NOTE ON WITT GROUP AND KO-THEORY OF COMPLEX GRASSMANNIANS

NOBUAKI YAGITA

Abstract. For a complex Grassmannian X, there is the isomorphism between the Balmer’s Witt group and the quotient of topological K-theories so that W^{∗}(X) ≅ KO^{2∗}(X)/KU^{2∗}(X).

1. Introduction

Let X be a smooth variety over a field k with 1/2. The Witt group W (X) is the quotient of the Grothendieck group of vector bundles with quadratic forms over X, by subbundles V with quadratic forms which admit Lagragian subbundles E (i.e., E is its own orthogonal complement in V).

Hence when k = ℂ, there is the natural map

W (X) → KO^{0}(X(ℂ))/KU^{0}(X(ℂ)).

Here KO^{0}(−) and KU^{0}(−) is the usual (topological) real and complex K-theories. One purpose of this paper is to show that this map is isomorphic when X = M_{m,n} the complex Grassmannian of m-planes in an m + n-plane. Moreover, we have the isomorphism

W^{∗}(X) ≅ KO^{2∗}(X(ℂ))/KU^{2∗}(X(ℂ))

where W^{∗}(X) is the Balmer’s Witt group with W^{0}(X) = W (X).

The right hand side of the above isomorphism is computed explicitly by Hara and Hara-Kono [Ha],[Ha-Ko] by using Atiyah-Hirzebruch spectral sequence, here the computation of Sq^{2} is most important.

On the other hand, W^{∗}(X) is given recently by Balmer-Calmes [Ba-Ca2] in complete general forms, using m × n-framed even Young diagrams. However we can also get the results (for k = ℂ) by using Pardon and Balmer-Walter spectral sequences by using the computation of Sq^{2} by Hara-Kono. The another purpose of this paper is to explain the relation between the results by Balmer-Calmes and Hara-Kono.

1991 Mathematics Subject Classification. Primary 19G12, 55N15; Secondary 55N20.

Key words and phrases. Grassmannian, Witt group, K-theory .

I would like to thank to Burt Totaro who suggested the above isomorphisms, to Paul Balmer who taught me the result of Calmes and him, and to Akihiro Ohshita who taught me results of Hara-Kono and him.

2. KO-theory

We explain the KO-theory of Garassmannian according to Hara [Ha] and Hara-Kono [Ha-Ko]. Let M_{m,n} be the complex Grassmannian GL_{m}(ℂ^{m+n}) of m-planes in ℂ^{m+n}. Then there is the homeomorphism

M_{m+n} ≅ U(m + n)/(U(m) × U(n)).

By using the Serre spectral sequence induced from the fiber sequence

U(m + n)/U(m) × U(n) → BU(m) × BU(n) → BU(m + n),

we get the cohomology for any field K

(2.1) H^{∗}(M_{m,n}; K) ≅ K[a_{1}, ..., a_{m}, b_{1}, ..., b_{n}]/(c_{1}, ..., c_{m+n})

where a_{i}, b_{j}, c_{k} are Chern classes induced from maps in the above fibering, and c_{i} = ∑ a_{i−j}b_{j}. (See also [Fu], [La] and the arguments in §4 bellow.)

Recall the coefficient ring of the (topological) KO^{∗}-theory is

KO^{∗} ≅ ℤ[μ, μ^{−1}, η, w]/(2η, η^{3}, w^{2} − 4μ)

with |μ| = −8, |w| = −2, |η| = −1. To compute KO^{∗}(M_{m,n}), we consider the Atiyah-Hirzebruch spectral sequence

E_{2}^{∗,∗′} ≅ H^{∗}(M_{m,n}; KO^{∗}) ⟹ KO^{∗}(M_{m,n}).

It is well known that the first differential is ([Ha])

d_{2}(x) = η ⊗ Sq^{2}(x̄)

where x̄ ∈ H^{∗}(M_{m,n}; ℤ/2) is the mod 2 reduction of x. The Squaring operation is well known (from Wu formula) [Ha]

Sq^{2}(a_{1}) = a_{1}^{2}, Sq^{2}(a_{2i}) = a_{2i+1} + a_{2i}a_{1} for i ≥ 1.

Let H(m, n) be the homology H(H^{∗}(M_{m,n}; ℤ/2); Sq^{2}) with the differential Sq^{2}. Hara and Kono get this homology

Theorem 2.1. (Hara-Kono [Ha-Ko]) Let B(k, l) be the graded algebra

B(k, l) = ℤ/2[a_{2}^{2}, ..., a_{2k}^{2}, b_{2}^{2}, ..., b_{2l}^{2})/(c_{2}^{2}, ..., c_{2k+2k}^{2}).

Then we have the isomorphism

H(m, n) ≅ \begin{cases}B(k, l) if (m, l) = (2k, 2k), (2k + 1, 2l), (2k, 2l + 1) \\ B(k, l) ⊕ B(k, l)\{a_{2k+1}b_{2l}\} if (m, l) = (2k + 1, 2l + 1). \end{cases}

WITT GROUP AND KO-THEORY 3

They also proved

Theorem 2.2. ([Ha-Ko]) The Atiyah-Hirebruch spectral sequence collapses from E_{3}^{∗,∗′}-term.

For the proof of this theorem, Hara and Kono used the natural maps U(n) → Sp(n) and Sp(n) → U(2n). Let us write N_{m,n} = Sp(m + n)/Sp(m) × Sp(n) and consider the maps

M_{m,n} →^{q} N_{m,n} →^{c′} M_{2m,2n}.

The cohomology is also computed as the case U(n),

H^{∗}(N_{n,m}; ℤ/2) ≅ ℤ/2[q_{1}, ..., q_{m}, r_{1}, ..., r_{n}]/(s_{1}, ..., s_{m+n}),

s_{i} = ∑ q_{i−j}r_{j} with q^{∗}q_{i} = a_{i}^{2} and c′^{∗}a_{2i} = q_{i} (c′^{∗}(a_{2i−1}) = 0).

Note KO^{odd} = KO^{8∗−1} and H^{∗}(N_{m,n}) = H^{4∗}(N_{m,n}). By the dimensional reason for differntial deg(d_{r}) = (r, −r + 1), we know the Atiyah-Hirzebruch spectral sequence for KO^{∗}(N_{m,n}) collapses from E_{2}^{∗,∗′}-term, that means

grKO^{∗}(N_{m,n}) ≅ KO^{∗} ⊗ H^{∗}(N_{m,n}; ℤ).

Proof of Theorem 2.2. By the naturality of the spectral sequence, the maps q^{∗}, c′^{∗} are defined as maps of spectral sequences. The fact q^{∗}q_{i} = a_{i}^{2} implies that d_{r}(a_{i}^{2}) = 0 for all r > 1. The fact c′^{∗}a_{2i} = q_{i} implies that a_{2i}^{2} ≠ 0 in E_{∞}^{∗,∗′} and moreover each nonzero element in B(k, l) is also nonzero in E_{∞}^{∗,∗′}. If d_{r}(a_{2k+1}b_{2l}) ≠ 0, then it is contained in B(k, l) by dimensional reason ; this is a contradiction to the preceding result. □

There is the well known exact sequence for topological space X

(1) → KO^{∗}(X) →^{η} KO^{∗}(X) →^{c} KU^{∗}(X) →^{r} KO^{∗}(X) →

where c is the complexification and r is the restriction maps. Therefore

KO^{∗}(X)/KU^{∗}(X) ≅ ηKO^{∗}(X).

From the above theorem, we see

E_{∞}^{odd,∗} ≅ E_{∞}^{8∗−1,∗} ≅ ℤ/2{η}[μ, μ^{−1}] ⊗ H(m, k).

Hence we have

Corollary 2.3. grKO^{2∗}(M_{m,n})/KU^{2∗}(M_{m,n}) ≅ H(m, n).

3. Balmer’s Witt group.

For a smooth X over a field k with 1/2 ∈ k, let W (X) denote the Witt group of X. Balmer defined the periodic Witt group W^{i}(X) ≅ W^{i+4}(X), (i ∈ ℤ) with W^{0}(X) = W (X).

Balmer and Walter [Ba-Wa] define the Gersten-Witt complex

0 → W (k(X)) → ⊕_{x∈X^{(1)}}W (k(x)) → ... ⊕_{x∈X^{(n)}} W (k(x)) → 0.

Let H^{∗}(W (X)) denote the cohomology group of the above cochain complex, with W (k(X)) places in degree 0. From the above complex, we have the (Balmer-Walter) spectral sequence

E(BW)_{2}^{r,s} ≅ \begin{cases}H^{r}(W (X)) (s = 4s′) ⟹ W^{r+s}(X) \\ 0 (s ≠ 0(mod(4)) \end{cases} .

By the affirmative answer of the Milnor conjecture of quadratic forms by Orlov-Vishik-Voevodsky [Or-Vi-Vo], we have the isomorphism of graded rings H^{∗}(k(x); ℤ/2) ≅ grW^{∗}(k(x)). Using this fact, Pardon ([Pa],[To]) defined the spectral sequence

E(P)_{2}^{r,s} ≅ H_{Zar}^{r}(X; H_{ℤ/2}^{s}) ⟹ H^{r}(W (X)) ≅ E(BW)_{2}^{r,4s}

so that the differential d_{r} has degree (1, r −1) for r ≥ 2. Here H_{ℤ/2}^{n} the Zarisky sheaf induced from the presheaf H_{et}^{n}(V; ℤ/2) for open subset V of X.

The above sheaf cohomology H_{Zar}^{r}(X; H_{ℤ/2}^{s}) relates the motivic cohomology H^{∗,∗′}(X; ℤ/2) ( see [Vo1-3]). Let τ ∈ H^{0,1}(Speck(k); ℤ/2) ≅ ℤ/2 be a generator. (Hence H^{∗,∗′}(Speck(ℂ); ℤ/2) ≅ ℤ/2[τ].) Then we get the long exact sequence from the solution of Beilinson-Lichtenbaum conjecture by Voevodsky [Or-Vi-Vo]

→ H^{m,n−1}(X; ℤ/2) →^{×τ} H^{m,n}(X; ℤ/2)

→ H_{Zar}^{m−n}(X; H_{ℤ/2}^{n}) → H^{m+1,n−1}(X; ℤ/2) →^{×τ} .

Therefore, we have

Lemma 3.1. E(P)_{2}^{m−n,n} ≅ H_{Zar}^{m−n}(X; H_{ℤ/2}^{n}) ≅

H^{m,n}(X; ℤ/2)/(τ) ⊕ Ker(τ)|H^{m+1,n−1}(X; ℤ/2).

In particular, E(P)_{2}^{m,m} ≅ H^{2m,m}(X; ℤ/2) ≅ CH^{m}(X)/2. Moreover Totaro proved

Lemma 3.2. (Totaro [To]) If x ∈ E(P)_{2}^{m,m} ≅ CH^{m}(X)/2, then d_{2}(x) = Sq^{2}(x).

Now we consider the case X = M_{m,n} and k = ℂ. Since M_{m,n} is cellular, we see

H^{2∗,∗′}(M_{m,n}; ℤ/2) ≅ ℤ/2[τ]⊗CH^{∗}(M_{m,n})/2 ≅ ℤ/2[τ]⊗H^{2∗}(M_{m,n}; ℤ/2).

Hence we have

E(P)_{2}^{∗,∗′} ≅ E(P)_{2}^{∗,∗} ≅ H^{2∗}(M_{m,n}; ℤ/2).

From the result of Totaro, we have

E(P)_{3}^{∗,∗′} ≅ H(H^{2∗}(M_{m,n}; ℤ/2); Sq^{2}) = H(m, n).

By dimensional reason of differential deg(d_{r}) = (1, r − 1), it is immediate that the spectral sequence collapses from E_{3}^{∗,∗′}-term, i.e., E(P)_{3}^{∗,∗′} ≅ E_{∞}^{∗,∗′}.

The Balmer-Walter spectral sequence also collapses, that is, we will prove

E(P)_{∞}^{∗,∗} ≅ E(BW)_{2}^{∗,r} ≅ E(BW)_{∞}^{∗,r}, r = 0 mod(4).

For this we consider the Sp(n)-version of above arguments. We consider spectral sequences for N_{m,n}. By dimensional reason, the Pardon and Balmer-Walter spectral sequences collapse from E_{2}-terms. (Note H^{∗}(N_{m,n}; ℤ/2) = 0 for ∗ ≠ 0 mod(4).) That is

grW^{∗}(M_{m,n}) ≅ E(BW)_{∞}^{∗,r} ≅ H^{2∗}(N_{m,n}; ℤ/2).

Then the arguments of the proof of Theorem 2 also work. Thus we see the collapseness of the Balmer-Walter spectral sequence for M_{m,n}. Therefore we have isomorphisms

Theorem 3.3.

grW^{∗}(M_{m,n}) ≅ H(m, n) ≅ grKO^{2∗}(M_{m,n})/KU^{2∗}(M_{m,n}).

4. Young diagram and Witt group

In this section we recall the result of Balmer-Calmes [Ba-Ca1,2], and consider relation to the result of Hara-Kono.

The cohomology H^{∗}(M_{m,n}) (or CH^{∗}(M_{m,n})) is also computed by induction on n, m. In fact the following exact sequence

(4.1) → H^{∗−|a_{n}|}(M_{m,n−1}) →^{g_{∗}} H^{∗}(M_{m,n}) →^{f^{∗}} H^{∗}(M_{m−1,n}) →^{∂}

becomes split since ∂ = 0. Here g_{∗} is the Gysin map for the embedding M_{m,n−1} ⊂ M_{m,n}. The map f^{∗} is induced from

M_{m−1,n} ←^{proj.} M_{m,n} − M_{m,n−1} ⊂ M_{m,n}.

(See [Ba-Ca1,2] or Laksov [La], [Fu]). Here note g_{∗}(x) = a_{m} · x.

It is well known that the cohomology of M_{m,n} is stated also by using Young diagram. The m × n-framed partition λ = (λ_{1}, ..., λ_{d}) means

n ≥ λ_{1} ≥ ... ≥ λ_{d} ≥ 1 and m ≥ d.

The partition λ corresponds a Young diagram, consisting of λ_{i} boxes in the i-th row from the top, lined up on the left. Then m × n-framed Young diagrams with d = |λ| = λ_{1}+...+λ_{d} form the basis of H^{d}(M_{m,n}), namely, H^{∗}(M_{m,n}) ≅ ⊕_{∗=|λ|}ℤ. This fact is shown as follows.

The Young diagram for the conjugate partition λ̃ of λ is obtained by interchanging rows and columns in the diagram. For a Young diagram λ, we can define the Schur polynomial (e.g. see [Fu]) by

Δ_{λ}(b) = det(b_{λ_{i}+j−i}) ∈ H^{∗}(M_{m,n}).

It is known (Lemma 14.5.1 in [Fu]) Δ_{λ}(b) = Δλ̃(a). Hence we have Δ_{(k)}(b) = b_{k} and Δ_{(k̃)=(1,...,1)}(b) = a_{k}. Moreover we see by the above definition of Δ_{λ},

Δ_{λ}(b) = b_{λ_{1}}...b_{λ_{d}} mod(F_{>λ}).

Here F_{>λ} is the filtration of elements b_{λ′}_{1}....b_{λ′}_{d′} with λ′ > λ by the lexicographical order.

We still know the above Δ_{λ}(b) make a basis [λ] of H^{∗}(M_{m,n}) from (2.1). However we can also get it by induction by using the short exact sequence (4.1) such that

g_{∗}([λ]) = [(1, ..., 1) + λ] = [(1 + λ_{1}, ..., 1 + λ_{d}, 1, ..., 1)],

(Indeed, Δ_{g_{∗}(λ)}(b) = a_{m} · Δ_{λ}(b) mod(F_{>g^{∗}(λ)}).) The induced map f^{∗} is given f^{∗}(λ) = λ for d < m, and = 0 for d = m.

Let us say that framed Young diagram λ is strongly even if all its segments have even length, namely all λ_{i}, λ̃_{i} are even. Then its Schur polynomial is written

Δ_{λ}(b) = (b_{λ_{1}}^{2}b_{λ_{3}}^{2}...b_{λ_{d−1}}^{2}) mod(F_{>λ}).

Hence if m or n is even, then set of strongly even m×n-framed diagrams make ℤ/2-base of the ring B(k, l) given in Theorem 2.1 by Hara-Kono.

Balmer and Calmes results generalize above arguments. We can consider the generalized Witt group W^{i}(X; L) for i ∈ ℤ/4 and L ∈ P ic(X)/2 such that the usual Witt group W^{i}(X) = W^{i}(X; O_{X}).

Let us say that framed Young diagram λ is even if all its segments have even length, which are strictly inside of the frame, namely all λ_{i} − λ_{i+1} for 1 ≤ i ≤ d − 1, λ̃_{i} − λ̃_{i+1} for 1 ≤ i ≤ d̃− 1 are even. Let t(λ) be half of the perimeter of λ.

Theorem 4.1. (Balmer-Camles [Ba-Ca2]) The total Witt group

W^{tot}(M_{m,n}) = ⊕_{i∈ℤ/4, L∈ℤ/2}W^{i}(M_{m,n}; L)

has ℤ/2-basis indexed by even Young diagrams λ. The corresponding base [λ] is in W^{|λ|}(M_{m,n}, t(λ)).

Remark. In [Ba-Ca2], the theorem is stated in very generalized situation.

Let m or m be even. Then when t(λ) = 0, it is easily seen that λ is even means strongly even. So the argument before the above theorem explains the relation of the results by Hara-Kono and Balmer-Calmes.

Next we consider the case (m, n) = (2k+1, 2l+1). Each even m×nframed diagram λ with t(λ) = 0 is easily seen strongly even [λ^{se}] or [Γλ^{se}] which is defined as

[(2l + 1, 1, ..., 1^{m}) + (0, λ^{se})] = [(2l + 1, λ_{1}^{se} + 1, ...., λ_{d^{se}}^{se} + 1, 1..., 1^{m})].

Note μ = (2l + 1, 1, .., 1) is even but not strongly even, and |μ| = odd, t(μ) = 0. We still know [λ^{se}] form the ℤ/2-basis of B(k, l) in Theorem 2.1. Note that

Δ_{μ}(b) = b_{2l+1}a_{2k} = a_{2k+1}b_{2l} mod(F_{>μ}).

Hence [Γλ^{se}] form a basis of B(k, l){a_{2k+1}b_{2l}}. Therefore we see even m × n-framed Young diagrams form the base of

B(k, l) ⊕ B(k, l){a_{2k+1}b_{2l}} = H(m, n).

Thus we can explain the relation between Hara-Kono and Balmer-Calmes.

Balmer and Camles prove their theorem by showing following (long) exact sequence. (They construct the Gysin and the boundary maps as the maps in W^{total}(X).) Let g : Z ⊂ X be a regular closed immersion of codim = c ≥ 2, and U = X − Z. Let ω_{g} be the relative canonical bundle (for the definition see [Ba-Ca1]). Then there is the natural exact sequence

→ W^{∗−c}(Z, ω_{g} ⊗ L|_{Z}) →^{g_{∗}} W^{∗}(X, L) →^{f^{∗}} W^{∗}(Z, L|_{Z}) → .^{∂}

In general ∂ ≠ 0. In fact, when X = M_{m,n}, it is proved (Figures 4-6 in [Ba-Ca2]).

g_{∗}([λ]) = \begin{cases}[(1, .., 1^{m}) + λ] if m − d : even \\ 0 otherwise, \end{cases} f^{∗}([λ]) = \begin{cases}[λ] if d < m \\ 0 otherwise, \end{cases}

∂([λ]) = \begin{cases}[λ − (1, ..., 1^{d})] if λ_{d} : odd \\ 0 otherwise. \end{cases}

In the 𝔸^{1}-homotopy category, Hornbostel [Ho] proved that W^{∗}(−) is represented as a ℙ^{1}-spectrum. This implies

W^{∗+1}(ℙ^{1} ∧ X) ≅ W^{∗}(X).

Therefore we can define the natural map

q : W^{∗}(X) → KO^{2∗}(X(ℂ))/KU^{2∗}(X(ℂ)) for all ∗ .

We will see that this map induces the isomorphism given preceding and this sections.

First we consider the cases (m, n) = (2k, 2l). Let g̃ : M_{m,n−2} ⊂ M_{m,n} and f̃ : M_{m,n} → M_{m−2,n}. (Note g̃_{∗}(x) = a_{m}^{2}x.) Then we have the commutative diagram.

0 ⟶ W^{0}(M_{m,n−2}) ⟶^{g̃_{∗}} W^{0}(M_{m,n}) ⟶^{f̃^{∗}} W^{0}(M_{m−2,n}) ⟶^{˜∂∗} 0
↓^{q_{1}}  ^{q_{2}}↓  ^{q_{3}}↓
0 ⟶ K^{0}(M_{m,n−2}) ⟶^{g̃_{∗}} K^{0}(M_{m,n}) ⟶^{f̃^{∗}} K^{0}(M_{m−2,n}) ⟶^{˜∂∗} 0

where K^{∗}(X) = KU^{∗}(X)/KO^{∗}(X). The exactness of rows follow from the isomorphism given in the preceding or this sections. (In fact this case W_{m,n}^{∗} = W_{m,n}^{0}.) By the induction and five lemma, we have the isomorphism of q_{2}. The case m or n even follows from the above result and the naturality.

The case (m, n) = (2k + 1, 2l + 1) is proved as follows. Let g′ : M_{2k,2l} → M_{2k+1,2l+1} and recall g′_{∗}([λ]) = [Γλ]. Consider the following diagram

\begin{matrix}W^{0}(M_{2k,2l}) & ⟶_{≅}^{g′_{∗}} & W^{1}(M_{2k+1,2l+1}) \\ ↓^{q_{1}} & ^{q_{2}}↓ & . \\ K^{0}(M_{2k,2l}) & ⟶_{≅}^{g′_{∗}} & K^{2}(M_{2k+1,2l+1}). \end{matrix}

We see q_{2} is isomorphic for this case. Similarly we consider f′ : M_{2k,2l} → M_{2k+1,2k+1}. By the isomorphism of preceding or this section, we have the isomorphism f′^{∗} : W^{0}(M_{2k+1,2k+1})≅W^{0}(M_{2k,2k}). This also induces the isomorphism of q for W^{0}(M_{m,n}) → K^{0}(M_{m,n}). Thus we can show

Theorem 4.2. The map q : W^{∗}(M_{m,n}) → KO^{2∗}(M_{m,n})/KU^{2∗}(M_{m,n}) induces the isomorphism.

Corollary 4.3. There is the isomorphism of graded rings

grW^{∗}(M_{m,n}) ≅ H(m, n).

In particular, we note that grW^{0}(M_{2k,2l}) ≅ H^{∗}(M_{k,l}; ℤ/2).

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Department of Mathematics, Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan

E-mail address: yagita@mx.ibaraki.ac.jp