Poincaré Duality for Algebraic De Rham Cohomology.

oincaré Duality for Algebraic De Rham ohomology. Francesco Baldassarri, aurizio ailotto, Luisa Fiorot Abstract. We discuss in some detail the algebraic notion of De Rham cohomology with compact supports for singular schemes over a field of characteristic zero. We prove oincaré duality with respect to De Rham homology as defined by Hartshorne [H.75], so providing a generalization of some results of that paper to the non proper case. n order to do this, we work in the setting of the categories introduced by Herrera and Lieberman [HL], and we interpret our cohomology groups as hyperext groups. We exhibit canonical morphisms of cospecialization from complex-analytic De Rham resp. rigid cohomology groups with compact supports to the algebraic ones. These morphisms, together with the specialization morphisms [H.75, V.1.2] resp. [BB, 1] going in the opposite direction, are shown to be compatible with our algebraic oincaré pairing and the analogous complex-analytic resp. rigid one resp. [B.97, 3.2]. ntroduction. n his paper on De Rham cohomology of algebraic varieties [H.75], Hartshorne defined De Rham cohomology and homology groups for singular schemes over a field of characteristic zero. He proved a global oincaré duality theorem in the proper case. Naturally, one expects to have a notion of algebraic De Rham cohomology with compact supports oincaré dual to De Rham homology, extending Hartshorne s De Rham cohomology and oincaré duality to not necessarily proper nor regular schemes. To be useful, such a theory should admit an independent i.e. not based on duality with homology and easily computable definition. The method for doing this in general has been well-known since Deligne s lectures on crystals in characteristic zero at HES in arch 1970. He sketched there an algebraic theory of the functor f! for crystals, providing the natural setting for a general answer to the above problem. An account of those lectures, however, has never been made available: we are not aware of any satisfactory reference for algebraic De Rham cohomology with compact supports, not even in the standard case of constant coefficients on a smooth open variety. n view of the generality of Deligne s lectures, the idea of publishing an independent account of the case of constant coefficients may seem obsolete. On the other hand, even the choice of a suitable level of generality for coefficients is embarassing, even if only because of the simultaneous need of pro-coherent and ind-coherent O-modules. We have preferred to introduce the minimal amount of technique necessary to give a good and flexible definition of algebraic cohomology with compact supports for singular -varieties. Hartshorne s choice in [H.75] was to ignore the open case altogether, concentrating on the proper singular case, despite his experience with Serre duality for open varieties over a field of characteristic zero [H.72]. The main point of this article is the following. The case of a proper singular -scheme, presents the big technical advantage that one can develop all considerations in the category of abelian sheaves on the Zariski space. n the open case, will have to be compactified to, and will be embedded as a closed subscheme of a scheme, smooth in a neighbourhood of. The category A b is suitable for the definition of H DR,c, = Ω H Ω \ but not so for proving that this definition is independent of the embeddings. We are aware of at least three methods for proving that the given definition of H DR,c is good. The most general one thrives in the previously mentioned context of crystals in characteristic zero. A second method is based on Grothendieck s linearization of differential operators [AB, Appendix D] dual to Saito s linearization [S]. A third approach is via the filtered refinement of Herrera and Lieberman s -odules [HL] proposed by Du Bois [DB.90]. We plan to investigate the full formalism of Grothendieck operations for De Rham coefficients following Du Bois, in the near future. For the limited scope of the present article however, 2000 athematics Subject lassification. rimary 14F. University of adova, taly 1 [baldassa,maurizio,fiorot]@math.unipd.it

we chose to simply revisit the original article of Herrera and Lieberman [HL], which has the virtue of making the relation between Serre and oincaré duality very explicit. This theory is of course not sufficiently flexible to permit the introduction of Grothendieck s operations in general. But constructions are very explicit and the comparison with analytic theories becomes natural. We hesitated on whether we should work with pro-objects of the category of coherent sheaves on a scheme or analytic space, as Deligne does, or take their derived inverse limits on a formal scheme or formal analytic space, as in Hartshorne [H.72], [H.75]. The point of a ittag-leffler type of result as [H.75,.4.5] see also lemma 1.3.1 below is that for suitable pro-objects, derived direct images commute with inverse limits, so that applying the functor lim to those pro-objects is harmless. As a consequence, in our situation we take indifferently one attitude or the other. t is worthwhile mentioning that when the base scheme is Spec, the procategory of finite dimensional -vector spaces is equivalent via lim to the category of linearly compact topological -vector spaces. So, Hartshorne s Serre duality of [H.72], may be regarded as a perfect topological pairing between a discrete topological -vector space and a linearly compact one both in general infinite dimensional. n the case of De Rham cohomology [H.75], this specializes to a duality of finite dimensional -vector spaces. We use this topological pairing when describing a duality of spectral sequences converging to oincaré duality. One should compare this with the analytic problem posed by Herrera and Lieberman [HL, 5]. We want to mention that very recently, hiarellotto and Le Stum have included in their wide-ranging article [LS] a short account of Deligne s method for defining De Rham cohomology with compact supports for open smooth -varieties. n the case of a smooth -scheme and of a connection E,, with E a coherent, hence locally free O -module, the De Rham cohomology groups with compact supports H q DR,c, E, were defined in [AB, Def. D.2.16] and oincaré duality was proved [AB, D.2.17], in fact in a more general relative situation. The definition of f! proposed by atz-laumon [L, section 7], Du Bois [DB.90, 6.9] and ebkhout [,.5.3], for f : a morphism of smooth -varieties and an object of Dhol b D, is for us less satisfactory, since duality is built into its definition. The experience with Dwork s dual theory [AB, appendix D], shows that an independent definition of f! is of great help in calculations and for arithmetic applications, and is therefore very desirable. We should mention here the crucial help we received from. Berthelot who provided us with his notes of Deligne s course and whose treatement of rigid cohomology was also very useful, somewhat paradoxically, to organize our discussion in the algebraic case. ontents ntroduction. 0. Notation and reliminaries. 1. De Rham ohomology with compact supports. 2. Hyperext functors and De Rham Homology. 3. Algebraic oincaré Duality. 4. ünneth formulae. 5. lassical comparison theorems. 6. ompatibility of rigid and algebraic oincaré duality. References. 0. Notation and reliminaries. 0.1. Schemes. Let be a field of characteristic 0. By scheme we will mean a separated -scheme of finite type. orphisms and products of schemes will be taken over Spec, unless otherwise specified. For a scheme, will denote the underlying topological space. 0.2. losed mmersions. f is a closed subscheme of the scheme, we indicate by, or simply by, the coherent deal of O associated to. 0.3. nfinitesimal neighborhoods. f i : is a closed immersion of schemes, we denote by the -th infinitesimal neighborhood of in, that is the scheme having as topological space University of adova, taly 2 [baldassa,maurizio,fiorot]@math.unipd.it

and O +1 restricted to as structural sheaf. Let be the formal completion of along, i.e. the formal scheme inductive limit of all infinitesimal neighborhoods of in ; its underlying topological space is and its structural sheaf is O := O = lim O. Following [EGA, 10.8], if F is a quasi-coherent O -odule, F will be the sheaf an O - odule of formal sections of F along i.e. the projective limit lim i F where i : is the canonical immersion; if F is coherent, F = i F where i : is the canonical morphism of ringed spaces, since our schemes are locally noetherian. 0.4. Differentials on infinitesimal neighborhoods. The closed immersion of in, of deal = = +1, gives rise to the exact sequence 2 d Ω 1 O O Ω 1 0. The exact sequence shows that lim 0 2 2 O 2+1 O 0 = 0. Therefore, the canonical morphism lim Ω 1 O O lim is an isomorphism. This isomorphism still holds in higher degrees, i.e. lim Ω i O O for all i. On the other hand, Ω i = lim Ω i, so that Ω i = lim Ω i = lim Ω 1 Ω i = Ω i O O = Ω i. 0.5. ategories of Differential Operators. Let be a scheme. We recall [HL] or [B.74,.5] that the category of complexes of differential operators of order less or equal to one is defined as follows: the objects of are complexes whose terms are O -odules and whose differentials are differential operators of order less or equal to one. orphisms between such complexes are O -linear maps of degree zero of graded O -modules, compatible with the differentials. We recall that for any complex F = F i di F F i+1 of abelian sheaves on and k Z, F [k] is defined by F [k] i = F i+k and d i F [k] = 1k d i+k F, for all i Z. f f = f : F G is such a morphism and k is an integer, f [k] : F [k] G [k] is usually defined by f [k] j = f j+k, for all j, and may therefore be identified with f. These conventions are used in particular for F and f an object and a morphism in. A homotopy between two morphisms in is a homotopy in the sense of the category of complexes of abelian sheaves, except that the homotopy operator of degree 1 is taken to be O -linear. We denote by c resp. qc the full subcategory of consisting of complexes with coherent resp. quasicoherent terms. We slightly generalize the previous definitions as follows. Let lim F α and lim α F β be two procoherent O -odules i.e. two objects of the pro-category rooh. A morphism f : lim β F α lim α G β β of roa b is a differential operator of order one if f factors via the commutative diagram 0.5.1 lim 1 α F α lim d α 1,Fα f f lim F α lim α G β β is the universal differential operator of order one of source F α, and f is a morphism of where d 1,F α rooh. Now, pc will denote the category of complexes of pro-coherent O -odules whose differentials are differential operators of order one. orphisms in pc will be maps of degree zero of, University of adova, taly 3 [baldassa,maurizio,fiorot]@math.unipd.it

graded objects of rooh, compatible with the differentials. There is a natural exact and faithful functor ro c pc compatible with homotopical equivalence. Similarly, there is a natural exact and faithful functor, compatible with homotopical equivalence, from the ind-category of c, nd c, to qc, since ndoh is equivalent to the category of quasi-coherent O -odules [H.RD, Appendix]. We point out that any object of qc resp. pc which is a bounded complex, is in fact in the essential image of nd c in qc resp. of ro c in pc. 0.5.2. This subsection wants to motivate the definition of Hom k, in [HL, 2]. We recall that for a graded left Ω -odule F and an integer k, one sets F [k] to be the graded left Ω -odule E such that E j = F j+k and such that for α a section of E j = F j+k and ϕ a section of Ω i, the scalar product ϕ α of ϕ and α in E is 1 ik times the scalar product ϕ α of ϕ and α in F. orphisms of graded E F left Ω -odules are meant to be of degree zero. They form a Γ, O -module Hom Ω, ; we denote by Hom Ω, the sheafified version. We point out that if ω is a section of Ωk, and F is a graded left Ω -odule, left multiplication by ω is a morphism F F [k]. Once again, if f is a morphism of graded left Ω -odules and k is an integer, f [k] identifies with f. For F and G as before, and k an integer, one sets Hom k Ω F, G = Hom Ω F, G [k] = Hom Ω F [ k], G and similarly for Hom k Ω F, G = Hom Ω F, G [k]. The skew-commutativity of Ω i.e. αβ = 1ij βα, for α Ω i and β Ω j permits to interpret any graded left or right Ω -odule F as two-sided. n fact, if F is a graded left Ω -odule, we can define a structure of graded right Ω -odule on it by setting α ϕ = 1 ij ϕ α, for ϕ a section of Ω i and α a section of F j. t is then clear that a morphism of graded left Ω -odules is also right Ω -linear, and the other way around. This is why the notation Homk Ω, does not carry any indication on whether left or right linearity is assumed. We observe that an element Φ of Hom k Ω F, G turns out to be a collection Φ = ϕ j j of maps of abelian sheaves ϕ j : F j G j+k satisfying ϕ i+j ω F α = 1ik ω G ϕ jα, for sections ω of Ω i and α of F j cf. [HL, loc. cit.]. The possibility of interchanging the left and right Ω -odule structure gives a meaning and a structure of Ω -odule to Ω N, for two Ω -modules and N. Here one uses the right resp. left Ω - odule structure on resp. N to take the tensor product, while the left resp. right Ω -odule structure on the tensor product is given by left resp. right Ω -odule structure of resp. N. Similarly, Hom Ω, N has a structure of Ω -module. 0.5.3. The category is equivalent to the category of graded left Ω -odules F, endowed with a morphism of graded abelian sheaves D = D F : F F [1] satisfying D F [1] D F = 0 and 0.5.4 D F ϕ α = ϕ D F α + d ϕ α F F [1] F for sections ϕ of Ω and α of F. f F and G are two objects of, a morphism f : F G is then a morphism of graded Ω -odules, such that f[1] D F = D G f. For any integer k, F [k] as an object of, is the graded left Ω -odule F [k], endowed with D F [k] = 1 k D F. The category is endowed with a natural tensor product Ω and with an internal hom Hom Ω,. One sets D F Ω G α β = D F α β + 1i α D G β, for sections α of F i and β of G. One also defines by D = D Hom Ω F,G : Hom Ω F, G Hom Ω F, G [1] DΦ = D G Φ Φ[1] D F [ k], University of adova, taly 4 [baldassa,maurizio,fiorot]@math.unipd.it

if Φ is a section of Hom k Ω F, G = Hom Ω F [ k], G. 0.5.5. Obviously, two morphisms f and g : F G in are homotopic via the homotopy operator ϑ : F [1] G a morphism of graded O -odules if and only if The previous formula means in fact that for any i g f = D G ϑ ϑ[1] D F [1]. g i f i = D i 1 G ϑ i + ϑ i+1 D i F. t is easy to check by induction on the degree of differential forms, that ϑ is automatically Ω -linear, and that the previous formula simply means that g f DHom 1 Ω F, G. Similarly, for f Hom 0 Ω F, G, Df = 0 is equivalent to f being a morphism F G in the category. So, cf. [HL, remark p.104] H 0 Hom Ω F, G is the group of -morphisms F G up to homotopy. 0.5.6. t is also possible to interpret as the category of graded left -odules where = Ω 1 D Ω is the mapping cylinder of the identity map of Ω. t is a graded O -Algebra, whose product is defined using the wedge product of Ω and D2 = 0, while the structure of complex is defined by Dα = dα 1 + 1 i α 2 D + dα 2 if α = α 1 D + α 2 with α 1 Ω i 1 and α 2 Ω i. Therefore, the category has enough injectives. 0.5.7. For a morphism π : of schemes, we have canonical morphisms of graded differential Rings T π : Ω π Ω, S π : π 1 Ω Ω. We deduce from this a pair of adjoint functors π :, π :. For an object F of, the complex of abelian sheaves π F coincides with the usual direct image of the complex of abelian sheaves F. For G in instead, π G = Ω π 1 Ω π 1 G and D π G ϕ π 1 α = d ϕ π 1 α + 1 i ϕ π 1 D G α, for ϕ a section of Ω i and α a section of G. 0.5.8. Lemma. Let π : be a morphism of schemes, and assume that two morphisms f and g : F G in are homotopic via the homotopy operator ϑ : F [1] G. Then π f and π g : π F π G in are homotopic via the homotopy operator π ϑ : π F [1] π G. roof. The main point is that, for any section ϕ of Ω i and any section α of F, 0.5.9 π ϑd ϕ π 1 α + 1 i d ϕ π 1 ϑα = 0 since more generally π ϑϕ π 1 α = 1 i ϕ π 1 ϑα, ϑ being a morphism of degree 1, so that D π G π ϑ + π ϑ D π F ϕ π 1 α = = D π G 1 i ϕ π 1 ϑα + π ϑd ϕ π 1 α + 1 i ϕ π 1 D F α = 1 i d ϕ π 1 ϑα + ϕ π 1 D G ϑα + 1 i+1 d ϕ π 1 ϑα + ϕ π 1 ϑd F α = ϕ π 1 D G ϑ + ϑd F α = ϕ π 1 g fα = π g π fϕ π 1 α. We point out that if π is a locally closed immersion, then π F coincides, as a graded O -odule, with the usual inverse image in the sense of graded O -odules. 0.5.10. Let F and G be two objects of, and let G J be an injective resolution of G in. The local resp. global hyperext functors of Herrera-Lieberman are defined in [B.74,.5.4.3] as the abelian sheaves resp. groups, for any p Z, Ext p F, G = H p Hom Ω Ω F, University of adova, taly 5 [baldassa,maurizio,fiorot]@math.unipd.it

resp. Ext p F, G = H p Hom Ω Ω F,, where an object of is naturally regarded as a complex of abelian sheaves on, and H p is taken in that sense. This definition is not exactly the original one of Herrera and Lieberman [HL, 3], but leads to isomorphic objects; this subtlety on the definition of hyperext functors will be explained in section 2 below. The local resp. global hyperext functors are limits of a spectral sequence resp. E p,q 1 = R q Hom p Ω F, G =: Ext p,q Ω F, G = E p+q = Ext p+q F, G Ω E p,q 1 = R q Hom p Ω F, G =: Ext p,q Ω F, G = E p+q = Ext p+q F, G, Ω obtained as the first spectral sequence of the bicomplex under consideration. As a particular case we obtain, for any object F, the functors and H p F = Ext p Ω Ω, F H p, F = Ext p Ω Ω, F. n the last case, the spectral sequence above is the usual first spectral sequence of hypercohomology The hyperext functors naturally extend to functors H q, F p = H p+q, F. Ext p Ω, : ro nd ndab and resp. and Ext p Ω, : nd ro roab Ext p Ω, : ro nd ndab Ext p Ω, : nd ro roab. 0.6. Direct image with compact supports Deligne. We recall that in the appendix to [H.RD], Deligne defines for an open immersion j : U of locally noetherian schemes the functor j! prolongement par zéro in the following way. Let F be a coherent O U -odule, and take F a coherent extension on ; let the coherent deal of O defining U. Then j! F is the pro-coherent sheaf on given by lim N F. N Deligne proves that the definition is independent of the choice of the coherent extension F. The functor j! : oho U rooho naturally extends to the category rooho U by the condition of commuting to all projective limits. The functor j! extends to the subcategory of pc U whose objects regarded as complexes are bounded below, with values in pc and preserves homotopical equivalence of morphisms. f f, g : E F are homotopically equivalent morphisms in pc, then we have R lim H i f = R lim H i g : R lim H i E R lim H i F maps of abelian sheaves and R lim H i, f = R lim H i, g : R lim H i, E R lim H i, E maps of abelian groups. 0.7. ousin complex. We recall from [H.RD, chap. V], that for any abelian sheaf F one functorially defines a complex E F the ousin complex of F uniquely defined by suitable conditions of support w.r.t. the stratification of by the codimension of its points see [H.RD, V.2.3] or [H.75,.2]. oreover there is a functorial augmentation morphism F E F, where F is regarded as a complex concentrated in degree zero. The functor restricts to a functor from the category of O -odules into complexes of such i.e. with O -linear differentials. University of adova, taly 6 [baldassa,maurizio,fiorot]@math.unipd.it

Under suitable conditions on the sheaf F see [H.RD,V.2.6], the ousin complex is a flabby resolution of F. oreover, if is smooth E O admits an explicit description see [H.RD, example p. 239] proving that it is an injective resolution of O. n particular if F is a locally free O -odule of finite type, we have a canonical isomorphism of complexes E F = E O O F. n fact these complexes both satisfy the conditions to be the ousin complex of F, so that by [H.RD, V.3.3] the canonical morphism is an isomorphism. As in [H.75] we extend the definition of ousin complex by associating to each complex F of O - odules with Z-linear differentials the al complex EF associated to the double complex E F defined by the ousin complexes of its components. 0.7.1. Notice that E r Ω, for any r N, and EΩ are naturally objects of qc. ore generally, if D : F G is a differential operator then, for any r, E r D : E r F E r G is a differential operator of the same order this is easly seen using [EGA V,16.8.8]. 1. De Rham ohomology with compact supports. 1.1. Setting. Let j : be an open immersion of the scheme into a proper scheme, and assume i : is a closed immersion in a scheme smooth in a neighborhood of. We have then the following embeddings j. Let be the complement of in endowed with some structure of closed subscheme of, and let h : be the closed immersion. We do not suppose that be dense in as the symbol may suggest, even if we can always reduce to that case, replacing by the closure of in. 1.1.1. Let W be an open smooth subscheme of containing as a closed subset. The closed immersion W can be used to calculate the De Rham cohomology and homology of, as defined by Hartshorne [H.75]. ore generally, any open subscheme W of containing as a closed subscheme e.g. the not necessarily smooth open subscheme would work for Hartshorne s computation of De Rham homology and cohomology. n fact, W would contain an open smooth subscheme W containing as a closed subscheme. The open immersion u : W W then induces isomorphisms of infinitesimal neighbourhoods N = W N W of in W and W, respectively. On the other hand, the trace map Tr u induces an isomorphism of ousin complexes Tr u : Γ EΩ W = Γ EΩ W. 1.1.2. onsider now the infinitesimal neighborhoods of and in W and, respectively. n the following diagram j h = W j i h the two squares are cartesian. f we put = W, = and =, the ideal of the closed immersion h is := = + +1 +1. 1.2. Definition. n the previous notation, we define the De Rham cohomology of with compact supports H DR,c as the hypercohomology of the simple complex of abelian sheaves on associated to the bicomplex Ω h Ω, where O sits in bidegree 0, 0. Namely, H DR,c =, Ω H h Ω. 1.3. roposition. We may rewrite the previous definition as H DR,c = H, Rlim N Ω N = H, Rlim j! Ω W = lim H N, N = lim H, j! Ω W Ω University of adova, taly 7 [baldassa,maurizio,fiorot]@math.unipd.it

where are the deals of O corresponding to the closed subschemes, respectively. We are using the compact notation N Ω for the complex 0 N N 1 Ω 1 1 Ω 1 N 2 Ω 2 2 Ω 2... and similarly for other complexes of this type. roof. For each N the short exact sequence of complexes 0 N Ω Ω Ω gives the exact sequence of projective systems of complexes 0 { N Ω } N {Ω } {Ω and the isomorphism in ro c lim N Ω N We apply the functor Rlim to get Rlim N N Ω = lim N Ω N 0 N } N 0 Ω N. = Ω Ω, where O sits in bidegree 0, 0. This proves the first isomorphism of the proposition. f we take hypercohomology first and then projective limits we obtain instead isomorphisms lim H, N Ω = lim H, Ω Ω N. N N We now have the following generalization of [H.75,.4.5] to suitable complexes of abelian sheaves, already used in the proof of [loc. cit.,.5.2]. For further use in the rigid-analytic context, we express the result for G-topological spaces. We recall that a base B for a G-topology on is a class of admissible open subsets such that for any admissible open subset there is an admissible covering with elements in B. 1.3.1. Lemma. Let F n n N be an inverse system of complexes in degrees 0 of abelian sheaves on the G-topological space. Let T be a functor on the category of complexes of abelian sheaves on, taking its values in an abelian category A, where arbitrary direct products exist. We assume that the functor T commutes with arbitrary direct products and that there is a base B for the G-topology of such that: a For each U B, the inverse system F nu n is surjective, b For each U B, H i U, F j n = 0 for all i > 0 and all j, n. Then, for each i, there is an exact sequence 0 lim 1 R i 1 T F n R i T lim F n α i lim R i T F n 0. n particular, if for some i, R i 1 T F n satisfies the ittag-leffler condition, then α i is an isomorphism. roof lemma. One may reason precisely as in the case of an ordinary topological space, and simply follow the proof of [H.75,.4.5], taking into account the structure of injectives in the category of complexes in degrees 0 over any abelian category [T,.2.4]. We apply the previous lemma to the topological space or, if one prefers, the functor Γ, and the simple complex lim Ω Ω N = lim Ω 2N Ω N. N ore precisely, one takes F N = Ω 2N Ω N in the lemma. We observe that, since the coherent sheaves appearing in the complex have support in the proper subscheme, the -vector spaces H, i Ω 2N Ω N are finite dimensional, so that they satisfy, for variable N, the ittag-leffler condition. Therefore lim H, Ω Ω N = H, lim Ω lim N Ω N N University of adova, taly 8 [baldassa,maurizio,fiorot]@math.unipd.it N

which is H DR,c by definition. This proves the isomorphisms in the first line of the statement. To check the isomorphism on the second line, we write in the notation of 1.1 lim j! Ω W = lim N N Ω = lim N N Ω. n order to prove that the given definition of De Rham cohomology with compact supports is good, we need some enhancements to proposition.1.1 of [H.75]. 1.4. Lemma. Let Z be a sequence of closed immersions of schemes. For any 0, we have a cartesian diagram of closed immersions 1.4.1 roof. The equality Z The square is cartesian because so is Z i i Z = Z follows from = Z. O +1 Z +1 +1 = O +1 Z. i Z Z. 1.5. n the situation of the previous lemma, for any section s : of, there is a unique section s : Z Z of i : Z Z fitting in a commutative diagram, necessarily cartesian, 1.5.1 β Z s s α Z. To the morphism i resp. s we associate the c -morphism T i : Ω i Ω resp. the c -morphism T s : Ω s Ω. Similarly, to the morphism i resp. s we associate the c Z -morphism T i : Ω Z resp. the c Z -morphism T s : Ω Z Since s i = id resp. s i = id Z resp., we have i Ω Z s Ω Z s T i T s = id Ω s T i T s = id Ω Z University of adova, taly 9 [baldassa,maurizio,fiorot]@math.unipd.it..

We will be interested in the composite c -morphism resp. c Z -morphism 1.5.2 s Ω resp. s T i Ω T s s Ω 1.5.3 s Ω Z s T i Ω Z T s s Ω Z. 1.5.4. Local situation. n the notation of the previous lemma, assume furthermore that and are affine and smooth. Let be the ideal of O corresponding to and assume moreover that the O- module 2 is free on the generators x 1,..., x n. Let us denote by D n := Spec [x 1,..., x n ]x 1,..., x n +1, the -th infinitesimal neighborhood of the origin in the affine -space of dimension n. Then a section s as in 1.5.1 certainly exists and that cartesian diagram can be identified with the standard diagram 1.5.5 β Z = D n pr 2 id α α D d = D n Z Z pr. 2 1.5.6. Lemma. n the local situation above, the composite morphism T s s 1.5.2 is homotopic to the identity of s roof. The canonical morphisms D n Ω Z σ in the category c Z. Spec D ι n, T i in formula where ι corresponds to the canonical augmentation [x 1,..., x n ]x 1,..., x n +1, fit in the diagram with cartesian squares 1.5.7 pr 1 D n s π σ Spec where π : Spec is the structural morphism. We have as usual a c D n -morphism and a morphism of -vector spaces We observe that Ω D n We have while So, σ T ι T σ = id, while T ι : Ω D n i pr 1 ι D n, ι = D n = T σ : σ Ω. D n is freely generated over = D by its global sections n {x α dx λ1 dx λp p + α }. T ι x α dx λ1 dx λp = 1.5.8 id σ Ω D n where d := d D n and T σ 1 = 1. { 1 if α = 0 and p = 0 0 otherwise T σ σ T ι = d ϑ + ϑ d ϑ : σ Ω D n σ Ω [ 1] D n University of adova, taly 10 [baldassa,maurizio,fiorot]@math.unipd.it

is the morphism of -vector spaces such that 0 if α = 0 and p = 0, ϑx α dx λ1 dx λp = 1 p 1 j+1 x α dx λ1 dx p + α λj dx λp otherwise. j=1 We now observe that π σ Ω D n = s Ω, that π T σ = T s and that π σ T σ = s T i and take ϑ := π ϑ : s Ω s Ω [ 1], a morphism of graded Ω -odules. Applying the functor π to 1.5.8 we conclude that id s Ω T s s T i = d by 0.5.9. This proves the claim. ϑ + ϑ d = d ϑ + ϑ d Now we can handle the case of smooth morphisms, which will be used in the proof of the theorem., 1.6. roposition. Let f : be a smooth morphism of smooth schemes. Let Z be a closed subscheme of such that the composition with f gives a closed immersion of Z in. Then the canonical map T f : Ω Z is locally a homotopic isomorphism in the category of Z f Ω Z -odules. roof. Since the morphism f : is smooth, and Z is a closed subset of both and, we may factorize the diagram Z i1 locally on and at the points of Z, as Z i 2 i 2 f i 1 f f i 2 where i 1 and i 2 are closed immersions such that i 1 = i 1 i 2, and f is an étale morphism see [H.75,.1.3]. We may and will insist that be affine and that the ideal of in satisfy the condition that 2 be a free O -odule. Taking the infinitesimal neighborhoods of Z we have Z f Z i 1 Z f where f is an isomorphism. The morphism i 1 admits therefore the retraction s = f 1 f, for which f s = f. Notice that, f being étale, the retraction s comes from the unique section s : of the canonical inclusion fitting in the first commutative square here below. The second square below is obtained by taking -th infinitesimal neighborhoods of Z in the objects of the first. s f = and Z = Z =Z Z s f Z the lower maps of the above diagrams are given by the composition f, the left vertical arrows are nilpotent closed immersions, while the right vertical maps are étale morphisms. University of adova, taly 11 [baldassa,maurizio,fiorot]@math.unipd.it

Now, lemma 1.5.6 applied to the closed immersion i 1 and its section s shows that the composite morphism s Ω Z s T i 1 s is homotopic to the identity of s Ω Z s i 1 Ω Z = ΩZ T s s in the category c Z, while T i 1 T s = id Ω. Z Taking direct images via f, and identifying f Ω Z with T f, we conclude that while f f T i 1 T s f is homotopically equivalent the identity of f with Ω Z f T s = id Ω, Z T i 1 : f Ω Z Ω. n other words Z Ω Z, so that f T s is identified f Ω, Z f T i 1 : f Ω Z = f s Ω Z is a homotopic inverse of the canonical morphism T f. f s T i 1 f Ω Z = Ω Z 1.7. Lemma. Let be a scheme and U = {U α } be an open covering. For an object F of pc we define the Čech co-complex U, F of F on U as follows: p U, F = j α! F U α α =p+1 where α = α 0,..., α p is a multi-index, U α = i U α i and j α is the inclusion of U α in. The differentials p U, F p 1 U, F are defined as usual by the simplicial structure. Then we have a canonical augmentation morphism 0 U, F F making U, F into a left resolution of F in the category pc. n other words, the sequence 1.7.1 2 U, F 1 U, F 0 U, F F 0 is exact. roof. n order to prove the exactness of 1.7.1, we have to prove that for any i the sequence in rooh given by U, F i is a left resolution of F i. Since all the constructions involved commute with the functor lim we may assume that each F i is a coherent O -odule. So, let F be in oh. The complex U, F F 0 is exact if and only if for any injective quasi-coherent O -odule the sequence Hom O U, F, Hom O F, 0 is exact. The exact functors Hom O, : oh Qoh, for an injective of Qoh, form in fact a conservative family. Using the adjunction of j α! and jα 1 see the Deligne appendix in [H.RD] the last sequence is just the usual Čech resolution of the sheaf Hom O F, Hom O j α! F Uα, = j α Hom OUα F Uα, Uα = j α Hom O F, Uα. α =p+1 α =p+1 α =p+1 University of adova, taly 12 [baldassa,maurizio,fiorot]@math.unipd.it

1.8. Theorem. The definition of H DR,c is independent of the choice of the compactification and of the closed immersion of in smooth around. roof. Given 1 1 and 2 2, and i = i i, as in the definition, we consider the closure of the diagonal immersion of in 1 2, and the product 1 2. So we have a diagram i 1 1 1 j 1 1 2 1 2 j 2 2 i 2 2 where all the horizontal maps are closed immersions, except which is an open one. oreover, since the natural morphisms from to 2 and 1 are closed immersions, we have that = is contained in 1 2. We then obtain diagrams 1 2 i i for i = 1, 2. Therefore we are reduced to the case of a proper morphism g : 1 2 with g 1 2 = 1 in particular g 1 2 and a morphism f : 1 2 restricting to a smooth morphism f : W 1 W 2, if W 1 is taken to be sufficiently small. We have then the commutative diagram j 1 j 2 1 i 1 g 1 f 2 i 2 2. As usual, h i : i i will denote the closed immersions for i = 1, 2. We have to prove that H 1, Ω 1 1 h 1 Ω 1 1 = H ore precisely we have to prove that the canonical morphism 1.8.1 Ω 2 2 h 2 Ω 2 2 Rg Ω 1 1 h 1 Ω 1 1 2, Ω 2 2 h 2 Ω 2 2. induces isomorphisms on the hypercohomology groups. We will show that the previous morphism is a quasiisomorphism of abelian sheaves. Taking infinitesimal neighborhoods of and i in W i and i respectively, we have diagrams j 1 W 1 1 1 f f W 2 j 2 2 2 where j 1 and j 2 are open immersions, f is smooth and f is proper. By proposition 1.3 and remark 0.3 we may study the morphism 1.8.2 lim N 2 Ω lim 2,N 2,N Rf N 1 Ω 1 1 corresponding to the morphism 1.8.1. We explicitly recall, without proof, the following enhanced version of proposition 5 of Deligne s appendix to [H.RD] whose proof depends on [EGA, rop. 3.3.1]. University of adova, taly 13 [baldassa,maurizio,fiorot]@math.unipd.it

1.8.3. roposition. Let f U f V, be a cartesian diagram of noetherian schemes, where the horizontal maps are open immersions, f and f are proper morphisms and f is acyclic. Let be an deal of O defining the closed subset \ V, and let J denote the extension of the O -deal to an deal of O. Then, for any coherent O -odules F i if k > 0, lim R k f n J n F = 0, ii if k = 0, for sufficiently big n, f J n+1 F = f J n F. From this proposition, we have that for any, i and any sufficiently big N, there is a canonical N 1 Ω i. We are then reduced to proving 1 1 isomorphism lim Rf N N+N 1 Ω i = lim 1 1 N N 2 f that, for any, the canonical morphism 1.8.4 lim N N 2 Ω lim 2 2 N N 2 f N 1 Ω 1 1 is a quasi-isomorphism. By 1.3 and flat base change, we may rewrite 1.8.4 as 1.8.5 j 2! Ω j W 2 2! f Ω W 1 and we know by proposition 1.6 that Ω f Ω is locally a homotopic isomorphism. So there W 2 W 1 exists an open covering U of W 2 such that for any i the canonical morphism i U, Ω W 2 i U, f Ω W 1 is a homotopic isomorphism. Therefore, applying the functor j 2!, which is exact in the category rooh, to the diagram U, Ω Ω 0 W 2 W 2 U, f Ω W 1 f Ω 0 W 1 we obtain that 1.8.5 is a quasi-isomorphism in the category roab. 1.9. Remark. n the setting 1.1 we may suppose that is smooth. n fact, if are immersions as in 1.1, we can take a resolution of singularities à la Hironaka π : we are in characteristic zero, which restricts to an isomorphism on any open smooth subscheme W of, and therefore on. Taking as the closure of in, we have a commutative diagram j j i where a closed subscheme of the inverse image by π of is proper and j is an open immersion. We may then calculate the De Rham cohomology with compact supports of using the first line of the diagram, i.e. we can assume smooth. Notice that this can also be proven independently of the previous theorem, in a simpler way. n fact, let W be any smooth open subscheme of containing as a closed subscheme. Then in the previous diagram W is also an open subscheme of with the same property. Now, π University of adova, taly 14 [baldassa,maurizio,fiorot]@math.unipd.it

for any, the proper map π induces a proper map π : Rπ j =! j!. We then have a morphism of spectral sequences H q, j H q, j! Ω p W! Ω p W = H p+q, j = H p+q, j! Ω W, and an isomorphism of functors! Ω W The left hand arrow is an isomorphism, since both source and target identify with the cohomology groups with compact supports for coherent sheaves Hc q W, Ωp as defined in [H.72, 2]. n view of regularity, W the limits of the two spectral sequences are also isomorphic. 1.10. roposition. The De Rham cohomology with compact supports is a contravariant functor w.r.t. proper morphisms and a covariant functor w.r.t. open immersions. roof. For an open immersion j : 1 2 we may calculate the De Rham cohomologies with compact supports of 1 and 2 using an open immersion of 2 into a proper scheme: j 1 2 j2 2 i2 2 since the diagonal arrow is again an open immersion. Now remark that 2 = 2 2 is contained in 1 = 2 1, so that 1 2. The canonical commutative diagram Ω 2 2 h 1 Ω 2 1 Ω 2 2 h 2 Ω 2 2 where h i is the closed immersion i 2, induces a natural morphism H DR,c 1 H DR,c 2. Let now h : 1 2 be a proper morphism. As in the first step of the proof of the theorem, we may complete a diagram as 1 1 1 h 2 2 2 using = the closure of the image of the canonical morphism 1 1 2, and the product = 1 2. So we are reduced to a diagram of the form. 1.10.1 1 1 1 h g f 2 2 2 where the first square is cartesian. n fact the canonical morphism 1 g 1 2 is clearly an open immersion and it is a proper morphism since its composition with g g 1 2 : g 1 2 2 is proper; therefore it is the identity. As a consequence we have that g 1 2. We then have a commutative diagram g Ω 1 1 g h 1 Ω 1 1 Ω 2 2 h 2 Ω 2 2 so that we deduce a natural map H DR,c 2 H DR,c 1. 1.10.2. Remark. Notice that if is a proper scheme we have H DR,c = H DR, since we may choose = in the setting 1.1 of our definition. n general, let j : be the open immersion in 1.1. By the proposition we have a canonical map H DR,c H DR,c = H DR the last equality by properness of. oreover, by the contravariant functoriality of De Rham cohomology without supports we have a University of adova, taly 15 [baldassa,maurizio,fiorot]@math.unipd.it

canonical morphism H DR H DR. Therefore, by composition, we have for any scheme a canonical morphism H DR,c H DR,c = H DR H DR. This morphism is induced by the canonical morphism of complexes Ω j Ω W which defines a morphism Ω h Ω j Ω W ; taking hypercohomology gives the canonical morphism between De Rham cohomologies. 1.11. roposition. Let j : U be an open immersion of schemes and i : Z = U be the closed immersion of the complement, endowed with some closed subscheme structure. There exists a long exact sequence H i 1 DR,c Z Hi DR,cU HDR,c i HDR,cZ i H i+1 DR,c U. roof. We can choose a compactification of and a closed immersion of in a scheme smooth around. We then construct the diagram j U i i Z U where we remark that U is closed in and contains Z as an open subset; moreover U Z =. Therefore, we are in the situation to calculate the three De Rham cohomologies with compact supports: H DR,c U = H, Ω Ω U = lim H, N U Ω,N H DR,c = H, Ω Ω = lim H, N Ω,N H DR,c Z = H U, Ω U Ω = lim H U, N Ω,N From the exact sequences 0 N U Ω N Ω N Ω 0 U we have immediatly the conclusion taking the long exact sequence of hypercohomology. U 1.11.1. Remark. t will follow from our oincaré duality theorem below that the long exact sequence of the proposition is dual of the exact sequence of a closed subset for homology, see [H.75,.3.3]. 1.11.2. Remark. f is a proper scheme, then also Z is proper and the exact sequence of the proposition can be written as H i 1 DR Z Hi DR,cU HDR i HDRZ i H i+1 DR,c U because of the remark 1.10.2. This will be one ingredient for the proof of the oincaré duality theorem, see below. 1.12. roposition. Let be the union of two closed subschemes 1 and 2 ; then there exists a ayer-vietoris exact sequence for the De Rham cohomology with compact supports H i 1 DR,c 1 2 HDR,c i HDR,c i 1 HDR,c i 2 HDR,c i 1 2 H i+1 DR,c roof. We can take a compactification of which is the union of closed subschemes 1 and 2, compactifications of 1 and 2, respectively. t suffices to replace by the union of the closure of 1 and of 2 in. Since the i are closed in, we have that = is the union of the i = i i. The. University of adova, taly 16 [baldassa,maurizio,fiorot]@math.unipd.it

proof of the ayer-vietoris sequence in De Rham cohomology without supports, [H.75,.4.1], gives the following diagram with exact rows 0 Ω Ω 1 Ω 2 Ω 1 2 0 0 Ω Ω 1 Ω 2 Ω 1 2 0. Then we can deduce the exact sequence of pro-complexes N Ω N 0 lim lim,n 1 Ω N lim,n 2 Ω N lim 1,N 1 2 Ω 0 2,N 1 2 from which the exact sequence of ayer-vietoris of De Rham cohomology with compact supports follows. 1.13. Example: De Rham cohomology with compact supports of affine spaces. f = A n, we can take = = n, and the exact sequence of the closed subset = = n 1 is H i 1 DR n 1 Hi DR,cA n HDR i n δ HDR i n 1 Hi+1 DR,c An Now, for projective spaces we have HDR i n = for 0 i 2n even, and 0 otherwise [H.75,.7.1]; moreover, the coboundary operator δ is an isomorphism. Then we deduce HDR,cA i n { 0 if i 2n = if i = 2n. 1.14. roposition. For any scheme of dimension n we have that HDR,c i = 0 for i > 2n. roof. n fact we can take the exact sequence of as an open subset of with closed complement, and apply the analogous results for ordinary De Rham cohomology of [H.75,.7.2]. 2. Hyperext functors and De Rham Homology. n this section we recall the notion of De Rham homology, which will be essential in the proof of our duality theorem in the next section. For simplicity, in this and the next section we will assume, in the setting 1.1, to have chosen a smooth see remark 1.9. n that case, we can use the classical ousin complex of Ω, rather than the more complicated dualizing complex of Du Bois [DB.90], and we can make explicit its relevance in the calculation of certain hyperext groups of Herrera and Lieberman. 2.1. n the notation of our setting 1.1, since W is a smooth scheme containing as a closed subscheme, the Hartshorne definition of De Rham homology [H.75,.3] may be expressed as H DR = H 2n W, Ω W where n is the dimension of W, and the right hand side indicates hypercohomology with support in of the complex Ω W. 2.1.1. Using the resolution of Ω W given by the ousin complex EΩ W, Hartshorne interpretes his definition as H DR = H 2n W, Γ EΩ W = H 2n ΓW, Γ EΩ W since Γ EΩ W is a complex of injective O W -odules [H.75,.3]. 2.2. n the following we will characterize the notion of De Rham homology in terms of the functors Hom Ω, which are important in order to understand the differentials of the complex and Hom O, which are related to the notion of support. 2.2.1. The ring change formula Hom O F, = Hom Ω F O Ω,, where F is an O -odule and is a graded Ω -odule, is an isomorphism of graded Ω -odules. f moreover is a -odule and F O Ω admits a structure of -odule for example if is smooth and F is a left D -odule, then the ring change formula gives a structure of -odule to N = Hom O F,. Notice that in general d i N is not induced by id Hom F, d i, since this last University of adova, taly 17 [baldassa,maurizio,fiorot]@math.unipd.it

operator does not preserve O -linearity. As a particular case, we see that if is a -odule and Z is closed in, then Γ Z := lim N Hom O O Z N, = lim N Hom Ω Ω N Z, is a -odule. 2.2.2. The same sort of phenomena are explored in detail in [HL], and [B.74,.5.2] for the other ring change formula. Namely, for F an O -odule and an Ω -odule, we have a canonical isomorphism of Ω -odules Hom Ω, Hom O Ω, F = Hom O, F. f moreover is a -odule and Hom O Ω, F admits a structure of -odule for example if is smooth and F is a right D -odule, then the second ring change formula gives a structure of -odule to Hom O, F. A special case of this isomorphism is implicitely used by [H.75] in the proof of the duality theorem. Assuming smooth, if F = Ω n[ n] where n = dim, Hom O Ω, Ωn [ n] = Ω, the isomorphism becomes Hom Ω, Ω = Hom O, Ω n [ n] and the terms have a structure of -odule if does see [HL, 2.9]. 2.3. On the definition of HyperExt functors. This section is meant to justify the useful sign convention of [HL, 3], and to modify some incorrect statements in that section 1. We also prove that the definition of hyperext groups of [B.74,.5.4.3], which we adopted here, is equivalent to the original definition of [HL, 3]. We start with an easy lemma on complexes of -odules. 2.3.1. Lemma. Let J, : J,q d,q J J,q+1 be a complex of,q -odules, and let d J indicate the differentials of each term J,q notice that d J is a differential operator, while d J is an O -linear map. We define a new complex J, of -odules in the following way: for any p and q let J p,q = J p,q, d p,q J = 1q d p,q p,q J and d = 1 J p p,q d J. Then the canonical map σ, J : J, J, defined by σ p,q J = 1pq id J p,q is an isomorphism of complexes of -odules. oreover, the functor sending J, to J, is an involution of the category of complexes of -odules. roof. learly, for any q, J,q with the differentials d,q J is a -odule. oreover we have the commutativity d d = d d, so that J, is a complex of J J J J -odules. Therefore we have only to prove that σ, J is a morphism of complexes of -odules, that is it commutes with the differentials, which is an easy exercise. We point out that the structure of Ω -odule on each J,q is given by if α Ω i and u J,q. α u = 1 iq α J u,,q J,q 2.3.2. orollary. Let F be a -odule and J, a complex of -odules as before. Then we have canonical isomorphisms of complexes of -odules Hom p Ω where σ J = Hom Ω id F, σ J, σ p,q Hom = σ Hom p Ω F, J,q σp,q J Hom p Ω F, J,q σp,q Hom Hom p Ω F, J,q F, Hom p Ω F, J,q p,q = and J,q Hom p Ω F, J,q p,q. 1 cf. the lines of loc. cit. preceding formula 3.1:... Ω -linearity, a property which is preserved by d and d. This statement is false. University of adova, taly 18 [baldassa,maurizio,fiorot]@math.unipd.it

roof. This is a consequence of the previous lemma. We make explicit the differentials in the objects of the corollary in view of the next result. differentials in the first term are given by d and d defined as usual: and d p,q : Hom p Ω F, J,q Hom p+1 Ω F, J,q d p,q Φ = d,q J Φ 1p Φ d F d p,q : Hom p Ω F, J,q Hom p Ω F, J,q+1 d p,q,q Φ = d J Φ. The second term also has the usual differentials, but using the differentials of and d p,q : Hom p Ω F, J,q Hom p+1 F, J,q d p,q : Hom p Ω F, J,q Hom p Ω F, J,q+1 where Ψ = Ψ a, with Ψ a : F a J a+p,q. Finally the differentials of the last object are given by and δ p,q : Hom p Ω F, J,q Hom p+1 F, J,q δ p,q : Hom p Ω F, J,q Hom p Ω F, J,q+1 Ω Ω d p,q Ψ = 1 q d,q J The J instead of J. So we have Ψ 1p Ψ d F d p,q Ψ a = 1 p+a,q d J Ψ a δ p,q Ψ = d,q J Ψ 1p+q Ψ d F δ p,q Ψ a = 1 a,q d J Ψ a. We note that the hyperext functors are defined by [HL] using the third bicomplex of the corollary, and by [B.74] using the first one; the corollary proves therefore that the definitions are equivalent. 2.3.3. roposition. Using the previous notation, we have a canonical identification of -odules Hom p Ω F, J,q = Hom Ω F, J between the al complex of Hom p Ω F, J,q indicates the al complex associated to J,. p,q p,q and the complex Hom Ω F, J, where J roof. We first point out that the al complex associated to a complex of -odules is canonically a -odule. Notice that, for any r, the r-th level of any of the two complexes appearing in the statement is described as collection of O -linear morphisms Ψ p,q a for varying a, p and q with p+q = r where Ψ p,q a : F a J a+p,q and satisfying the following linearity w.r.t. sections α Ω i Ψ p,q a+i αu = 1ip+q αψ p,q a u for any u section of F a. Therefore we only have to prove that the differentials in the two complexes coincide. We can prove that they are given by D r Ψ p,q a = d a+p 1,q J Ψ p 1,q a + 1 r+1 Ψ p 1,q a+1 df a + 1 p+a+1 d a+p,q 1 J Ψ a+p,q 1 a where Ψ = Ψ p,q a with Ψ p,q a : F a J a+p,q for p+q = r, and D r Ψ p,q a : F a J a+p,q for p+q = r+1. n fact, using the previous notation, the al differential r in the first case is defined by r Ψ p,q a = δ Ψ p,q a = d a+p 1,q J + 1 p+1 δ Ψ p,q a Ψ p 1,q a 1 r Ψ p 1,q a+1 df a + 1 p+1 1 a d while the differential D r in the second complex is given by D r Ψ p,q a = d J Ψ p,q a 1 r Ψ d F p,q a = d a+p 1,q J Ψ p 1,q a a+p,q 1 J Ψ a+p,q 1 a, + 1 p+a+1 a+p,q 1 d J Ψ a+p,q 1 a 1 r Ψ p 1,q a+1 df a. University of adova, taly 19 [baldassa,maurizio,fiorot]@math.unipd.it

2.4. f is a smooth scheme of dimension n, then the sheaves of differentials Ω i O -odules, so that we have the following isomorphisms are locally free Hom Ω F, E r Ω = Hom Ω F, E r O O Ω = Hom Ω F, E r O O Hom O Ω, Ω n [ n] 2.4.1 = Hom Ω F, Hom O Ω, E r O O Ω n [ n] = Hom O F, E r O O Ω n [ n] = Hom O F, E r Ω n [ n] using once more the contravariant ring change formula. Similarly, 2.4.2 Hom Ω F, E r Ω = Hom O F, E r Ω n [ n] and, in particular, Hom Ω F, E r Ω = Hom O F, E r Ω n [ n]. Since E r Ω n is an injective O -odule for any r, we conclude that E r Ω is an injective Ω -odule. 2.4.3. roposition. We have a canonical isomorphism for any graded Ω -odule. Hom Ω, EΩ Hom O, E Ω n [ n] roof. Follows immediately from 2.3.3 and 2.4.2. 2.5. Definition. Let be a smooth scheme of pure dimension n. For any -odule, we define its dual -odule as := Hom Ω, EΩ [2n]. Notice that, as a graded O -odule, = Hom O, E Ω n [n], so that this notion of dual is compatible with duality in the derived category of O -odules. f N = is a dual, then for any i, the O -odule N i = n j=0 Hom O j n i, E j Ω n is flabby, since Ej Ω n is an injective O -odule. The functor is exact and naturally extends to a functor ro nd. 2.6. De Rham Homology. oming back to the notation of 1.1, we rewrite the definition of De Rham homology for an easier construction of the duality morphism in the following section. 2.6.1. roposition. Let j W : W be the open immersion of W in, and assume that is smooth. Then lim N Ω = jw Γ EΩ W N and we have canonical isomorphisms = H, lim N Ω. H DR N University of adova, taly 20 [baldassa,maurizio,fiorot]@math.unipd.it

roof. We compute j W Γ EΩ W = lim N N Γ EΩ = lim N Hom O O, EΩ,N = lim Hom O N, Hom O O, EΩ,N = lim Hom O N N N, EΩ = Hom O lim N, EΩ N = Hom Ω lim N Ω, EΩ. Since Γ EΩ W is a complex of flabby abelian sheaves the objects are injective O W -odules, we have, in the derived category of abelian sheaves on, j W Γ EΩ W = Rj W Γ EΩ W and H W, Ω W = H W, Γ EΩ W = H, j W Γ EΩ W. 2.6.2. orollary. n the previous notation, let be a -odule. There are canonical isomorphisms Ext p, Ω Ω = H p Hom Ω, EΩ and Ext p, Ω Ω = H p Hom Ω, EΩ. oreover, H DR p = Ext 2n p Ω lim N N Ω, Ω = lim N Ext 2n p Ω N Ω, Ω. roof. By 2.4.1 and [B.74,.5.4.8], the ousin resolution E Ω of Ω permits the computation of the local and global hyperext functors of [HL]. So we have Ext p, Ω Ω = H p Hom Ω, E Ω. roposition 2.3.3 gives H p Hom Ω, E Ω = H p Hom Ω, E Ω which proves the first formula. The second formula follows immediately from the first. The last assertion of the statement combines the first one with proposition 2.6.1. 3. Algebraic oincaré Duality. Our first result is the construction of a duality morphism, which will be compatible with the duality morphisms constructed in the proper case in [H.75,.5]. As already said, in the body of this section, we suppose that the scheme of the setting 1.1 be smooth. The oincaré duality theorem will be proved using the long exact sequences for a closed subset in De Rham cohomology with compact supports and De Rham homology. We also give an alternative proof, with a strategy similar to the original one of [H.75] for the proof of the duality theorem in the proper case: the canonical duality morphism is induced by a morphism of spectral sequences, and we can use the results of [H.72] cohomology with compact supports for coherent sheaves to prove that it is an isomorphism. For the construction of the morphism of spectral sequences we make University of adova, taly 21 [baldassa,maurizio,fiorot]@math.unipd.it