Commun. Math. Phys. 211, 359 363 2000) Communications in Mathematical Physics Springer-Verlag 2000 Determinant Bundle in a Family of Curves, after A. Beilinson and V. Schechtman Hélène snault 1, I-Hsun Tsai 2 1 FB6, Mathematik, Universität ssen, 45117 ssen, Germany. -mail: esnault@uni-essen.de 2 Department of Mathematics, National Taiwan University, Taipei, Taiwan. -mail: ihtsai@math.ntu.edu.tw Received: 17 October 1999 / Accepted: 29 November 1999 Abstract: Let π : X S be a smooth projective family of curves over a smooth base S over a field of characteristic 0, together with a bundle on X. Then A. Beilinson and V. Schechtman define in [1] a beautiful trace complex tr A on X, the 0th relative cohomology of which describes the Atiyah algebra of the determinant bundle of on S. Their proof reduces the general case to the acyclic one. In particular, one needs a comparison of Rπ tr A F ) for F = and F = D), where D is étale over S see Theorem 2.3.1, reduction ii) in [1]). In this note, we analyze this reduction in more detail and correct a point. 1. Introduction Let π : X S be a smooth projective morphism of relative dimension 1 over a smooth base S over a field k of characteristic 0. One denotes by T X and the tangent sheaf over k, byt X/S the relative tangent sheaf, and by ω X/S the relative dualizing sheaf. For an algebraic vector bundle on X, one writes = OX ω X/S. Let Diff, ) resp. Diff/S, /S) Diff, )) be the sheaf of first order resp. relative) differential operators on and ɛ : Diff, ) nd) OX T X be the symbol map. The Atiyah algebra A := {a Diff, ) ɛa) id OX T X } of is the subalgebra of Diff, ) consisting of the differential operators for which the symbolic part is a homothety. Similarly the relative Atiyah algebra A /S A of consists of those differential operators with symbol in id OX T X/S and A,π A with symbols in T π = dπ 1 π 1 T S ) T X. Let X S X be the diagonal. Then there is a canonical sheaf isomorphism Diff/S, /S) = 2 ) see [1], Sect. 2) which is locally written as follows. Let x be a local coordinate of X at a point p, and x, y) be the induced local coordinates on X S X at p, p), such that the equation of becomes
360 H. snault, I-H. Tsai x = y. Let e i be a local basis of, ej be its local dual basis. Then the action of P = i,j e i e j P ij x, y) x y) 2 dy on s = l s ly)e l is Ps) = e i P 1) ij x, 0)s j x) + P ij x, x)s 1) j x, 0)), 1.1) i j where P ij x, y) = P ij x, x) + y x)p 1) ij x, y x), s j y) = s j x) + y x)s 1) j x, y x). Beginning with 0 ) 2 ) Diff/S, /S) 0, 1.2) ) restricting to A /S Diff, ), and pushing forward by the trace map ) ω = ωx/s, yields an exact sequence 0 ω X/S tr A 1 γ A /S 0. 1.3) One defines the trace complex tr A by A,π for i = 0, tr A 1 for i = 1, O X for i = 2 and 0 else, with differentials d 1 := γ and d 2 equal to the relative Kähler differential see [1], Sect. 2). One has an exact sequence of complexes 0 X/S [2] tr A T X/S T π )[1] 0, 1.4) where X/S is the relative de Rham complex of π. Taking relative cohomology, one obtains the exact sequence 0 O S R 0 π tr A ) T S 0. 1.5) Furthermore R 0 π tr A ) is a sheaf of algebras [1], 1.2.3). One denotes by πtr A ) the sheaf on S together with its algebra structure. Finally, let B i,i = 1, 2 be two sheaves of algebras on S, with an exact sequence of sheaves of algebras 0 O S B i T S 0. 1.6) One defines B 1 +B 2 by taking the subalgebra of B 1 B 2, inverse image B 1 TS B 2 of the diagonal embedding T S T S T S, and its push out via the trace map O S O S O S. The aim of this note is to prove Theorem 1.1. Let D X be a divisor, étale over S. One has a canonical isomorphism π tr A ) = π tr A D) ) + A det π D ) 1.7) This is [1] Theorem 2.3.1, ii). We explain in more details the proof given there and correct a point in it.
Determinant Bundle 361 2. Proof of Theorem 1.1 The proof uses the construction of a complex L, together with maps L tr A and L tr A D) i D A D inducing isomorphisms from R 0 π L with the left and the right-hand side of Theorem 1.1. We make the construction of L and the maps explicit, and show that the induced morphisms are surjective, with the same non-vanishing) kernel. We first recall the definition of the sub-complex L tr A see [1, Theorem 2.3.1, ii)]): L 0 tr A 0 consists of the differential operators P with ɛp) T π < D >, where T π < D >= T π T X < D >and T X < D >= Hom OX 1 X <D>,O X), where 1 X <D>denotes the sheaf of 1-forms with log poles along D. In particular, d 1 ) 1 L 0 ) tr A maps to i D A D, and L 1 d 1 ) 1 L 0 ) is defined as the kernel. Then L 2 = O X. The product struture on tr A is defined in [1], 2.1.1.2, and coincides with the Lie algebra structure on tr A 0 = A,π. Since L 2 = tr A 2, to see that the product structure stabilizes L, one just has to see that L 0 tr A 0 is a subalgebra, which is obvious, and that L 0 L 1 tr A 1 takes values in L 1, which is a consequence of Proposition 2.2. As in Sect. 1, we denote by A /S the relative Atiyah algebra of, with symbolic part T X/S and by A,π Beilinson s subalgebra of the global Atiyah algebra with symbolic part T π.ifι : F is a vector bundle, isomorphic to away of D, then one has an injection of differential operators induced by ι on the second argument, and an injection induced by ι on the first argument. One has Definition 2.1. Diff, F ) i Diff, ) 2.1) Diff, F ) j DiffF, F ) 2.2) A /S,F/S) := A /S i Diff, F ) = A F/S j Diff, F ) Recall γ : tr A 1 A /S denotes the map coming from the filtration by the order of poles of O X X ) on tr A 1. One has Proposition 2.2. Proof. One considers γ 1 A /S, D)/S) = γ 1 D) A /S, D)/S) = L 1. = [ D) 2 ) D) ) D) 2 ) + ) ) ] [ / D) D) ) ] 2.3)
362 H. snault, I-H. Tsai which, via the natural inclusion to is the inverse image γ 1 Diff, D)) 2 ) 2.4) ) ) here we abuse notations, still denoting by γ the map coming from the filtration), and via the map coming from the natural inclusion D) 2 ) D) ) D) D)2 ) D) D) ) 2.5) and the identification with the first term of the filtration on D) D)2 ) D) D) ) ) D) D) = D) D) ) ) is the inverse image γ 1 D) Diff, D)). 2.6) The filtration induced by the order of poles of O X X D) induces the exact sequences 0 Hom, D)) A /S, D)/S) T X/S D) 0, 2.7) 0 nd) A /S T X/S 0, 2.8) 0 nd) A D)/S T X/S 0. 2.9) Now, as one has an injection L tr A with cokernel Q, and again by looking at the filtration by the order of poles on the sheaf in degree -1), one obtains and Theorem 2.3. One has an exact sequence Q = nd) D [1] 2.10) 0 R 0 π nd) D ) R 0 π L ) R 0 π tr A ) 0. On the other hand, one has an injection L tr A D) i D A D with cokernel P, and, as L injects into tr A D), one has an exact sequence 0 i D A D [0] P [ tr A D) /L ] 0, 2.11) where i D : D X is the closed embedding. We see that the induced filtration on the sheaf in degree -1) of [ tr A D) /L ] has graded pieces 0, nd D ), T X/S D ), whereas the filtration on the sheaf in degree 0) has graded pieces 0,T π /T π < D > = T X/S D ). This last point comes from the obvious Lemma 2.4. {P Diff, ), P D)) D)} = {P Diff md), md)), ɛp ) nd) T< D >} for any m Z, where ɛ is the symbol map.
Determinant Bundle 363 So Lemma 2.5. [ tr A D) /L ] is quasi-isomorphic to nd D )[1]. The connecting morphism R 1 π [ tr A D) /L ] R 0 π i D A D )[0] is just the natural embedding π nd D )) π i D A D ) with cokernel π π 1 D T S.IfD is irreducible, one has π π 1 D T S = T S, and therefore Proposition 2.6. If D is irreducible, one has an exact sequence 0 R 0 π L R 0 π [ tr A D) ] R0 π [i D A D ] T S 0 and the image of R 0 π L is obtained from the direct sum by taking the pull back under the diagonal embedding T S T S T S. On the other hand, still assuming D irreducible, one has the exact sequence 0 π nd D ) R 0 π [i D A D ] T S = π π 1 D T S 0 2.12) and the Atiyah algebra A detπ D ) is the push out of R 0 π [i D A D ] by the trace map π nd D ) O S. Defining K := Ker O S π nd D ) id Tr ) O S = π nd D ), 2.13) one thus obtains Theorem 2.7. If D is irreducible, one has an exact sequence 0 K R 0 π L π tr A D) ) + A detπ D ) 0. It can be easily shown that the embedding π nd D ) R 0 π L in Theorems 2.3 and 2.7 is the same embedding of a subsheaf of ideals. It finishes the proof of Theorem 1.1 when D is irreducible. In general, since D is étale over S, its irreducible components are disjoint, thus one proves Theorem 1.1 by adding one component at a time. References 1. Beilinson, A., Schechtman, V.: Determinant Bundles and Virasoro Algebras. Commun. Math. Phys. 118, 651 701 1988) Communicated by A. Connes