivalent to an empty product. We refer to X([1]) as the underlying space of the ${\mathbf {E}}_\infty$-monoid X, and will sometimes confuse X with X([1]). It is pointed in the sense that it comes equipped with a map $X([0])\rightarrow X([1])$ from a weakly contractible space; the space X([0]) is the ’unit’ of X([1]). Moreover, using the map $[2]\rightarrow [1]$ sending only 0 to 0, we get a natural map

$$\begin{aligned} X([1])\times X([1])\simeq X([2])\rightarrow X([1]) \end{aligned}$$

which gives an addition law on X([1]). The higher data precisely ensure that the unit is unital, and that the addition law is commutative and associative ’up to coherent homotopy’. In particular, for an ${\mathbf {E}}_\infty$-monoid X, the set of connected components $\pi _0 X {:=} \pi _0 X([1])$ forms a commutative monoid.

Construction 12.5

Let C be a symmetric monoidal groupoid. We define an ${\mathbf {E}}_\infty$-monoid N(C) in the following way. For each finite pointed set $(S,s)\in {\mathrm {Fin}}_*$, let

$$\begin{aligned} N(C)(S) = N\left( \{(X_T)_{T\subset S{\setminus } \{s\}}, X_{T\sqcup T^\prime }\cong X_T\otimes X_{T^\prime }\}\right) \end{aligned}$$

be the nerve of the groupoid of objects $X_T\in C$ for all $T\subset S{\setminus } \{s\}$ which are equipped with compatible isomorphisms $X_{T\sqcup T^\prime }\cong X_T\otimes X_{T^\prime }$ for any disjoint subsets $T, T^\prime \subset S{\setminus } \{s\}$. For any morphism $f: (S,s)\rightarrow (S^\prime ,s^\prime )$, let the map

$$\begin{aligned} N(C)(S)\rightarrow N(C)(S^\prime ) \end{aligned}$$

be given by

$$\begin{aligned} (X_T)_{T\subset S{\setminus } \{s\}}\mapsto (X_{f^{-1}(T^\prime )})_{T^\prime \subset S^\prime {\setminus } \{s^\prime \}}. \end{aligned}$$

Remark 12.6

Here, we have defined an actual functor from ${\mathrm {Fin}}_*$ to the category of Kan complexes. The naive definition of N(C)(S) would be $N(C)(S) = N(C)^{S{\setminus } \{s\}}$, with the map

$$\begin{aligned} N(C)^{S{\setminus } \{s\}}\rightarrow N(C)^{S^\prime {\setminus } \{s^\prime \}} \end{aligned}$$

for a map $f: (S,s)\rightarrow (S^\prime ,s^\prime )$ of pointed sets being given by

$$\begin{aligned} (X_t)_{t\in S{\setminus } \{s\}}\mapsto (\otimes _{t\in f^{-1}(t^\prime )} X_t)_{t^\prime \in S^\prime {\setminus } \{s^\prime \}}. \end{aligned}$$

This definition is not compatible with composition on the nose, but can be made into a functor of $\infty$-categories $N(C): {\mathrm {Fin}}_*\rightarrow {\mathcal {S}}$. However, the verification of this fact is most easily done by constructing a weakly equivalent strict functor, as above.

We leave it to the reader to spell out the definition of compatibility of the isomorphisms $X_{T\sqcup T^\prime }\cong X_T\otimes X_{T^\prime }$; verifying that $N(C)([n])\rightarrow N(C)^n$ is an equivalence uses the precise axioms of a symmetric monoidal category.

Definition 12.7

An ${\mathbf {E}}_\infty$-monoid X is called grouplike if the commutative monoid $\pi _0(X)$ is a group. Let ${\mathrm {Mon}}_{{\mathbf {E}}_\infty }^{\mathrm {gp}}\subset {\mathrm {Mon}}_{{\mathbf {E}}_\infty }$ denote the full subcategory of grouplike ${\mathbf {E}}_\infty$-monoids.

Grouplike ${\mathbf {E}}_\infty$-monoids are equivalent to connective spectra.

Definition 12.8

The $\infty$-category of connective spectra ${\mathrm {Sp}}^{\ge 0}$ is given by the limit of

$$\begin{aligned} {\mathcal {S}}_*\buildrel \Omega \over \leftarrow {\mathcal {S}}_*^{\ge 1}\buildrel \Omega \over \leftarrow {\mathcal {S}}_*^{\ge 2}\buildrel \Omega \over \leftarrow \cdots \end{aligned}$$

in the $\infty$-category of $\infty$-categories, where ${\mathcal {S}}_*$ is the $\infty$-category of pointed spaces, ${\mathcal {S}}_*^{\ge i}\subset {\mathcal {S}}_*$ is the full subcategory of i-connected spaces²⁹, and $\Omega : {\mathcal {S}}_*\rightarrow {\mathcal {S}}_*$, $X\mapsto *\times _X *$ is the loop space functor.

The following theorem is due to Segal, [48, Proposition 3.4], cf. [35, Theorem 5.2.6.10, Remark 5.2.6.26]. The functor ${\mathrm {Mon}}_{{\mathbf {E}}_\infty }^{\mathrm {gp}}\rightarrow {\mathrm {Sp}}^{\ge 0}$ is usually called “the infinite loop space machine”.

Theorem 12.9

There is a natural equivalence of $\infty$-categories

$$\begin{aligned} {\mathrm {Mon}}_{{\mathbf {E}}_\infty }^{\mathrm {gp}}\simeq {\mathrm {Sp}}^{\ge 0}. \end{aligned}$$

May–Thomason [42], showed that there is essentially only one such equivalence (although there are many constructions).³⁰ Let us briefly sketch the definition of the functors in either direction. Let $X\in {\mathrm {Sp}}^{\ge 0}$ be a connective spectrum. Then the natural map

$$\begin{aligned} X\sqcup X\rightarrow X\times X \end{aligned}$$

from the coproduct to the product, taken in the $\infty$-category of connective spectra (equivalently, of spectra), is an equivalence, cf. [35, Lemma 1.1.2.10]. Thus, the obvious map $X\sqcup X\rightarrow X$ extends to a map $X\times X\rightarrow X$, which is the ’addition law’ of the spectrum. Repeating the same arguments for arbitrary finite (co)products shows that the forgetful functor

$$\begin{aligned} {\mathrm {Mon}}_{{\mathbf {E}}_\infty }({\mathrm {Sp}}^{\ge 0})\rightarrow {\mathrm {Sp}}^{\ge 0} \end{aligned}$$

from the $\infty$-category of ${\mathbf {E}}_\infty$-monoids in ${\mathrm {Sp}}^{\ge 0}$ to ${\mathrm {Sp}}^{\ge 0}$, is an equivalence, i.e. any (connective) spectrum comes equipped with a canonical ${\mathbf {E}}_\infty$-monoid structure. Composing with the forgetful map

$$\begin{aligned} {\mathrm {Mon}}_{{\mathbf {E}}_\infty }({\mathrm {Sp}}^{\ge 0})\rightarrow {\mathrm {Mon}}_{{\mathbf {E}}_\infty }({\mathcal {S}}) = {\mathrm {Mon}}_{{\mathbf {E}}_\infty } \end{aligned}$$

gives a functor ${\mathrm {Sp}}^{\ge 0}\rightarrow {\mathrm {Mon}}_{{\mathbf {E}}_\infty }$, preserving all $\pi _i$. As all $\pi _i$, including $\pi _0$, of a spectrum are groups, it follows that the functor takes values in ${\mathrm {Mon}}_{{\mathbf {E}}_\infty }^{\mathrm {gp}}$.

Conversely, start with an ${\mathbf {E}}_\infty$-monoid $X\in {\mathrm {Mon}}_{{\mathbf {E}}_\infty }$ (not necessarily grouplike). Then X admits a classifying space BX. Indeed, there is a natural functor $\Delta ^{\mathrm {op}}\rightarrow {\mathrm {Fin}}_*$ (cf. [48, p. 295]) taking the m-simplex $\Delta ^m$ to [m]. This allows one to view X as a simplicial space $X^\prime : N(\Delta ^{\mathrm {op}})\rightarrow N({\mathrm {Fin}}_*)\rightarrow {\mathcal {S}}$, which can be turned into a space $BX = \mathop {{\mathrm {colim}}}_{\Delta ^{\mathrm {op}}} X^\prime \in {\mathcal {S}}$. In fact, one checks directly that BX still carries a canonical ${\mathbf {E}}_\infty$-monoid structure, cf. [48, Definition 1.3]. This gives a functor

$$\begin{aligned} B: {\mathrm {Mon}}_{{\mathbf {E}}_\infty }\rightarrow {\mathrm {Mon}}_{{\mathbf {E}}_\infty }. \end{aligned}$$

There is a natural map $X\rightarrow \Omega BX$ of ${\mathbf {E}}_\infty$-monoids.³¹ The following is [48, Proposition 1.4].

Proposition 12.10

If $X\in {\mathrm {Mon}}_{{\mathbf {E}}_\infty }$ is k-connected, then BX is $(k+1)$-connected. Moreover, the natural map

$$\begin{aligned} X\rightarrow \Omega BX \end{aligned}$$

is an equivalence if and only if X is grouplike.

Remark 12.11

As $\pi _0(\Omega BX) = \pi _1(X)$ is a group, $\Omega BX$ is always a grouplike ${\mathbf {E}}_\infty$-monoid. Also, BX is always 1-connected, and thus grouplike (as $\pi _0 BX=0$ is a group).

Thus, if $X\in {\mathrm {Mon}}_{{\mathbf {E}}_\infty }^{\mathrm {gp}}$ is a grouplike ${\mathbf {E}}_\infty$-monoid, the sequence of spaces $X,BX,B^2X,\ldots$ forms a connective spectrum, giving a functor

$$\begin{aligned} {\mathrm {Mon}}_{{\mathbf {E}}_\infty }^{\mathrm {gp}}\rightarrow {\mathrm {Sp}}^{\ge 0}. \end{aligned}$$

In particular, the construction shows that as with classical monoids, one can form the group completion.

Corollary 12.12

The full subcategory ${\mathrm {Mon}}_{{\mathbf {E}}_\infty }^{\mathrm {gp}}\subset {\mathrm {Mon}}_{{\mathbf {E}}_\infty }$ is a reflective subcategory, cf. [33, Remark 5.2.7.9], i.e. it admits a left adjoint, called the group completion,

$$\begin{aligned} {\mathrm {Mon}}_{{\mathbf {E}}_\infty }\rightarrow {\mathrm {Mon}}_{{\mathbf {E}}_\infty }^{\mathrm {gp}}: X\mapsto X^{\mathrm {gp}}=\Omega BX\ . \end{aligned}$$

Recall that $K_0(C)$ is the group completion of the commutative monoid of objects of C up to isomorphism. Using the above machinery, we can erase the words “up to isomorphism”, and arrive at the definition of higher algebraic K-theory:

Definition 12.13

Let C be a symmetric monoidal category. The K-theory spectrum $K(C)\in {\mathrm {Sp}}^{\ge 0}$ is defined to be the image under ${\mathrm {Mon}}_{{\mathbf {E}}_\infty }^{\mathrm {gp}}\simeq {\mathrm {Sp}}^{\ge 0}$ of the group completion $N(C^\simeq )^{\mathrm {gp}}$ of the ${\mathbf {E}}_\infty$-monoid $N(C^\simeq )$ from Construction 12.5.

Clearly, this construction is functorial in C. There is a certain situation in which the group completion is unnecessary.

Definition 12.14

A symmetric monoidal category C is called a Picard groupoid if C is a groupoid, and any object $X\in C$ admits an inverse $X^{-1}\in C$ such that $X\otimes X^{-1}\cong 1$.

The groupoids ${\mathcal {P}}{\mathrm {ic}}(R)$ and ${\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R)$ are examples of Picard groupoids (explaining the name), as one can form the inverse of a line bundle.

Proposition 12.15

A symmetric monoidal groupoid C is a Picard groupoid if and only if the ${\mathbf {E}}_\infty$-monoid N(C) is grouplike.

Proof

One has an identification of $\pi _0 N(C) = C/\simeq$ with the monoid (under $\otimes$) of objects of C up to isomorphism. But, by definition, C is a Picard groupoid if and only if $C/\simeq$ is a group. $\square$

In particular, for a Picard groupoid, the space underlying K(C) is just the nerve N(C), which is 1-truncated. One can show that this induces an equivalence between Picard groupoids and 1-truncated connective spectra, cf. [43, §3]. The idea is that both can be identified with grouplike ${\mathbf {E}}_\infty$-monoids in groupoids.

Because of this equivalence, we will often confuse a Picard groupoid with its K-theory spectrum, and in particular we continue to write ${\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R)$ for the corresponding (1-truncated) connective spectrum.

Definition 12.16

Let R be a commutative ring. The K-theory spectrum of R is $K(R) = K({\mathcal {V}}{\mathrm {ect}}(R))$.

Corollary 12.17

There is a natural functorial map

$$\begin{aligned} {\mathrm {det}}: K(R)\rightarrow {\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R) \end{aligned}$$

of connective spectra.

Proof

Apply the K-theory functor to

$$\begin{aligned} {\mathrm {det}}: {\mathcal {V}}{\mathrm {ect}}(R)\rightarrow {\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R). \end{aligned}$$

$\square$

As ${\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R)$ is 1-truncated, the map $K(R)\rightarrow {\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R)$ factors canonically over the 1-truncation $\tau _{\le 1}K(R)$. In fact, ${\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R)$ is, at least after Zariski sheafification, precisely the 1-truncation of K(R).

Proposition 12.18

The natural map

$$\begin{aligned} {\mathrm {det}}: \tau _{\le 1} K(R)\rightarrow {\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R) \end{aligned}$$

of presheaves of groupoids on the category of affine schemes becomes an isomorphism after Zariski sheafification.

Proof

We need to prove that the maps

$$\begin{aligned} K_1(R) = \pi _1 K(R)\rightarrow \pi _1 {\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R) \end{aligned}$$

and

$$\begin{aligned} K_0(R) = \pi _0 K(R)\rightarrow \pi _0 {\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R) \end{aligned}$$

are isomorphisms after Zariski sheafification; equivalently, for local rings R. But if R is local, $K_1(R) = R^\times = \pi _1 {\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R)$, and $K_0(R) = {\mathbf {Z}}= \pi _0 {\mathcal {P}}{\mathrm {ic}}^{\mathbf {Z}}(R)$. $\square$