Math. J. Okayama Univ. 62 (2020), 1–25

A REPRESENTATION FOR ALGEBRAIC K-THEORY OFQUASI-COHERENT MODULES OVER AFFINE SPECTRALSCHEMESMariko Ohara
Abstract. In this paper, we study K-theory of spectral schemes byusing locally free sheaves. Let us regard the K-theory as a functor K
on affine spectral schemes. Then, we prove that the group completionΩB G (B G GL) represents the sheafification of K with respect to Zariski(resp. Nisnevich) topology G, where B G GL is a classifying space of acolimit of affine spectral schemes GLn .

1. Introduction
An ∞-category is a notion of categories up to coherent homotopy. The
spectral algebraic geometry in terms of ∞-category, introduced by Lurie [15][16], is a generalization of algebraic geometry.
In this paper, we study the K-theory on spectral schemes. For the Ktheory of a category of projective modules of finite rank in the sense of spectral algebraic geometry, we construct an object in the ∞-category which
represents the K-theory. The main theorem (Theorem 1.1) is a generalization of the representation theorem of the K-theory in the classical algebraic
geometry proved by Morel and Voevodsky [20, Proposition 3.9].
1.1. Statement of the main theorem. By an ∞-category with w∞ cofibrations we mean a pointed ∞-category with a class of morphisms (called
w∞ -cofibrations) satisfying certain conditions (see Definition 5.1 for details).
Let C be an ∞-category with w∞ -cofibrations. We define the algebraic Ktheory space of C by
K(C) = Ω|S• (C)|,where S• is the S-construction (cf. [6]) and | − | is the geometric realization.
We say that a spectrum E is connective if πn E ≃ 0 for n < 0. Let R bea connective E∞ -ring, and CAlgcn the ∞-category of connective E∞ -rings.
be an ∞-category of projective R-modules of finite rank which
Let Mod∞projR
we recall in Definition 2.1. It becomes an ∞-category with w∞ -cofibrations
by Definition 5.3.
We denote by CAlgG an opposite ∞-category (CAlgcn )op equipped witheither the Zariski topology or the Nisnevich topology G, and by Spec G R
an object in the essential image of Yoneda functor CAlgG → Shv(CAlgG ),Mathematics Subject Classification. Primary 18E99; Secondary 19D10.
Key words and phrases. Infinity category, Derived algebraic geometry, K-theory.
12

M. OHARA

where Shv(CAlgG ) is the ∞-category of sheaves on CAlgG given in Section
2. Here, we denote by (−)op the opposite ∞-category.
Let Sb be the ∞-category of not-necessary small spaces. We define afunctor(1.1)
K : (CAlgG )op → Sb).
which carries a spectral scheme Spec G R to the K-theory K(Mod∞proj
RG
g the sheafification from the ∞-category of functors on
We denote by (−)(CAlgG )op to Shv(CAlgG ) which we recall in Definition 2.4.
Let GLn be the affine groupscheme of general linear group in` spectral
G
GG
Shv(CAlg ). Let B GL = n∈N B GLn be the coproduct of the classifying
sheaf B G GLn of GLn , where B G is a functor given by taking classifying sheaf
which we recall in Section 4. Let ΩB G be a functor on Shv(CAlgG ) definedin Definition 3.15. We denote by ΩB G (B G GL) the group completion (cf.
Definition 4.15).
Theorem 1.1 (cf. Theorem 6.2). Let MapShv b(CAlgG ) (−, −) denote theGSmapping space of the ∞-category ShvSb(CAlg ) which we recall in Definition 3.8. There is an equivalencee G (Mod∞proj ).
MapShv b(CAlgG ) (Spec G R, ΩB G (B G GL)) ≃ K
RSe G is representable
Remarks for Theorem 1.1. It is known that the sheaf K
by an object in ShvSb(CAlgG ) by the brown representability theorem in thesense of ∞-category [11, Proposition 5.5.2.7]. The importance is that wee G.
give a concrete object which represents K
Morel and Voevodsky used the cofinality theorem in Quillen’s K-theory
in the proof of [20, Proposition 3.9]. Since the cofinality theorem is notestablished in the K-theory of ∞-categories, their proof cannot be applied toour case directly. To avoid this problem, we treat an ∞-category Mod∞proj
R
instead of the finitely generated projective modules.
1.2. Outline of this paper. This paper is organized as follows. In Section
2, we introduce the terminology of an ∞-category endowed with Grothendiecktopology and sheaves on it. In Section 3, we characterize the group completion functor on Shv(CAlgG ) explicitly in Proposition 3.16, which is oneof the important lemmas in this paper. In Section 4, we show the correspondence between the value of the affine group scheme GLn on S and theautomorphisms of S n , where S is an E∞ -ring in Proposition 4.8. We also
demonstrate that the classifying sheaf B G GLn is equivalent to the sheafof projective modules of finite rank in Proposition 4.21. We also define
the notion of Zariski connectedness and relate a certain mapping space ofA REPRESENTATION FOR ALGEBRAIC K-THEORY

3B G GLn is equivalent to the sheaf of projective modules of finite rank in
Proposition 4.26. In Section 5, we recall the several notion and proposition
with respect to K-theory. In Section 6, by using these results, we prove
Theorem 6.2.
Acknowledgement. The author would like to express deeply her thanks toProfessor Nobuo Tsuzuki for his valuable advice and checking this paper.
The author would like to express her thanks to Professor Satoshi Mochizuki
for his valuable comments on algebraic K-theory, especially the resolution
theorem and the cofinality, and for reading this paper. The author also
would like to express her thanks to Professor Yuki Kato for valuable comments to the author. Finally, the author would like to thank the anonymousreferees for detailed comments and corrections which substantially improved
this paper.
2. Preliminary
We fix the universe U such that N ∈ U. We define the Cat∞ by U-small∞-category, which is locally U-small.
Although there are a lot of languages of higher category theory, we use
the same notation in Lurie’s book [11] and papers [12], [15] and [13].
Let Fun(−, −) be an ∞-category of functors. For a pair of functors f :C → D and g : D → C between ∞-categories, according to [11, Proposition
5.2.2.8], we say that the functor f is a left adjoint to g (resp. g is a rightadjoint to f ) if there exists a unit map u : idC → g ◦ f given in [11, Defnition
5.2.2.7].
For an E∞ -ring R, we have the ∞-category ModR of R-modules [12, Section 4.2]. Since the tensor product on the ∞-category of spectra is compatible with the geometric realizations [12, Corollary 4.8.2.19], ModR becomes
the symmetric monoidal ∞-category by [12, Theorem 4.5.2.1]. We denoteby ⊗R the tensor product on ModR .
Let R be an E∞ -ring and a ∈ π0 R an element. The localization of R withrespect to a, which is denoted by R[a−1 ], is an E∞ -ring (see [16, Remark
2.9] and [12, 7.2.4]).
Definition 2.1. Let R be a connective E∞ -ring and M an R-module.
(i) We say that M is free of rank n if M ≃ R⊕n .
(ii) We say that M is finitely generated projective if there exists n ∈ Nsuch that M is a retract free R-module of rank n.
(iii) We say that M is projective of rank n if it is finitely generated projective and we can choose elements x1 , · · · , xm ∈ π0 R such that theygenerate the unit ideal and all of the localization M [x−1i ] are free mod−1
ules of rank n over R[xi ].

4

M. OHARA(iv) We say that M is projective of finite rank if there exists n ∈ N suchthat M is projective of rank n.
ree) an ∞-category of free(resp. Modnproj
We denote by Modnf
R
R(resp.
(resp. projective) R-modules of rank n. We denote by Modproj
R∞proj) the ∞-category of finitely generated projective R-modules
ModR(resp. projective R-modules of finite rank).
is not equivalent.
and Modproj
Remark 2.2. In general, Mod∞proj
RR
As in [11], [12] and [4], to treat certain size of limits and colimits, we needto enlarge the universe and fix the size of universes properly.
We adopt the axiom of universes which allows us to consider that every
cardinal can be strictly upper bounded by a strongly inaccessible cardinal.
Then, there exists a bijection between strongly inaccessible cardinals anduniverses, and thus we can take a suitable enlargement of universe which weneed.
By the axiom of universe, there exists an enlargement of universes U ∈ Vsuch that every U-small object is also V-small, so that Cat∞ becomes aV-small category.
In this paper, sometimes we need to treat the ∞-category CAlg as small,and to regard the maximal ∞-groupoid (ModR )≃ of ModR as a space. We
also need to treat the size of limits and colimits. In these cases, we enlarge
the universe as the following definition and proceed the arguments.
Definition 2.3. Let S be the ∞-category of U-small spaces. Throughout
this paper, we enlarge the universe U ∈ V such that ModR is a V-small ∞\category. We use the notation Sb for S and Cat∞ for Cat∞ after changing
the universe from U to V.
2.1. Sheaves and spectral affine schemes. Let us take X as an ∞category equipped with a Grothendieck topology as in [11, Definition 6.2.2.1]
and C an ∞-category which admits limits. We denote by ShvC (X ) an ∞category of C-valued sheaves on X in [11, Definition 6.2.2.6].
g : Fun(X op , S) → ShvS (X ) be an localization deDefinition 2.4. Let (−)
fined in [11, Definition 6.2.2.6], which is a left adjoint of the inclusion. For
an object F of Fun(X , S), we say that Fe is a sheafification of F .
Remark 2.5 ([11] Construction 6.2.2.9, Remark 6.2.2.12). For an ∞-category
C equipped with a Grothendieck topology, and a presheaf F on C, the sheafification Fe of F is given by the following formula: for any C ∈ C,
Fe (C) = colimC (0) limC ′ ∈C (0) F (C ′ ),/C/CA REPRESENTATION FOR ALGEBRAIC K-THEORY

5

where the first limit is the limit of the simplicial diagram associated to
C ′ → C [11, Corollary 6.2.3.5] and the second colimit is taken over thecollection of covering sieves on C.
Here is a list of morphisms of E∞ -rings.
• Let f : A → B be a morphism of E∞ -rings. Recall that f : A →B is said to be flat if π0 B is flat π0 A-module and the underlying
map of commutative rings π0 A → π0 B induces the isomorphism(πi A) ⊗π0 A (π0 B) ≃ πi B for every integer i.
• Let f : A → B be a morphism of E∞ -rings. f : A → B is said to beétale if it is flat and the underlying map of commutative rings π0 A →π0 B is étale. The class of étale morphisms on (CAlgcn )op satisfies
the axiom of admissible morphisms by [15, Proposition 2.4.17].
• We say that a morphism f : A → B of E∞ -rings is faithfully flat if
it is flat morphism and the underlying map of commutative rings
π0 A → π0 B is faithfully flat. By a class of faithfully flat morphisms, Grothendieck topology is defined on CAlgop , which is called
flat topology [16, Definition 5.2, Proposition 5.4].
Let S be a sphere spectrum. Recall the Grothendieck topology givenby the admissibility on covering sieves from [15, Definition 1.2.1]. LetSp(S) be an ∞-category with Grothendieck topology given as [16, DefiniGZar
Sp(S)op )op equipped
tion2.10]. We denote by CAlgZar an ∞-category Ind(GZar
with Zariski topology [15, Notation 2.2.6]. Moreover, this Zariski geometry is finitary [15, Remark 2.2.8] by the definition. (For the base change
assumption, see [15, Remark 1.2.4]).
Let A be a connective E∞ -ring. Let CAlget
A be an ∞-category spanned by
op be a sieve containing
connective étale algebras over A. Let C ⊂ (CAlgetA)
A. We say that C is a Nisnevich covering sieve on A if it contains a collection of morphisms A → Aa such that their underlying maps of commutative
rings π0 A → π0 Aa determine a Nisnevich covering defined in [14, Definition
1.1]. Let A be a connective E∞ -ring. We define the admissible morphisms in(CAlgcn )op by the morphisms corresponding to étale morphisms in CAlgcn ,and the collection of admissible morphisms A → Aa generates a covering
sieve on A if and only if it corresponds to a Nisnevich covering sieve. Then,it generates a Grothendieck topology on (CAlgcn )op , which is called the Nisnevich topology. We denote by CAlgN is an opposite ∞-category (CAlgcn )op
endowed with the Nisnevich topology.
Let CAlgG denote either CAlgZar or CAlgN is . Let us denote an object inthe essential image of Yoneda functor CAlgG → ShvS (CAlgG ) by Spec G R,
and we call the object an affine spectral scheme.
The following proposition is a special case of [16, Proposition 5.7].

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M. OHARA

Proposition 2.6. Let us consider CAlgop endowed with the flat topology.
Then, in the case of ∞-topos CAlgop with flat topology, a functor F is a sheaf
if, it preserves finite products and for any covering X → Y in (CAlg)op ,F (Y ) → lim∆ F (X• )is an equivalence. Here, X• → Y is the simplicial object associated to X →Y , and the notation lim∆ in the right hand side is a limit taken over thesimplicial diagram X• .
Lemma 2.7. (i) A Zariski covering sieve is a Nisnevich covering sieveon (CAlgcn )op .
(ii) A Zariski covering sieve and a Nisnevich covering sieve are covering
sieves on the flat topology on (CAlgcn )op .
Proof. Let R be a connective E∞ -ring and f1 , · · · , fn elements in π0 R which
generate the unit ideal of π0 R. To prove (i), we show that each R → R[fi−1 ]is flat for all n and {π0 R → π0 R[fi−1 ]} is an ordinary Nisnevich coveringof π0 R. Since each R → R[fi−1 ] is flat by [16, Remark 2.9] and {π0 R →π0 R[fi−1 ]} is an ordinary Zariski covering of π0 R, so that it is a Nisnevich
covering of π0 R [14, Remark 1.13].
To show (ii), we show that, for any collection of morphisms {R → Ra }such that their underlying Q
maps of commutative rings {π0 R → π0 Ra } forma Nisnevich covering, R → α Rα is faithfully flat. Note that we can assume
that this Nisnevich covering is a finite collection of morphisms [14, Remark
1.6].
Q
Since each R → Rα is étale, it is flat. Q
It follows Q
that R → α Rα isalso flat. By [14, Remark 1.12], π0 R → π0 ( α Rα ) ∼= α π0 Rα is faithfullyflat.

3. Group completion in an ∞-topos
Definition 3.1. Let C be an ∞-category with finite products, and O⊗ be either the ∞-operad N∆ (Assoc⊗ ) given in [12, Definition 4.1.1.3] or N∆ (Fin∗ ).
Note that we have a natural map ρ : N∆ (Assoc⊗ ) → N∆ (Fin∗ ) by forgetting
linear orderings (cf. [12, Remark 4.1.1.4]).
(i) An ∞-monoid in C is a functor M : N∆ (Assoc⊗ ) → C such that themorphism M (hni) → M (h1i) induced by the inert maps σ i : hni →h1i with linear orderings in each 1 ≤ i ≤ n induces an equivalence
M (hni) ≃ M (h1i)n for all n ∈ N.
(ii) A commutative ∞-monoid in C is a functor M : N∆ (Fin∗ ) → C suchthat the morphism M (hni) → M (h1i) induced by the inert maps σ i :hni → h1i induces an equivalence M (hni) ≃ M (h1i)n for all n ∈ N.

A REPRESENTATION FOR ALGEBRAIC K-THEORY

7(iii) An ∞-monoid is an ∞-group if its image in hC is a group object.
Also, a commutative ∞-monoid is a commutative ∞-group if its image
in hC is a commutative group object. We denote by M on(C) andnM on(C) an ∞-category of commutative ∞-monoids in C and notneccesary commutative ∞-monoids in C respectively. We also denoteby Gp(C) and nGp(C) an ∞-category of commutative ∞-groups in C
and not-neccesary commutative ∞-groups in C respectively (cf. [9,Proposition 1.1]).
Note that we have a diagram
ρ∗

Gp(C)


M on(C)ρ∗/ nGp(C)/ nM on(C),where the horizontal morphisms are induced from ρ : N∆ (Assoc⊗ ) → N∆ (Fin∗ )and the vertical morphisms are obtained by regarding ∞-groups as ∞monoids.
Remark 3.2. In terminology of [9], a commutative ∞-monoid is called
an E∞ -monoid, and a commutative ∞-group is called an E∞ -group by[9, Proposition 1.1]. We adopt the terminology, and also describe thenot-necessary commutative case in the terminology since we apply the notnecessary commutative case to the group completion on the ∞-category S
of spaces in the following Lemma 3.9.
3.1. The group completion functor. We consider morphisms which go
to the other direction to the vertical morphisms in the diagram in Definition 3.1.
Definition 3.3. Let C be an ∞-category with finite products as in Definition 3.1. We say that a functor M on(C) → Gp(C) is group completion on C
if it is the left adjoint to the inclusion functor Gp(C) → M on(C) given in the
diagram in Definition 3.1. We also say that a functor nM on(C) → nGp(C)is group completion on C if it is the left adjoint to the inclusion functor
nGp(C) → nM on(C). By the universal property of adjoint functors, thegroup completion functor is uniquely determined up to equivalence [11, Remark 5.2.2.2].
Q

Definition 3.4. (i) Let Fun (C, D) denote an ∞-category of functors
which preserve the finite products.
Q(ii) We define the product on Fun (C, D) by taking the objectwise product
in D.

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M. OHARA

Lemma 3.5 ([9] Lemma
Q 1.6). Let C and D are ∞-categories with finite products. Let us take Fun (C, D) as an ∞-category of functors which preserve
the finite products. Then, we have
Q

Q

Q

Q

M on(Fun (C, D)) ≃ Fun (C, M on(D)),
andGp(Fun (C, D)) ≃ Fun (C, Gp(D)).

Definition 3.6 (cf. [11] 6.1.2.7). Let C be an ∞-topos. For an ∞-monoid
object G, there exists a colimit of the simplicial homotopy diagram by [12,Lemma 5.2.2.6]···// G × G × G/// G × G

/G

// 1.

where the face map is given by the multiplication on G and the degeneracy
map is given by the unit morphism. We define BG is a colimit of thesimplicial homotopy diagram.
Remark 3.7. Let S be the symmetric cartesian monoidal ∞-category of
spaces. In terminology of topology, an object M in S is grouplike if π0 M is agroup object in hS [12, Example 5.2.6.4]. A grouplike object in S is just an∞-group in S by Definition 3.1. If C is a symmetric cartesian monoidal model
category, an ∞-monoid object can be regarded as an associative algebra
object by [12, Proposition 2.4.2.5].
3.2. Characterization of the group completion on S. For a simplicial
set S and its vertexes x and y, recall that a simplicial set MapS (x, y) isdefined as follows. This construction is due to Joyal.
Definition 3.8. [cf. [11] 1.2.2.2, Corollary 4.2.1.8] Let Set∆ be the category1of simplicial sets. Let S be a simplicial set and S ∆ a simplicial set which
sends [n] to HomSet∆ (∆1 × ∆n , S), where ∆n is the n-simplex for n ≥ 0 and
HomSet∆ (−, −) is the hom-set of Set∆ . Let s, t : [0] → [1] be maps definedby s(0) = 0 and t(0) = 1.
(i) Take vertexes x, y ∈ S. The mapping space MapS (x, y) from x to y isdefined by the pullback
MapS (x, y)/ S ∆1
(s,t)



∗

(x,y)



/ S × S,

where the morphism (s, t) is induced by s and t, e.g., it sends HomSet∆ ({0}×∆n , S) to the n-simplices Sn of the first factor and HomSet∆ ({1} ×

A REPRESENTATION FOR ALGEBRAIC K-THEORY

9∆n , S) to the n-simplices Sn of the second factor, and the morphism(x, y) sends ∗ to (x, y) ∈ S × S.
(ii) If S is an ∞-category, MapS (x, y) becomes a Kan complex by [11,Proposition 1.2.2.3]. For an ∞-category S and objects x, y ∈ S, we saythat MapS (x, y) is the mapping space between x and y.
Let S be the ∞-category of spaces, and S∗ the pointed ∞-category of
spaces (cf. [12, Notation 5.2.6.11]).
For a space X, we define the loop space of X by Map(S 1 , X), where S 1
is a simplicial circle ∆1 /∆0 .
There is the group completion theorem on simplicial monoids due to
Quillen, which is given by taking classifying space and loop space. Bytaking the simplicial localization [7] of categories, we have an equivalencefrom the ∞-category of connected spaces to the ∞-category of not-neccesarycommutative ∞-groups in S obtained by restricting the adjoint functors
B : nM on(S) ⇄ S∗ : Ω, which includes the case arising from commutative
topological monoids [19]. There is also an equivalence from the ∞-category
Spcn to Gp(S) obtained by B and Ω. Moreover, we have the following
lemma.
Lemma 3.9. Let S be an ∞-category of spaces. Let i : nGp(S) → nM on(S)be the forgetfull functor. With the previous notations, we have the adjunctionΩB : nM on(S) ⇄ nGp(S) : i,where i is the forgetfull functor. In other words, ΩB is the group completionon S.
We can also take the left adjoint of the inclusion i : Gp(S) → M on(S)restricted in commutative case, so that it is also the restriction of the left
adjoint in Lemma 3.9. Note that, since sifted colimits commute with finite
products, ΩB commutes with finite products by the definition of B.
Qb3.3. Characterization of the group completion on
Fun ((CAlgG )op , S).
Qb by usWe characterize the group completion functor on Fun ((CAlgG )op , S)
ing Ω and B.
Definition 3.10. (i) In this paper, we will also denote by ΩB the left
adjoint functor given by ΩB : M on(S) ⇄ Gp(S) after Lemma 3.9.
(ii) We also denote by ΩB the functorb → Fun((CAlgG )op , S)b
Fun((CAlgG )op , M on(S))

which sends a presheaf F to a presheaf given by R 7→ ΩB(F (R)).

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M. OHARA

We will give an explicit description of the group completion functor for
the ∞-category of presheaves which preserves finite products.
Definition 3.11. We define a functor
Q
Q
Q
G op bbΩB : M on(Fun ((CAlg ) , S)) → Gp(Fun ((CAlgG )op , S))
by the restriction of ΩB, i.e., ΩB
commutative diagramQ

is the functor which satisfies the following

G op bM on(Fun((CAlg) , S))
O
Q

ΩB

b ΩB
M on(Fun ((CAlg ) , S))
G op

Q

b ,
/ Gp(Fun((CAlg G )op , S))O
Qb/ Gp(Fun ((CAlgG )op , S))

where the vertical morphisms are inclusions.
Q

Lemma 3.12.
gives the group completion on the ∞Q The functor ΩB
G op bcategory Fun ((CAlg ) , S) up to equivalences.
Q

b be an ∞-category of presheaves on CAlgG
Proof. Let Fun ((CAlgG )op , S)
which preserve the finite products. This ∞-category has the pointwise finite
products. By Lemma 3.5, we have the equivalences
Q
Q
G op bb
M on(Fun ((CAlg ) , S)) ≃ Fun ((CAlgG )op , M on(S))
and

Q

Q

b ≃ Fun ((CAlgG )op , Gp(S)).
b
Gp(Fun ((CAlg ) , S))
G op

Q

b is deTherefore, the group completion functor on Fun ((CAlgG )op , S)
termined by the point-wise values in S. The problem is reduced to thegroup completion functor nM on(S) → nGp(S) for S, which is ΩB by
Lemma 3.9.

3.4. Characterization of the group completion on ShvSb(CAlgG ).
Definition 3.13 (cf. [16] Proposition 1.15). We define the product on
ShvSb(CAlgG ) by the pointwise product, i.e., the product induced from theformation of product in S under the sheafification.
Since a finite limit of sheaves are again a sheaf, the Qpointwise productb in
on ShvSb(CAlgG ) is the restriction of the product on Fun ((CAlgG )op , S)
Definition 3.4.
GG
Lemma 3.14. We have ShvGp(S)b (CAlg ) ⊂ Gp(ShvSb(CAlg )).

Proof. By definition, the objectwise products of presheaves becomes pointwise products of sheaves after sheafification. Since the sheafification is left
exact [11, Definition 5.3.2.1], it commutes with the finite products. ThereGG
fore, we have ShvGp(S)b (CAlg ) ⊂ Gp(ShvSb(CAlg )).

A REPRESENTATION FOR ALGEBRAIC K-THEORY

11

Definition 3.15. We define the functorΩB G : M on(ShvSb(CAlgG )) → Gp(ShvSb(CAlgG )by the composition of the inclusionQbi′ : M on(ShvSb(CAlgG )) → M on(Fun ((CAlgG )op , S))
with
Q

Q

b → Gp(Fun ((CAlgG )op , S))
b → Shv b (CAlgG ),M on(Fun ((CAlg ) , S))
Gp(S)
G op

where the first functor
Q which is induced from the
Q pointwise
Q is the functor
G op bbgroup completion ΩB : M on(Fun ((CAlg ) , S)) → Gp(Fun ((CAlgG )op , S)),and the second functor is obtainedby the equivalence in Lemma 3.5 andQG˜ : Fun ((CAlg )op , Gp(S))
b → Shv b (CAlgG ). Notethe sheafification (−)
Gp(S)
G
G
that, we have ShvGp(S)b (CAlg ) ⊂ Gp(ShvSb(CAlg )) by Lemma 3.14.

Proposition 3.16. Let CAlgG be the ∞-category equipped with the Grothendiecktopology which is defined in Section 2. Then, ΩB G is the group completionon ShvSb(CAlgG ).
Proof. Let us take F and G from M on(ShvSb(CAlgG )) and Gp(ShvSb(CAlgG ))respectively. By the adjunction in Lemma 3.9 and the definition of ΩB G ,we have
Map(ΩB G F, G) ≃ Map((−)≃ ◦ ΩB

Q

◦ i′ (F ), G) ≃ Map(ΩBiF, iG) ≃ Map(F, iG),

G
where the left two mappingQspaces are taken in Gp(ShvSb(CAlg)), the thirdQ
G op bband the last are in Gp(Fun ((CAlg ) , S)) and M on(Fun ((CAlgG )op , S))
respectively. Since F and G are already sheaves and the sheafification commutes with finite products, we can regard the last mapping space as thattaken in M on(ShvSb(CAlgG )).


4. Classifying sheaf of GL and projective modules of finite
rank
4.1. The affine spectral scheme GLn .
Definition 4.1 ([12] Notation 3.1.3.8). Let R be an E∞ -ring, and CAlgR
the ∞-category of R-algebras.
(i) We define a functor SymR : ModR → CAlgR the left adjoint of theforgetful functor CAlgR → ModR which sends an R-algebra S to an
R-module S.

12

M. OHARA22(ii) For a free R-module R⊕n of rank n2 , π0 SymR R⊕n is isomorphic
to the polynomial ring (π0 R)[x11 , · · · , xnn ] over π0 R. Let us denote
2
SymR R⊕n by R{x11 , · · · , xnn }, where each xij is the representative of
the indeterminants of the polynomial ring over π0 R.
Definition 4.2. (i) For an object X in a symmetric monoidal ∞-category,an endomorphism object End(X) is an object equipped with the evaluation morphism e : End(X) ⊗ X → X which induces a weak homotopy
equivalence Map(Y ⊗ X, X) ≃ Map(Y, End(X)) for every Y .
(ii) Let R be an E∞ -ring. We have the endomorphism object of R⊕n in
ModR . Let us denote it by EndR (Rn ).
The following lemma is explained in [12, Remark 7.1.2.2].
Lemma 4.3. We have an isomorphismπ∗ EndR (R⊕n ) ∼= π∗ Map

ModR (R⊕n, R⊕n )

of ModR satisfies the universal property of endomor22
phism, we have an equivalence R⊕n ≃ EndR (R⊕n ). Since we have R⊕n ≃EndR (R⊕n ), we have an equivalence SymR EndR (R⊕n ) ≃ R{x11 , · · · , xnn }by Definition 4.1 (ii).
2
The object R⊕n

Definition 4.4. Let R be an E∞ -ring. Let us denote Spec Zar SymR EndR (R⊕n )by Mn,R in ShvS (CAlgZar ).
(i) Under the equivalence SymR EndR (R⊕n ) ≃ R{x11 , · · · , xnn }, we defineQthe element (det) ∈ π0 Mn,R by the determinant relation Στ ∈Sn sgn(τ ) xi,τ (i)of xij s.
(ii) We define an affine scheme GLn,R by inverting the determinant elementof Mn,R .
(iii) In the case that the base scheme R is the sphere spectrum S, we denote
Mn,R and GLn,S by Mn and GLn .
Let us take S as an R-algebra. We denote by Mn,R (S) and GLn,R (S) the
mapping spaces
MapCAlgR (SymR EndR (R⊕n ), S)
and
MapCAlgR (SymR EndR (R⊕n )[(det)−1 ], S)
respectively.
Remark 4.5. Since GLn is corepresented by an E∞ -ring, it is flat sheafby [16, VII, Theorem 5.15], so that it is already a Nisnevich sheaf. We also
use the notation GLn for the image in ShvS (CAlgN is ) under the sheafification.

A REPRESENTATION FOR ALGEBRAIC K-THEORY

13

4.2. The equivalence GLn (R) ≃ Aut(Rn ) as ∞-groups.
Definition 4.6. We define the space AutR (S n ) by the following pullbackof simplicial sets
AutR (S n )/ Map

ModR (Sn, Sn)π0

(π0 MapModR (S n , S n ))×/ π0 Map

ModR (Sn , S n ),where we regard (π0 MapModR (Rn , Rn ))× and π0 MapModR (Rn , Rn ) as constant simplicial sets and (π0 MapModR (Rn , Rn ))× is the invertible objects in
π0 MapModR (Rn , Rn ).
Remark 4.7. Note that the constant simplicial sets which appear in the
diagram of Definition 4.6 are also Kan complexes since the homotopy setof the mapping space is made from the same mapping space by replacing
1-simplices with isomorphism. (Note that any 1-simplex in a Kan complex
is invertible. )Proposition 4.8. For an R-algebra S, we have an equivalence GLn,R (S) ≃AutR (S n ) as ∞-groups in S, which is functorial with respect to R-algebra
S.
Proof. Note that the right vertical morphism π0 in the diagram in Definition 4.6 is a Kan fibration.
Since GLn,R (S) is formulated by the following pullback of simplicial sets
(4.1)

GLn,R (S)/ Mn,R (S)





π0

(π0 Mn,R (S))×/ π0 Mn,R (S),where we regard π0 Mn,R (S) as a constant simplicial set and (π0 Mn,R (S))×
is the invertible objects in π0 Mn,R (S), and by the coglueing lemma (cf. [11]A.2.4.3), it suffices to construct the two morphisms Mn,R (S) → MapModR (S n , S n )and π0 Mn,R (S) → π0 MapModR (S n , S n ) which preserve the multiplicationand show that the following diagram is commutative:(4.2)/ π0 Mn,R (S) o(π0 Mn,R (S))×
Mn,R (S)π0

(π0 MapModR (S n , S n ))×/ π0 Map

ModR (Sn, Sn) oπ0

MapModR (S n , S n ).

Note that objects in the diagram (4.1) are Kan complexes by the same reasonof Remark 4.7

14

M. OHARA

An element of Mn,R (S) is a morphism SymR EndR (R⊕n ) → S of Ralgebras for an R-algebra S. By the adjointness of SymR , the morphismof R-algebras is corresponding to a morphism of R-modules
2R⊕n ≃ EndR (R⊕n ) → S.
2

On the other hand, we have an equivalence R⊕n ≃ EndR (R⊕n ) obtained by
2
evaluating the each factor of Rn . Therefore, if we regard S ⊗R R⊕n as an
2
S-module, the above morphism corresponds to a morphism S ⊗R R⊕n → S
of S-modules. This gives an identification Mn,R (S) ≃ EndS (S ⊕n ) as Smodules.
To show that GLn,R (S) → AutR (S n ) is a morphism of ∞-groups, we fix
22
the choice of the second equivalence : MapModS (S ⊕n , S) ≃ MapModS (S, S)⊕n ≃MapModS (S n , S n ). Then, by the composition of S-module endomorphisms
on S ⊕n , EndS (S ⊕n ) has a canonical ∞-monoid structure for each S definedin Definition 3.1, the spectral scheme Mn,R is an ∞-monoid.
We can identify the discrete group π0 Mn,R (S)× with the class of π0 Salgebra morphisms π0 SymR EndR (R⊕n )[det−1 ] → π0 S. Here, det is the determinant element given by the determinant relation of xij s in π0 R[x11 , · · · , xnn ].
By ordinary theory of affine group schemes, these morphisms correspond to2(π0 EndS (S ⊕n ))× by a similar choice Hom(π0 S, π0 S)⊕n ∼= Hom(π0 S n , π0 S n )of isomorphism. Since 2-morphisms are invertible, this induces an equivalence between AutR (S n ) and SymR EndR (R⊕n )[det−1 ] → S by the above
pullback.
Since we fix the choice of equivalences in the proof, GLn,R is an ∞-groupscheme with respect to the monoid structure of MapModR (Rn , Rn ).

\
4.3. The Cat∞ -valued functor (nP roj). In this subsection, we show that\the functor (nP roj) : CAlgcn → Cat∞ defined by sending an E∞ -ring R toof projective modules ofthe maximal Kan complex of ∞-category Modnproj
R
rank n is a sheaf.
Definition 4.9 ([13] Definition 2.6.14). Let P be a property for objects(A, M ) in an ∞-category CAlgcn ×CAlg Mod. We say that P is local for theflat topology if the following conditions are satisfied:(i) Let f : A → B be a flat morphism of connective E∞ -rings, and M
an A-module. If (A, M ) has the property P , (B, B ⊗A M ) has theproperty P . If f is faithfully flat, the converse holds.
(ii) For any finite collection (Ai , Mi ) of the objects in CAlgcnQ×CAlgQMod
such that each (Ai , Mi ) has the property P , the product ( Ai , Mi )has the property P .

A REPRESENTATION FOR ALGEBRAIC K-THEORY

15

Theorem 4.10 ([16] Corollary 6.13, Lemma 6.17). A functor CAlgcn →\Cat∞ given by R 7→ ModR is a sheaf.
conSince (nP roj) is a subfunctor of the sheaf R 7→ ModR , if Modnproj
Rsisting of objects which satisfy the conditions in Definition 4.9 for eachconnective E∞ -ring, we note that (nP roj) is also a sheaf by the similarargument in [13, cf. Remark 1.5.3].
\
Lemma 4.11. Let (nP roj) : CAlgcn → Cat∞ be a functor given by R 7→
nproj
ModR . Then, the functor (nP roj) is a sheaf with respect to flat topology.
\
Proof. Since a functor CAlgcn → Cat∞ given by R 7→ ModR is already asheaf By Theorem 4.10, it suffices to check that the projective modules ofrank n satisfy the condition of Definition 4.9 (i) and (ii).
By [13, Proposition 2.6.15 (1), (6), (9)], the condition of finitely generated
projective is flat local property. Since the tensor product preserves rank, theprojective modules of rank n satisfy the condition of Definition 4.9 (i).
We will check the condition (ii) of Definition 4.9. Assume that (Ai , Mi ) isa pair such that Ai is connective E∞ -ring
Q and Mi is a projective
Q Ai -moduleof rank n for 1 ≤ i ≤ m. To show that i Mi is a projective i Ai -moduleofthere exists
set {ga }a of objects in
Q rank n, it suffices toQshow thatQ a finite−1−1
i Ai such that eachi Mi [ga ] is a freei Ai [ga ]-module of rank n. We
choose such {ga }a as follows.
For each i, we take fi1 , · · · , fik ∈ π0 Ai such that (Mi )[fil−1 ] is a freeQ
Ai [fil−1 ]-module of rank n for 1 ≤ l ≤ k.QSince i Ai → Ai is flat and theessential image
Q is generated by the form (Ai ⊗A Aj )⊗Aj M and Ai ⊗A Aj ≃Q
0, Aj ⊗ i Ai i Mi ≃ Mj .
Q
Q
Q
From this, we have the equivalence i Ai [fil−1 ]⊗Qi Ai i Mi ≃ i Mi [fil−1 ],Q
Qwhere i Mi [fil−1 ] is regarded as a i Ai [fil−1 ]-module and is free of rankn.
Corollary 4.12. (nP roj) is a sheaf in Nisnevich topology and Zariski topology.
Proof. It follows from Lemma 2.7 and Lemma 4.11.



Definition 4.13 ([11] Theorem 3.1.5.1). Let Sb be the ∞-category of spaces\
and Cat∞ the ∞-category of ∞-categories defined in Definition 2.3 after
enlarging the universe respectively. We regard the Kan complexes as ∞\categories, so that we have an inclusion functor i : Sb → Cat∞.
Since the inclusion preserves small colimits, by [11, Corollary 5.5.2.9],
there is an adjunction
≃
\
(4.3)
i : Sb ⇄ Cat∞ : (−) .

16

M. OHARA

Note that the right adjoint (−)≃ is given by taking the maximal ∞groupoid.
\
Lemma 4.14. The right adjoint in (4.3) induces a functor from Cat∞bvalued sheaves to S-valued sheaves.

Proof. Since (−)≃ is a right adjoint, it preserves limits. Especially, it preserves the limits of simplicial diagrams in Proposition 2.6. Therefore, (−)≃
b\sends Cat∞ -valued sheaves to S-valued sheaves.

ree ≃) . Recall that the
4.4. Comparison between B(GLn (R)) to (Modnf
R
notion of classifying object from Definition 3.6. We denote by B G the classifying functor on ShvSb(CAlgG ). Acording to Remark 4.5, we have GLn in
ShvS (CAlgG ).

Definition 4.15. (i) We use the notation B(GLn (R)) for the classifying
space of the value GLn (R) in S, and (B G GLn )(R) for the value at Rof the classifying sheaf of GLn in ShvS (CAlgG ).
ree ≃cn→ Sb be a functor which sends R to the(ii) We let (Modnf(−) ) : CAlgree ≃) of free R-modules of rank n.
∞-groupoid (Modnf
R

Note that GLn (R) ≃ AutR (R⊕n ), and B(GLn (R)) is not equal to (B G GLn )(R).
Remark 4.16. Apparently, the classifying sheaf B G GLn of GLn in ShvS (CAlgG )in Definition 4.15 depends the Grothendieck topology on CAlgG , but consequently, we show that it is equivalent to the flat sheaf (nP roj) in Proposition 4.21 below, B G GLn in Zariski topology is equivalent to that in Nisnevich
topology.
ree ≃) as a simLemma 4.17 (cf. [1] Section B.3). Let us regard (ModnfR
nf ree ≃plicial set, and take a vertex x ∈ (ModR ) . Then, we have a morphism
ree ≃) of simplicial sets.
B(AutR (x)) → (Modnf
R

Proof. Since we have an equivalence
AutR (x) → MapModnf ree (x, x)≃ ,Rwhich is induced from the inclusion in Definition 4.6. We note that this morphism factors through the equivalence Map(Modnf ree )≃ (x, x) ≃ MapModnf ree (x, x)≃
R

R

ree ≃
and Map(Modnf ree )≃ (x, x) is 1-simplices of (Modnf) by taking x = y andRRidentifying the image in S × S with S in Definition 3.8. Since the morphism
AutR (x) → Map(Modnf ree )≃ (x, x) is obtained by the inclusion, it preserves
R∞-groupoid structure, so that it preserves the monoid structure. Sinceree ≃) are connected, the induced morphism
both B(AutR (x)) and (ModnfR
nf ree ≃
B(AutR (x)) → (ModR ) is an equivalence.

A REPRESENTATION FOR ALGEBRAIC K-THEORY

17

Let B(GLn (−)) : CAlgcn → Sb be a functor which sends R to the space
B(GLn (R)) defined in Definition 4.15. We will compare the functor B(GLn (−))
with a functor given in Definition 4.15(ii). The following proposition is obtained by the similar arguments in the case [1, Section b.3] of rank 1.
Proposition 4.18. The functor B(GLn (−)) is equivalent to the functor
ree ≃(Modnf(−) ) defined in Definition 4.15(ii).
Proof. By taking x = y = Rn in Lemma 4.17, we have a morphism fromree ≃) for each R ∈ CAlgcn .
B(GLn (R)) → (ModnfR
By composing a morphism B(GLn (R)) ≃ BAutS (Rn ) which we obtainedree ≃) obtained by Lemma 4.17,
by Proposition 4.8 with B(AutS (R)) → (ModnfR
nf ree ≃we have a morphism B(GLn (R)) → (ModR ) . It suffices to show that aree ≃) is an equivalence and functorial withmorphism B(GLn (R)) → (Modnf
R
respect to R.
By construction, we identify B(GLn (R)) with a full ∞-subgroupoid ofree ≃) such that the object is only R⊕n and the class of morphisms(ModnfR
is identified with GLn (R) ≃ AutS (Rn ) by Proposition 4.8, which is fully
faithfull functorial assignment with respect to R.
ree ≃)We show that it is essentially surjective. It suffices to show that (ModnfR
nf reeis connected as a simplicial set. This is obvious since, for M ∈ ModR ,we have M ≃ R⊕n .

4.5. A natural transformation from B(GLn (−)) to (B G GLn )(−). We
construct a natural transformation from B(GLn (−)) to (B G GLn )(−) byusing the comparison between the sheaf of projective modules of rank n andthe sheafification of the functor of free modules of rank n.
bFor each R, we have the following colimit in S:// ∗GLn (R)/ B(GLn (R)).

Note that the functor corresponds to this cofiber is not a sheaf since thesecond condition of [13, Definition 2.6.14 (1)] fails. According to the notation^in Definition 2.4, we write B(GLn (−)) for the sheafification of B(GLn (−)).
Remark 4.19. In the stable ∞-category of spectra, taking B is equivalent
to taking Σ by stability. So, although the notation Σ of the suspention canbe used, we use the notation B in the colimit of the above coequalizer since
objects appearing in the coequalizer are grouplike and can be regarded asconnective spectra via the equivalence from Gp(S) to Spcn which preserves
sifted colimits.
Lemma 4.20. We have the following cofiber sequence
GLn

//

∗

^
/ B(GLn (−))

18

M. OHARA

in ShvSb(CAlgG ).

Proof. Since GLn is corepresented by an E∞ -ring, it is flat sheaf by [16,VII, Theorem 5.15], so that it is a Zariski (resp. Nisnevich) sheaf. Since
sheafification is left adjoint, it commutes with the cofiber sequence.
For each A → R in CAlgcn , the natural inclusions induce the commutative
diagramree/ Modnproj
Modnf
A
A



/ Modnproj ,

ree
Modnf
R

R

\
in Cat∞ , so that we have a natural transformation
(4.4)

ree
→ Modnproj
Modnf
(−) .
(−)

^
By the adjointness of sheafification, we have a morphism B(GLn (−)) →≃b(nP roj) of S-valued sheaves.
4.6. An equivalence B G GLn ≃ (nP roj)≃ . The weak homotopy eqivalence in the following statement is written in [2, Proposition 5.1]. However,
we show it by a fortiori argument using morphisms induced by previous
subsection.
Proposition 4.21. The morphism (4.4) in the previous subsection gives anequivalence B G GLn ≃ (nP roj)≃ in ShvSb(CAlgG ).
≃^Proof. From adjointness of a morphism f(−) : B(GLn (−)) → (nP roj) ofb
S-valuedsheaves, we have the following homotopy commutative diagram ofb
S-valuedpresheaves:^B(GL
7 n (−))❖♥♥

L ♥♥♥♥

♥
♥♥♥
♥
♥
♥

B(GLn (−))

i

❖❖❖
❖❖❖f
❖❖❖
❖❖'
/ (nP roj)≃ ,where L is the morphism associated to the sheafification and i is induced
from the inclusion.
Let us denote the limits of the following simplicial diagrams by B(GLn (A))R:///B(GLn (A ⊗R A))
B(GLn (A))// · · · ,where R → A is a faithfully flat morphism. By Remark 2.5, the sheafification^B(GLn (−)) is described by the term of B(GLn (−)), i.e., it is the colimit ofthose B(GLn (A))R taken over the every covering sieve R → A .

A REPRESENTATION FOR ALGEBRAIC K-THEORY

19≃^
For fR : B(GLn (R)) → (nP roj) (R) and an object Q, i.e., a vertex in(nP roj)≃ (R), let (Q)≃ denote the full ∞-subgroupoid spanned by Q. Since
ShvSb(CAlgG ) is locally presentable, there exists a corresponding combinatorial simplicial model category up to Quillen equivalences. We take it under^
the consideration, and construct hR,Q : (Q)≃ → B(GLn (R)) as follows.
We take a Zariski local A of R such that an object Q ∈ (nP roj)≃ (R) is
trivialized on A. Then, we naturally identified the ∞-groupoid spanned by^
Q ⊗ A with B(GLn (A)), so that we have an object in B(GLn (R)). This is
the assignment under the morphisms of ∞-groupoids. By f ◦ L ≃ i in theabove diagram, we have fR ◦ hR,Q is homotopic to identity.
^
Conversely, for an object P in B(GLn (R)), we take its value on the covering sieve R → R under the morphisms associated to the colimit, and denoteby P ′ . Then, by an extension of the coefficients of P ′ gives an object in^B(GLn (R)), which is equivalent to P by the construction. This shows that
hR,f (P ) ◦ fR is homotopic to identity.
Since fR is obviously a Kan fibration and, by the above arguments, any
fiber of fR is contractible. By [11, Lemma 4.1.3.2, Corollary 4.1.2.6], it is aweak homotopy equivalence.


4.7. Zariski connected E∞ -rings. Next, we consider the condition on aconnective E∞ -ring R such that any finitely generated projective R-module
has finite constant rank.
Definition 4.22. (i) We say that an E∞ -ring is Zariski non-connected if
there exists those objects f, g ∈ π0 R such that R[f −1 ] ⊗R R[g−1 ] ≃ 0
and R ≃ R[f −1 ] × R[g−1 ].
(ii) We say that an E∞ -ring is Zariski connected if it is not Zariski nonconnected.
Lemma 4.23. The following conditions are equivalent:(i) R be an E∞ -ring such that π0 R has no non-trivial idempotent element.
(ii) R is Zariski connected.
has finite constant rank.
(iii) Any P ∈ Modproj
R
Proof. For proving (iii) from (i), let R be an E∞ -ring such that π0 R has
no non-trivial idempotent, and P ∈ Modproj
R . Note that π0 P is a finitely
generated projective π0 R-module [12, Remark 7.2.2.20]. Then, there exists
f1 , · · · , fm which generate the unit ideal of π0 R such that each π0 P [fi−1 ] isfree of finite rank ni over π0 R[fi−1 ]. For an ordinary commutative ring π0 R,
in this the condition (i) is equivalent to that ni s are constant, and let us denote it by n. Since finitely generated free π0 R[fi−1 ]-modules π0 P [fi−1 ] can belifted by finitely generated R[fi−1 ]-modules P [fi−1 ] and a morphism between

20

M. OHARA

flat R-modules is an equivalence if and only if it induces an isomorphism on
π0 [12, Lemma 7.2.2.17], locally P has a constant rank.
Conversely, if π0 R has a non-trivial idempotent element, we have that therank ni are not constant. In this case, the above argument shows that (iii)
implies (i).
We show that (i) implies (ii). Since π0 preserves finite products, if R is
Zariski non-connected, by passing to π0 and applying the ordinary commutative ring theory, π0 R has a non-trivial idempotent. Conversely, if π0 R has
a non-trivial idempotent e, it also a non-trivial idempotent of π∗ R. Since
the localization of an E∞ -ring with one element commutes with π∗ , we haveπ∗ (R[e−1 ]) ∼= (π∗ R)[(1 − e)−1 ]. Since π∗= (π∗ R)[e−1 ] and π∗ (R[(1 − e)−1 ]) ∼commutes with finite products, by taking π∗ of R[e−1 ] × R[(1 − e)−1 ], we
conclude that R is equivalent to R[e−1 ] × R[(1 − e)−1 ], so that R is Zariski
non-connected. Thus, (ii) implies (i).

. Now, we will prove the4.8. Comparison between B G GL and Mod∞proj
Rmain proposition in this section by applying the following proposition.
Proposition 4.24 ([16] Lemma 3.21). Let C be an ∞-topos, I an index
set and`{Xi }i∈I a collection of objects in C. For every subset J ⊂ I, let
XJ ≃ i∈J Xi . Let C ∈ C be an object such that every covering of C has afinite subcovering [16, Definition 3.1]. Then, the canonical morphism
colim MapC (C, XJ ) → MapC (C, XI )J⊂I

induces a homotopy equivalence. Here, the colimit in the left hand side is
run through the all finite subsets J ⊂ I.
Lemma 4.25. Let I ⊂ N be a finite index set. If R is Zariski connected,
the canonical morphismaa
B G GLi ),MapShv b(CAlgG ) (Spec G R, B G GLi ) → MapShv b(CAlgG ) (Spec G R,
S

S

i∈I

i∈I

is an equivalence.
Proof. To check that the canonical morphism is an equivalence,to`it suffices
GG
show that, for any morphism φ in MapShv b(CAlgG ) (Spec R, i∈I B GLi ),S
there exists an index j ∈ N which is uniquely determined by φ, the morphism
φ has the unique factorizationaSpec G R → B G GLj →B G GLi .
i∈I

A REPRESENTATION FOR ALGEBRAIC K-THEORY

21`
Since the sheafification commutes with coproducts, i∈I B G GLi is equiv`
alent to the sheafification of the presheaf given by R 7→ i∈I (B G GLi (R)).
(Note that the coproduct of sheaves are the sheafification of the objectwise
coproduct as presheaves, and an functor obtained by objectwise
coproduct`of sheaves is not always a sheaf.) Therefore, an object of ( i∈I B G GLi )(R)
can be represented by a finitely generated projective R-module whose local
ranks are contained in I.
be an ∞-category of finitely generated projective modLet IModproj
Rules whose local rank is contained in I. Since R is Zariski connected, by`roj. Therefore,≃ {i∈I} ModiP
Lemma 4.23(iii), we deduce that IModproj
RR`
for any object of ( i∈I B G GLi )(R), there exists a unique j ∈ I such thatthe object can be represented by a finitely generated projective R-moduleof rank j.
By applying` the Yoneda embedding [11, Section 5.1.3], we have that an
object of ( i∈I B G GLi )(R) represented by a finitely generated projective`
R-module of rank j corresponds to the morphism Spec G R → i∈I B G GLi
which has the unique factorizationaSpec G R → B G GLj →B G GLi .
i∈I
We consider a decomposition of an E∞ -ring R by using the idempotent
element in π0 R.
Proposition 4.26. Let Spec G R : (CAlgG )op → Sb be a spectral scheme
and Spec G R → B G GLn a morphism in ShvSb(CAlgG ). Let B G GL be a``sheaf n∈N B G GLn . For I ⊂ J ⊂ N, we have the system i∈I B G GLi →`Gj∈J B GLj given by inclusions. Then there is an equivalence of ∞-groupoids;)≃ .
MapShv b(CAlgG ) (Spec G R, B G GL) ≃ (Mod∞proj
R
S

Proof. By Proposition 4.24, we havea
B G GLi ) ≃ MapShv b(CAlgG ) (Spec G R, B G GL).
colim MapShv b(CAlgG ) (Spec G R,
I⊂N

S

S

i∈I

By decomposing R with the Zariski connected E∞ -rings (given by thecorresponding irreducible decomposition on π0 ) and applying Lemma 4.25,
for I = {1, · · · , n}, we have the equivalenceaa
B G GLi )MapShv b(CAlgG ) (Spec G R, B G GLi ) ≃ colim MapShv b(CAlgG ) (Spec G R,
colim
I⊂N

S

i∈I

I⊂N

S

i∈I

22

M. OHARA

By Proposition 4.21, we identify the left hand side with
roj ≃
) .
)≃ ≃ (Mod∞P
colim(Modiproj
R
R
i∈N


5. K-theory of Mod∞proj
R
Definition 5.1 (cf. [4] 1.2, [6] Definition 2.1). Let C be a pointed ∞category. A class of w∞ -cofibrations is a class of morphisms in C which
satisfies the following conditions:(i) ∗ → X is a w∞ -cofibration for any object X,(ii) The class of w∞ -cofibrations includes weak equivalences,(iii) Any composition of w∞ -cofibrations is a w∞ -cofibration,(iv) For a w∞ -cofibration X → Y and a morphism X → Z, there exists a
pushout square
/Y
X


Z

f


/ W,

in which the morphism f is a w∞ -cofibration.
We call such a pair of C and a w∞ -cofibrations an ∞-category with w∞ cofibrations.
In terminology of [5], a w∞ -cofibration is called a cofibration. In terminology of [3], a w∞ -cofibration is called an igressive morphism.
Remark 5.2. We say that a w∞ -cofibration is a split w∞ -cofibration if itis split as a morphism.
to be an ∞-category with w∞ Definition 5.3. We make into Mod∞proj
Rcofibrations as follows.
to be a w∞ -cofibration if
Declare a morphism f : P1 → P2 in Mod∞proj
R.
it is a morphism in ModR and the cofiber of f is an object in Mod∞proj
R
∞proj
proj
) is an ∞-category with w∞ -cofibrations.
Then, ModR (resp. ModR
is always split, e.g., it isLemma 5.4. (i) A w∞ -cofibration in Mod∞projR∞a split w -cofibration in Remark 5.2.
(ii) Let R1 → R2 be a morphism of connective E∞ -rings. Then, a functor
given by the extension of coefficients is an→ Mod∞proj
Mod∞proj
R2
R1
exact functor in the sence of K-theory.
Proof. By [12, Proposition 7.2.2.6 (5)], a cofiber sequence of finitely generated projective modules is always split up to homotopy. Thus, (i) holds.

A REPRESENTATION FOR ALGEBRAIC K-THEORY

23

Since the tensor product commutes with the cofiber sequences and preserves rank, the extension of coefficients is exact.
Remark 5.5. In this paper, we consider the connective K-theory. We canregard the K-theory spectrum as a K-theory space.
Let C be a pointed additive ∞-category.
Recall that the maximal ∞-groupoid C ≃ of the additive ∞-category C
inherits the structure of ∞-monoid space by applying [9, Corollary 6.6] tothe functor Cat∞ → S given by taking maximal ∞-groupoid. Especially,)≃ is a commutative ∞-monoid.
the additive ∞-category (Mod∞proj
R
Note that the composition of the two lax symmetric monoidal functors
M on(Cat∞ ) → M on(S) and the group completion on the ∞-category of
spaces M on(S) → Gp(S) in Lemma 3.9 is factored by the group completionof Cat∞ with the natural inclusion Gp(S) → Gp(Cat∞ ) by [9, Proposition
8.14].
There is an ∞-version of the additivity theorem [21, Theorem 1.8.7] given
by [8]. We state the theorem as in Lurie’s unpublished note [17].
Recall the notion of a split w∞ -cofibration from Remark 5.2. We willwhose w∞ -cofibrations defined inapply the following theorem for Mod∞proj
R
Definition 5.3 are split w∞ -cofibrations by Lemma 5.4.
Theorem 5.6 ([8], cf. [17] Theorem 10). Let C be a pointed additive ∞category with split w∞ -cofibrations, and C ≃ be the maximal ∞-groupoid ofC as in (4.3). Let K(C) be the algebraic K-theory and ΩB(C ≃ ) the groupcompletion of C ≃ .
Then, there is an equivalence(5.1)ΩB(C ≃ ) → K(C)of spases. Here, Ω and B are defined in Section 4.
).
)≃ ) ≃ K(Mod∞proj
Corollary 5.7. We have ΩB((Mod∞proj
RR(resp. Modproj
Proof. The ∞-category Mod∞proj
R ) is additive, pointed by 0,R∞with split w -cofibrations. Thus, Theorem 5.6 can be applied.

6. Proof of Theorem 1.1
Let us keep the notation explained in the previous sections. Now we prove
Theorem 1.1.
`
Let B G GL = n∈N B G GLn . Note that, the sheafification preserves coproducts and finite products. By Proposition 4.26, we can regard B G GL as
is a symmet)≃ . Since, for an E∞ -ring R, Mod∞proj
the sheaf (Mod∞projR(−)
ric monoidal ∞-category with direct sum and with tensor product which
preserves direct sum in each variable separately. Moreover, since the direct

24

M. OHARA

sum commutes with the sheafification, the juxtaposition of B G GL arises
from that of B(GL(−)) which is obtained in objectwise manner. So, we
obtain the following proposition.
`
Proposition 6.1. Let B G GL = n∈N B G GLn . Then, B G GL becomes acommutative ∞-monoid.
Theorem 6.2. Let CAlgG be the ∞-category (CAlgcn )op equippedwith Zariski`
or Nisnevich topology defined in Section 2. Let B G GL = n∈N B G GLn andΩB G (B G GL) the group completion as a sheaf on CAlgG . Note that B G GL
becomes a commutative ∞-monoid as in Proposition 6.1.
There is an equivalence of ∞-groupoids:e G (Mod∞proj ),MapShv b(CAlgG ) (Spec G R, ΩB G (B G GL)) ≃ K
RSe G is the sheafification of K defined by (1.1).
where K
Proof. By Proposition 4.21, we have an equivalence)≃ ),ΩB(B G GL(R)) ≃ ΩB((Mod∞proj
Rwhere (−)≃ denotes the maximal ∞-groupoid (4.3). Since all w∞ -cofibrations
is additive, by
are split and the homotopy category of Modproj
in Mod∞proj
RR∞proj ≃) ) is equivalent to the algeCorollary 5.7, we obtain that ΩB((ModR∞proj) given by S• construction.
braic K-theory K(ModR
On the other hand, we have an equivalence induced by Yoneda embedding
MapShv b(CAlgG ) (Spec G R, ΩB G (B G GL)) ≃ (ΩB G B G GL)(R).
S

By virtue of Proposition 6.1, we can apply Proposition 3.16 to the commutative ∞-monoid sheaf B G GL. The sheafification of the objectwise groupcompletion functor R 7→ ΩB(B G GL(R)) is equivalent to the group completion of the sheaf given by the assignment R 7→ (ΩB G B G GL)(R). Thus, wehave an equivalence (ΩB G B G GL)(R) ≃ ΩB(B G GL(R)) after the sheafification.
Remark 6.3. Since Yoneda embedding and the sheafification preserve finite products, and since the functor (−)≃ taking maximal ∞-groupoid islax monoidal [9, Remark 8.7], the equivalence in Theorem 6.2 preservescommutative ∞-group structure.
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Mariko Ohara
Department of Mathematical Sciences
Shinshu University
Matsumoto, 390-8621 Japane-mail address: ohara12m@shinshu-u.ac.jp(Received October 4, 2017 )(Accepted January 30, 2019 )