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On the De Rham Cohomology of Algebraic Varieties PUBLICATIONS MATHÉMATIQUES DE L’I.H.É.S. ROBIN HARTSHORNE On the de Rham cohomology of algebraic varieties Publications mathématiques de l’I.H.É.S., tome 45 (1975), p. 5-99 <http://www.numdam.org/item?id=PMIHES_1975__45__5_0> © Publications mathématiques de l’I.H.É.S., 1975, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES by ROBIN HARTSHORNE CONTENTS INTRODUCTION .............................................................................. 6 CHAPTER I. — Preliminaries .............................................................. n § i. Cohomology theories ............................................................... 9 § 2. Direct and inverse images .......................................................... 10 § 3. Hypercohomology ................................................................. 11 § 4. Inverse limits...................................................................... 12 § 5. Completions ...................................................................... 18 § 6. Associated analytic spaces........................................................... 19 § 7. Functorial maps on De Rham cohomology ........................................... 22 CHAPTER II. — Global Algebraic Theory .................................................. 23 § i. Algebraic De Rham cohomology .................................................... 24 § 2. The canonical resolution of the De Rham complex ................................... 30 § 3. Algebraic De Rham homology ...................................................... 36 § 4. Mayer-Vietoris and the exact sequence of a proper birational morphism ................ 41 § 5. Duality ........................................................................... 47 § 6. Finiteness ......................................................................... 51 § 7. Further developments .............................................................. 52 CHAPTER III. — Local and Relative Theory ............................................... 64 § i. Local invariants ................................................................... 65 § 2. Finiteness and duality theorems ..................................................... 68 § 3. Relations between local and global cohomology ...................................... 71 § 4. Relative cohomology ............................................................... 73 § 5. Cohomology and base extension .................................................... 77 § 6. Cohomology at a non closed point .................................................. 83 § 7. The Lefschetz theorems ............................................................ 85 CHAPTER IV. — The Comparison Theorems ............................................... 88 § i. The global case ................................................................... 88 § 2. The formal analytic Poincare lemma ................................................ 92 § 3. The local case..................................................................... 95 § 4. The relative case .................................................................. 96 BIBLIOGRAPHY .............................................................................. 98 INTRODUCTION i. Historical. The idea of using differential forms and their integrals to define numerical invariants of algebraic varieties goes back to Picard and Lefschetz. More recently, Atiyah and Hodge [2] and Grothendieck [16] showed that algebraic differential forms could be used to calculate the singular cohomology of a smooth scheme over C. This algebraic- topological comparison theorem has been generalized by Lieberman and Herrera [29] and by Deligne (unpublished) to include the case of singular schemes over C. Lieberman and Herrera also proved a duality theorem for first order differential operators, of which our Theorem (II. 5. i) is a special case. See also Lieberman [49] for another comparison theorem. In another context, algebraic differential forms have proved to be useful in the study of the monodromy of a family of complex varieties, using the Gauss-Manin connection. SeeKatzandOda [34],Katz ([32] and [33]), Deligne [10], and Brieskorn [8]. In the purely analytic context, holomorphic differentials on a singular analytic space have been studied by Reiffen [44] and Bloom and Herrera [5]. Finally in the study of varieties in characteristic p, algebraic differential forms are important in Monsky's formal cohomology [38], and as motivation for Grothendieck's crystalline cohomology ([18] and [4]). Our interest in De Rham cohomology dates from 1967, when we first found an algebraic proof of Poincard duality for a smooth proper scheme over a field. Our purpose in the present paper is to lay the foundations of a purely algebraic theory of algebraic De Rham cohomology and homology for schemes with arbitrary singularities over a field of characteristic zero. 2. Main results. The main results of this paper have been outlined in the announcement [27], to which we refer. See also [24, III, § 7.8], and [25]. Not mentioned in the announce- ment however, are the Thom-Gysin sequence of a vector bundle (II, § 7.9), the theory of relative cohomology and base change (HI, § 4? 5)5 and the formal-analytic Poincard lemma (IV, § 2). Please note also some changes necessary in the results of the announcement. The theorem [27, i .2 d] needs a slight further hypothesis (see (11.4.4)). The local finiteness ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES theorem [27, 2.2] is now valid more generally (III. 2.1). The field of representatives in [27, 2.4] must be assumed to be "good" (see III, § 6). The results of [27, § 3] as stated are valid only if Y is connected and rational over k (see (III. 3.2)). 3. Comments. Whereas the comparison theorems of Chapter IV have been proved before, our paper is original in that the properties of algebraic De Rham cohomology developed in Chapters II and III are proved by purely algebraic techniques. Our presentation is also original in the systematic use of a homology theory, analogous to the theory of homology with locally compact supports ofBorel and Moore [6]. When dealing with smooth proper schemes, Poincard duality eliminates the need for a homology theory. But in dealing with non-proper schemes, and with singularities, one needs something more. One can develop the theory of De Rham cohomology with compact supports, using the techniques of [26]. However we have found it preferable to develop a homology theory. The connection is that for any scheme Y of finite type over k, H^(Y) is the dual ofH,(Y). Using this homology theory, the Gysin sequence [24, III. 8.3] of our earlier treatment of De Rham cohomology now reappears as the exact sequence of homology of a closed subset (II. 3.3). 4» Applications. The most important applications of the theory so far, which also provided the motivation for writing this paper, are contained in the work of Ogus [43]. He was able to find algebraic proofs of the striking theorems of Earth [3], while at the same time eliminating the hypotheses of non-singularity. He was also able to determine completely the cohomological dimension of projective space P" minus a subscheme Y, in terms of the De Rham cohomology invariants of Y. This answers problems raised in [23] and [24, III, § 5]. We will discuss some of Ogus' results, in particular the Lefschetz theorem on hyperplane sections, in Chapter III, § 7. Another application of our theory is to give some necessary conditions for a singular variety to be deformed into a smooth variety. These will be discussed in our forthcoming paper [28]. 5. Problems* Of the many questions which come to mind, perhaps the most persistent is <( What about characteristic p? " Of course our results depend on characteristic zero, not only for the resolution of singularities which is not yet known in characteristic p, but also for the integration used in proving the invariance of our definitions, which is false 8 ROBIN HARTSHORNE in characteristic p. However, one can hope that the more formal aspects of our theory, in particular the systematic use of homology and its exact sequences, may be useful in characteristic p. Another interesting topic for research is the algebraic treatment of monodromy and the cohomology of special fibres of a family. Here one can hope that the introduction of the sheaves of relative cohomology in the singular case (HI, § 4, 5) will be useful. In particular, one can ask, do the De Rham cohomology groups of the fibres of a flat proper morphism obey any semi-continuity analogous to the semi-continuity theorems for coherent sheaves [EGA III, § 6, 7] ? A third problem is to study various filtrations on the De Rham cohomology. For instance, we do not know an algebraic proof of the degeneration of the first spectral sequence, which begins Ef3 == H^X, Q^) for X smooth and proper over C. For a scheme Y with arbitrary singularities, since the complex Q^ used to define the cohomology of Y, where Y is embedded in the smooth scheme
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