Hodge-Arakelov-theoretic Evaluation I

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Hodge-Arakelov-theoretic Evaluation I

Yuichiro Hoshi

RIMS, Kyoto University

December 10, 2015

Notation and Terminology

“∆”: the geometric portion of “Π”,

i.e., the kernel of the outer surjection from “Π” to the arithmetic quotient of “Π”

d(−): the profinite completion of (−) For a topological groupG,

∞Hⁱ(G, A)^def= lim−→^H⊆G:open subgps of finite indexHⁱ(H, A)

§ 2 Galois-theoretic Theta Evaluation

In§2, §2¹₂, and §3: Fix a v ∈V^bad.

Π_v ^def= Π^tp_X

v

∃func’l alg’m⇒

Π^tp_Y¨

v

−−−→ Π^tp_Y

v −−−→ Πv = Π^tp_X

 v

y y y Π^tp_¨

Yv −−−→ Π^tp_Y

v −−−→ Π^±_v ^def= Π^tp_X

 v

y

Π^cor_v ^def= Π^tp_Cv M^Θ_∗ = (M^ΘM)_M: a projective system of mono-theta environments F_v: the tempered Frobenioid determined by X

v

Goal: Θ

v

evaluation labeled by⇒ t q^t2

v (t ∈LabCusp^± atv)

Problem 1 A conjugacy indeterminacy in a situation related to the theta value “q^t2

v” at|t| ∈ |Fl|depends, a priori, on the label |t| ∈ |Fl|. That is to say, various objects at |t| ∈ |Fl|is well-defined up to conjugation which is, a priori, independent of the label|t| ∈ |Fl|. On the other hand, we want to establish a suitable Kummer theory for such theta values.

⇒We have to synchronizes conjugacy indeterminacies in situations related to the theta values at various |t| ∈ |Fl|.

Problem 2 By using the structure of a Hodge-theater, we synchronized globally the various “LabCusp^±_v” by means of — relative to^†D_≻,v

through0

→ ^†D^⊚± (^through← ^0†D_≻,w) — the var. bij.

{cuspidal inertia subgps ofπ₁^tp(X

v)}/Inn(π₁^tp(X

v))

→ {∼ cuspidal inertia subgps of π₁^´et(X_K)}/Inn(π^´et₁ (X_K)).

Moreover, the crucial globalF^⋊±_l -symmetry arises from a profinite conjugation (cf. π^´et₁ (C_F)/π₁^´et(X_F)→^∼ π₁^´et(C_K)/π^´et₁(X_K)∼=F^⋊±_l ).

⇒We have to discuss the comparison between tempered conjugation and profinite conjugation.

Γ₍₋₎: the dual graph of the special fiber of (−)_v

Irr(−): the set of irreducible components of the special fiber of (−)v

Fix an inversion automorphism ι^def= ιY¨ of Y¨

v

⇒ι_X,ιY¨, and Z→^∼ Irr( ¨Y) (∼= Irr(Y)∼= Irr( ¨Y)∼= Irr(Y)), (Note: ι↷Irr ⇝ −1↷ Z)

⇒a subgraph Γ^▶₍₋₎ ⊆Γ₍₋₎ for (−)∈ {X, X,Y ,¨ Y¨}, i.e.,

−l^⋇

•−−−−−^−l⋇•−− · · · −−^{+1 −}•−−−−−¹ •−−−−−⁰ •−− · · · −−^{1 l⋇}•−−−−−^{−1 l}•^⋇ t∈LabCusp^±(Πv)

⇒a subgraph Γ^•₍₋₎^t ⊆Γ^▶₍₋₎ for (−)∈ {X, X,Y ,¨ Y¨}, i.e.,

•t ⊆ ⁻•−−−−−^{l⋇ −l⋇}•−− · · · −−^{+1 −}•−−−−−¹ •−−−−−⁰ •−− · · · −−^{1 l⋇}•−−−−−^{−1 l}•^⋇

“•” ^def= “•0” (i.e., for instance, Γ^•₍₋₎ ^def= Γ^•₍₋₎⁰ )

□∈ {•t,▶}

Π_v□ ⊆Π_v: a decomposition subgroup ofΓ^□_X ⊆Γ_X, i.e., “Π^tp_X,Γ□

X

” Π^±_v□ ^def= N_Π±

v(Π_v□) ⊆ Π^±_v Π_v□¨

def= Π_v□∩Π^tp_¨

Yv

⊆ Π^tp_¨

Yv

Thus, for instance:

Π^±_v□/Π_v□ →^∼ Π^±_v/Π_v →^∼ Gal(X

v/X_v) (∼=µ_l), Π^±_v□∩Π_v = Π_v□ [Π_v□ : Π_v□¨] = 2, [Π^±_v□ : Π_v□] =l, ...

Then:

Π_v ^∃func’l⇒

alg’m (Π_v• ⊆Π_v▶ ⊆Π_v, ι) well-defined up to Π_v-conj.

(Recall: Π_v ^∃func’l⇒

alg’m θ(Π_v)⊆_∞θ(Π_v)⊆_∞H¹(Π^tp_¨

Y(Π_v),(l·∆_Θ)(Π_v))) Thus:

Π_v ^∃func’l⇒

alg’m θ^ι(Π_v)⊆θ(Π_v), _∞θ^ι(Π_v)⊆_∞θ(Π_v): µ_2l-, µ-torsors Moreover: Π_v ^∃func’l⇒

alg’m

(l·∆_Θ)(Π_v▶¨): the subquotient of Π_v▶¨ det’d by (l·∆_Θ)(Π_v) Π_v ↠G_v(Π_v): the arithmetic quotient of Π_v

Π_v▶¨ ↠G_v(Π_v▶¨): the arithmetic quotient of Π_v▶¨

.Key Lemma (Comparison Between Temp’d Conj. and Prof. Conj.) ..

...

I_t ⊆Π_v: an inertia subgp ass’d to the cusp lab’d by t s.t. I_t ⊆∆_v□ γ,γ^′ ∈∆b^±_v

Then the following three conditions are equivalent:

• γ^′ ∈∆^±_v□ • I_t^(γ·γ′) ⊆Π^γ_v□ • I_t^(γ·γ′) ⊆(Π^±_v□)^γ where(−)^{γ def}= γ·(−)·γ⁻¹

Key Lemma follows from the theory of semi-graphs of anabelioids.

In the situation of Key Lemma, write δ^def= γ·γ^′ ∈∆b^±_v. (^Lemma⇒ I_t^δ =I_t^(γ·γ′) ⊆Π^γ_v□ = Π^δ_v□)

By the theory of semi-graphs of anabelioids,

one can construct, from the inclusions I_t^δ =I_t^(γ·γ′) ⊆Π^γ_v□ = Π^δ_v□: (a) the dec. gpD^δ_t ^def= N_Π^δ_v(I_t^δ)⊆Π^δ_v that contains I_t^δ

(b) a dec. gpD^δ_µ− ⊆Π^δ_v▶¨, well-def’d up to (Π^±_v▶¨)^δ-conj., ass’d to µ₋ (c) a dec. gpD^δ_t,µ− ⊆Π^δ_v□¨, well-def’d up to (Π^±_v□¨)^δ-conj., ass’d to the µ₋-translation of the cusp that det. I_t^δ (i.e., an ev. pt lab’d by t) Moreover, this construction is compatible w/:

• the ∆b^±_v-conjugation • the inclusion Π_v•t⊆Π_v▶ If, moreover, □=•t, then the construction of (a) and (c) is compatible w/ theΠb^cor_v -conjugation.

(Recall: the F^⋊±_l -symmetry arises from theΠb^cor_v -conjugation.)

. ...

I_t^δ =I_t^(γ·γ′) ⊆Π^δ_v▶¨ ⊆Π^γ_v▶ = Π^δ_v▶

By restrictingθ^ι(Π^γ_v)⊆∞θ^ι(Π^γ_v) to Π^γ_v▶¨ ⊆Π^tp_¨

Y (Π^γ_v), we obtain µ_2l-, µ-torsors

θ^ι(Π^γ_v▶¨) ⊆ _∞θ^ι(Π^γ_v▶¨) ⊆ _∞H¹(Π^γ_v▶¨,(l·∆_Θ)(Π^γ_v▶¨)).

Thus, by restricting them to(Gv(Π^γ_v▶¨)←^∼)D^δ_t,µ− ⊆Π^γ_v▶¨, we obtain θ^t(Π^γ_v▶¨) ⊆ ∞θ^t(Π^γ_v▶¨) ⊆ ∞H¹(Gv(Π^γ_v▶¨),(l·∆Θ)(Π^γ_v▶¨)), i.e., “µ_2l·q^t2

v ⊆µ·q^t2

v ”, where −l^⋇ ≤t ≤l^⋇ is det’d by t.

(∞)θ^|t|(Π^γ_v▶¨) ^def= _(∞)θ^t(Π^γ_v▶¨) (= _(∞)θ^−t(Π^γ_v▶¨))

In summary:

Π_v ^∃func’l⇒

alg’m {θ^|t|(Π^γ_v▶¨)}|t|∈|Fl|, {∞θ^|t|(Π^γ_v▶¨)}|t|∈|Fl| arising from

• Π^γ_v▶¨: an arbitrary ∆b^±_v-conjugate of Π_v▶¨

• I_t^δ: an arbitrary ∆b^±_v-conjugate of It s.t. I_t^δ ⊆Π^γ_v▶¨ (t ranges over the elements of LabCusp^±(Π^γ_v)

γ-conj.

→∼ LabCusp^±(Π_v)) Moreover, this alg’m is compatible w/ the independent conj. actions of ∆b^±_v on the sets of (not temp’d but) prof. conj. {Π^γ_v▶¨}γ and {I_t^δ}δ.

.Remark ..

...

A conjugacy indeterminacy in a situation related to the theta value

“_(∞)θ^|t|” at |t| ∈ |Fl| depends, a priori, on the label |t| ∈ |Fl|. That is to say, various objects at |t| ∈ |Fl|is well-defined up to conjugation which is, a priori, independent of the label|t| ∈ |Fl|. However, our resulting theta values

(∞)θ^|t|(Π^γ_v▶¨)⊆_∞H¹(G_v(Π^γ_v▶¨),(l·∆_Θ)(Π^γ_v▶¨))

for various|t| ∈ |Fl| are computed relative to “label-independent”

G_v(Π^γ_v▶¨) and (l·∆_Θ)(Π^γ_v▶¨). conjugate synchronization (⇒ One may apply Kummer theory related to theta values.)

Recall: Π_v ^∃func’l⇒

alg’m M_TM^× (Π_v)⊆(M_TM^× ·θ^ι)(Π_v)⊆(M_TM^× ·∞θ^ι)(Π_v) in _∞H¹(Π_v,(l·∆_Θ)(Π_v))

(Recall: “M_TM^× ” is an isomorph of “O^×_F

v”.) By restricting them to Π^γ_v▶¨ ⊆Π_v, we obtain

M_TM^× (Π^γ_v▶¨)⊆(M_TM^× ·θ^ι)(Π^γ_v▶¨)⊆(M_TM^× ·∞θ^ι)(Π^γ_v▶¨) in _∞H¹(Π^γ_v▶¨,(l·∆_Θ)(Π^γ_v▶¨))

Thus, by the nat’l “_∞θ^ι(Π^γ_v▶¨)↠∞θ⁰(Π^γ_v▶¨)”, we obtain a splitting (M_TM^× ·∞θ^ι)(Π^γ_v▶¨)/M_TM^µ (Π^γ_v▶¨) = M_TM^×µ(Π^γ_v▶¨)×(

∞θ^ι(Π^γ_v▶¨)/M_TM^µ (Π^γ_v▶¨) )

. (Recall:“M_TM^µ ”(resp.“M_TM^×µ”) is an isomorph of “O^µ_F

v”(resp.“O_F^×µ

v”).)

In the remainder of§2, suppose: Π^tp_X(M^Θ_∗) = Π_v

⇒1→Π_µ(M^Θ_∗)→Π_M^Θ_∗ →Π^tp_X(M^Θ_∗)

By base-chan’g Π_v▶¨ ⊆Π_v▶ ⊆Π_v = Π^tp_X(M^Θ_∗)via Π_M^Θ_∗ →Π^tp_X(M^Θ_∗), we obtain closed subgroupsΠ_MΘ

∗▶¨ ⊆Π_MΘ

∗▶ ⊆Π_MΘ

∗. Π_µ(M^Θ_∗▶¨), (l·∆_Θ)(M^Θ_∗▶¨),Π_v▶¨(M^Θ_∗▶¨), G_v(M^Θ_∗▶¨):

the respective “corresponding subquotients” of Π_MΘ

∗▶¨

⇒ ∃a cyclotomic rigidity isomorphism (l·∆_Θ)(M^Θ_∗▶¨)→^∼ Π_µ(M^Θ_∗▶¨)

By applying the cycl. rig. isom. (l·∆_Θ)((M^Θ_∗▶¨)^γ)→^∼ Π_µ((M^Θ_∗▶¨)^γ) arising from the “γ-conjugate” (M^Θ_∗)^γ of M^Θ_∗ (where γ ∈∆b^±_v) to

θ^ι(Π^γ_v▶¨)⊆∞θ^ι(Π^γ_v▶¨)⊆∞H¹(Π^γ_v▶¨,(l·∆Θ)(Π^γ_v▶¨)),

θ^|t|(Π^γ_v▶¨)⊆_∞θ^|t|(Π^γ_v▶¨)⊆_∞H¹(G_v(Π^γ_v▶¨),(l·∆_Θ)(Π^γ_v▶¨)), and

(M_TM^× ·∞θ^ι)(Π^γ_v▶¨)/M_TM^µ (Π^γ_v▶¨) =M_TM^×µ(Π^γ_v▶¨)×(_∞θ^ι(Π^γ_v▶¨)/M_TM^µ (Π^γ_v▶¨)),

we obtain

θ^ι_env((M^Θ_∗▶¨)^γ)⊆∞θ^ι_env((M^Θ_∗▶¨)^γ)⊆∞H¹(Π_v▶¨((M^Θ_∗▶¨)^γ),Πµ((M^Θ_∗▶¨)^γ), θ^|t|

env((M^Θ_∗▶¨)^γ)⊆_∞θ^|t|

env((M^Θ_∗▶¨)^γ)⊆_∞H¹(Gv((M^Θ_∗▶¨)^γ),Πµ((M^Θ_∗▶¨)^γ)), (M_TM^× ·∞θ^ι

env)((M^Θ_∗▶¨)^γ)/M_TM^µ ((M^Θ_∗▶¨)^γ)

=M_TM^×µ((M^Θ_∗▶¨)^γ)×(

∞θ^ι

env((M^Θ_∗▶¨)^γ)/M_TM^µ ((M^Θ_∗▶¨)^γ) )

.

In a similar vein, by applying the cycl. rig. isom.

(l·∆_Θ)((M^Θ_∗▶¨)^γ)→^∼ µ_Zb(G_v((M^Θ_∗▶¨)^γ)), we obtain:

θ^ι

bs((M^Θ_∗▶¨)^γ)⊆_∞θ^ι

bs((M^Θ_∗▶¨)^γ)

⊆_∞H¹(Π_v▶¨((M^Θ_∗▶¨)^γ),µ_bZ(G_v((M^Θ_∗▶¨)^γ))),

θ^|t|

bs((M^Θ_∗▶¨)^γ)⊆∞θ^|t|

bs((M^Θ_∗▶¨)^γ)

⊆_∞H¹(Gv((M^Θ_∗▶¨)^γ),µ_bZ(Gv((M^Θ_∗▶¨)^γ))),

(M_TM^× ·∞θ^ι_bs)((M^Θ_∗▶¨)^γ)/M_TM^µ ((M^Θ_∗▶¨)^γ)bs

=M_TM^×µ((M^Θ_∗▶¨)^γ)bs×(

∞θ^ι_bs((M^Θ_∗▶¨)^γ)/M_TM^µ ((M^Θ_∗▶¨)^γ)bs

) .

§ 2

Multiradial Kummer-detachment of Theta Monoids

Goal: “multiradial Kmm-detach.” of theta monoids, i.e., “O_F^×_v ·Θ

v” Strategy: By the final assertion of §1, we have:

Π_v multiradial

alg’m⇝ ´etale-like O^×_F

v·Θ

v via multiradial

cycl. rig.⇝ mono-theta O^×_F

v ·Θ

v , i.e., labeled by “env”

Thus, by applying the Kummer theory for theta functions:

Frobenius-like O_F^×

v ·Θ

v

Kummer theory⇝ (O_F^×

v·Θ

v)_env

• Recall: F_v ^∃func’l⇒

alg’m µ_2l(T^÷Y¨

v

)·Θ

v ⊆ O^×(T^÷Y¨

v

) (which determines the monoid O_C^▷Θ

v(A^Θ_∞) =O^×_CΘ

v(A^Θ_∞)·Θ^N

v|A^Θ_∞) Ψ_Fv^Θ =

{

Ψ_Fv^Θ_,α^def= O_C^×Θ

v(A^Θ_∞)·(Θ^α

v)^N|A^Θ_∞

}

α∈Aut_Dv( ¨Y

v)

∞Ψ_Fv^Θ = {

∞Ψ_Fv^Θ_,α^def= O_C^×Θ

v(A^Θ_∞)·(Θ^α

v)^Q≥0|A^Θ_∞

}

α∈Aut_Dv( ¨Y

v)

• Recall: F_v ^∃func’l⇒

alg’m the base-theoretic hull Cv ⊆ F_v

(Πv ↷) Ψ_Cv ^def= O_C^▷_v(A^Θ_∞)(well-defined up to Πv-conjugation) .

...

(∞)Ψ_Fv^Θ: the Frobenius-like theta monoid Ψ_Cv: the Frobenius-like constant monoid

• Recall: M^Θ_∗ ^∃func’l⇒

alg’m M_TM^× (M^Θ_∗), θ

env(M^Θ_∗)⊆_∞θ

env(M^Θ_∗) in _∞H¹(Π^tp_¨

Y (M^Θ_∗),Π_µ(M^Θ_∗)) Ψ_env(M^Θ_∗)^def=

{

Ψ^ι_env(M^Θ_∗)^def= M_TM^× (M^Θ_∗)·θ^ι

env(M^Θ_∗)^N }

ι:inv. autom.

∞Ψ_env(M^Θ_∗)^def= {

∞Ψ^ι_env(M^Θ_∗)^def= M_TM^× (M^Θ_∗)·_∞θ^ι

env(M^Θ_∗)^Q≥0 }

ι:inv. aut.

• Recall: M^Θ_∗ ^∃func’l⇒

alg’m G_v(M^Θ_∗), µ_bZ(G_v(M^Θ_∗))→^∼ Π_µ(M^Θ_∗)

∃func’l

alg’m⇒ M_TM(M^Θ_∗)⊆∞H¹(Π^tp_¨

Y(M^Θ_∗),Π_µ(M^Θ_∗)) (Recall: “M_TM” is an isomorph of “O^▷_F

v”.) (Π^tp_X(M^Θ_∗)↷) Ψ_cns(M^Θ_∗)^def= M_TM(M^Θ_∗) .

...

(∞)Ψ_env(M^Θ_∗): the mono-theta-theoretic theta monoid Ψ_cns(M^Θ_∗): the mono-theta-theoretic constant monoid

In particular, by applying the above algorithm toM^Θ_∗(Π_v):

. ...

(∞)Ψ_env(M^Θ_∗(Π_v)): the ´etale-like theta monoid Ψ_cns(M^Θ_∗(Π_v)): the ´etale-like constant monoid

In order to obtain “multiradial Kummer-detachment” of “Θ

v”, let us relate

Frobenius-like/mono-theta-theoretic/´etale-like theta monoids.

In the remainder of§2¹₂, suppose: M^Θ_∗(F_v) = M^Θ_∗

Then, by applying the Kummer theory, relative to a suitable assign’t

“ι 7→α”, we obtain an isomorphism _(∞)Ψ_FΘ v,α

→∼ (∞)Ψ^ι_env(M^Θ_∗)

(cf. the Kummer theory of theta functions in tempered Frobenioids).

Write

(∞)Ψ_Fv^Θ −→^∼ (∞)Ψ_env(M^Θ_∗) for the collection of the above isomorphisms.

Moreover, again by applying the Kummer theory, we obtain an isomorphism

Ψ_Cv −→^∼ Ψcns(M^Θ_∗).

Thus, every isomorphism M^Θ_∗(Π_v)→^∼ M^Θ_∗ =M^Θ_∗(F_v) determines:

(´etale) (mono-theta) (Frobenius) Π_v = Π^tp_X(M^Θ_∗(Π_v)) →^∼ Π^tp_X(M^Θ_∗) = Π^tp_X(M^Θ_∗)

↷ ↷ ↷

∞Ψ_env(M^Θ_∗(Π_v)) →^∼ ∞Ψ_env(M^Θ_∗) ←^∼ ∞Ψ_FΘ v

∪ ∪ ∪

Ψ_env(M^Θ_∗(Π_v)) →^∼ Ψ_env(M^Θ_∗) ←^∼ Ψ_FΘ v

G_v =G_v(M^Θ_∗(Π_v)) →^∼ G_v(M^Θ_∗) = G_v(M^Θ_∗)

↷ ↷ ↷

Ψ_env(M^Θ_∗(Π_v))^× →^∼ Ψ_env(M^Θ_∗)^× ←^∼ Ψ^×_FΘ v

Moreover, every isom. M^Θ_∗(Π_v)→^∼ M^Θ_∗ =M^Θ_∗(F_v) also determines:

(´etale) (mono-theta) (Frobenius) Π_v = Π^tp_X(M^Θ_∗(Π_v)) →^∼ Π^tp_X(M^Θ_∗) = Π^tp_X(M^Θ_∗)

↷ ↷ ↷

Ψcns(M^Θ_∗(Πv)) →^∼ Ψcns(M^Θ_∗) ←^∼ Ψ_Cv

Gv =Gv(M^Θ_∗(Πv)) →^∼ Gv(M^Θ_∗) = Gv(M^Θ_∗)

↷ ↷ ↷

Ψcns(M^Θ_∗(Πv))^× →^∼ Ψcns(M^Θ_∗)^× ←^∼ Ψ^×_Cv That is to say, we obtain various Kummer isomorphisms.

Thus, by the final assertion of§1, we obtain

“multiradial Kummer-detachment” of theta monoids:

Π_v multiradial

alg’m (cf.⇝ §1)(∞)Ψ_env(M^Θ_∗(Π_v))

via multiradial

←∼

cycl. rig. (∞)Ψ_FΘ v

.Remark ..

...

On the other hand, the above discussion only gives

“uniradial Kummer-detachment” of constant monoids.

(cf. “(Π_v ↷Ψ_Cv)⇝(G_v(M^Θ_∗)↷Ψ^×_C

v)”: a uniradial environment) (⇒the theory of log-shells)

§ 2

Definition (used in § 4)

(1) v ∈V^non ⇒ Fv^⊢×: the Fro’d “corresponding to”Gv ↷O^×_F

v

(omit the case ofv ∈V^arc)

an F^⊢×-prime-strip ^def⇔ {an isomorph of Fv^⊢×}v∈V

(2) v ∈V^non ⇒ Fv^⊢×µ: the×µ-Kummer Frobenioid “corresponding to”G_v ↷O^×_F^µ

v equipped with the×µ-Kummer structure, i.e., {Im(

(O^×_F

v)^H =O^×

F^H_v ,→ O^×_F

v ↠O_F^×µ

v

)}H⊆Gv:open subgps

(omit the case ofv ∈V^arc)

an F^⊢×µ-prime-strip ^def⇔ {an isomorph of Fv^⊢×µ}v∈V

(3) ^†F^⊢ ={^†Fv^⊢}v∈V: an F^⊢-prime-strip v ∈V^bad ⇒^†F_v^⊢ “corresponds to” G_v ↷(O_F^×

v×q^N

v

modµ_2l

←- q^N

v)

†Fv^⊢▶×µ: the split-×µ-Kummer Frobenioid “corresponding to”

G_v ↷(O_F^×µ

v ×(µ_2l·q^N

v/µ_2l)←-(µ_2l·q^N

v/µ_2l))w/ ×µ-Kmm str.

v ∈V^good∩V^non ⇒ ^†F_v^⊢ “corresponds to” G_v ↷(O_F^×

v ×p^N_v ←- p^N_v)

†Fv^⊢▶×µ: the split-×µ-Kummer Frobenioid “corresponding to”

G_v ↷(O_F^×µ

v ×p^N_v ←- p^N_v)w/ ×µ-Kmm str.

(omit the case ofv ∈V^arc)

an F^⊢▶×µ-prime-strip ^def⇔ {an isomorph of F_v^⊢▶×µ}v∈V

(4) anF^⊩▶×µ-prime strip ^def⇔ a suitable collection of data (^†C^⊩,Prime(^†C^⊩)→^∼ V,^†F^⊢▶×µ,{^†ρ_v}v∈V), i.e., a collection of data obtained by replacing the “^†F^⊢” of an F^⊩-prime strip “(^†C^⊩,Prime(^†C^⊩)→^∼ V,^†F^⊢,{^†ρ_v}v∈V)” by an F^⊢▶×µ-prime-strip ^†F^⊢▶×µ

Thus:

†F^⊢-prime-strip ^∃func’l⇒

alg’m

†F^⊢×-prime-strip ^∃func’l⇒

alg’m

†F^⊢×µ-prime-strip

†F^⊢-prime-strip ^∃func’l⇒

alg’m

†F^⊢▶×µ-prime-strip

†F^⊩-prime-strip ^∃func’l⇒

alg’m

†F^⊩▶×µ-prime-strip