Lines crossing a tetrahedron and the Bloch group
aa r X i v : . [ m a t h . N T ] A p r Lines crossing a tetrahedron and the Bloch group
Kevin Hutchinson and Masha VlasenkoFebruary 27, 2018
According to B. Totaro ([7]), there is a hope that the Chow groups of a field k can be computed using a very small class of affine algebraic varieties (linearspaces in the right coordinates), whereas the current definition uses essentiallyall algebraic cycles in affine space. In this note we consider a simple modificationof CH (Spec( k ) ,
3) using only linear subvarieties in affine spaces and show thatit maps surjectively to the Bloch group B ( k ) for any infinite field k . We alsodescribe the kernel of this map.The second autor is grateful to Anton Mellit, who taught her the idea ofpassing from linear subspaces to configurations (Lemma 1 below) and pointedout the K-theoretical meaning of Menelaus’ theorem, and to the organizers ofIMPANGA summer school on algebraic geometry for their incredible hospitalityand friendly atmosphere. Let k be an arbitrary infinite field. Consider the projective spaces P n ( k ) withfixed sets of homogenous coordinates ( t : t : · · · : t n ) ∈ P n ( k ). We call asubspace L ⊂ P n ( k ) of codimension r admissible ifcodim (cid:16) L ∩ { t i = · · · = t i s = 0 } (cid:17) = r + sfor every s and distinct i , . . . , i s . (Here codim(X) > n means X = ∅ .) Let C rn = Z h admissible L ⊂ P n ( k ) , codim(L) = r i be the free abelian group generated by all admissible subspaces of P n ( k ) ofcodimension r . Then for every r we have a complex . . . d −→ C rr +2 d −→ C rr +1 d −→ C rr −→ −→ . . . (we assume that C rn = 0 when n < r ) with the differential d [ L ] = X ( − i [ L ∩ { t i = 0 } ] (1)where every { t i = 0 } ⊂ P n ( k ) is naturally identified with P n − ( k ) by throwingaway the coordinate t i . We are interested in the homology groups of thesecomplexes H rn = H n ( C r • ). 1or example, one can easily see that H ∼ = k ∗ . Indeed, a hyperplane { P α i t i = 0 } is admissible whenever all the coefficients α i are nonzero, andif we identify C ∼ = Z [ k ∗ ] [ { α t + α t = 0 } ] h α α i C ∼ = Z [ k ∗ × k ∗ ] [ { α t + α t + α t = 0 } ] h(cid:0) α α , α α (cid:1)i (2)then the differential d : C −→ C turns into[( x, y )] [ x ] − [ xy ] + [ y ] . (one can recognize Menelaus’ theorem from plane geometry behind this simplecomputation). Hence we have H ∼ = Z [ k ∗ ] .(cid:8) [ x ] − [ xy ] + [ y ] : x, y ∈ k ∗ (cid:9) ∼ = k ∗ . Continuing the identifications of (2), C • turns into the bar complex for thegroup k ∗ (with the term of degree 0 thrown away) and therefore H n = H n ( k ∗ , Z ) , n ≥ . Now we switch to r = 2 and try to compute H . The four hyperplanes { t i = 0 } form a tetrahedron ∆ in the 3-dimensional projective space P ( k ) andthe line ℓ is admissible if it1) intersects every face of ∆ transversely, i.e. at one point P i = ℓ ∩ { t i = 0 } ;2) doesn’t intersect edges { t i = t i = 0 } of ∆, i.e. all four points P , . . . , P ∈ ℓ are different .Therefore it is natural to associate with ℓ a number, the cross-ratio of the fourpoints P , . . . , P on ℓ . Namely, there is a unique way to identify ℓ with P ( k )so that P , P and P become 0, ∞ and 1 respectively, and we denote the imageof P by λ ( ℓ ) ∈ P ( k ) r { , ∞ , } = k ∗ r { } . We extend λ linearly to a map C λ −→ Z [ k ∗ r { } ] X n i [ ℓ i ] X n i [ λ ( ℓ i )] Theorem 1.
Let σ : k ∗ ⊗ k ∗ −→ k ∗ ⊗ k ∗ be the involution σ ( x ⊗ y ) = − y ⊗ x .(i) If d ( P n i [ ℓ i ]) = 0 then P n i λ ( ℓ i ) ⊗ (1 − λ ( ℓ i )) = 0 in ( k ∗ ⊗ k ∗ ) σ .(ii) Let L ⊂ P ( k ) be an admissible plane and ℓ i = L ∩ { t i = 0 } , i = 0 , . . . , .If we denote x = λ ( ℓ ) and y = λ ( ℓ ) then λ ( ℓ ) = yx , λ ( ℓ ) = 1 − x − − y − and λ ( ℓ ) = 1 − x − y . (iii) The map induced by λ on homology λ ∗ : H −→ B ( k ) (3)2 s surjective, where B ( k ) = Ker (cid:16) Z [ k ∗ r { } ] −→ ( k ∗ ⊗ k ∗ ) σ [ a ] a ⊗ (1 − a ) (cid:17)D [ x ] − [ y ] + h yx i − h − x − − y − i + h − x − y i , x = y E is the Bloch group of k ([5]).(iv) We have H ∼ = H (GL ( k )) /H ( k ∗ ) and the kernel of (3) K = Ker (cid:0) H λ ∗ −→ B ( k ) (cid:1) fits into the exact sequence −→ Tor( k ∗ , k ∗ ) ∼ −→ K/T ( k ) −→ k ∗ ⊗ K ( k ) −→ K M ( k ) / −→ , (4) where Tor( k ∗ , k ∗ ) ∼ is the unique nontrivial extension of Tor( k ∗ , k ∗ ) by Z / , and T ( k ) is a -torsion abelian group (conjectured to be trivial). We remark that Tor( k ∗ , k ∗ ) = Tor( µ ( k ) , µ ( k )) is a finite abelian group if k is a finitely-generated field. Furthermore, it is proved in [5] that B ( k ) hasthe following relation to K ( k ): let K ind3 ( k ) be the cokernel of the map fromMilnor’s K-theory K M ( k ) −→ K ( k ), then there is an exact sequence0 −→ Tor( k ∗ , k ∗ ) ∼ −→ K ind3 ( k ) −→ B ( k ) −→ k is a number field then as a consequence of (5) and Borel’stheorem ([1]) we have dim B ( k ) ⊗ Q = r , where r is the number of pairs of complex conjugate embeddings of k into C . Proof of (i) and (ii).
One can check that the diagram C λ (cid:15) (cid:15) d / / C t : t : t ] t ⊗ ( − t ) + ( − t ) ⊗ t + t ⊗ t + t ⊗ t (cid:15) (cid:15) Z [ k ∗ r { } ] [ a ] a ⊗ (1 − a ) / / ( k ∗ ⊗ k ∗ ) σ is commutative, and therefore (i) follows. It is another tedious computation tocheck (ii).In the next section we will prove the remaining claims (iii) and (iv) and alsoshow that H n ∼ = H n (GL ( k ) , Z ) /H n ( k ∗ , Z ) n ≥ . (6)3 Complexes of configurations
We say that n + 1 vectors v , . . . , v n ∈ k r are in general position if every ≤ r ofthem are linearly independent. Let C ( r, n ) be the free abelian group generatedby ( n + 1)-tuples of vectors in k r in general position. For fixed r we have acomplex . . . d −→ C ( r, d −→ C ( r, d −→ C ( r, d [ v , . . . , v n ] = X ( − i [ v , . . . , ˇ v i , . . . , v n ] (7)The augmented complex C ( r, • ) −→ Z −→ d (cid:16)X n i [ v i , . . . , v in ] (cid:17) = 0and v ∈ k r is such that all ( n + 2)-tuples [ v, v i , . . . , v in ] are in general position(such vectors v exist since k is infinite) then X n i [ v i , . . . , v in ] = d (cid:16)X n i [ v, v i , . . . , v in ] (cid:17) . Lemma 1. C rn ∼ = C ( r, n ) GL r ( k ) for the diagonal action of GL r ( k ) on tuplesof vectors. Moreover, the complex C r • is isomorphic to the truncated complex C ( r, • ) GL r ( k ) , •≥ r .Proof. For n ≥ r there is a bijective correspondence between subspaces of codi-mension r in P n ( k ) and GL r ( k )-orbits on ( n + 1)-tuples [ v , . . . , v n ] of vectorsin k r satisfying the condition that v i span k r . It is given by L ⊂ P n [ v , . . . , v n ] , v i = image of e i in k n +1 / e L ∼ = k r [ v , . . . , v n ] e L = Ker[ v , . . . , v n ] T ⊂ k n +1 where e L is the unique lift of L to a linear subspace in k n +1 and e , . . . , e n is astandard basis in k n +1 .An admissible point in P r ( k ) is a point which doesn’t belong to any of the r + 1 hyperplanes { t i = 0 } , and for the corresponding vectors [ v , . . . , v r ] itmeans that every r of them are linearly independent. For n > r a subspace L of codimension r in P n ( k ) is admissible whenever all the intersections L ∩ { t i =0 } are admissible in P n − ( k ). Hence it follows by induction that admissiblesubspaces correspond exactly to GL r ( k )-orbits of tuples “in general position”.Obviously, differential (1) is precisely (7) for tuples.The tuples of vectors in general position in k r modulo the diagonal actionof GL r ( k ) are called configurations , so C ( r, n ) GL r ( k ) is the free abelian groupgenerated by configurations of n + 1 vectors in k r . Proof of (iii) in Theorem 1.
For brevity we denote C (2 , n ) by C n and GL ( k )by G . Since the complex of G -modules C • is quasi-isomorphic to Z we havethe hypercohomology spectral sequence with E pq = H q ( G, C p ) ⇒ H p + q ( G, Z ).Since all modules C p with p > E pq = 0 for p, q > E p = ( C p ) G . If G ⊂ G is the stabilizer of (cid:0) (cid:1) then E q = H q ( G, Z [ G/G ]) =4 q ( G , Z ) by Shapiro’s lemma. We have k ∗ ⊂ G and H q ( k ∗ , Z ) = H q ( G , Z )(see Section 1 in [6]), so E q = H q ( k ∗ , Z ). Further, E p = H p (( C • ) G ) and E q = H q ( k ∗ , Z ). This spectral sequence degenerates on the second term. Indeed, theembedding k ∗ ֒ → Gα (cid:18) α (cid:19) is split by determinant, and therefore all maps H q ( k ∗ , Z ) −→ H q ( G, Z ) areinjective. Consequently, E ∞ pq = E pq and for every n ≥ −→ H n ( k ∗ , Z ) −→ H n ( G, Z ) −→ H n (cid:0) ( C • ) G (cid:1) −→ . It follows from Lemma 1 that H n = H n (cid:0) ( C • ) G (cid:1) = H n ( G, Z ) /H n ( k ∗ , Z ) , n ≥ . Let D n be the free abelian group generated by ( n + 1)-tuples of distinctpoints in P ( k ). Again we have the differential like (7) on D • and the augmentedcomplex D • −→ Z −→ C • to D • since a non-zero vector in k defines a point in P ( k ) and the group actionagrees. The spectral sequence e E pq = H q ( G, D p ) ⇒ H p + q ( G, Z ) was consideredin [5]. In particular, e E p = ( D p ) G is the free abelian group generated by ( p − G -orbit of every ( p + 1)-tuple contains a uniqueelement of the form (0 , ∞ , , x , . . . , x p − ), and the differential d : e E −→ e E is given by [ x, y ] [ x ] − [ y ] + h yx i − h − x − − y − i + h − x − y i . (8)According to [5], terms e E pq with small indices are H ( k ∗ ⊕ k ∗ ) H ( k ∗ ) ⊕ ( k ∗ ⊗ k ∗ ) σ ( k ∗ ⊗ k ∗ ) σ k ∗ Z p ( k )where p ( k ) is the quotient of Z [ k ∗ r { } ] by all 5-term relations as in right-handside of (8), and the only non-trivial differential starting from p ( k ) is d : p ( k ) −→ H ( k ∗ ) ⊕ ( k ∗ ⊗ k ∗ ) σ = Λ ( k ∗ ) ⊕ ( k ∗ ⊗ k ∗ ) σ [ x ] x ∧ (1 − x ) − x ⊗ (1 − x )5herefore e E = e E ∞ = B ( k ) and we have a commutative triangle H ( G ) & & & & LLLLLLLLLL / / / / E ∞ = H (cid:15) (cid:15) e E ∞ = B ( k )where both maps from H ( G ) are surjective, hence the vertical arrow is alsosurjective. It remains to check that the vertical arrow coincides with λ ∗ . A line ℓ in P ( k ) is given by two linear equations and for an admissible line it is alwayspossible to chose them in the form ( t + x t + x t = 0 ,t + y t + y t = 0 . This line corresponds to the tuple of vectors (cid:18) (cid:19) , (cid:18) (cid:19) , (cid:18) x y (cid:19) , (cid:18) x y (cid:19) which can be mapped to the points 0 , ∞ , , x y y x in P ( k ), hence the verticalarrow maps it to [ x y y x ] (actually we need to consider a linear combination of lineswhich vanishes under d but for every line the result is given by this expression).On the other hand, four points of its intersection with the hyperplanes are P = (0 : y x − y x : − x : x ) P = ( y x − y x : 0 : − y : y ) P = ( − x : − y : 0 : 1) P = ( − x : − y : 1 : 0)and if we represent every point on ℓ as αP + βP then the corresponding ratios βα will be 0 , ∞ , − x y , − x y . Hence λ ( ℓ ) = x y y x again and (iii) follows.To prove (iv) we first observe that the Hochschild-Serre spectral sequenceassociated to 1 −→ SL ( k ) −→ GL ( k ) det −→ k ∗ −→ −→ H (cid:16) k ∗ , H (SL ( k ) , Z ) (cid:17) −→ Ker (cid:16) H (GL ( k ) , Z ) det −→ H ( k ∗ , Z ) (cid:17) −→ H (cid:16) k ∗ , H (SL ( k ) , Z ) (cid:17) −→ . (9)The first term here maps surjectively to K ind3 ( k ) (see the last section of [2]),and the map is conjectured by Suslin to be an isomorphism (see Sah [4]). It isknown that its kernel is at worst 2-torsion (see Mirzaii [3]).Thus we let T ( k ) := Ker (cid:0) H ( k ∗ , H (SL ( k ) , Z )) −→ K ind3 ( k ) (cid:1) .
6y the preceeding remarks, this is a 2-torsion abelian group. Since the em-bedding k ∗ −→ GL ( k ) is split by the determinant, the middle term in (9) isisomorphic to H . Then applying the snake lemma to the diagram0 / / T ( k ) / / (cid:127) _ (cid:15) (cid:15) H (cid:16) k ∗ , H (SL ( k ) , Z ) (cid:17) / / (cid:127) _ (cid:15) (cid:15) K ind3 ( k ) / / (cid:15) (cid:15) (cid:15) (cid:15) / / K / / H / / B ( k ) / / −→ Tor( k ∗ , k ∗ ) ∼ −→ K/T ( k ) −→ H (cid:16) k ∗ , H (SL ( k ) , Z ) (cid:17) −→ . Finally, it follows from [2] that there is a natural short exact sequence0 −→ H (cid:16) k ∗ , H (SL ( k ) , Z ) (cid:17) −→ k ∗ ⊗ K M ( k ) −→ K M ( k ) / −→ . This proves (4).
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