Bredon motivic cohomology of the complex numbers

Over the complex numbers, we compute the C 2 -equivariant Bredon motivic cohomology ring with Z / 2 coeﬃcients. By rigidity, this extends Suslin’s calculation of the motivic cohomology ring of algebraically closed ﬁelds of characteristic zero to the C 2 -equivariant motivic setting.


Introduction
Bredon motivic cohomology (introduced in [7] and [8]) is a generalization of motivic cohomology to the setting of smooth varieties with finite group action.Part of a larger group of motivic C 2 -invariants, such as Hermitian K-theory and motivic real cobordism, it plays an essential role in equivariant motivic homotopy theory.One distinguishing feature is that Bredon motivic cohomology appears as the zero slice of the equivariant motivic sphere [6].
Bredon motivic cohomology is ready for concrete computations, which will be crucial for applications of the theory to other motivic and topological invariants.In this paper, we compute the Bredon motivic cohomology ring with Z/2-coefficients.The usual methods [13], [19], [20] generalize the computations to an algebraically closed field of characteristic zero.These can be seen as a first step in understanding the largely unknown and difficult to compute Bredon cohomology ring for an arbitrary field k (for partial results in this direction see [18]) as well as the C 2 -equivariant motivic Steenrod algebra of cohomology operations.
Our computations are organized via modules over Bredon cohomology of a point.Before presenting our computations, we recall this ring and introduce some notation used to explain our results.

Bredon cohomology
In equivariant topology, Bredon cohomology plays the role that singular cohomology plays in ordinary topology.Some of its key features are that it takes a Mackey functor as coefficients, it is graded by representations, and is represented by an equivariant Eilenberg-MacLane spectrum, see [11] for details.The case of interest to us is G = C 2 , in which case we write σ for the sign representation.The group RO(C 2 ) is identified with Z ⊕ Z{σ}.We adopt the convention that ⋆ stands for an RO(C 2 )-grading and we use * for an integer grading.For an abelian group A, the Bredon cohomology with coefficients in the constant Mackey functor A, of a C 2 spectrum X is written H i+pσ Br (X, A).If X = Σ ∞ X + we simply write H i+pσ Br (X, A) := H i+pσ Br (Σ ∞ X + , A).The Bredon cohomology ring of a point with Z/2-coefficients was originally computed by Stong in unpublished work.Written accounts can be found in [1,Appendix] and [10,Proposition 6.2].For the corresponding computation with Z-coefficients see [3,Theorem 2.8] or [5, Section 2] for recent discussion of these computations.We write M C2 2 := H ⋆ Br (pt, Z/2).Let Z/2[a, u] be the polynomial ring generated by elements whose degrees are |a| = σ and |u| = −1 + σ.Consider Z/2[a −1 , u −1 ] as a graded Z/2[a, u]-module and write where Σ m+nσ M denotes the shifted graded module given by (Σ m+nσ M ) a+pσ = M a−m+(p−n)σ .From Figure 1, one sees that M C2 2 consists of two cones; NC is the "negative cone".The Bredon cohomology ring of a point is where the multiplicative structure is determined by the action of Z/2[a, u] on NC and all products between elements in NC are trivial.Writing θ ∈ NC for the element which corresponds to 1 ∈ Z/2[a −1 , u −1 ], we express elements of NC in the form θ a m u n , for m, n ≥ 0. We introduce some auxiliary M C2 2 -modules.See Figure 1 for graphical depictions.Recall the universal free C 2 -space EC 2 ; a geometric model is S(∞σ) = colim n S(nσ), where S(nσ) is the unit sphere in the n-dimensional real sign representation.The space EC 2 is defined to be the unreduced suspension of EC 2 , which by definition fits into the cofiber sequence of based C 2 -spaces, EC 2+ → S 0 → EC 2 .
The Bredon cohomology ring of EC 2 is where |u| = −1 + σ and |a| = σ, see e.g., [1,Lemma 27].Then is the localization map.In other words, it is the map which sends u → u, a → a and maps NC to 0.
The Bredon cohomology of EC 2 is see e.g., [1,Lemma 28].The right-hand side is a Z/2[a, u]-module and hence an M C2 2 -module (elements of NC act by 0) and this isomorphism is an induced by S 0 → EC 2 , is this quotient followed by the inclusion of the negative cone into M C2 2 .Explicitly, it is the map This is a Z/2[a, u]-module and hence an M C2 2 -module (elements of NC act by 0).Note that there is an identification For i ≤ 0 we set B i = 0.There are canonical M C2 2 -module quotients B i+1 → B i .Moreover, there are also (1.5) Lastly we note that there are M C2 2 -module maps defined by composing the quotient map with multiplication by u 2 , Explicitly it is the map else.

Our computation
We describe the main computations.Bredon motivic cohomology is graded by a 4-tuple of integers, written as (a + pσ, b + qσ); this 4-tuple is viewed as a pair of C 2 -representations (here σ denotes the sign representation), the first one is the cohomological degree and the second representation is the weight.The grading by 4-tuples presents an organizational problem.Our solution is to organize Bredon motivic cohomology into M C2 2 -modules, which we now explain.If X is a complex variety with C 2 -action, Betti realization induces a comparison homomorphism Re : between Bredon motivic cohomology of X and the Bredon cohomology of the C 2 -topological space X(C).When X = Spec(C), this induces an isomorphism of bigraded rings by Proposition 2.6, In particular, we can view H ⋆,b+qσ C2 (X, Z/2) as an M C2 2 -module, for each b, q.The free motivic C 2 -space EC 2 can be modeled as A(∞σ) \ 0, where A(nσ) is the n-dimensional sign representation.There is a motivic isotropy separation sequence EC 2+ → S 0 → EC 2 , where EC 2 is defined so that this a cofiber sequence (see Section 2.1 for details), which breaks the problem of computing H ⋆,⋆ C2 (C, Z/2) into pieces.Each of EC 2 and EC 2 determine a region of H ⋆,⋆ C2 (C, Z/2) and Betti realization determines the remaining nonzero region, see Theorem 4.1.These regions are shown in Figure 3.In this picture we have projected onto the plane determined by the weight.In particular, the displayed elements do not all live the same cohomological degree.In integer bidegrees, the Bredon motivic cohomology of EC 2 agrees with ordinary motivic cohomology of EC 2 /C 2 = BC 2 .The motivic cohomology of BC 2 was computed by Voevodsky [16,Theorem 6.10].In our case, where the base field is C, his computation takes the form where |e 1 | = (1, 1), |e 2 | = (2, 1), and |τ | = (0, 1).In Section 3, we leverage Voevodsky's computation, Betti realization, and that the cohomology of EC 2 is (−2 + 2σ, −1 + σ)-periodic, to find an equivariant lift of τ to an element τ σ ∈ H 0,σ C2 (EC 2 ) such that multiplication by τ σ is an isomorphism whenever b + q ≥ 0. Thus we find that The cohomology of EC 2 is both (σ, 0) and (0, σ)-periodic and Betti realization identifies We keep track of weights via the elements τ σ and µ, where Having determined the M C2 2 -module structures in all of the regions in Figure 3, we determine the multiplicative structure in Theorem 4.7, where we find there is an isomorphism of M C2 2 -algebras The left hand summand comes from the regions determined by EC 2 and Betti realization in Figure 3.
The right hand summand arises from the region determined by EC 2 .The multiplicative structure involving elements in this region is determined as follows. (i) (iv) a τ σ -multiplication starting in weights i−(i+1)σ, is the map (1.5).These are exactly the multiplications crossing the border from the region determined by EC 2 into the region determined by Betti realization, see Figure 3. (v) All products in the right hand summand are trivial.
Outline.A brief outline of the paper is as follows.Sections 1 and 2 are devoted to the introduction and preliminaries.The main computations of Bredon motivic cohomology are carried out in Sections 3 and 4. In the last section, we generalize the results to any algebraically closed field of characteristic zero via a rigidity result for Bredon motivic cohomology.

Notation.
• H a+pσ,b+qσ C2 (X, A) is the Bredon motivic cohomology of a C 2 -smooth scheme, with coefficients A. • H n,q (X, A) is motivic cohomology of a smooth scheme X.
• H a+bσ Br (X, A) is the Bredon cohomology of a C 2 -topological space X with coefficients in the constant Mackey functor A.
• We write ⋆ for an RO(C 2 )-grading and * for a Z-grading.
For example, • S σ is the topological sphere associated to the real sign representation σ.
• All C 2 -varieties are over C and we view C 2 as a group scheme by Acknowledgements.The authors wish to thank Institut Mittag-Leffler, Stockholm, where the research of this paper started in 2017 during the program on Algebro-Geometric and Homotopical Methods.Heller and Østvaer thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program on K-theory, algebraic cycles and motivic homotopy theory in 2020.We are grateful to the referee for useful comments on a previous draft of this paper.

Preliminaries
We record some background on Bredon motivic cohomology.

Equivariant motivic homotopy
The stable equivariant motivic homotopy category SH C2 (k) is the stabilization of Voevodsky's category of equivariant motivic spaces [2], with respect to Thom spaces of representations.We recall a few key facts and the notation we use in the case G = C 2 .See [9], [8], or [4] for details.
Let V = a + pσ be a C 2 -representation, where a denotes the a-dimensional trivial representation and pσ is the p-dimensional sign representation.We write A(V ) and P(V ) for the C 2 -schemes A dim(V ) and P dim(V )−1 equipped with the corresponding action coming from V .The associated motivic representation sphere is Indexing is based on the following four spheres.There are two topological spheres S 1 , S σ and two algebro-geometric spheres S t = (A 1 \ {0}, 1) equipped with trivial action, and

Bredon motivic cohomology
In this indexing, we have T ≃ S 2,1 and T σ ≃ S 2σ,σ .The stable equivariant motivic homotopy category SH C2 (k) is the stabilization of (based) C 2 -motivic spaces with respect to the motivic sphere T ρ corresponding to the regular representation ρ = 1 + σ.We make use of two fundamental cofiber sequences in SH C2 (k).The first is The second is Here, EC 2 is the universal free motivic C 2 -space.It has a geometric model, ) /C 2 is the geometric classifying space BC 2 constructed by Morel-Voevodsky [12] and Totaro [14].Note also that EC 2 = colim n S 2nσ,nσ .In particular, the maps S 0 → T σ and S 0 → S σ induce equivalences

Bredon motivic cohomology
Bredon motivic cohomology is represented in SH C2 (k) by the spectrum M A associated to an abelian group A, where M A n = A tr,C2 (T nρ ) is the free presheaf with equivariant transfers, see [8] for details.

Definition 2.3 ([8]
).The Bredon motivic cohomology of a motivic C 2 -spectrum E with coefficients in an abelian group A is defined by If X ∈ Sm C2 k we typically write When A is a ring, then H ⋆,⋆ C2 (X, A) is a graded commutative ring by [8,Proposition 3.24].Specifically this means that if x ∈ H a+pσ,b+qσ C2 (X, A) and y ∈ H c+sσ,d+tσ C2 (X, A), then x ∪ y = (−1) ac+ps y ∪ x.
A few features of this theory, which we use are the following (see [8]).
• If E is in the image of SH(k) → SH C2 (k), i.e. it has "trivial action", then there is an isomorphism in integral bidegrees with ordinary motivic cohomology, • If X has free action, then there is an isomorphism in integral bidegrees with ordinary motivic cohomology,

Betti realization
The map of sites Sm C2 C → Top C2 , given by X → X(C), where the set of complex points is equipped with the analytic topology, extends to a functor Re : SH C2 (C) → SH C2 between the stable equivariant motivic homotopy category over C and the classical stable equivariant homotopy category.We refer to this functor as the Betti realization.
The indexing of the spheres above was chosen to interact well with complex Betti realization; we have Re(S a+pσ,b+qσ ) ≃ S a+pσ .
By [8,Theorem A.29], Re(M A) ≃ HA, where HA is the equivariant Eilenberg-MacLane spectrum associated to the constant Mackey functor A. In particular, for any smooth C 2 -scheme over C there is a map Re : Betti realization takes the cofiber sequences (2.1) and (2.2) to the corresponding ones in SH C2 .Given X in SH C2 (C), we obtain a comparison of long exact sequences Using the Beilinson-Lichtenbaum theorem proved by Voevodsky and Rost [15], [17], it is shown in [8] that Betti realization is an isomorphism in a suitable range, on Bredon cohomology of smooth schemes.In Section 4, we will see that a stronger result holds for X = Spec(C).For the moment, we note that in nonnegative integer weights, we always have an isomorphism for finite coefficients.In particular, Betti realization induces an isomorphism of rings H ⋆,0 C2 (C, Z/n) ∼ = M C2 n and so H ⋆,⋆ C2 (X, Z/n) is a module over M C2 n .In fact, by [18] or the same argument below, Betti realization is an isomorphism in weight zero even with Z-coefficients.Proof.A) is an isomorphism for all a.In particular the result holds for p = 0. Using the comparison long exact sequence (2.4) and the five lemma, the result holds for all p by induction.

Vanishing of Bredon motivic cohomology
An important feature of motivic cohomology is its vanishing regions.If X ∈ Sm k then H a,b (X, Z/n) = 0 in any of the following cases (1) a > 2b, (2) a > b + dim(X), or (3) b < 0. The vanishing regions for H ⋆,⋆ C2 are more complicated.In this subsection k is a field with char(k) = 2.
Proposition 2.7. (1) ) and (0, σ)-periodic, (2) follows from (1).The first statement follows from the long exact sequence induced by (2.2).Indeed, by [8,Proposition 3.16] the map k and suppose that b + q < 0 and b < 0. Then Proof.Since b < 0, we have H a+pσ,b+qσ C2 ( EC 2 ∧ X + ) = 0. Using the cofiber sequence (2.2), we obtain (EC 2 ×X), it suffices to see that H a+pσ,n C2 (EC 2 ×X) = 0 for n < 0. This follows from the case p = 0, by induction using (2.1).Proposition 2.9.If a ≥ 2b + 2 then for any Proof.The two statements are equivalent using (2.2).Therefore, we will establish the first one.Since we can assume that p = q = 0. We can assume that X has trivial action, since . This group vanishes when a > 2b.To conclude the proposition, we use the long exact sequence obtained from (2.1) and induction on p ≥ 2q.
The following example shows that the general vanishing range in the previous proposition can't be improved.
Example 2.11.For any p, we have To see this, we first note that H ⋆,1−2σ C2 (P 1 × EC 2 , Z/n) = 0 (see the proof of Proposition 3.3) and therefore from (2.2) we see that H 2+pσ,1−2σ 3 Bredon motivic cohomology of EC ).The lemma follows from the comparison of long exact sequences induced by this cofiber sequence, the five lemma, the Thom isomorphism, and that Re : Furthermore, Betti realization is an isomorphism if a ≤ 2b.It is multiplication by 2 if a ≥ 2b + 2, p = −a, and a is even.All other Betti realizations are zero.[8,Theorem 5.4] and the statement of the proposition is compatible with this periodicity, it suffices to treat the case q = 0. We now assume that q = 0.
When p = 0, then We suppress the coefficient group for typographical simplicity and proceed by induction on p.To begin with, we use the comparison of exact sequences (2.4) This establishes the result in case p ≥ 0. Now we establish the result for p ≤ 0. Using the comparison of exact sequences (2.4) and the five lemma, we find that if the map ) is an isomorphism for all i ≤ 2b when n = p + 1, then this map is also an isomorphism for all i ≤ 2b when n = p.By downward induction on p, starting with p = 0, we deduce the computation for a ≤ 2b.Now assume that H 2b+1+nσ,b (EC 2 ) = 0 for n = p + 1.If p ≥ −(2b + 1), if follows from the exact sequence induced by (2.1) that this group vanishes for n = p as well.Thus downward induction implies the result for p < −(2b + 1) once we treat the case p = −(2b + 1).Consider the comparison of exact sequences That the bottom left horizontal arrow is an isomorphism can be seen by noting that this map can be identified with the restriction to the fiber homomorphism H 2b sing (Th(γ), Z/n) → H 2b sing (S 2b , Z/n), where γ is the vector bundle on BC 2 determined by the b-dimensional complex sign representation.This map is an isomorphism because γ is orientable.It follows that the map labeled φ is an isomorphism and so The case a ≥ 2b + 2 thus follows from Proposition 2.6.
For the last statement about Betti realization, we have already checked that it is an isomorphism if a ≤ 2b.The remaining part of the statement follows from the commutative diagram, where 2b To see that the bottom arrow is multiplication by 2, note that for 2 ≤ a, H a+pσ Br (pt, Z) ∼ = H Br −a−pσ (EC 2 , Z), see e.g., [5] for details, and under this identification the lower arrow is induced by the norm map HZ hC2 → HZ hC2 .
(EC 2 , Z/n).Using the vanishing H a+2q,b+q C2 (EC 2 , Z/n) ∼ = H a+2q,b+q (BC 2 , Z/n) = 0 together with the exact sequence induced by (2.1), the result follows by induction.Notation 3.4.We introduce certain elements in the cohomology of Spec(C) and EC 2 .The stated isomorphisms between the cohomology of Spec(C) and EC 2 all follow from the exact sequences associated to (2.2) together with the vanishing of the Bredon motivic cohomology of EC 2 in the relevant degrees, see Proposition 2.7.
•  (EC 2 , Z/n) is injective.But it follows from Proposition 3.2 that either both of these groups are Z/n or both are 0. Thus the map is an isomorphism.Theorem 3.6.Let n ≥ 2. The canonical map is an isomorphism of rings Thus, together with periodicity, we have an isomorphism The result now follows from Lemma 3.5.We end this section with a determination of H ⋆,⋆ C2 ( EC 2 , Z/2).The M C2 2 -submodules of H ⋆ Br ( EC 2 , Z/2), defined by B i := Σ 2−2σ Z/2[a ±1 , u −1 ]/(u −2i ) were introduced in Remark 1.3.In the following proposition µ and τ σ are formal variables which serve the purpose of placing B i into the correct weight.The names of these formal variables are chosen to indicate the H ⋆,⋆ C2 (C, Z/2)-module structure, which will be determined in the next section.

Proposition 3.8. There is an isomorphism of
where |τ σ | = (0, σ) and |µ| = (0, 1 − σ). Proof , multiplication by u 2 .We'll see in the next section that this describes the action of H ⋆,⋆ C2 (C).The cohomology of EC 2 , together with the module structure just described, is displayed in the following figure.

Bredon motivic cohomology of algebraically closed fields
In this section we consider an algebraically closed field k and a natural number n > 1 coprime to char(k).Let V be a C 2 -equivariant smooth scheme over k.First we note a rigidity theorem for rational points: Theorem 5.1.For a connected smooth scheme X over k and k-rational points x 0 , x 1 of X, (x 0 ) * = (x 1 ) * : H ⋆,⋆ C2 (V × X, Z/n) → H ⋆,⋆ C2 (V, Z/n).
According to [19], Theorem 5.1 follows if the functor F (−) = H ⋆,⋆ C2 (V × −, Z/n) is a homotopy invariant presheaf on Sm/k with weak transfers in the sense of [20].The four conditions that need to be fulfilled according to [19] are: 1) Additivity: For X = X 0 ⊔ X 1 with corresponding embeddings i m : X m ֒→ X for m = 0, 1 and f : X → Y a map in Sm/k, we have f * = (f i 0 ) * i * 0 + (f i 1 ) * i * 2) Base change: For every finite flat map f , closed embedding g, and cartesian diagram: we have g * f * = f 1 * g 1 * 3) Normalization: If f is the identity map on k then f * = id H ⋆,⋆ C 2 (V,Z/n) 4) Homotopy invariance: The rational points 0 and 1 of the affine line A 1 k with trivial C 2 -action yield equal pullback maps 0 * = 1 * : H ⋆,⋆ C2 (V × k A 1 k ) → H ⋆,⋆ C2 (V ).The functor F fulfills all four conditions above as it is a homotopy invariant presheaf with equivariant transfers ( [8]).Moreover, because it is a homotopy invariant presheaf with equivariant transfers, according to [13], [19], [20], we have the following theorem: Theorem 5.2.Suppose k ⊂ K is an extension of algebraically closed fields and X is a smooth C 2equivariant scheme.If n is coprime to char(k), then π : Spec(K) → Spec(k) induces an isomorphism: Proof.We can write Spec(K) = lim U (U ), where U is an affine smooth variety over Spec(k).There is an induced map so if π * (x) = 0 then there exists a map φ : U → Spec(k) such that φ * (x) = 0.Because U has a k-rational point, φ yields a splitting and φ * is injective.This implies x = 0 so π * is injective.

Figure 2 :
Figure 2: The modules B i and the maps (1.4) and (1.6) between them.The shaded region indicates the kernel of •u 2 : B m+1 → B m .

Proposition 2 . 6 .
Let A be a finite abelian group and b ≥ 0. Betti realization induces an isomorphism for any a, p H a+pσ n is a generator.Next we compute the multiplicative structure.The M C2 n -modules H ⋆,b+qσ C2 (EC 2 , Z/n) together with multiplicative structure are displayed in Figure4below.

Figure 5 :
Figure 5: H ⋆,b+qσ C2 ( EC 2 , Z/2).Vertical green lines are multiplication by τ σ , diagonal blue lines are multiplication by µ.The curved red lines indicate the action of ξ, which acts as multiplication by u 2 .
4) as follows.The top row of this sequence is obtained by smashing X with (2.1) and the bottom is obtained similarly via Betti realization.Here we use the identifications H a+pσ,b+qσ H a+p sing (Re(X)) and the Re is compatible with these identifications, see [8, Proposition 3.14].Smashing X with (2.2) we obtain the comparison of long exact sequences C2(C 2+ ∧ X) ∼ = H a+p,b+q (X) and H a+pσ Br (Re(X)) ∼ = and if a > 2b this last group is zero and so the proposition follows.

2
In this section, we compute the Bredon motivic cohomology ring of EC 2 .Betti realization plays a key role in our determination of this ring and we start by leveraging motivic cohomology of BC 2 .From [8, Proposition 3.16], we have an isomorphism H a,b C2 (EC 2 , A) ∼ = H a,b (BC 2 , A) and this isomorphism fits into the commutative diagram.Proof.If A = Z/2, this can be read off of Voevodsky's computation (1.7), since Betti realization of e 1 is the generator of H * sing (BC 2 , Z/2).In general, we use that BC 2+ sits in the cofiber sequence, see [16, Section 6], BC 2+ → P ∞ + → Th(O(−2) . It follows from Proposition 2.7 and Lemma 3.1 that Betti realization