In this paper, we study p-adic Hodge theory for non-commutative algebraic varieties. Firstly, we propose a conjecture that for K a complete discretely valued nonarchimedean extension K of ℚp with perfect residue field k and 𝒯 an 𝒪K-linear idempotent-complete, small smooth proper stable ∞-category, there exists a Bcrys-coefficient isomorphism preserving additional structures between the K(1)-local K-theory on the generic fiber of 𝒯 and the topological periodic cyclic homology on the special fiber. This conjecture can be regarded as a non-commutative analog of the crystalline comparison theorem. We then proceed to prove the following results: the topological negative cyclic homology πiTC−(𝒯/𝕊[z];ℤp) admits a Breuil–Kisin module structure, and the non-commutative analog of Bhatt–Morrow–Scholze’s comparison theorems holds. Additionally, we demonstrate that the ℤp[GK]-module obtained from the topological negative cyclic homology is a ℤp-lattice of a crystalline representation. Finally, we show that when the generic fiber of 𝒯 admits a geometric realization in the sense of Orlov, the non-commutative analog of the crystalline comparison theorem proposed by the author holds.
Mathematics Subject Classification 202014F30 (primary)14A22 (secondary)Keywordsp-adic representationnon-commutative algebraic geometryJSPS KAKENHI22K13898JSPS KAKENHI25KJ0210This work was supported by JSPS KAKENHI Grant Numbers 22K13898 and 25KJ0210.Introduction
The aim of this paper is to study p-adic Hodge theory for non-commutative algebraic varieties. Specifically, we propose a conjecture and prove several results related to the relationship between K(1)-local K-theory and topological periodic homology in this non-commutative setting.
Notation 1.1.
Fix a prime p. Let K be a complete discretely valued nonarchimedean extension K of ℚp with perfect residue field k. Here, 𝒪K is the ring of integers of K and π∈𝒪K is a uniformizer. We write 𝒞 to denote the completion K¯^ of K¯ endowed with its unique absolute value extending the given absolute value on K, let W be the Witt ring of k, and let K0 be the fraction field of W. Let 𝔪 be the maximal ideal of 𝒪K. For a spectrum S, let LK(1)S be the Bousfield localization of complex K-theory at prime p.
Cohomology theories such as de Rham cohomology RΓdR(−/K), Hodge cohomology RΓZar(−,Ω−/K∗), l-adic cohomology RΓét(−,ℤl) and crystalline cohomology RΓcry(−/W(k)) are important tools in the study of algebraic geometry and arithmetic geometry. On the other hand, homological invariants such as (topological) periodic homology, (topological) cyclic homology and K-theory play an important role in the study of C∗-algebra, Lie algebra and non-commutative geometry. There are deep and subtle links between cohomology invariants and homological invariants. One of the most well-known examples is the Atiyah–Hirzebruch spectral sequence. For a finite-dimensional CW-complex M, Atiyah–Hirzebruch [4] prove that there is a spectral sequence
E2i,j={HSingi(M,ℤ),j even0,j odd⟹Kj−itop(M).
The Thomason spectral sequence is an arithmetic-geometrical analog of the Atiyah–Hirzebruch spectral sequence. For a smooth variety X over a field of characteristic 0, Thomason [36] shows that there exists a spectral sequence
E2i,j={Héti(X,ℤp(l)),j=2l0,j odd⟹πj−iLK(1)K(X),
and this spectral sequence degenerates after tensoring ℚp. Besides, Hesselholt [21] shows a close relation between p-adic cohomology theory and topological cyclic homology.
Bondal–Kapranov [10], Orlov [34] and Kontsevich [27] introduce non-commutative algebraic geometry in which a dg-category (or a stable ∞-category) is studied as a non-commutative space. Nowadays, non-commutative algebraic geometry plays an important role in research on mirror symmetry, mathematical physics and algebraic geometry. Besides, homological invariants are, in general, well defined for dg-categories and stable ∞-categories. It has been known that some comparison theorems between cohomology groups can be formulated naturally for dg-categories and stable ∞-categories: instead of cohomology theories, one can consider homological invariants. For a smooth proper dg-category 𝒯 over ℂ, Kaledin [25] proves that there is an isomorphism HPn(𝒯/ℂ)≃⨁i∈ℤHHn+2i(𝒯/ℂ), which has been conjectured by Kontsevich–Soibelman [28] and can be regarded as a non-commutative Hodge decomposition via Connes [12], Feigin–Tsygan [18] and Hochschild–Kostant–Rosenberg [24]. Besides, Blanc [9] conjectured that there is an equivalence Chtop∧𝕊Hℂ:Ktop(𝒯)∧𝕊Hℂ→HP(𝒯/ℂ), which can be regarded as the non-commutative de Rham comparison theorem, and in some cases, the equivalence is proved by A. A. Khan [26]. For a smooth proper stable ∞-category 𝒯 over W, Scholze proves that there is an isomorphism of W-modules πnTP(𝒯k;ℤp)≃πiHP(𝒯/W) (it has been unpublished yet), and Petrov–Vologodsky obtain the same result for any stable ∞-category [35].
In the study of p-adic cohomology theories, crystalline comparison theory [16] states that for a smooth proper variety X over 𝒪K, the p-adic étale cohomology Héti(X𝒞,ℚp)⊗ℚpBcrys is isomorphic to Hcryi(Xk/W)⊗WBcrys, and the isomorphism is compatible with GK-action, Frobenius endomorphism and filtration. We study a non-commutative version of the crystalline comparison theorem. For a commutative ring R, we will refer to R-linear idempotent-complete, small stable ∞-categories simply as R-linear categories. For an 𝒪K-linear category 𝒯, GK acts continuously on ℤp-module πiLK(1)K(𝒯𝒞), and there is a Frobenius operator Fr:πiTP(𝒯k;ℤp)[1p]≃canπiTC−(𝒯k;ℤp)[1p]→𝜑πiTP(𝒯k;ℤp)[1p] (see [2]). Inspired by Petrov and Vologodsky’s work on non-commutative crystalline cohomology theory [35], and the study of motivic filtration of the K(1)-local K-theory [36] and the topological periodic homology [8], we predict the following conjecture.
Conjecture 1.2 (Non-commutative version of crystalline comparison theorem [<xref rid="j_JNCG661_ref_016" ref-type="bibr">16</xref>, <xref rid="j_JNCG661_ref_037" ref-type="bibr">37</xref>]).
Let 𝒯 be a smooth proper 𝒪K-linear category. Then there is an isomorphism of Bcrys-module
πiTP(𝒯k;ℤp)⊗WBcrys≃πiLK(1)K(𝒯𝒞)⊗ℤpBcrys
which is compatible with GK-action and Frobenius endomorphism. In particular, the p-adic representation πiLK(1)K(𝒯𝒞)⊗ℤpℚp is crystalline.
Remark 1.3.
Assume 𝒪K=W. Then the isomorphism of W-modules
πnTP(𝒯k;ℤp)→∼πnHP(𝒯/W)
has been proved independently by Scholze (unpublished) and by Petrov–Vologodsky [35] for p>2 2\)]]>. For all primes p and for smooth proper dg-categories, this result has since been established by Devalapurkar–Raksit [13] and Mao [32]. Consequently, the non-commutative Hodge–de Rham filtration on πnHP(𝒯/W) induces a natural filtration on TP(𝒯k;ℤp). It is further expected that for a general complete DVR 𝒪K of mixed characteristic, the base change
πnTP(𝒯k;ℤp)⊗WK
inherits a filtration arising from the non-commutative Hodge–de Rham filtration on πnHP(𝒯/𝒪K). Moreover, we expect the isomorphism in Conjecture 1.2 to be compatible with these filtrations.
For a smooth proper variety X over 𝒪K, by [36] one has a GK-equivariant isomorphism πiLK(1)K(𝒯𝒞)⊗ℤpℚp≃⨁n∈ℤHéti+2n(X𝒞,ℤp(n))⊗ℤpℚp, and similarly by [7] one has an isomorphism πiTP(𝒯k;ℤp)[1p]≃⨁n∈ℤHcryi+2n(Xk/W[1p])(n) of isocrystals, thus the conjecture holds for 𝒯=perf(X) via crystalline comparison theorem [37]. In this paper, we approach this conjecture via K-theoretical version of Bhatt–Morrow–Scholze’s comparison theorem [8]. We will use the language of stable ∞-categories, following Lurie [31].
Definition 1.4.
For an 𝔼∞-ring R, we let Cat∞perf(R) denote the ∞-category of R-linear categories, where the morphisms are exact functors.
Definition 1.5.
For an 𝔼∞-ring R, we let Cat∞,satperf(R) denote the ∞-category of smooth proper R-linear categories, where the morphisms are exact functors.
Definition 1.6
(See also [34, Section 4.1]). For an 𝔼∞-ring R, an R-linear category 𝒯admits a geometric realization if there is a derived scheme X over R such that the truncation π0(X) is separated scheme of finite type over π0(R) and there is a fully faithful admissible inclusion 𝒯⊂perf(X) of R-linear categories (see [1, Definition 3.1]).
In many cases, a stable ∞-category is known to admit a geometric realization. For example, the derived Fukaya category of a symplectic manifold is known or expected to admit a geometric realization from the study of mirror symmetry. Besides, Lunts–Bergh–Schnürer [5] proved that the stable infinity category of perfect complexes on a smooth proper Deligne–Mumford admits a geometric realization.
Remark 1.7.
Assume R is an algebraically closed field of characteristic 0. In [34, Question 4.4], Orlov asked if there exist R-linear idempotent-complete, small smooth proper stable ∞-categories which do not admit a geometric realization. This is still an important open problem.
Remark 1.8.
If a smooth proper R-linear category 𝒯 admits a geometric realization 𝒯↪perf(X), the dual 𝒯op∈Cat∞,satperf also admits a geometric realization 𝒯op↪perf(X)op=perf(X).
First, we fix a sequence of elements ζn∈K¯ inductively such that ζ0= 1 and (ζn+1)p=ζn. Given this sequence, let ε=(ζ0,ζ1,ζ2,…)∈lim←Frob𝒪ℂ/p be the corresponding element in the tilt, and let [ε]∈𝔸inf be its Teichmüller lift. Let 𝔖=W⟦z⟧, and let θ~:𝔖→𝒪K be the usual map whose kernel is generated by Eisenstein polynomial E of π. Let ϕ:𝔖→Ainf be the W-linear map that sends z to [π♭]. We write ξ=ϕ(E). We write μ=[ε]−1. Let φ:𝔖→𝔖 be a Frobenius endomorphism which is Frobenius on W and sends z to zp. We prove a non-commutative version of Bhatt–Morrow–Scholze’s Breuil–Kisin cohomology theory RΓ𝔖(−) [8].
Theorem 1.9
(Theorem 2.17, non-commutative version of [8]). Let 𝒯 be an 𝒪K-linear smooth proper category. Then there is a natural number n such that the following holds:
For any
i≥n,
πiTC−(𝒯/𝕊[z];ℤp)
has a natural structure of a Breuil–Kisin module.
For any
i≥n,
πiTC−(𝒯/𝕊[z];ℤp)∨
has a natural structure of a Breuil–Kisin module of finite
E-height.
(K(1)-local K-theory comparison) Assume
𝒯𝒞
admits a geometric realization. For any
i≥n, after scalar extension along
ϕ¯:𝔖→Ainf
which sends
z
to
[π♭]p
and is the Frobenius on
W, one recovers
K(1)-local K-theory of the generic fiber
(Topological periodic homology theory comparison) For any
i≥n, after scalar extension along the map
ϕ~:𝔖→W which is the Frobenius on
W and sends
z to
0, one recovers topological periodic homology theory of the special fiber
Bhatt–Morrow–Scholze’s Breuil–Kisin cohomology theory [8, Theorem 1.2] implies the crystalline comparison theorem. On the other hand, Theorem 1.9 does not imply Conjecture 1.2. This difference arises as follows. On Breuil–Kisin cohomology theory, there is a GK-equivariant isomorphism (see [7, Theorem 1.8 (iii)])
RΓAinf(X𝒪𝒞)⊗AinfAcry≃RΓcry(X𝒪𝒞/p/Acry).
By combining (1.3) with [7, Theorem 1.8 (iv)], we obtain a canonical (GK,φ)-equivariant isomorphism
RΓét(X𝒞,ℤp)⊗ℤpAcry[1pμ]≃RΓcry(X𝒪𝒞/p/Acry)[1pμ].
The isomorphism induces the crystalline comparison theorem (see [7, Theorem 14.4]). On the other hand, the non-commutative analog of (1.3) becomes the following GK-equivariant isomorphism:
πiTP(𝒯𝒪𝒞;ℤp)⊗AinfAcry^≃πiTP(𝒯𝒪𝒞/p;ℤp),
where Acry^ is the completion of Acry with respect to the Nygaard filtration (see [8, Definition 8.9]). The problem is that μ is a 0-divisor in Acry^ (see [6, Corollaries 2.11 and 2.12]). Thus the non-commutative analog of (1.4) becomes the trivial equation. Therefore, we will study πiTC−(𝒯/𝕊[z];ℤp) in more detail. In Section 3, we will show that the dual Breuil–Kisin module πiTC−(𝒯/𝕊[z];ℤp)∨ admits a Breuil–Kisin GK-module structure in the sense of Gao [19]. In Section 4, using Du–Liu’s work on (φ,G^)-module [14], we will prove the following.
Remark 1.10.
As pointed out by the referee, the issue that only the Nygaard-completed Acrys arises naturally from THH has recently been resolved by Mao [32, Proposition 1.8], who constructs a decompleted version of TP(𝒯𝒪𝒞/p;ℤp). Although we have not reworked the present arguments in that formalism, we expect that Mao’s approach could simplify some of the proofs.
Theorem 1.11.
Let 𝒯 be a smooth and proper 𝒪K-linear category. Then there exists an integer n≥0 such that, for all i≥n, the following statements hold:
The
ℤp[GK]-module
TAinf(πiTC−(𝒯/𝕊[z];ℤp)∨) is a
ℤp-lattice of a crystalline representation, and there is an isomorphism of
Bcrys-module
which is compatible with
GK-action and Frobenius endomorphism.
If
𝒯𝒞 admits a geometric realization, then there is a
GK-equivariant isomorphism
TAinf(πiTC−(𝒯/𝕊[z];ℤp)∨)≃πiLK(1)K(𝒯𝒞)∨
of
ℤp-modules.
We prove the following as a corollary.
Theorem 1.12
(Main Theorem, Theorem 4.19). Let 𝒯 be a smooth proper 𝒪K-linear category. If 𝒯𝒞 admits a geometric realization, then Conjecture 1.2 holds for 𝒯.
Remark 1.13.
In the case 𝒯 embeds fully faithfully into perf(X) with X smooth proper over 𝒪K, Theorem 1.11 can be deduced from the crystalline comparison for smooth proper schemes via a Fourier–Mukai kernel argument.
Non-commutative version of Breuil–Kisin cohomologyBreuil–Kisin modules and Breuil–Kisin cohomology theory <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:mi>R</mml:mi><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mi>𝔖</mml:mi></mml:msub></mml:mrow></mml:math><tex-math><![CDATA[\(R\Gamma_{\mathfrak{S}}\)]]></tex-math></alternatives></inline-formula>
Let us start by recalling the theory of Breuil–Kisin modules.
Definition 2.1.
A Breuil–Kisin module is a finitely generated 𝔖-module M equipped with an 𝔖-linear isomorphism
φM:M⊗𝔖,φ𝔖[1E]≃M[1E].
For a Breuil–Kisin module 𝔐, let us denote 𝔐∨=Hom𝔖(𝔐,𝔖). We note 𝔐∨=Hom𝔖(𝔐,𝔖) is a Breuil–Kisin module whose Frobenius map φ𝔐∨ is given by
where we use facts that φ:𝔖→𝔖 and 𝔖→𝔖[1E] are flat. We note that 𝔐∨ is finite free 𝔖-module; this follows from the facts that gl.dim𝔖=2.
Lemma 2.2
([17, Corollaire 11.1.14] and [7, Lemma 4.27]). Let ϕ¯:𝔖→Ainf be the map that sends z to [π♭]p and is Frobenius on W. Let M be a Breuil–Kisin module, and let MAinf=M⊗𝔖,ϕ¯Ainf. Then M′[1p]=MAinf[1p]⊗Ainf[1p]K0 is a finite free K0-module equipped with a Frobenius automorphism. Fix a section k→𝒪𝒞/p, then there is a (noncanonical) φ-equivariant isomorphism
MAinf⊗AinfBcrys≃M′[1p]⊗K0Bcrys.
In [8], Bhatt–Morrow–Scholze constructed a cohomology theory valued in Breuil–Kisin modules for smooth proper formal schemes over 𝒪K. Let φAinf:Ainf→Ainf be the Frobenius endomorphism of Ainf. Let ϕ:𝔖→Ainf be the W-linear map that sends z to [π♭]. Note that the following diagram commutes:
Theorem 2.3
([8]). Let X/𝒪K be a smooth proper formal scheme. Then there exists an 𝔖-linear cohomology theory RΓ𝔖(X) equipped with a φ-semi-linear map, with the following properties:
All
H𝔖i(X):=Hi(RΓ𝔖(X)) are Breuil–Kisin modules.
(étale comparison) After scalar extension along
ϕ¯:𝔖→Ainf, one recovers étale cohomology of the generic fiber
RΓ𝔖(X)⊗𝔖Ainf[1μ]p∧≃RΓét(X𝒞,ℤp)⊗ℤpAinf[1μ]p∧.
(Crystalline comparison) After scalar extension along the map
𝔖→W, which is the Frobenius on
W and sends
z to
0, one recovers crystalline cohomology of the special fiber
RΓ𝔖(X)⊗𝔖𝕃W≃RΓcry(Xk/W).
(de Rham comparison) After scalar extension along the map
θ~∘φ:𝔖→𝒪K, one recovers de Rham cohomology
RΓ𝔖(X)⊗𝔖𝕃𝒪K≃RΓdR(X/𝒪K).
Perfect modules and Künneth formula
Let (𝒜,⊗,1𝒜) be a symmetric monoidal, stable ∞-category with biexact tensor product. Firstly, we recall from [2, Section 1].
Definition 2.4
([2, Definition 1.2]). An object X∈A is perfect if it belongs to the thick subcategory generated by the unit.
For a lax symmetric monoidal, exact ∞-functor F:𝒜→Sp, F(1𝒜) is naturally an 𝔼∞-ring. For any X,Y∈𝒜, we have a natural map
F(X)⊗F(1𝒜)F(Y)→F(X⊗Y).
Since F is exact, if X is perfect, then the map (2.3) is an equivalence, and F(X) is a perfect F(1𝒜)-module.
We regard 𝒪K as an 𝕊[z]-algebra via z↦π. There is a symmetric monoidal ∞-functor
Since a morphism of homotopy groups π∗(𝕊[z]p∧)→π∗(𝕊[z1/p∞]p∧) is faithfully flat and there is an isomorphism π∗THH(𝒪K[π1/p∞];ℤp)≃𝒪K[π1/p∞][u], where degu=2 (see [8, Section 6]), we obtain an isomorphism
π∗THH(𝒪K/𝕊[z];ℤp)≃𝒪K[u],
where u has degree 2 (see [8, Proposition 11.10]).
Proposition 2.5.
Any dualizable object in Mod–THH(𝒪K/𝕊[z];ℤp)(SpBS1) is perfect.
Proof.
By the isomorphism (2.5), π∗THH(𝒪K/𝕊[z];ℤp) is a regular noetherian ring of finite Krull dimension concentrated in even degrees. We obtain the claim by [2, Theorem 2.15].∎
Proposition 2.6.
Suppose 𝒯1,𝒯2 are 𝒪K-linear categories, and suppose 𝒯1 is smooth and proper. Then THH(𝒯1/𝕊[z];ℤp) is perfect in Mod–THH(𝒪K/𝕊[z];ℤp)(SpBS1), and TC−(𝒯1/𝕊[z];ℤp) (resp. TP(𝒪K/𝕊[z];ℤp)) is a perfect TC−(𝒯1/𝕊[z];ℤp)-module (resp. TP(𝒪K/𝕊[z];ℤp)-module) and the natural map
are lax symmetric monoidal and exact (see [33, Corollary I.4.3]). It is enough to show that THH(𝒯1/𝕊[z];ℤp) is perfect in Mod–THH(𝒪K/𝕊[z];ℤp)(SpBS1). Since the ∞-functor
is symmetric monoidal, and 𝒯1 is dualizable in Cat∞perf(𝒪K) (cf. [31, Chapter 11]), it follows that THH(𝒯1/𝕊[z];ℤp) is dualizable in Mod–THH(𝒪K/𝕊[z];ℤp)(SpBS1). By Proposition 2.5, we obtain the claim.∎
Breuil–Kisin module of stable <inline-formula><alternatives><mml:math display="inline"><mml:mi mathvariant="normal">∞</mml:mi></mml:math><tex-math><![CDATA[\(\infty\)]]></tex-math></alternatives></inline-formula>-categories
Let us recall Antieau–Mathew–Nikolaus’s comparison theorem of symmetric monoidal ∞-functors from [2].
Proposition 2.7
([2, Proposition 4.6]). Let 𝒯,𝒯^ be symmetric monoidal ∞-categories. Let F1,F2:𝒯→𝒯^ be symmetric monoidal functors, and let t:F1⇒F2 be a symmetric monoidal natural transformation. Suppose every object of 𝒯 is dualizable. Then t is an equivalence.
In [8, Proposition 11.10], Bhatt–Morrow–Scholze showed the following: on homotopy groups, there exist isomorphisms
π∗TC−(𝒪K/𝕊[z];ℤp)≃𝔖[u,v]/(uv−E),
where u is of degree 2 and v is of degree −2, and
π∗TP(𝒪K/𝕊[z];ℤp)≃𝔖[σ±],
where σ is of degree 2. Let φ be the endomorphism of 𝔖 determined by the Frobenius on W and z↦zp. Nikolaus–Scholze [33] construct two maps
can,φ𝒯hS1:TC−(𝒯/𝕊[z];ℤp)⇉TP(𝒯/𝕊[z];ℤp).
In [8, Proposition 11.10], Bhatt–Morrow–Scholze also showed the morphism
can𝒪K:π∗TC−(𝒪K/𝕊[z];ℤp)→π∗TP(𝒪K/𝕊[z];ℤp)
is 𝔖-linear and sends u to Eσ and v to σ−1, and the morphism
φ𝒪KhS1:π∗TC−(𝒪K/𝕊[z];ℤp)→π∗TP(𝒪K/𝕊[z];ℤp)
is φ-linear and sends u to σ and v to φ(E)σ−1. Since σ is an invertible element in TP(𝒪K/𝕊[z];ℤp), φ𝒪KhS1 induces a map
φ~𝒯hS1:TC−(𝒯/𝕊[z];ℤp)[1u]→TP(𝒯/𝕊[z];ℤp)
for an 𝒪K-linear stable ∞-category 𝒯. We note that on homotopy groups, the morphism
φ~𝒪KhS1:π∗TC−(𝒪K/𝕊[z];ℤp)[1u]→π∗TP(𝒪K/𝕊[z];ℤp)
is given by 𝔖[u±]→𝔖[σ±] which is φ-semi-linear and sends u to σ.
Lemma 2.8.
Let 𝒯 be a smooth proper 𝒪K-linear category. Then the following statements hold:
π∗TC−(𝒯/𝕊[z];ℤp) is a finitely generated
π∗TC−(𝒪K/𝕊[z];ℤp)-module, thus
πiTC−(𝒯/𝕊[z];ℤp) is a finitely generated
𝔖-module for all
i.
There is a natural number
n satisfying that for any
j≥n,
πjTC−(𝒯/𝕊[z];ℤp)→u⋅πj+2TC−(𝒯/𝕊[z];ℤp) is an isomorphism.
Proof.
Claim (1) directly follows from Proposition 2.6. Moreover, for any j≥0, by the calculation of π∗TC−(𝒪K/𝕊[z];ℤp) (see [8, Proposition 11.10]), we know that the map πjTC−(𝒪K/𝕊[z];ℤp)→u⋅πj+2TC−(𝒪K/𝕊[z];ℤp) is an isomorphism. This yields claim (2) by claim (1).∎
Theorem 2.9.
Let 𝒯 be a smooth proper 𝒪K-linear category. Then there is a natural number n such that the homotopy group πiTC−(𝒯/𝕊[z];ℤp) has a natural structure of a Breuil–Kisin module for any i≥n, and the dual πiTC−(𝒯/𝕊[z];ℤp)∨ also has a natural structure of a Breuil–Kisin module of finite E-height for any i≥n.
Proof.
The morphism φ~𝒯hS1 induces a morphism of TP(𝒪K/𝕊[z];ℤp)-modules
By Proposition 2.6, both sides of the map (2.7) yield symmetric monoidal functors from Cat∞,satperf(𝒪K) to Mod–TP(𝒪K/𝕊[z];ℤp)(Sp), and the map (2.7) yields a symmetric monoidal natural transformation between them. By Proposition 2.7, the morphism (2.7) is an equivalence. On 0-th homotopy group, φ~𝒪KhS1 is given by φ:𝔖→𝔖. Since φ is flat, one has an isomorphism
πiTC−(𝒯/𝕊[z];ℤp)[1u]⊗𝔖,φ𝔖≃πiTP(𝒯/𝕊[z];ℤp)
for any i. By Lemma 2.8 (2), one obtains an isomorphism
πiTC−(𝒯/𝕊[z];ℤp)≃πiTC−(𝒯/𝕊[z];ℤp)[1u]
for any i≥n, thus we have an 𝔖-linear isomorphism on homotopy groups
πiTC−(𝒯/𝕊[z];ℤp)⊗𝔖,φ𝔖≃πiTP(𝒯/𝕊[z];ℤp)
for any i≥n.
After inverting E∈𝔖≃π0TC−(𝒪K/𝕊[z];ℤp), the morphism can𝒯 induces a morphism of TP(𝒪K/𝕊[z];ℤp)[1E]-module
Both sides of the map (2.10) yield symmetric monoidal functors from Cat∞,satperf(𝒪K) to Mod–TP(𝒪K/𝕊[z];ℤp)[1E](Sp), and the map (2.10) yields a symmetric monoidal natural transformation between them. Thus the morphism (2.10) is an equivalence. Note that the morphism
After localization by E, the morphism (2.13) coincides with the dual of (2.12).∎
The comparison between <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>TC</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo></mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒯</mml:mi><mml:mo>/</mml:mo><mml:mi>𝕊</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mrow><mml:mo>;</mml:mo><mml:msub><mml:mi>ℤ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math><![CDATA[\(\operatorname{TC}^{-}(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_{p})\)]]></tex-math></alternatives></inline-formula> and <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:mi>TP</mml:mi><mml:mo></mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>𝒯</mml:mi><mml:msub><mml:mi>𝒪</mml:mi><mml:mi>𝒞</mml:mi></mml:msub></mml:msub><mml:mo>;</mml:mo><mml:msub><mml:mi>ℤ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math><![CDATA[\(\operatorname{TP}(\mathcal{T}_{\mathcal{O}_{\mathcal{C}}};\mathbb{Z}_{p})\)]]></tex-math></alternatives></inline-formula>
In this section, for a smooth proper 𝒪K-linear category 𝒯, we will carefully compare TC−(𝒯/𝕊[z];ℤp) with TP(𝒯𝒪𝒞;ℤp). Denote K∞=K(π1/p∞), and let 𝒪K∞ be the ring of integers of K∞.
Lemma 2.10
([8, Corollary 11.8]). For any 𝒪K-linear stable ∞-category 𝒯, the natural map
THH(𝒯𝒪𝒞;ℤp)→THH(𝒯𝒪𝒞/𝕊[z1/p∞];ℤp)
is an equivalence which is compatible with S1-action and GK∞-action. In particular, the natural map
TP(𝒯𝒪𝒞;ℤp)→TP(𝒯𝒪𝒞/𝕊[z1/p∞];ℤp)
is an equivalence which is compatible with GK∞-action.
Proof.
The morphism 𝕊[z1/p∞]→𝒪K∞∣z1/pn↦π1/pn fits into the following commutative diagram:
which is compatible with S1-action. According to [8, Proposition 11.7], the natural map THH(𝕊[z1/p∞];ℤp)→𝕊[z1/p∞]p∧ is an equivalence. Since π1/pn is in K∞ for any n, thus the equivalence is GK∞-equivariant.∎
Proposition 2.11.
For a smooth proper 𝒪K-linear category 𝒯, THH(𝒯𝒪𝒞/𝕊[z1/p∞];ℤp) is perfect in Mod–THH(𝒪𝒞/𝕊[z1/p∞];ℤp)(SpBS1). In particular, TP(𝒯𝒪𝒞/𝕊[z1/p∞];ℤp) is perfect in Mod–TP(𝒪𝒞/𝕊[z1/p∞];ℤp)(Sp).
Proof.
We already know THH(𝒯/𝕊[z];ℤp) is perfect in Mod–THH(𝒪K/𝕊[z];ℤp)(SpBS1) (see Proposition 2.6). Besides, the functor
is exact and sends THH(𝒯𝒪𝒞/𝕊[z];ℤp) to THH(𝒯𝒪𝒞/𝕊[z1/p∞];ℤp), it follows that THH(𝒯𝒪𝒞/𝕊[z1/p∞];ℤp) is a perfect object in Mod–THH(𝒪𝒞/𝕊[z1/p∞];ℤp)(SpBS1).∎
Lemma 2.10 and Proposition 2.11 imply the following.
Proposition 2.12.
For a smooth proper 𝒪K-linear category 𝒯, THH(𝒯𝒪𝒞;ℤp) is a perfect object in Mod–THH(𝒪𝒞;ℤp)(SpBS1).
The following is a non-commutative version of the comparison theorem between RΓ𝔖 and RΓAinf in [8, Theorem 1.2 (1)].
Theorem 2.13.
Let 𝒯 be a smooth proper 𝒪K-linear category. Then there is a natural number n satisfying that there is a GK∞-equivariant isomorphism
which is compatible with GK∞-action. By Proposition 2.6, the left side of the map (2.16) yields a symmetric monoidal functor from Cat∞,satperf(𝒪K) to Mod–TP(𝒪𝒞/𝕊[z1/p∞];ℤp)(Sp). By Proposition 2.11, the right side of the map (2.16) also yields a symmetric monoidal functor from Cat∞,satperf(𝒪K) to Mod–TP(𝒪𝒞/𝕊[z1/p∞];ℤp)(Sp). By [8, Section 11], on homotopy groups, the morphism
for any i. By Lemma 2.8 (2), we obtain the claim.∎
The comparison between <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>TC</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo></mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒯</mml:mi><mml:mo>/</mml:mo><mml:mi>𝕊</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mrow><mml:mo>;</mml:mo><mml:msub><mml:mi>ℤ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math><![CDATA[\(\operatorname{TC}^{-}(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_{p})\)]]></tex-math></alternatives></inline-formula> and <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:mi>TP</mml:mi><mml:mo></mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>𝒯</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>;</mml:mo><mml:msub><mml:mi>ℤ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math><![CDATA[\(\operatorname{TP}(\mathcal{T}_{k};\mathbb{Z}_{p})\)]]></tex-math></alternatives></inline-formula>
In this section, for a smooth and proper 𝒪K-linear category 𝒯, we will carefully compare TC−(𝒯/𝕊[z];ℤp) with TP(𝒯k;ℤp). There exists a Cartesian diagram of 𝔼∞-ring below.
For an 𝒪K-linear stable ∞-category 𝒯, the diagram yields morphisms
TC−(𝒯/𝕊[z];ℤp)→TC−(𝒯k;ℤp)
and
TC−(𝒯/𝕊[z];ℤp)→TC−(𝒯k;ℤp)→φkhS1TP(𝒯k;ℤp).
Theorem 2.14.
Let 𝒯 be a smooth proper 𝒪K-linear category. Then there is a natural number n such that the morphism (2.17) is an isomorphism
πiTC−(𝒯/𝕊[z];ℤp)[1p]⊗𝔖[1p],ϕ~K0→πiTP(𝒯k;ℤp)[1p]
for any i≥n, where ϕ~ is the map which sends z to 0 and Frobenius on W.
Proof.
The morphism (2.17) induces a commutative diagram
Firstly, we prove that the morphism TP(𝒯/𝕊[z];ℤp)→TP(𝒯k;ℤp) induces an isomorphism
πiTP(𝒯/𝕊[z];ℤp)[1p]⊗𝔖[1p],γK0→πiTP(𝒯k;ℤp)[1p]
for any i, where γ is a W-linear and sends z to 0. By Theorem 2.9 and [7, Proposition 4.3], there is n such that for any i≥n, the homotopy group πiTC−(𝒯/𝕊[z];ℤp) is a Breuil–Kisin module, and πiTC−(𝒯/𝕊[z];ℤp)[1p] is a finite free 𝔖[1p]-module. By the isomorphism (2.9) and the fact that TP(𝒯/𝕊[z];ℤp) is 2-periodic, we obtain that πiTP(𝒯/𝕊[z];ℤp)[1p] is a finite free 𝔖[1p]-module for any i. Thus, on homotopy groups, one obtains an isomorphism
is an equivalence. Both sides of the map (2.19) yield symmetric monoidal functors from Cat∞,satperf(𝒪K) to Mod–TP(k;ℤp)[1E](Sp), thus by Proposition 2.7, we know the map (2.19) is an equivalence. By the isomorphism (2.18) and the fact that TP(𝒯𝒪𝒞;ℤp) is 2-periodic, on homotopy groups, we have an isomorphism
πiTP(𝒯/𝕊[z];ℤp)[1p]⊗𝔖[1p],γK0→πiTP(𝒯k;ℤp)[1p]
for any i. Combining the isomorphism (2.9) with an equality γ∘φ=ϕ~, we obtain an isomorphism
The comparison theorem between <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:msup><mml:mi>TC</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:mo></mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒯</mml:mi><mml:mo>/</mml:mo><mml:mi>𝕊</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mrow><mml:mo>;</mml:mo><mml:msub><mml:mi>ℤ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math><![CDATA[\(\operatorname{TC}^{-}(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_{p})\)]]></tex-math></alternatives></inline-formula> and <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi>K</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:msub><mml:mi>K</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>𝒯</mml:mi><mml:mi>𝒞</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math><![CDATA[\({L_{K(1)}}K(\mathcal{T}_{\mathcal{C}})\)]]></tex-math></alternatives></inline-formula>
Firstly, we prove Künneth formula of K(1)-local K-theory for 𝒞-linear categories that admit a geometric realization.
Proposition 2.15.
For 𝒞-linear categories 𝒯1, 𝒯2 which admit geometric realization, the natural map
LK(1)K(𝒯1)⊗LK(1)K(𝒞)LK(1)K(𝒯2)→LK(1)(𝒯1⊗𝒞𝒯2)
is an equivalence.
Proof.
At first, we prove the claim in the case that there exist smooth proper varieties X1 and X2 so that 𝒯1=perf(X1) and 𝒯2=perf(X2). As a preliminary reduction, we first establish the claim over 𝒞=ℂp, the completion of ℚ¯p with respect to |⋅|p. We fix an isomorphism of fields σ:𝒞≃ℂ. We denote Xi×Spec𝒞Specℂ by Xi,ℂ. Due to Thomason [36], we have an equivalence LK(1)K(Xi)≃Ktop(Xi,ℂan)p∧. Due to Atiyah [3], since a CW complex which comes from a compact complex manifold is finite, we know a natural map
is an equivalence. The isomorphism of fields σ:𝒞≃ℂ induces an equivalence of ring spectra Ktop(pt)p∧≃LK(1)K(ℂ). We see that there exists a commutative diagram
and we obtain the claim. We now treat the general case. We fix an embedding of fields ℂp↪𝒞. There exists a subfield ℂp⊂L⊂𝒞 such that L is finitely generated over ℂp and X and Y are defined over L. Take a variety T over ℂp such that
the function field K(T) is isomorphic to L. Then we can take varieties X′,Y′ over ℂp and morphisms X′→T,Y′→T over ℂp such that the generic fibers of X′,Y′ tensored with 𝒞 are X,Y, respectively. Then, by shrinking T, we may assume X′,Y′ are proper and smooth over T. Choose a ℂp-value point t on T. In that case, by using proper smooth base change theorems in étale cohomology and Thomason’s spectral sequence [36], we have canonical equivalences
Secondly, we prove the claim in the case when 𝒯1=perf(X) and 𝒯2=perf(Y) for a smooth variety X and a smooth proper variety Y. Choose a good compactification (X¯,Σi=1nDi) of X where each Di is irreducible. For n≤i, we write Xi=X−⋃j=1iDj. The Verdier quotient perfDi∩Xi−1(Xi−1)⊂perf(Xi−1)→perf(Xi) induces a fiber sequence
LK(1)K(Y)→LK(1)K(Y)→LK(1)K(Y),
and the Verdier quotient perf(Di∩Xi−1)×Y(Xi−1×Y)=perfDi∩Xi−1(Xi−1)⊗perf(Y)⊂perf(Xi−1)⊗perf(Y)→perf(Xi)⊗perf(Y) induces a fiber sequence
LK(1)K((Di∩Xi−1)×Y)→LK(1)K(Xi−1×Y)→LK(1)K(Xi×Y).
There is a morphism of fiber sequence LK(1)K(Di∩Xi−1)⊗LK(1)K(𝒞) (2.22) → (2.23). Note that Di∩Xi−1 have a good compactification (Di,Σl=1i−1Di∩Dl). By the induction on i, we obtain the claim. The same argument as above proves in the case that X and Y are smooth varieties.
Thirdly, we prove the claim in the case that 𝒯1=perf(X) and 𝒯2=perf(Y) for a separated and finite type scheme X over 𝒞 and a smooth variety Y. Choose a sequence of closed subschemes of X
∅=Z0⊂Z1⊂Z2⊂⋯⊂Zn=X
such that Zi\|Zi−1| is smooth over 𝒞 for any i. For any variety Z over 𝒞, since 𝒞 is characteristic 0, we have an equivalence LK(1)K(Z)≃LK(1)HK(Z) (see [38]), where HK(Z) is the homotopy K-theory of Z. Using Quillen’s localization theorem, we have fiber sequences
and there is a morphism of fiber sequences LK(1)K(Y)⊗LK(1)K(𝒞) (2.24) → (2.25). By the induction on i, we obtain the claim. The same argument as above shows the claim in the case that X and Y are smooth varieties.
For a 𝒞-dg-algebra B concentrated in non-positive degrees, using the Dundas–Goodwillie–McCarthy theorem [15], we have a homotopy pullback square
Since 𝒞 has characteristic 0, we have LK(1)TC(B)=LK(1)TC(H0(B))=0. Thus the natural map LK(1)K(B)→LK(1)K(H0(B)) is an equivalence. For affine derived schemes X and Y whose truncations are separated and of finite type over 𝒞, the claim holds for 𝒯1=perf(X) and 𝒯2=perf(Y). For derived schemes X and Y whose truncations are smooth over 𝒞, the claim follows from the case that X and Y are affine derived schemes by [11, Theorem A.4]. For 𝒞-linear categories 𝒯1, 𝒯2 which admit geometric realization 𝒯1↪perf(X) and 𝒯2↪perf(Y), the claim follows from the fact that LK(1)K(𝒯1) (resp. LK(1)K(𝒯2)) is a retract of LK(1)K(X) (resp. LK(1)K(Y)) (see [1, Proposition 3.4]).∎
We let Cat∞,satperf(𝒪K)geom denote the ∞-category of smooth proper 𝒪K-linear categories 𝒯 such that 𝒯𝒞 admits a geometric realization.
Let 𝒯 be a smooth proper 𝒪K-linear category. Using [6, Theorem 2.16], one obtains an equivalence LK(1)K(𝒯𝒪𝒞)≃LK(1)K(𝒯𝒞) of ring spectra. We recall some results about topological Hochschild homology theory from [6, 8, 22]. On homotopy groups, there is an isomorphism
π∗TP(𝒪𝒞;ℤp)≃Ainf[σ,σ−1],
where σ is a generator σ∈TP2(𝒪𝒞;ℤp). Let β∈K2(𝒞) be the Bott element. For any 𝒪𝒞-linear stable ∞-category 𝒟, we have an identification LK(1)K(𝒟)≃LK(1)K(𝒟𝒞) (see [6, Theorem 2.16]). The cyclotomic trace map tr:K(𝒪𝒞)→TP(𝒪𝒞;ℤp) sends β to a ℤp∗-multiple of ([ε]−1)σ (see [22] and [23, Theorem 1.3.6]); the cyclotomic trace map induces a morphism
By applying Propositions 2.12 and 2.15, we see that both sides of the map (2.27) naturally yield symmetric monoidal functors from Catk,satperf to ModLK(1)TP(C/ℤp)(Sp), and thus the map (2.27) indeed induces an equivalence. On homotopy groups, the morphism LK(1)tr:π∗LK(1)K(𝒞)→π∗LK(1)TP(𝒪𝒞;ℤp) is given by ℤp[β±]→Ainf[1[ε]−1]p∧[σ±] which sends β to ([ε]−1)σ, and is flat. Thus we obtain the claim.∎
Non-commutative version of Bhatt–Morrow–Scholze’s comparison theorem
In this section, as a summary of the previous several sections, we prove a non-commutative version of Theorem 2.3.
Theorem 2.17.
Let 𝒯 be a smooth proper 𝒪K-linear category. Then there is n satisfying the following statements:
For any
i≥n,
πiTC−(𝒯/𝕊[z];ℤp) is a Breuil–Kisin module.
For any
i≥n,
πiTC−(𝒯/𝕊[z];ℤp)∨ is a Breuil–Kisin module of finite
E-height.
(K(1)-local K-theory comparison) Assume
𝒯𝒞 admits a geometric realization. For any
i≥n, after scalar extension along
ϕ¯:𝔖→Ainf, one recovers
K(1)-local K-theory of generic fiber
(Topological periodic homology theory comparison) For any
i≥n, after scalar extension along the map
ϕ~:𝔖→W which is the Frobenius on
W and sends
z to
0, one recovers topological periodic homology theory of the special fiber
Claims (1) and (1′) follow from Theorem 2.9. Claim (2) follows from Theorems 2.13 and 2.16. Claim (3) follows from Theorem 2.14.∎
Corollary 2.18.
For a smooth proper 𝒪K-linear category 𝒯 which admits a geometric realization and an integer i, there is an isomorphism of Bcrys-modules
πiTP(𝒯k;ℤp)⊗WBcrys≃πiLK(1)K(𝒯𝒞)⊗ℤpBcrys.
Proof.
The claim follows from Theorem 2.17 and Lemma 2.2.∎
Breuil–Kisin module and semi-stable representationBreuil–Kisin <inline-formula><alternatives><mml:math display="inline"><mml:msub><mml:mi>G</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:math><tex-math><![CDATA[\(G_{K}\)]]></tex-math></alternatives></inline-formula>-module
Let us recall the work of H. Gao on Breuil–Kisin GK-module [19]. Look at the following diagram:
We note that ϕ sends E to ξ. Let (𝔐,φ𝔐) be a Breuil–Kisin module. We write 𝔐^ to denote 𝔐⊗𝔖,ϕAinf. Upon taking the tensor product −⊗𝒮,ϕ𝔸inf, the induced 𝒮[1E]-linear isomorphism 𝔐⊗𝒮,φ𝒮[1E]→φ𝔐𝔐[1E] then gives rise to the isomorphism
φ^𝔐:𝔐^⊗Ainf,φAinfAinf[1ξ]≃𝔐^[1ξ].
Definition 3.1
([19, Definition 7.1.1]). Let (𝔐,φ𝔐) be a finite free Breuil–Kisin module of finite E-height, equipped with a continuous Ainf-semi-linear GK-action on 𝔐^=𝔐⊗𝔖,ϕAinf. We call (𝔐,φ𝔐) a Breuil–Kisin GK-module if this GK-action satisfies the following conditions:
GK commutes with φ^𝔐.
𝔐⊂𝔐^GK∞ via the embedding 𝔐⊂𝔐^.
𝔐/z𝔐⊂(𝔐^⊗AinfW(k¯))GK via the embedding 𝔐/z𝔐⊂𝔐^⊗AinfW(k¯).
The GK-equivariant surjective map of rings 𝒪𝒞→k¯ induces a morphism of ring spectra
TP(𝒯𝒪𝒞;ℤp)→TP(𝒯k¯;ℤp).
Combining an equivalence THH(𝒯k¯;ℤp)≃THH(k¯;ℤp)⊗THH(𝒪𝒞;ℤp)THH(𝒯𝒪𝒞;ℤp) with Proposition 2.12, we obtain the following proposition.
Proposition 3.2.
For a smooth proper 𝒪K-linear category 𝒯, THH(𝒯k¯;ℤp) is perfect in Mod–THH(k¯;ℤp)(SpBS1).
Let us recall the calculation of TP(𝒪𝒞;ℤp) [8]. On homotopy groups,
φ𝒪𝒞hS1:π∗TC−(𝒪𝒞;ℤp)→π∗TP(𝒪𝒞;ℤp)
is given by
Ainf[u,v]/(uv−ξ)→Ainf[σ±]
which is φAinf-linear and sends u to σ and v to φAinf(ξ)σ−1, where u and σ have degree 2 and v has degree −2. Similarly, on homotopy groups,
φk¯hS1:π∗TC−(k¯;ℤp)→π∗TP(k¯;ℤp)
is given by
W(k¯)[u,v]/(uv−p)→W(k¯)[σ±]
which is φW(k¯)-linear and sends u to σ and v to pσ−1.
Theorem 3.3.
Let 𝒯 be a smooth proper 𝒪K-linear category. The morphism (3.2) induces an isomorphism
By Propositions 2.12 and 3.2, both sides of the map (3.2) yield symmetric monoidal functors from Cat∞,satperf(𝒪K) to Mod–TP(k¯;ℤp)(Sp), and the map (3.5) yields a symmetric monoidal natural transformation between them. Thus the morphism (3.5) is an equivalence. By Theorem 2.9 and [7, Proposition 4.3], there is n such that for any i≥n, the homotopy group πiTC−(𝒯/𝕊[z];ℤp) is a Breuil–Kisin module, and πiTC−(𝒯/𝕊[z];ℤp)[1p] is a finite free 𝔖[1p]-module. By Theorem 2.13 and the fact that TP(𝒯𝒪𝒞;ℤp) is 2-periodic, πiTP(𝒯𝒪𝒞;ℤp)[1p] is a finite free Ainf[1p]-module for any i. Thus π∗TP(𝒯𝒪𝒞;ℤp)[1p] is a flat graded π∗TP(𝒪𝒞;ℤp)[1p]-module, and we obtain the claim.∎
Lemma 3.4.
Let 𝒯 be a smooth proper 𝒪K-linear category. Then the following statements hold:
π∗TC−(𝒯𝒪𝒞;ℤp) is a finitely generated
π∗TC−(𝒪𝒞/𝕊[z];ℤp)-module, thus
πiTC−(𝒯𝒪𝒞;ℤp) is a finitely generated
Ainf-module for any
i.
There is a natural number
n such that for any
j≥n,
πjTC−(𝒯𝒪𝒞;ℤp)→u⋅πj+2TC−(𝒯𝒪𝒞;ℤp) is an isomorphism.
Proof.
Claim (1) directly follows from Proposition 2.12. For any j≥0, by the calculation of π∗TC−(𝒪𝒞;ℤp) (see [8, Proposition 11.10]), we know that πjTC−(𝒪𝒞;ℤp)→u⋅πj+2TC−(𝒪𝒞;ℤp) is an isomorphism. By claim (1), we see claim (2).∎
Theorem 3.5.
Let 𝒯 be a smooth proper 𝒪K-linear category. There is a natural number n such that the cyclotomic Frobenius morphism φ𝒯𝒪𝒞hS1:TC−(𝒯𝒪𝒞;ℤp)→TP(𝒯𝒪𝒞;ℤp) induces a GK-equivariant isomorphism
πiTC−(𝒯𝒪𝒞;ℤp)⊗Ainf,φAinfAinf≃πiTP(𝒯𝒪𝒞;ℤp)
for any i≥n.
Proof.
Cyclotomic Frobenius map φ𝒯𝒪𝒞hS1:TC−(𝒯𝒪𝒞;ℤp)→TP(𝒯𝒪𝒞;ℤp) yields a morphism
φ¯𝒯𝒪𝒞hS1:TC−(𝒯𝒪𝒞;ℤp)[1u]→TP(𝒯𝒪𝒞;ℤp).
The morphism φ¯𝒯𝒪𝒞hS1 induces a morphism of TP(𝒪𝒞;ℤp)-module
By Proposition 2.12, both sides of the map (3.7) yield symmetric monoidal functors from Cat∞,satperf(𝒪K) to Mod–TP(𝒪𝒞;ℤp)(Sp), thus the map (3.7) is an equivalence. On homotopy groups, φ¯𝒪𝒞hS1:π∗TC−(𝒪𝒞;ℤp)[1u]→π∗TP(𝒪𝒞;ℤp) is given by φAinf-linear map
Ainf[u±]→Ainf[σ±],
thus this is an isomorphism. Thus we obtain the isomorphism
πiTC−(𝒯𝒪𝒞;ℤp)[1u]⊗Ainf,φAinfπiTP(𝒯𝒪𝒞;ℤp)
for any i. By Lemma 3.4, there is a natural number n such that πiTC−(𝒯𝒪𝒞;ℤp)≃πiTC−(𝒯𝒪𝒞;ℤp)[1u] for any i≥n.∎
For a smooth proper 𝒪K-linear category 𝒯, we will prove that the free Breuil–Kisin module πiTC−(𝒯/𝕊[z];ℤp)∨ is a Breuil–Kisin GK-module.
Theorem 3.6.
Let 𝒯 be a smooth proper 𝒪K-linear category. There is an n such that for any i≥n, the following statements hold:
There is a continuous
Ainf-semi-linear
GK-action on
πiTC−(𝒯/𝕊[z];ℤp)∨^=πiTC−(𝒯/𝕊[z];ℤp)∨⊗𝔖,ϕAinf.
GK commutes with
φ^πiTC−(𝒯/𝕊[z];ℤp)∨.
One has the inclusion
πiTC−(𝒯/𝕊[z];ℤp)∨⊂(πiTC−(𝒯/𝕊[z];ℤp)∨^)GK∞ induced via the embedding
πiTC−(𝒯/𝕊[z];ℤp)∨⊂πiTC−(𝒯/𝕊[z];ℤp)∨^.
πiTC−(𝒯𝒪𝒞;ℤp)∨[1p]⊗Ainf[1p]W(k¯)[1p] is fixed by
GKur.
In particular, πiTC−(𝒯/𝕊[z];ℤp)∨ is a Breuil–Kisin GK-module.
Proof.
After localization by ξ∈Ainf≃π0TC−(𝒪𝒞;ℤp), the morphism can𝒯 induces a morphism of TP(𝒪𝒞;ℤp)[1ξ]-module
Both sides of the map (3.8) yield symmetric monoidal functors from Cat∞,satperf(𝒪K) to Mod–TP(𝒪𝒪𝒞;ℤp)[1ξ](Sp), and the map (3.8) yields a symmetric monoidal natural transformation between them. Thus the morphism (3.8) is an equivalence. Note that the morphism
can𝒪𝒞:π∗TC−(𝒪𝒞;ℤp)[1ξ]→π∗TP(𝒪𝒞;ℤp)[1ξ]
is an isomorphism (see [8, Proposition 11.10]). This yields a GK-equivariant isomorphism
can𝒯𝒪𝒞[1ξ]:π∗TC−(𝒯𝒪𝒞;ℤp)[1ξ]≃π∗TP(𝒯𝒪𝒞;ℤp)[1ξ].
Combined with (3.6), this yields a GK-equivariant isomorphism
for any i≥n. Since φAinf is an isomorphism, by Theorems 2.13 and 3.5, we have a GK∞-equivariant isomorphism
πiTC−(𝒯/𝕊[z];ℤp)⊗𝔖,ϕAinf≃πiTC−(𝒯𝒪𝒞;ℤp),
and we see the isomorphism (3.10) is the same as φ^πiTC−(𝒯/𝕊[z];ℤp). The group GK naturally acts on πiTC−(𝒯𝒪𝒞;ℤp), which is π0TC−(𝒪𝒞;ℤp)≃Ainf-semi-linear. Since φ:𝔖→Ainf is flat, the dual of (3.11) becomes the following GK∞-equivariant isomorphism:
πiTC−(𝒯/𝕊[z];ℤp)∨⊗𝔖,ϕAinf≃πiTC−(𝒯𝒪𝒞;ℤp)∨.
Via this isomorphism, πiTC−(𝒯/𝕊[z];ℤp)∨⊗𝔖,ϕAinf admits a continuous Ainf-semi-linear GK-action.
Since (3.10) is GK-equivariant, GK commutes with φ^πiTC−(𝒯/𝕊[z];ℤp)∨, yielding claims (1) and (2).
Consider the GK∞-equivariant isomorphism (3.12). Since g∈GK∞ acts on 1⊗g on πiTC−(𝒯/𝕊[z];ℤp)∨⊗𝔖,ϕAinf, we obtain claim (3).
Since φAinf:Ainf→Ainf is flat, we have GK-equivariant isomorphism
πiTC−(𝒯𝒪𝒞;ℤp)∨⊗Ainf,φAinfAinf≃πiTP(𝒯𝒪𝒞;ℤp)∨
by Theorem 3.5. Combined with Theorem 3.3, we see that πiTC−(𝒯𝒪𝒞;ℤp)∨[1p]⊗Ainf[1p]W(k¯)[1p] is fixed by GKur.
Since πiTC−(𝒯𝒪𝒞;ℤp)∨ is finite free 𝔖-module, there is a natural inclusion πiTC−(𝒯𝒪𝒞;ℤp)∨↪πiTC−(𝒯𝒪𝒞;ℤp)∨[1p], and it induces a natural inclusion
Thus πiTC−(𝒯𝒪𝒞;ℤp)∨⊗AinfW(k¯) is fixed by GKur. The last claim follows from claims (1)–(4) and [19, Lemma 7.1.2].∎
For a Breuil–Kisin module (𝔐,φ𝔐), let φ𝔐¯ denote the composite
φ𝔐¯:𝔐⊗𝔖W(𝒞♭)→φ𝔐′⊗φ𝒞𝔐[1E]⊗𝔖[1E]W(𝒞♭)=𝔐⊗𝔖W(𝒞♭),
where φ𝔐′ is the composite 𝔐→𝜑𝔐⊗𝔖,φ𝔖[1E]→φ𝔐𝔐[1E], and φ𝒞 is the Frobenius of W(𝒞♭). Let us define a ℤp-module
TAinf(𝔐):=(𝔐^⊗AinfW(𝒞♭))φ𝔐¯=1.
If (𝔐,φ𝔐) is equipped with a Breuil–Kisin GK-module structure, then TAinf(𝔐) inherits a natural structure of a ℤp[GK]-module.
Theorem 3.7.
The following statements hold:
For a smooth proper
𝒪K-linear category
𝒯 and a sufficiently large
i, the
ℤp[GK]-module
TAinf(πiTC−(𝒯/𝕊[z];ℤp)∨) is a
ℤp-lattice of a semi-stable representation.
We assume
𝒯𝒞 admits a geometric realization. Then there is a
GK-equivariant isomorphism
πiLK(1)K(𝒯𝒞)∨≃TAinf(πiTC−(𝒯/𝕊[z];ℤp)∨).
In particular,
πiLK(1)K(𝒯𝒞)⊗ℤpℚp is a semi-stable representation.
Proof.
(1) By Theorem [19, Theorem 7.1.7], we obtain the claim.
(2) Let 𝒯 be a smooth proper 𝒪K-linear category. We assume 𝒯𝒞 admits a geometric realization. By the same discussion as in the proof of Theorem 2.16, we have an equivalence
By the functionality of cyclotomic Frobenius φ−hS1:TC−(−;ℤp)→TP(−;ℤp), we have the following commutative diagram:
We note ξ|μ in Ainf. Since φ𝒪𝒞hS1:π0LK(1)TC−(𝒪𝒞;ℤp)[1ξ]→π0LK(1)TP(𝒪𝒞;ℤp)[1ξ~] is given by the Frobenius φAinf:Ainf[1μ]p∧→Ainf[1φ(μ)]p∧, on homotopy groups, we have GK-equivariant diagram
After the base change along Ainf[1μ]p∧→W(𝒞♭), the above diagram becomes the following GK-equivariant diagram:
Thus the isomorphism πiLK(1)K(𝒯𝒞)⊗ℤpW(𝒞♭)≃πiTC−(𝒯𝒪𝒞;ℤp)⊗AinfW(𝒞♭) is Frobenius equivariant. Since ℤp→Ainf and Ainf→W(𝒞♭) are flat and compatible with Frobenius automorphisms, the dual of the isomorphism becomes the Frobenius-equivariant isomorphism
πiLK(1)K(𝒯𝒞)∨⊗ℤpW(𝒞♭)≃πiTC−(𝒯𝒪𝒞;ℤp)∨⊗AinfW(𝒞♭),
and we obtain the claim.∎
Remark 3.8.
Similarly, we can obtain that there is an isomorphism of ℤp[GK]-modules
πiLK(1)K(𝒯𝒞)∨∨≃TAinf(πiTC−(𝒯/𝕊[z];ℤp)∨∨).
<inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>φ</mml:mi><mml:mo>,</mml:mo><mml:mover accent="true"><mml:mi>G</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mtext>-</mml:mtext></mml:math><tex-math><![CDATA[\(({\varphi},\hat{G})\)]]></tex-math></alternatives></inline-formula>modules and crystalline representations
In this section, we will prove Theorem 1.12. We would like to thank H. Gao [20] for sharing the strategy and key constructions to prove Theorem 1.12.
Breuil–Kisin <inline-formula><alternatives><mml:math display="inline"><mml:msub><mml:mi>G</mml:mi><mml:mi>K</mml:mi></mml:msub></mml:math><tex-math><![CDATA[\(G_{K}\)]]></tex-math></alternatives></inline-formula>-modules and <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>φ</mml:mi><mml:mo>,</mml:mo><mml:mover accent="true"><mml:mi>G</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math><tex-math><![CDATA[\(({\varphi},\hat{G})\)]]></tex-math></alternatives></inline-formula>-modules
Let us recall (φ,G^)-module from [14]. Let L:=⋃n=1∞K∞(ζn), G^:=Gal(L/K) and HK:=Gal(L/K∞). Du–Liu construct 𝔖-algebras 𝔖(2) and 𝔖st(2) as the following [14, Sections 2.1 and 2.3]. We set 𝔖⊗ˆ2:=𝔖⟦y−z⟧=W⟦z,y−z⟧. Then 𝔖⊗ˆ2 is 𝔖⊗ℤp𝔖-algebra via z⊗1↦z, 1⊗z↦y=(y−z)+z; in this way, one can extend Frobenius action on 𝔖 to 𝔖⊗ˆ2 which is Frobenius on W and sends z to zp and y−z to yp−zp. Let 𝔖(2) be 𝔖⟦y−z⟧{y−zE}δ∧, where {−}δ∧ means freely adjoining elements in the category of (p,E)-completed δ- 𝔖-algebras [14, Sections 2.1 and 4.1]. Then 𝔖(2) is an 𝔖-algebra via u↦u. The structure map is flat (see [14, Proposition 2.2.7]), and 𝔖(2) is a sub- 𝔖-algebra of Ainf, where we regard Ainf as an 𝔖-algebra via ϕ (see [14, Section 2.4]). Let us recall the definition of 𝔖st(2) from [14, Section 2.3]. Define a Frobenius action on W⟦z,𝔶⟧ which sends x to xp and 𝔶 to (1+𝔶)p−1 and set 𝔴=𝔶E. Let 𝔖st(2)=𝔖⟦z,𝔶⟧{𝔴}δ∧, and it is a sub- 𝔖-algebra of Ainf, where we regard Ainf as an 𝔖-algebra via ϕ (see [14, Section 2.4]). There is a natural inclusion of 𝔖-algebra W⟦z,y−z⟧⊂W⟦z,𝔶⟧ which sends y to z(𝔶+1); it is a δ-ring map. By construction, we thus obtain an inclusion of sub- 𝔖-algebras 𝔖(2)↪𝔖st(2) of Ainf. Let us denote θ0:𝔖→𝔖(2)∣z↦z, and denote θ1:𝔖→𝔖(2)∣z↦y. We note that there is a commutative diagram (see [14, Corollary 2.4.5])
where ι sends z to [π♭] and y to [ε]⋅[π♭]. We regard 𝔖(2) and 𝔖st(2) as 𝔖-algebras via the diagram unless otherwise stated.
Definition 4.1
([14, Definition 3.3.2]). Let (𝔐,φ𝔐) be a finite free Breuil–Kisin module of finite E-height. We call (𝔐,φ𝔐)(φ,G^)-module if it satisfies the following conditions:
There is a continuous 𝔖st(2)-semi-linear G^-action on 𝔐⊗𝔖𝔖st(2).
G^ commutes with φ on 𝔐⊗𝔖𝔖st(2).
𝔐⊂(𝔐⊗𝔖𝔖st(2))HK.
G^ acts on (𝔐⊗𝔖𝔖st(2))⊗𝔖st(2)W(k) trivially.
Remark 4.2.
For a Breuil–Kisin GK-module (𝔐,φ𝔐,GK↷𝔐⊗𝔖,ϕAinf), the p-adic representation TAinf(𝔐) is a ℤp-lattice of semi-stable representation of non-negative Hodge–Tate weight [19, Theorem 7.1.7]. Du–Liu proved that the sub-modules 𝔐⊗𝔖𝔖st(2)⊂𝔐⊗𝔖,ϕAinf⊂TAinf(𝔐)⊗ℤpAinf are GK-stable [14, Theorem 3.3.3] (see also [30, Proposition 3.1.3]). By Definition 3.1 (3), GL=Gal(K¯/L)(⊂GK∞) acts on 𝔐⊂𝔐⊗𝔖,ϕAinf trivially, thus GK-action on 𝔐⊗𝔖𝔖st(2) factors through G^. The G^-action on 𝔐⊗𝔖𝔖st(2) induces the (φ,G^)-module structure on (𝔐,φ𝔐) [14, Theorem 3.3.3].
The comparison theorem between <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:mi>TP</mml:mi><mml:mo></mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒯</mml:mi><mml:mo>/</mml:mo><mml:mi>𝕊</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mrow><mml:mo>;</mml:mo><mml:msub><mml:mi>ℤ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math><![CDATA[\(\operatorname{TP}(\mathcal{T}/\mathbb{S}[z];\mathbb{Z}_{p})\)]]></tex-math></alternatives></inline-formula> and <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:mi>TP</mml:mi><mml:mo></mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒯</mml:mi><mml:mo>/</mml:mo><mml:mi>𝕊</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mrow><mml:mo>;</mml:mo><mml:msub><mml:mi>ℤ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math><![CDATA[\(\operatorname{TP}(\mathcal{T}/\mathbb{S}[z_{0},z_{1}];\mathbb{Z}_{p})\)]]></tex-math></alternatives></inline-formula>
We regard 𝒪K as an 𝕊[z0,z1]-algebra via the map z0,z1↦π. In [29], Liu–Wang revealed the structure of TP(𝒪K/𝕊[z0,z1];ℤp): there is an isomorphism
π0TP(𝒪K/𝕊[z0,z1];ℤp)≃𝔖(2),
where we identify z0=z and z1=y (see [29, Theorem 1.3]). In this section, we will compare TP(𝒯/𝕊[z];ℤp) with TP(𝒯/𝕊[z0,z1];ℤp) for an 𝒪K-linear category 𝒯. We note that the functor
is a lax symmetric monoidal; THH(𝒪K/𝕊[z0,z1];ℤp) admits an 𝔼∞-ring structure, and TP(𝒪K/𝕊[z0,z1];ℤp) also admits an 𝔼∞-ring structure. For each i=0,1, we consider the map of ring spectra 𝕊[z]→z↦zi𝕊[z0,z1], which in turn induces a morphism of 𝔼∞-ring spectra
ei:TP(𝒪K/𝕊[z];ℤp)→z↦ziTP(𝒪K/𝕊[z0,z1];ℤp).
For ♡∈{THH,TP,TC−} and an 𝒪K-linear category 𝒯, the left unit e0,𝒯 and right unit e1,𝒯 are the maps
♡(𝒯/𝕊[z];ℤp)→♡(𝒯/𝕊[z0,z1];ℤp)
induced by z↦z0 and z↦z1, respectively. Let us denote by THH(𝕊[z0,z1]/𝕊0[z]) (resp. THH(𝕊[z0,z1]/𝕊1[z])) the relative topological Hochschild homology of 𝕊[z]→𝕊[z0,z1]∣z→z0 (resp. 𝕊[z]→𝕊[z0,z1]∣z→z1). There is a commutative diagram in Mod–THH(𝕊[z])(SpBS1)
Since the bottom-left square and the bottom square are pushout squares (see [29, Lemma 2.3]), and similarly, since the right square is a pushout square, the top-right square is also a pushout square. Thus, by the transitivity property of relative THH, we obtain the following equivalence:
Applying 𝒯=perf(𝒪K) to the equivalence (4.3), we see that the bottom and total squares are pushout squares. We know that the top square is a pushout square. Thus we have the following S1-equivariant equivalence:
from ModTHH(𝒪K/𝕊[z];ℤp)(SpBS1) to ModTHH(𝒪K/𝕊[z0,z1];ℤp)(SpBS1). In the case where the 𝒪K-linear category 𝒯 is smooth and proper, it follows that THH(𝒯/𝕊[z];ℤp) is dualizable and perfect in ModTHH(𝒪K/𝕊[z];ℤp)(SpBS1). Since the base change functor
−⨂THH(𝒪K/𝕊[z];ℤp)THH(𝒪K/𝕊[z0,z1];ℤp)
is an exact symmetric monoidal functor, we consequently obtain the following lemma:
Lemma 4.3.
For a smooth proper 𝒪K-linear category 𝒯, THH(𝒯/𝕊[z0,z1];ℤp) is dualizable and perfect in Mod–THH(𝒪K/𝕊[z0,z1];ℤp)(SpBS1).
Proposition 4.4.
Let 𝒯1,𝒯2 be smooth proper 𝒪K-linear categories. There are equivalences
By Propositions 2.6 and 4.4, both sides of the map (4.5) yield symmetric monoidal functors from Cat∞,satperf(𝒪K) to Mod–♡(𝒪K/𝕊[z0,z1];ℤp)(Sp). Since every smooth proper 𝒪K-linear category is dualizable in Cat∞,satperf, by [2, Proposition 4.6] we obtain the claim.∎
There is an equivalence of 𝔼∞-ring spectra ex:𝕊[z0,z1]→≃𝕊[z0,z1]∣z0↦z1,z1↦z0, and it induces a commutative diagram
and an isomorphism of 𝔖-algebra (see also [14, Section 4])
𝔖(2)⊗θ0,𝔖𝔖≃𝔖(2)⊗θ1,𝔖𝔖.
By [29, Corollary 3.7], we see that TP(𝒪K/𝕊[z0,z1];ℤp) is 2-periodic and there is an isomorphism π∗TP(𝒪K/𝕊[z0,z1];ℤp)=𝔖(2)[σ,σ−1] by choosing a generator σ∈π2TP(𝒪K/𝕊[z0,z1];ℤp). For i=0,1, on homotopy groups, the morphism
π∗TP(𝒪K/𝕊[z];ℤp)→eiπ∗TP(𝒪K/𝕊[z0,z1];ℤp)
is given by
𝔖[u,u−1]→𝔖(2)[σ,σ−1]
which is a θi-linear map sending u to ai⋅σ for some units ai∈(𝔖(2))∗.
Proposition 4.6.
There is an isomorphism
πjTP(𝒯/𝕊[z];ℤp)⊗𝔖,θi𝔖(2)≃πjTP(𝒯/𝕊[z0,z1];ℤp)
for any j and i=0,1.
Proof.
If i=0, by [14, Proposition 2.2.7] the graded-ring morphism (4.7) is flat. We obtain the claim by Proposition 4.5. By the isomorphism (4.6), θ1 is also flat. Thus the morphism (4.7) is also flat for i=1.∎
The comparison theorem between <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:mi>TP</mml:mi><mml:mo></mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:mi>𝒯</mml:mi><mml:mo>/</mml:mo><mml:mi>𝕊</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>z</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:mrow><mml:mo>;</mml:mo><mml:msub><mml:mi>ℤ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math><![CDATA[\(\operatorname{TP}(\mathcal{T}/\mathbb{S}[z_{0},z_{1}];\mathbb{Z}_{p})\)]]></tex-math></alternatives></inline-formula> and <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:mi>TP</mml:mi><mml:mo></mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mi>𝒯</mml:mi><mml:msub><mml:mi>𝒪</mml:mi><mml:mi>𝒞</mml:mi></mml:msub></mml:msub><mml:mo>/</mml:mo><mml:mi>𝕊</mml:mi></mml:mrow><mml:mrow><mml:mo maxsize="120%" minsize="120%">[</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mn>0</mml:mn><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mi mathvariant="normal">∞</mml:mi></mml:msup></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mi>z</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msup><mml:mi>p</mml:mi><mml:mi mathvariant="normal">∞</mml:mi></mml:msup></mml:mrow></mml:msubsup><mml:mo maxsize="120%" minsize="120%">]</mml:mo></mml:mrow></mml:mrow><mml:mo>;</mml:mo><mml:msub><mml:mi>ℤ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math><![CDATA[\(\operatorname{TP}(\mathcal{T}_{\mathcal{O}_{\mathcal{C}}}/\mathbb{S}\bigl{[}z_{0}^{1/p^{\infty}},z_{1}^{1/p^{\infty}}\bigr{]};\mathbb{Z}_{p})\)]]></tex-math></alternatives></inline-formula>
We regard 𝒪𝒞 as an 𝕊[z01/p∞,z11/p∞]-algebra via the map z01/pn↦π1/pn,z11/pn↦ζnπ1/pn. We will prove the comparison theorem between TP(𝒯/𝕊[z0,z1];ℤp) and TP(𝒯𝒪𝒞/𝕊[z01/p∞,z11/p∞];ℤp) for a smooth proper 𝒪K-linear category 𝒯. The following lemma is inspired by H. Gao in [20].
Lemma 4.7.
The natural map THH(𝒯𝒪𝒞;ℤp)→THH(𝒯𝒪𝒞/𝕊[z01/p∞,z11/p∞];ℤp) is an equivalence which is compatible with S1-action. In particular, the natural map
which is compatible with S1-action. For i=0,1, we have an S1-equivariant equivalence THH(𝕊[zi1/p∞];ℤp)≃𝕊[zi1/p∞]p∧ [8, Proposition 11.7], and we obtain the claim.∎
Lemma 4.8.
The natural map 𝕊[z0,z1]→𝕊[z01/p∞,z11/p∞] induces an isomorphism
The morphism of 𝔼∞-ring spectra e0:𝕊[z]→𝕊[z0,z1] naturally induces an isomorphism πiTP(𝒯/𝕊[z];ℤp)⊗𝒮,θ0𝒮z0,z1≃πiTP(𝒯/𝕊[z0,z1];ℤp). Similarly, the composite map 𝕊[z]→e0𝕊[z0,z1]→𝕊[z01/p∞,z11/p∞] also induces an isomorphism
Since there is an equation of morphisms ϕ=ι∘θ0, we obtain the claim.∎
<inline-formula><alternatives><mml:math display="inline"><mml:mi>τ</mml:mi></mml:math><tex-math><![CDATA[\(\tau\)]]></tex-math></alternatives></inline-formula>-action on <inline-formula><alternatives><mml:math display="inline"><mml:mrow><mml:mi>TP</mml:mi><mml:mo></mml:mo><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>𝒯</mml:mi><mml:msub><mml:mi>𝒪</mml:mi><mml:mi>𝒞</mml:mi></mml:msub></mml:msub><mml:mo>;</mml:mo><mml:msub><mml:mi>ℤ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math><tex-math><![CDATA[\(\operatorname{TP}(\mathcal{T}_{\mathcal{O}_{\mathcal{C}}};\mathbb{Z}_{p})\)]]></tex-math></alternatives></inline-formula>
The construction in this section is inspired by H. Gao [20]. Let τ∈Gal(L/⋃n=1∞K(ζn)) be a
topological generator and τ~∈GK be the lift of τ satisfying τ~(π1/pn)=ζnπ1/pn. Consider a map η:𝕊[z01/p∞,z11/p∞]→𝒪𝒞 which sends z01/pn to π1/pn and z11/pn to ζnπ1/pn.
We will use the following notations associated with a smooth proper 𝒪K-linear category 𝒯:
e0∞:TP(𝒯𝒪𝒞/𝕊[z1/p∞];ℤp)→≃TP(𝒯𝒪𝒞/𝕊[z01/p∞,z11/p∞];ℤp) the morphism given by z1/p∞↦z01/p∞
a:TP(𝒯/𝕊[z];ℤp)→TP(𝒯𝒪𝒞/𝕊[z1/p∞];ℤp) the morphism given by the commutative diagram
b:TP(𝒯/𝕊[z];ℤp)→TP(𝒯𝒪𝒞/𝕊[z01/p∞,z11/p∞];ℤp) the morphism given by commutative diagram
c:TP(𝒯/𝕊[z0,z1];ℤp)→TP(𝒯𝒪𝒞/𝕊[z01/p∞,z11/p∞];ℤp) the morphism given by commutative diagram
τ~1:TP(𝒯𝒪𝒞/𝕊[z1/p∞];ℤp)→TP(𝒯𝒪𝒞/𝕊[z01/p∞,z11/p∞];ℤp) the morphism given by the commutative diagram
Lemma 4.9.
The following diagram commutes:
Proof.
Since diagram (4.9) is compatible with 𝕊-ring spectrum structure, we obtain the claim.∎
The following theorem, inspired by H. Gao [20], plays the most important role in the proof of Theorem 1.12.
Proposition 4.10.
The following diagram commutes:
Proof.
Diagram (4.9) fits into the following commutative diagram:
Thus we see that the left square of (4.10) commutes. Look at the following diagram:
then we see that the right square of (4.10) commutes.∎
Fix an integer i≫0. By Proposition 2.6, πiTP(𝒯/𝕊[z];ℤp) is a finitely generated 𝔖-module. Choose an 𝔖-generator 𝔪1,𝔪2,…,𝔪n of πiTP(𝒯/𝕊[z];ℤp). We have an equation ϕ¯=ϕ∘φ:𝔖→Ainf (see diagram (2.2)); combining isomorphism (2.9) with Theorem 2.13, we see that a induces an isomorphism πiTP(𝒯/𝕊[z];ℤp)⊗𝔖,ϕAinf≃πiTP(𝒯𝒪𝒞/𝕊[z1/p∞];ℤp). For any element x∈πiTP(𝒯/𝕊[z];ℤp), let us write x¯ for the image of x under the sequence of maps πiTP(𝒯/𝕊[z];ℤp)→𝑎πiTP(𝒯𝒪ℂ/𝕊[z1/p∞];ℤp)←≃πiTP(𝒯𝒪ℂ;ℤp). Then it follows that the elements 𝔪1¯,𝔪2¯,…,𝔪n¯ form a set of 𝔸inf-generators for πiTP(𝒯𝒪ℂ;ℤp). Choose elements ax1,ax2,…,axn of Ainf so that
τ~(x¯)=ax1𝔪1¯+ax2𝔪2¯+⋯+axn𝔪n¯.
Proposition 4.11.
For any x, any choice of 𝒮-generators {𝔪1,…,𝔪n}, and any choice of elements axl∈𝔸inf, it follows that axl is contained in 𝒮z0,z1 via the inclusion ι:𝒮z0,z1↪𝔸inf.
Proof.
Combining Proposition 4.10 with Lemma 4.9, we see that the following diagram commutes:
We note that the map πiTP(𝒯𝒪𝒞;ℤp)→πiTP(𝒯𝒪𝒞/𝕊[z01/p∞,z11/p∞];ℤp) is 𝔸inf-linear. According to the results in Proposition 4.6 and Lemma 4.8, the morphism c:πiTP(𝒯/𝕊[z0,z1];ℤp)→πiTP(𝒯/𝕊[z01/p∞,z11/p∞];ℤp) is explicitly given by the map
By Proposition 4.6, the morphism e0,𝒯:πiTP(𝒯/𝕊[z];ℤp)→TP(𝒯/𝕊[z0,z1];ℤp) is given by the natural map πiTP(𝒯/𝕊[z];ℤp)→πiTP(𝒯/𝕊[z];ℤp)⊗𝒮,θ0𝒮z0,z1, and consequently, the set {e0,𝒯(𝔪1),e0,𝒯(𝔪2),…,e0,𝒯(𝔪n)} forms an 𝒮z0,z1-generator set of TP(𝒯/𝕊[z0,z1];ℤp). Therefore, for x∈πiTP(𝒯/𝕊[z];ℤp), there are ax1′,ax2′,…,axn′∈𝔖(2)↪𝜄Ainf such that
e1,𝒯(x)=Σl=1naxl′e0,𝒯(𝔪l).
Look at the diagram, then we see an equation τ~1(a(x))=Σaxl′c(e0,𝒯(𝔪l)), and we obtain the claim.∎
Remark 4.12.
We can apply the same procedure to τ~−1, and we obtain that there are elements bxj of 𝔖(2) so that τ~−1(x¯)=bx1𝔪1¯+bx2𝔪2¯+⋯+bxn𝔪n¯, where instead of η we use a morphism 𝕊[z01/p∞,z11/p∞]→𝒪𝒞 which sends z01/pn to π1/pn and z11/pn to ζn−1π1/pn.
Let τ~∗ denote the dual action of τ~ on the dual module πiTP(𝒯𝒪ℂ;ℤp)∨, which is defined as Hom𝔸inf(πiTP(𝒯𝒪ℂ;ℤp),𝔸inf). Choose an 𝔖-basis 𝔫1,𝔫2,…,𝔫d of πiTP(𝒯/𝕊[z];ℤp)∨. Combining (2.9) with Theorem 2.13, we obtain an isomorphism πiTP(𝒯/𝕊[z];ℤp)∨⊗𝔖,ϕAinf≃πiTP(𝒯𝒪𝒞;ℤp)∨. For any element y∈πiTP(𝒯𝒪ℂ;ℤp)∨, let us denote by y¯ the image of y under the natural inclusion πiTP(𝒯/𝕊[z];ℤp)∨↪πiTP(𝒯/𝕊[z];ℤp)∨⊗𝒮,ϕ𝔸inf≃πiTP(𝒯𝒪ℂ;ℤp)∨. Choose elements by1,by2,…,byd of Ainf so that
τ~∗(y¯)=by1𝔫l¯+by2𝔫2¯+⋯+byd𝔫d¯.
Remark 4.12 directly induces the following.
Corollary 4.13.
For any y, by1,by2,…,byd are contained in 𝔖(2) under the inclusion 𝔖(2)↪𝜄Ainf.
<inline-formula><alternatives><mml:math display="inline"><mml:mover accent="true"><mml:mi>τ</mml:mi><mml:mo>~</mml:mo></mml:mover></mml:math><tex-math><![CDATA[\(\tilde{\tau}\)]]></tex-math></alternatives></inline-formula>-action and crystalline representations
Let ϕ1:𝔖→Ainf be a W-linear map which sends z to [ε][π♭]. The following diagram commutes:
Let E1 denote θ1(E), E0 denote θ0(E) and ξ1 denote τ~(ξ). We note that 𝔖(2) is an integral domain (see [14, Proof of Lemma 2.3.2]), and ι(E1)=ξ1. Firstly, we prove the following lemma.
Lemma 4.14.
As sub-rings of Ainf[1ξ1], there is an equation
𝔖(2)=𝔖(2)[1E1]∩Ainf
of sub-algebra Ainf[1ξ1], where we regard 𝔖(2)[1E1] as a sub-ring of Ainf[1ξ1] via ι[1E1]:𝔖(2)[1E1]↪Ainf[1ξ1].
Proof.
It suffices to show that 𝔖(2) contains 𝔖(2)[1E1]∩Ainf. The map ι induces a morphism of W-algebra ι¯:𝔖(2)/(E1)→Ainf/(ξ1) and θ1 induces a morphism of W-algebra θ1¯:𝔖/(E)→𝔖(2)/(E1). The isomorphism (4.6) induces an isomorphism of 𝔖-algebra 𝔖⊗𝔖,θ0𝔖(2)≃𝔖⊗𝔖,θ1𝔖(2) which sends E0 to E1. By [14, Lemma 2.2.8 (2)], θ0 induces an isomorphism 𝒪K≃𝔖/(E)≃𝔖(2)/(E0)≃𝔖(2)/(E1) of W-algebras. Thus we obtain that ι¯ is given by natural W-algebra 𝒪K↪𝒪𝒞. We assume that there exists an element x of 𝔖(2)[1E1]∩Ainf⊂Ainf[1ξ1] such that x is not contained in 𝔖(2). From the fact that x∈𝒮z0,z1[1E1], it follows that, there exists a natural number n≥1 such that E1n⋅x is in 𝔖(2) and E1n−1⋅x is not in 𝔖(2). Since 𝔖(2) is an integral domain, E1n⋅x∉(E1)⊂𝔖(2). Thus the class E1n⋅x¯ is not zero in 𝔖(2)/(E1). On the other hand, since x∈Ainf, E1n⋅x∈(ξ1)⊂Ainf. Thus the class E1n⋅x is zero in Ainf/(ξ1). This is contradictory to the fact that ι¯:𝔖(2)/(E1)→Ainf/(ξ1) is injective.∎
Fix a smooth proper 𝒪K-linear category 𝒯 and an integer i≫0. The finite free 𝔖-module πiTC−(𝒯/𝕊[z];ℤp)∨ carries a natural (φ,G^)-module structure by Theorem 3.6 and Remark 4.2. By functoriality of can(−), we obtain the following commutative diagram:
Note that can𝒯∨ is 𝔖-linear, and can𝒯∨⊗idAinf is Ainf-linear.
We choose an 𝒮-basis {𝔫1,…,𝔫d} for the module πiTCn(𝒯/𝕊[z];ℤp)∨. For any element x in this module, we let x¯ denote its image under the natural inclusion
πiTC−(𝒯/𝕊[z];ℤp)∨↪πiTC−(𝒯/𝕊[z];ℤp)∨⊗𝔖,ϕAinf.
Since {𝔫1¯,…,𝔫d¯} is an Ainf-basis of πiTC−(𝒯/𝕊[z];ℤp)∨⊗𝔖,ϕAinf, for each x∈πiTC−(𝒯/𝕊[z];ℤp)∨ there exist elements ax1,…,axd∈Ainf such that
τ¯∗(x¯)=ax1𝔫1¯+ax2𝔫2¯+⋯+axd𝔫d¯.
Proposition 4.15.
For any x, the elements ax1,ax2,…,axd lie in 𝔖(2) under the inclusion 𝔖(2)↪𝜄Ainf.
Proof.
After inverting E, both can𝒯∨ and the inclusion
πiTC−(𝒯/𝕊[z];ℤp)∨↪πiTC−(𝒯/𝕊[z];ℤp)∨
become isomorphisms (see (2.11)). Hence, for any x∈πiTC−(𝒯/𝕊[z];ℤp)∨, there exists y∈πiTP(𝒯/𝕊[z];ℤp)∨ such that
f(y)=Em⋅xinπiTC−(𝒯/𝕊[z];ℤp)∨.
Choose an 𝔖-basis 𝔫1,…,𝔫d of πiTP(𝒯/𝕊[z];ℤp)∨. There exist elements cjl∈𝔖 (for 1≤j≤d, 1≤l≤n) such that
can𝒯∨(𝔫j)=cj1𝔪1+cj2𝔪2+⋯+cjn𝔪n
in πiTC−(𝒯/𝕊[z];ℤp)∨. Since the upper square of (4.13) is 𝔖-linear, we have
in Ainf. By diagram (4.1) and Corollary 4.13, the right-hand side lies in 𝔖(2). Hence axl∈𝔖(2)[1/E1]∩Ainf=𝔖(2).∎
Corollary 4.16.
The p-adic representation TAinf(πiTC−(𝒯/𝕊[z];ℤp)∨) is a ℤp-lattice of a crystalline representation.
Proof.
By [14, Corollary 3.3.4], it suffices to show that τ~∗(πiTC−(𝒯/𝕊[z];ℤp)∨) is contained in πiTC−(𝒯/𝕊[z];ℤp)∨⊗𝔖,θ0𝔖(2). By Proposition 4.5, we obtain the claim.∎
Remark 4.17.
We expect a more geometric proof of the corollary by showing that, for each i, the group πiTC−(𝒯/𝕊[z];ℤp) underlies a prismatic F-crystal. In particular, the associated descent datum πiTC−(𝒯/𝕊[z];ℤp)⇉πiTC−(𝒯/𝕊[z];ℤp)⊗𝔖𝔖(2)⇶πiTC−(𝒯/𝕊[z];ℤp)⊗𝔖𝔖(3) should be induced by the canonical maps
However, we do not currently know whether the natural base change map
πiTC−(𝒯/𝕊[z];ℤp)⊗𝔖𝔖(3)→∼πiTC−(𝒯/𝕊[z0,z1,z2];ℤp)
is an isomorphism. This result depends on the structure of π∗TC−(𝒪K/𝕊[z0,z1,z2];ℤp), which is currently not fully understood.
The proof of Main Theorems
The following theorem is the summary of Sections 3 and 4.
Theorem 4.18.
Let 𝒯 be a smooth proper 𝒪K-linear category. Then there exists an integer n≥0 such that, for all i≥n, the following statements hold:
Breuil–Kisin module πiTC−(𝒯/𝕊[z];ℤp)∨ has a Breuil–Kisin GK-module structure in the sense of [19].
Breuil–Kisin module πiTC−(𝒯/𝕊[z];ℤp)∨ has a (φ,G^)-module structure in the sense of [14].
The
ℤp[GK]-module
TAinf(πiTC−(𝒯/𝕊[z];ℤp)∨) is a
ℤp-lattice of a crystalline representation.
If
𝒯𝒞 admits a geometric realization, then there is a
GK-equivariant isomorphism
TAinf(πiTC−(𝒯/𝕊[z];ℤp)∨)≃πiLK(1)K(𝒯𝒞)∨
of
ℤp-modules.
Proof.
(1) follows from Theorem 3.6. (2) follows from Remark 4.2. (3) follows from Corollary 4.16. (4) follows from Theorem 3.7.∎
Theorem 4.19
(Main Theorem). Let 𝒯 be a smooth proper 𝒪K-linear category. If 𝒯𝒞 admits a geometric realization, then Conjecture 1.2 holds for 𝒯, that is, there is an isomorphism of Bcrys-module
πiTP(𝒯k;ℤp)⊗WBcrys≃πiLK(1)K(𝒯𝒞)⊗ℤpBcrys
which is compatible with GK-action and Frobenius endomorphism.
Proof.
Firstly, we will prove the claim when i is large enough. By Remark 3.8, we have a GK-equivariant isomorphism
πiLK(1)K(𝒯𝒞)∨∨≃TAinf(πiTC−(𝒯/𝕊[z];ℤp)∨∨),
thus we have an isomorphism
πiLK(1)K(𝒯𝒞)∨∨⊗ℤpBcrys≃πiTC−(𝒯/𝕊[z];ℤp)∨∨⊗𝔖Bcrys
which is GK-equivariant and compatible with Frobenius endomorphism, and there is an identification of rational Dieudonné modules (see [7, Remark 4.5])
and we have an isomorphism π2TP(k;ℤp)[1p]≃W[1p](1) (see [8]), where the twist refers to twisting the Frobenius. Thus we obtain
πiTP(𝒯k;ℤp)[1p](1)≃πi+2TP(𝒯k;ℤp)[1p].
Since we already proved the claim when i is large enough, we obtain the claim.∎
Acknowledgements
The author would like to thank Federico Binda, Lars Hesselholt, Buntaro Kakinoki, Shunsuke Kano, Hyungseop Kim and Hiroyasu Miyazaki for helpful discussions related to this subject. The author would also like to thank Ryomei Iwasa for comments on an earlier draft, Alexender Petrov for useful comments on a draft, Tasuki Kinjo for helpful discussions about derived schemes and Isamu Iwanari, Atsushi Takahashi and Shinnosuke Okawa for helpful discussions about non-commutative algebraic geometry. The author is deeply grateful to Hui Gao for helpful discussions about (φ,G^)-modules and for sharing his ideas about Theorems 1.11 and 1.12 with us. The author is also grateful to the anonymous referee for a careful reading of the manuscript and for many insightful comments and suggestions, which greatly improved the paper.
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