RATIONAL CIRCLE-EQUIVARIANT ELLIPTIC COHOMOLOGY OF CPV MATTEO BARUCCO Abstract. We compute rational T-equivariant elliptic cohomology of CPV where Tis the circle

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RATIONAL CIRCLE-EQUIVARIANT ELLIPTIC COHOMOLOGY OF CP(V)

MATTEO BARUCCO

Abstract.

We compute rational

-equivariant elliptic cohomology of

CP(V)

, where

is the circle

group, and

CP(V)

is the

-space of complex lines for a ﬁnite dimensional complex

-representation

. Starting from an elliptic curve

over

and a coordinate data around the identity, we achieve

this computation by proving that the

-equivariant elliptic cohomology theory

ECT

built in [Gre05],

and the

-equivariant elliptic cohomology theory

ECT2

built in [Bar22] are

-split, for the

subgroup

H1= 1 ×T<T2

. This result allows us to reduce the computation of

ECT(CP(V))

to the

computation of

-elliptic cohomology of spheres of complex representations, already performed in

[Bar22, Theorem 1.4].

Contents

1. Introduction 1

2. Elliptic cohomology of CP(V) 6

3. The circle case revisited 10

4. Building the map 12

5. Proving the H-equivalence 15

Appendix A. Algebraic models 23

References 29

1. Introduction

1.1.

Motivation.

Two of the most prominent examples of topological cohomology theories have a

one dimensional formal group associated: ordinary cohomology is associated to the additive formal

group, while complex

-theory is associated to the multiplicative one. More generally one can

associate to every complex orientable cohomology theory a one dimensional formal group [Qui69].

A great source of formal groups is the formal completion of a one dimensional algebraic group at

the identity. If we are working over an algebraically closed ﬁeld for example, we are soon in short

supply of one dimensional algebraic groups, except for the additive and multiplicative ones all the

others are elliptic curves. These more exotic theories whose associated formal group arise as the

formal completion of an elliptic curve

around the identity are called elliptic cohomology theories,

and were ﬁrst constructed in [Lan88] [LRS95].

Given a compact Lie group

, it is only natural to seek for a

-equivariant counterpart of

elliptic cohomology. If one takes complex K-theory as a model, than we can expect a good theory

of equivariant elliptic cohomology to encode the full algebraic group

, in contrast with the non-

equivariant theory simply encoding the formal completion of

around the identity. This is in

analogy to how equivariant K-theory works. By the Atiyah-Segal completion theorem [AS69] if we

complete the equivariant K-theory of the point at its augmentation ideal, we obtain the K-theory of

BG, the classifying space of G:

KUG(∗)∧

I∼

=KU (BG).

arXiv:2210.05527v1 [math.AT] 11 Oct 2022

Equivariant elliptic cohomology was ﬁrst constructed by Grojnowski in [Gro07] with complex

coeﬃcients, and the speciﬁc motivation to study certain elliptic algebras. The subject has thrived

since then and has found numerous applications to representation theory (for

connected compact

Lie group [Gro07], [GKV95]), to organize moonshine phenomena (especially for

ﬁnite [Dev96],

[Dev98]), and also quite surprisingly to physics [ST11], [Ber21].

Given the complexity of deﬁning Equivariant elliptic cohomology ([Lur18a], [Lur18b], [Lur19],

[GM20]), and its algebraic origin also in the non-equivariant case, it is quite diﬃcult to compute

-equivariant elliptic cohomology of meaningful

-spaces. Computations are often limited to ﬁxed

-spaces or spheres of complex representations [Gre05], [Bar22] to check that the theory deﬁned is

indeed elliptic cohomology. This paper has the goal to enhance computations at least for rational

-equivariant elliptic cohomology, and in particular given an elliptic curve

and a ﬁnite dimensional

complex

-representation

to compute rational

-equivariant elliptic cohomology of

(

): the

T-space of complex lines in V.

More precisely we will prove the following which is the main theorem of the paper.

Theorem 1.1.

For every elliptic curve

over

, if

ECT(3.14)

is the rational

-equivariant elliptic

cohomology theory built in [Gre05], and

is a ﬁnite dimensional complex representation of

, then:

(1) If Vhas one isotypic component, V=αznwith α≥1,

ECk

T(CP(V)) ∼

=Cα−1

for every k∈Z.

(2) If Vhas more than one isotypic component, V=Lnαnzn,

ECk

T(CP(V)) ∼

=(0keven

Cdkodd.

where d=Pi<j αiαj(i−j)2, and zis the natural representation of T.

Part (1) of this theorem simply checks our methods on previously done computations since

(

αzn

)

∼

=CPα−1

with the trivial

-action, and can be found in the paper as Lemma 2.13. In

contrast, part (2) is the main computational result, can be found in the paper as Theorem 2.26 and

to the knowledge of the author it is the ﬁrst time it appears in the literature. We refer to Remark

5.36 for a discussion of the geometry of this Theorem. We revise the construction of

ECT

in Section

Notice that there are diﬀerent representations, with diﬀerent complex projective spaces that

nonetheless have the same value for

and therefore rational

-equivariant elliptic cohomology does

not distinguish them. For example if

with the trivial action,

ε⊕

and

ε⊕z2

have

= 4, or

ε⊕

ε⊕z4

and

W00

ε⊕z⊕

have

= 16. Therefore they have

the same elliptic cohomology even if their complex projective spaces are quite diﬀerent.

In Theorem 1.1 we use the construction of rational

-equivariant elliptic cohomology of Greenlees

[Gre05, Theorem 1.1] with algebraic models. Algebraic models have the advantage to simplify

the picture and to maintain a more evident bond with the geometry of the curve

, which are

essential aspects when performing computations. This construction of rational equivariant elliptic

cohomology using algebraic models has been generalized for the 2-torus

[Bar22], and we will

make essential use of the T2-theory as well in the computation.

1.2.

Algebraic Models.

Algebraic models are a useful tool to study equivariant cohomology

theories: equivariant

-theory, equivariant cobordism, Bredon and Borel cohomology just to name

a few examples. For every compact Lie group

one can construct a category of

-spectra where

every cohomology theory

E∗

(

)is represented by a

-spectrum

, in the sense that for any based

-space

we have

E∗

(

) = [Σ

∞X, E

]

∗

. To simplify the picture one can restrict the focus to

rational

-equivariant cohomology theories, represented by rational

-spectra: the localization of

G-spectra at the rational sphere spectrum.

The main idea of algebraic models [Gre99] is for

a compact Lie group to deﬁne an abelian

category A(G)and an homology functor

(1.2) πA

∗:G-SpectraQ→ A(G)

together with a strongly convergent Adams spectral sequence to compute stable maps in rational

G-spectra:

(1.3) Ext∗,∗

A(πA

∗(X), πA

∗(Y)) =⇒[X, Y ]G

∗

Furthermore for certain groups the homology functor πA

∗can be lifted to a Quillen-equivalence

(1.4) G-SpectraQ'QdA(G)

between rational

-spectra and the category

(

)of diﬀerential graded objects in

(

). The

Adams spectral sequence

(1.3)

is a powerful tool to compute values of known theories, while the

Quillen-equivalence

(1.4)

can be used to build entirely new theories, simply constructing objects in

(

). This latest method is precisely the one used in [Gre05] and [Bar22]: building an object

ECG

(

)and using the Quillen-equivalence

(1.4)

for the circle

and the 2-torus

deﬁne a

-equivariant elliptic cohomology theory. This method works since Greenlees and Shipley

have extended the Quillen-equivalence (1.4) to tori of any rank Tr[GS18, Theorem 1.1].

1.3.

The method of this paper.

We start from an elliptic curve

over

together with a

coordinate function

in the local ring at the identity

. This gives us a rational

-equivariant

elliptic cohomology theory

ECT∈ A

(

)[Gre05, Theorem 1.1], as well as a rational

-equivariant

elliptic cohomology theory

ECT2∈ A

(

)[Bar22, Theorem 1.4]. We have also the complex abelian

surface X=C × C.

Given a complex

-representation

the main goal of this paper is to compute the reduced

cohomology of the pointed space:

EC∗

T(CP(V)).

We start pointing out the isomorphism of T-spaces:

(1.5) CP(V)∼

=S(V⊗Cw)/H1

where

is the natural complex representation of

= 1

×T<T2

and

(

V⊗Cw

)is the

-space

of vectors of unit norm in the complex vector space

V⊗Cw

. Notice that

V⊗Cw

is a complex

representation of

of the same dimension of

, where

acts on the second factor of the tensor

product. The computation is made possible since

ECT

and

ECT2

are

-split, where we view

the quotient group T2/H1:

Theorem 1.6. There is a natural transformation of T2-cohomology theories

(1.7) ε: InfT2

TECT−→ ECT2

which induces an isomorphism

(1.8) [T2/H+,InfT2

TECT]T2

∗∼

=[T2/H+, ECT2]T2

∗

for every subgroup Hof T2such that H∩H1={1}.

We prove this Theorem in Sections 4and 5. More precisely in Section 4we build the map

while

in Section 5we prove the H-equivalence (1.8). As an immediate Corollary:

Corollary 1.9. For any H1-free T2-space X:

(1.10) EC∗

T2(X)∼

=EC∗

T(X/H1)

We can apply this corollary to

(1.5)

, reducing the computation of

-equivariant elliptic cohomology

to a computation of T2-equivariant elliptic cohomology:

(1.11) EC∗

T(CP(V)+)∼

=EC∗

T(S(V⊗w)+/H1)∼

=EC∗

T2(S(V⊗w)+).

The

-equivariant elliptic cohomology of

(

V⊗w

)

is easier to compute in view of the coﬁbre

sequence of T2-spaces:

S(V⊗w)+−→ S0−→ SV⊗w

inducing a long exact sequence:

(1.12) EC∗

T2(SV⊗w)−→ EC∗

T2(S0)−→ EC∗

T2(S(V⊗w)+).

The ﬁrst two terms of

(1.12)

are computed in [Bar22, Theorem 1.4] that we recall here for easy

reference:

Theorem 1.13.

For every elliptic curve

over

and coordinate

te∈ OC,e

, there exists an object

ECT2∈ A

(

)whose associated rational

-equivariant cohomology theory

EC∗

(

)is 2-periodic.

The value on the one point compactiﬁcation

for a complex

-representation

with no ﬁxed

points is given in terms of the sheaf cohomology of a line bundle O(−DW)over X=C × C:

(1.14) ECn

T2(SW)∼

=(H0(X,O(−DW)) ⊕H2(X,O(−DW)) neven

H1(X,O(−DW)) nodd.

The associated divisor

is deﬁned as follows. First our elliptic curve

deﬁnes a functor

from compact abelian Lie groups to complex manifolds. If

is a compact abelian Lie group and

our ﬁxed elliptic curve, deﬁne

X(H) := HomAb(H∗,C).

Where we are considering group homomorphisms, and

H∗

HomLie

(

H, T

)is the character group

. Note this is an exact functor inducing an embedding

(

)

→X

(

)for every embedding

K →H

. The fundamental algebraic surface we are working on is

(

)

∼

=C × C

. Note

(

)

is a subvariety of

of the same dimension of

. For every non-trivial character

consider

the codimension 1subgroup

Ker

(

), then

(

Ker

(

)) is a codimension 1subvariety of

that is the

associated divisor of z0. If W=Pz0αz0z0deﬁne

(1.15) DW:= X

αz0X(Ker(z0)),

where in both sums

runs over all the non-trivial characters of

, and

αz0

is a non-negative

integer.

Remark 1.16.

One might argue that to compute rational

-equivariant elliptic cohomology of

(

)a more straightforward approach can be taken. Namely we have

ECT∈ A

(

)and if we

explicitly compute the algebraic model

πA(T)

∗

(

)

), then we can use the Adams spectral sequence

for the circle to compute directly

EC∗

(

)

). Originally this was the method tried on this

project, with a double motivation: ﬁrst as a warm-up in the circle case before generalizing to

higher tori, and second, once

ECT2

has been constructed, compute

EC∗

(

V⊗w

)

), and see if

the two values match. If indeed

EC∗

(

)

∼

=EC∗

(

V⊗w

)

), then one might expect that

ECT

and

ECT2

are

-split (Theorem 1.6). The problem encountered with this method is that after

computing the algebraic model

πA(T)

∗

(

)

)then computing maps in

(

)from this object

into an injective resolution of

ECT

still retains a lot of complexity. More precisely it is still diﬃcult

to compute

-module maps between arbitrary

-module when none of them is a suspension of

the base ring

. Therefore this more conceptual approach of proving

-splitness and computing

EC∗

(

V⊗w

)

)has been taken. Moreover we hope to replicate and generalise this to other spaces

such as Grassmannians of n-planes Grn(V)of a complex T-representation.

1.4.

Outline of the paper.

The paper is organized as follows. In the interest of clarity we start

in Section 2with the main computational result: namely we compute rational

-equivariant elliptic

cohomology of

(

)(Lemma 2.13 and Theorem 2.26) assuming

-splitness (Theorem 2.2), and

we devote the remaining of the paper in proving

-splitness. In Section 3we revise the construction

of circle equivariant elliptic cohomology from [Gre05] to make it ﬁt nicely with the construction of

the

-equivariant theory from [Bar22]. In Section 4we construct the map

(Lemma 4.21) between

the inﬂated

ECT

and

ECT2

, inducing the

-splitness. To do so we will also treat the inﬂation

functor in the algebraic models for general tori, proving a simple construction of the functor in the

algebraic models (Proposition 4.1). In Section 5we conclude the proof of

-splitness by proving

the isomorphism

(1.8)

for every subgroup

intersecting

trivially. This is done in Theorem

5.26 by proving that

induces an isomorphisms between the

Ext

-groups of the two Adams spectral

sequences. To obtain this result will be fundamental to compute the algebraic model for natural

cells (Lemma 5.3 and Lemma 5.11), and to build an injective resolution of the inﬂated

ECT(5.17)

We end with an Appendix on algebraic models recalling from the literature the deﬁnition of the

category A(G), and especially injective objects in it.

1.5.

Notation and Conventions.

The most important convention of the paper is that everything

is rationalized without comment. In particular all spectra are meant localized at the rational sphere

spectrum

S0Q

and all the homology and cohomology is meant with

coeﬃcients. Tensor products

⊗

are meant over

, or over the graded ring with only

in degree zero and zero elsewhere. By

subgroup of a compact Lie group we always mean closed subgroup. With the symbol

we mean the

torus of rank

: compact connected abelian Lie group of dimension

. We denote

the 2-torus

in all the paper (except where explicitly state otherwise like in the Appendix). The collection of

connected closed codimension 1subgroups of

{Hi}i≥1

indexed with

i≥

1, and with

= 1

×T

and

T×

1being the two privileged subgroups. We denote

G/H1∼

=H2∼

the quotient

group. We denote

the subgroup of

with

connected components and identity component

: we will refer to the subgroups with identity component

as being along the

-direction. We

denote with

a generic ﬁnite subgroup of

and with

and

generic closed subgroups of

any dimension. Given a module

we will denote

the two periodic version of

: it is a graded

module with

in each even degree and zero in odd degrees. We denote elements in direct sums

and products in the following way:

{xi}i∈Li≥1Mi

: this identiﬁes the element

in the direct

sum that has i-th component xi∈Mi.

We will freely use the standard notation of schemes from Algebraic Geometry. We denote

our

ﬁxed elliptic curve over

is the identity of the elliptic curve and for a positive integer

[

]is

the subgroup of elements of

-torsion, while

Chni

is the subset of elements of exact order

. We will

use

to denote a point of

of ﬁnite order. We have the complex abelian surface

C × C

, and

denote η(C)the generic point of a closed set C.

We will use some essential notation involving

ECT2

from [Bar22]. Following [Bar22, (2.7), (2.8)],

for every

i≥

1we pick a character

G→T

having

as kernel. If

(

x, y

) =

xλiyµi

we denote

πi

X → C

the corresponding projection:

πi

(

P, Q

) =

λiP

µiQ

. Then for every

i, j ≥

1we can

deﬁne the codimension 1 subvariety

Dij

π−1

(

Chji

)of

. The most important ring for

ECT2

: the ring of meromorphic functions of

with poles only in the collection

{Dij}ij

[Bar22, (3.8)].

We use

ODij

to denote the subring of

of those functions regular at the closed set

Dij

, while

mij <ODij is the ideal of those functions vanishing at Dij .

We will freely use the standard notation for algebraic models , and we recall it in the appendix

A. In particular

(

)is an abelian category with graded objects and no diﬀerentials, while

(

)

is the category of objects of

(

)with diﬀerentials. Cohomology is unreduced unless indicated

to the contrary with a tilde, so that

H∗

(

BG/H

) =

H∗

(

BG/H+

)is the unreduced cohomology

ring. To ease the notation sometimes we will omit the base ring we are taking the tensor product

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RATIONALCIRCLE-EQUIVARIANTELLIPTICCOHOMOLOGYOFCP(V)MATTEOBARUCCOAbstract.WecomputerationalT-equivariantellipticcohomologyofCP(V),whereTisthecirclegroup,andCP(V)istheT-spaceofcomplexlinesforanitedimensionalcomplexT-representationV.StartingfromanellipticcurveCoverCandacoordinatedataaroundtheidentity,...

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RATIONAL CIRCLE-EQUIVARIANT ELLIPTIC COHOMOLOGY OF CPV MATTEO BARUCCO Abstract. We compute rational T-equivariant elliptic cohomology of CPV where Tis the circle

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