Milnor-Witt motivic cohomology of complements of hyperplane arrangements

In this paper, we compute the (total) Milnor-Witt motivic cohomology of the complement of a hyperplane arrangement in an affine space as an algebra with given generators and relations. We also obtain some corollaries by realization to classical cohomology.


Introduction
Let K be a perfect field of characteristic different from 2, and let U ⊂ A N K be the complement of a finite union of hyperplanes.For K = R the cohomology ring H * sing (U (R), Z) is just the direct sum of Z corresponding to each regions(connected components), and those regions form a poset.In the special case when the hyperplanes arise from a root system, the resulting poset is the corresponding Weyl group with the weak Bruhat order.In general, the poset of regions is ranked by the number of separating hyperplanes and its Möbius function has been computed [1].
For any essentially smooth scheme X over K and any integers p, q ∈ Z, one can define the Milnor-Witt motivic cohomology groups H p,q MW (X, Z) introduced in [2].There are homomorphisms (functorial in X) for any p, q ∈ Z H p,q MW (X, Z) → H p,q M (X, Z) where the right-hand side denotes the ordinary motivic cohomology defined by Voevodsky.
As illustrated by the list of properties in the following section, the Milnor-Witt motivic cohomology groups behave in a fashion similar to ordinary motivic cohomology groups, but there are central differences (for instance, there are no reasonable Chern classes).
In this paper, we compute the total Milnor-Witt cohomology ring of the complement of a hyperplane arrangement in affine spaces H MW (U ) using methods very similar to [3], with some necessary modifications.To state our main result, we first recall a few facts.
Let R be a commutative ring.The Milnor-Witt K-theory of R is defined to be the graded algebra freely generated by elements of degree 1 of the form [a], a ∈ R × and an element η in degree −1, subject to the relations (1) [a][1 − a] = 0 for any a such that a, 1 − a ∈ R × \ {1}.
( It defines a presheaf on the category of schemes over a perfect field K via X → K MW * (O(X)).On the other hand, one can also consider the Milnor-Witt motivic cohomology (bigraded) presheaf X → H MW (X) := ⊕ p,q H p,q MW (X, Z) By [4, Theorem 4.2.2],there is a morphism of presheaves which specializes to the above isomorphism if X = Spec(F ) where F is a finitely generated field extension of K [5].
Theorem 1.1.Let K be a perfect field of characteristic different from 2 and let U ⊂ A N K be the complement of a finite union of hyperplanes.There is an isomorphism of H MW (K)-algebras MW (U, Z) corresponding to f under s.Here, H MW (K){G m (U )} is the free (associative) graded H MW (K)-algebra generated by G m (U ) in degree (1,1) and J U is the ideal generated by the following elements: As indicated above, this theorem and its proof are inspired from the computation of the (ordinary) motivic cohomology of U in [3].We can recover the main theorem [3,Theorem 3.5] of the motivic cohomology case by taking η = 0.As a corollary, we obtain the following result.

Corollary 1.2. Let U ⊂ A N K be the complement of a finite union of hyperplanes. The isomorphism of Theorem 1.1 induces an isomorphism
We do not know if the left-hand side coincides with K MW Conventions.The base field K is assumed to be perfect and of characteristic not 2.For a scheme X over K, we write H MW (X) for the total MW-motivic cohomology ring p,q∈Z H p,q MW (X, Z).For all f ∈ G m (U ), we use (f ) to indicate the corresponding generator in the corresponding free algebras (e.g.K MW n (K){G m (U )}) and [f ] to indicate the corresponding elements in the cohomology groups (e.g.H 1,1  MW (U, Z)).

Milnor-Witt Motivic Cohomology
In this section, we define Milnor-Witt motivic cohomology and state some properties that will be used in the proof of Theorem 1.1.We start with the (big) category of motives DM(K) := DM Nis (K, Z) defined in [4,Definition 3.3.2]and the functor M : Sm/K → DM(K).
The category DM(K) is symmetric monoidal [4,Proposition 3.3.4]with unit ½ = M(Spec(K)).For any integers p, q ∈ Z, we obtain MW-motivic cohomology groups By [4,Proposition 4.1.2],motivic cohomology groups can be computed as the Zariski hypercohomology groups of explicit complexes of sheaves.
We will make use of the following property of DM(K).First, we note that DM(K) is also a triangulated category.Proposition 2.1 (Gysin Triangle).Let X be a smooth K-scheme, let Z ⊂ X be a smooth closed subscheme of codimension c and let U = X \ Z. Suppose that the normal cone N X Z admits a trivialization φ : where the last two arrows depend on the choice of φ.
Futhermore, the Milnor-Witt motivic cohomology groups satisfy most of the formal properties of ordinary motivic cohomology and were computed in a couple of situations: (1) If q ≤ 1, there are canonical isomorphisms where K MW q is the unramified Milnor-Witt K-theory sheaf (in weight q) introduced in [9].
(2) If L/K is a finitely generated field extension there are isomorphisms where K M n (L) is the (n-th) Milnor K-theory group of L, the bottom horizontal map is the isomorphism of Suslin-Nesterenko-Totaro, and the right-hand vertical map is the natural homomorphism from Milnor-Witt K-theory to Milnor K-theory.This result has the following consequence: The Milnor-Witt motivic cohomology groups are computed via an explicit complex of Nisnevich sheaves Z(q) for any integer q ∈ Z.The above result shows that there is a morphism of complexes of sheaves where the right-hand side is the complex whose only non-trivial sheaf is K MW q in degree −q.For any essentially smooth scheme X over K, this yields group homomorphisms H p,q MW (X, Z) → H p−q (X, K MW q ) which are compatible with the ring structure on both sides.In the particular case p = 2n, q = n for some n ∈ Z, we obtain isomorphisms (functorial in X) where the right-hand term is the n-th Chow-Witt group of X (defined in [10] and [11]).Again, these isomorphisms fit into commutative diagrams where the right-hand vertical homomorphism is the natural map from Chow-Witt groups to Chow groups.
(3) The total Milnor-Witt motivic cohomology has Borel classes for symplectic bundles [12] but in general the projective bundle theorem fails [13].(4) If X is a smooth scheme over R, there are two interesting realization maps.On the one hand, one may consider the composite where the right-hand map is the complex realization map.On the other hand, one may also consider the following composite where I q is the unramified sheaf associated to the q-th power of the fundamental ideal in the Witt ring, K MW q → I q is the canonical projection and H p−q (X, I q ) → H p−q sing (X(R), Z) is Jacobson's signature map [14].
We note here that these two realization maps show that Milnor-Witt motivic cohomology is in some sense the analogue of both the singular cohomology of the complex and the real points of X.

Basic structure of the cohomology ring
Let V be an affine space, i.e.V ∼ = A N K for some N ∈ N. We consider finite families I of hyperplanes in V (that we suppose are distinct).We denote by |I| the cardinality of I and set For any hyperplane Y , we put Proposition 3.1.Let V and I be as above.We have for some set J and integers n j ≥ 0.
Proof.We proceed by induction on the dimension N of V and |I|.The Gysin triangle reads as If φ = 0, then the triangle is split and consequently we obtain an isomorphism Since |I − {Y }| < |I| and dim(Y ) = dim(V ) − 1 we conclude by induction that the right-hand side has the correct form.We are then reduced to show that φ = 0.
By induction, for some integers n j , m k ≥ 0, and it suffices to prove that As an immediate corollary, we obtain the following result.

Corollary 3.2. The motivic cohomology H
To obtain more precise results, we now study the Gysin (split) triangle (1) in more detail.We can rewrite it as and therefore we obtain the following short (split) exact sequence in which the morphisms are induced by the first two morphisms in the triangle (3) given by the morphism Proof.The commutative diagram of schemes [1] is just the projection.We conclude by observing that the middle vertical composite is just (α We may now prove the main result of this section.Proposition 3.4.The cohomology ring H MW (U ) is generated by the classes of units in U as an H MW (K) algebra.In particular, the homomorphism Proof.We still prove the result by induction on |I| and the dimension of V , the case |I| = 0 being obvious.Suppose then that the result holds for U Y I Y and U V I−{Y } and consider the split sequence (3).For any ) and the result now follows from the fact that α * is just induced by the inclusion

Relations in the cohomology ring
The purpose of this section is to prove that the relations of Theorem 1.1 hold in H MW (U ).The first two relations are obviously satisfied since the homomorphism is induced by the ring homomorphism Recall now that the last two relations are 3.
We will prove that they are equal to 0 in H MW (U ).Actually, it will be more convenient to work with the following relations 3 ′ .R(f 0 , . . ., f t ) defined by Proof.We first assume that 3. and 4. are satisfied.Since 1. and 2. are satisfied, we have . and the anti-commutativity law, we obtain (by 2.) (4) Conversely, suppose that 3 ′ and 4 ′ hold.A direct calculation shows that we have and consequently that 3. also holds.For every field K = F 2 , we have 1 Remark 1. Observe that the following properties of the relations R and anti-commutativity hold: (2) For any f 0 , . . ., f t ∈ G m (U ), by direct computation, we have for some polynomial P .We shall use the anti-commutativity and the fact that The following lemma will prove useful in the proof of the main theorem.

Lemma 4.2. Any morphism
is trivial for every Y ∈ I factors through M(K), i.e. there is a morphism ψ : M(K) − → T such that the following diagram Proof.We prove as usual the result by induction on |I|, the result being For H ∈ I ′ = I − {Y }, we have an associated Gysin morphism in which the morphism α Y (1) [1] on the left is split surjective.It follows that φ We conclude by induction.

M(U
The vertical composite being the identity, ψ = R(λ 0 , . . ., λ t ) and the latter is trivial by the relations in Milnor-Witt K-theory.
Applying lemma 4.1, we obtain the following corollary.
Corollary 4.4.Let S be an essentially smooth smooth K-scheme.

Proof of the main theorem
In this section, we prove Theorem 1.1.We denote by ( Now, choose linear polynomials φ 1 , . . ., φ s that define the hyperplanes Y i ∈ I and let J ′ U ⊂ H MW (K){G m (U )} be the ideal generated by the relations (1), ( 2), (3 ′ ) and (4 ′ ) for elements of the form f j = λ j φ i j or f j = λ j for λ j ∈ K × and φ i j ∈ {φ 1 , . . ., φ s }.We have a string of surjective morphisms of H MW (K)-algebras whose composite we denote by ρ ′ .
Theorem 5.1.The morphism of H MW (K) algebra is an isomorphism.
Proof.It suffices to prove that ρ ′ is an isomorphism.To see this, we work again by induction on |I|.If |I| = 0, we have U ∼ = A N K for some N ∈ N. By homotopy invariance, we have to prove that the map is an isomorphism.Now, the morphism of H MW (K)-algebras is surjective by Relation 1.Its composite with ρ ′ is the identity and we conclude in that case.Assume now that Y ∈ I is defined by φ 1 = 0 and that we have isomorphisms The morphism in which β is the unique lift of β Y * • ρ and the bottom row is exact.We are thus reduced to prove that the top sequence is short exact to conclude.It is straightforward to check that α is injective and β is surjective.Moreover, the commutativity of the diagram and the fact that β Y * • α Y * = 0 imply that β • α = 0, so we are left to prove exactness in the middle. Let and φ 1 , we may use relations ( 2) and ( 4) to see that . Consequently, we need to prove that if ι(x 1 ) = 0 then (φ 1 )α(x 1 ) is in the image of α.With this in mind, we now prove that the kernel of ι is generated by elements of the form R(f 0 , . . ., f t ) where f j = λφ i j with i j > 1 or } generated by such elements.By construction, the restriction induces a homomorphism which is surjective.Indeed, relations (1), ( 2) and (3 ′ ) can be lifted using the fact that the map G m (U ) is surjective, while an element satisfying relation (4) with every f j of the form f j = λ j φ i j or f j = λ j for λ j ∈ K × (with i j = 1) lifts to an element in L ′ .As in [3, proof of Theorem 3.5], we see that the kernel of the group homomorphism . We deduce that ker(ι) = L ′ .We now conclude.If ι(x 1 ) = 0, then x 1 ∈ L ′ and we may suppose that

Corollary 5.2. The graded ring isomorphism of Theorem 5.1 induces an isomorphism
Proof.Notice that the ideal J U of Theorem 5.1 is homogeneous, and it follows that n∈Z H n,n MW (U, Z) can be computed as H * , * MW (K){G m (U )}/J U , where H * , * MW (K) is the diagonal of H MW (K).

Combinatorial description
In this section, we fix an affine space V = A N K , a family of hyperplanes I and we set U := U N I .We let Q(U ) be the cokernel of the group homomorphism G m (K) → G m (U ), and we observe that the divisor map We consider the exterior algebra Λ Z Q(U ) and write Λ Z[η]/2η Q(U ) := Z[η]/2η⊗ Z Λ Z Q(U ).The abelian group Q(U ) being free, the Z[η]/2η-module Λ Z[η]/2η Q(U ) is also free, with usual basis.To provide a combinatorial description of H MW (U ), we will have to slightly modify the definition of the divisor map above, in order to incorporate the action of η.We then define a map as follows: (1) If f = λφ or f = λ where λ ∈ G m (K) and φ is a linear polynomial as above, then div(f Lemma 6.1.The map div is well-defined. Proof.We first notice that div(f g) = div(gf ), since div which allows to conclude by induction on the number of non-trivial factors in the decomposition of f 1 g 1 .
Let now L U ⊂ Λ Z[η]/2η Q(U ) be the ideal generated by the following elements: (1) As a consequence of Lemma 6.1, the map div induces a morphism of Z[η]/2η-algebras

It is now time to introduce the ring
Proof.We first prove that Ψ is well-defined, which amounts to show that the image of and div(f ) = 0, showing that the first relation is satisfied.The second relation is satisfied by definition of div, while relation (3 ′ ) is satisfied as Λ Z[η]/2η Q(U )/L U is an exterior algebra.As in the proof of Theorem 5.1, we are then left with elements of J ′ U , i.e. elements of the form R(f 0 , . . ., f t ) for t i=0 f i = 0, where is an element of L U .Note that if there are more than two constant functions among f j , α would be trivial.Suppose that f 0 = λ 0 is the only constant, and let f j = λ j φ j with kernel Y j ∈ I, so that α = Y 1 ∧. ..∧Y t .Since t j=1 λ j φ j = −λ 0 = 0, we can easily get that Y In the case where none of the f j are constant, α = t i=0 (−1) i Y 0 ∧ . . .∧ Y i ∧ . . .∧ Y t .And for every i, we have t j=0,j =i λ j φ j = −λ i φ i , which means which just fits the condition (2) of L U .This proves that Ψ is well-defined.
To prove that Ψ is an isomorphism, we construct the inverse map by and prove that it is well-defined.As above, we just need to discuss elements of . This shows that the inverse map is well-defined.
The following corollary shows that the rank of the free H MW (K)-module H MW (U ) is exactly the same as the rank of the free H M (K)-module H M (U ) [3,Proposition 3.11].

I-cohomology and singular cohomology
In ordinary motivic cohomology theory, we have a realization functor to the topological cohomology of complex points.This yields the following comparative result.Proposition 7.1.[3, Proposition 3.9] In the case K = C, there is an isomorphism of rings: − → ⊕ n H n sing (U (C), Q).In this section, we provide an analogue for the singular cohomology of the real points of the complement of a hyperplane arrangement defined over R. We start with some results about the I-cohomology [11].
As recalled in Section 2, we have natural homomorphisms from Milnor-Witt motivic cohomology to I * -cohomology H p,q MW (X, Z) → H p−q (X, K MW q ) → H p−q (X, I q ) which induce a ring homomorphism H MW (X) → ⊕ r,q H r (X, I q ) (where I q = K MW q = W for q < 0).In case X = Spec(K), we obtain in particular a ring homomorphism H MW (K) → ⊕ r,q H r (K, I q ) = ⊕ q∈Z I q (K) Proposition 7.2.The morphism of ⊕ q∈Z I q (K)-algebras j : H MW (U ) ⊗ H MW (K) (⊕ q∈Z I q (K)) → ⊕ r,q H r (U, I q ) is an isomorphism.Moreover H r (U, I q ) = 0, for r = 0.
Proof.Throughout the proof, we write H MW (U ) ⊗ I for the graded ring H MW (U ) ⊗ H MW (K) (⊕ q I q (K)).We follow the same induction process as in the proof of the main theorem.When |I| = 0, we only need to consider Spec(K) by homotopy invariance, and the result is trivial.
Assume now that Y ∈ I and that we have isomorphisms for U V I ′ and U Y I Y .Notice that for I-cohomology we still have a Gysin long exact sequence [11,Remarque 9.3.5].The proof of the main theorem yields the following commutative diagram: ⊕ q H −1 (U Y IY , I q−1 ) / / ⊕ q H 0 (U V I ′ , I q ) / / ⊕ q H 0 (U V I , I q ) / / ⊕ q H 0 (U Y IY , I q−1 ) / / ⊕ q H 1 (U V I ′ , I q ) By our assumption, H −1 (U Y I Y , I q−1 ) and H 1 (U V I ′ , I q ) are both 0, so the second line is also short exact.We conclude that j is an isomorphism as well.The same argument implies that H r (U V I , I q ) = 0 for r = 0.
The analogue of Corollary 3.2 in this setting then reads as follows.
Corollary 7.3.There is a finite set J and integers n j ≥ 0 for any j ∈ J such that H 0 (U V I , I q ) ∼ = ⊕ j∈J I q−n j (K)b j as a free ⊕ q I q (K)-module with basis b j ∈ H 0 (U V I , I n j ).Proof.Every step is the same as in Proposition 3.1, except the splitting, which comes from the identification with H MW (U V I ) ⊗ I.
and we are done.Let then |I| ≥ 1 and Y ∈ I.

Lemma 4 . 1 .
The two groups of relations are equivalent in H MW (U ).