%0 Journal Article %1 citeulike:11071789 %A Serre, Jean-Pierre %D 1955 %J Annals of Mathematics %K 13-02-commutative-algebra-research-exposition, 13d45-commutative-algebra-local-cohomology, 14-02-algebraic-geometry-research-exposition %N 2 %P 197--278 %T Faisceaux algébriques cohérents %U http://www.jstor.org/stable/1969915 %V 61 %X MR: This paper contains the basic material for the application of cohomology theory to algebraic geometry; the specific applications themselves are to appear in a forthcoming paper. Chapter I is concerned with the general theory of sheaves. A sheaf on a space X is a space F with a mapping p (the projection) of F into X which is a local homeomorphism. A continuous mapping s of an open set U of X into F such that p(s(x))=x for x∈U is called a section of F above U. In all algebraic applications, the sets Fx=p−1(x) have algebraic structures whose laws of composition are assumed to "vary continuously'' with x, in a sense easily made precise; the most interesting case is that of a sheaf of modules, where each Fx is a module over a ring Ax, the Ax's forming themselves a sheaf of rings. The notions introduced in sheaf theory are either of an algebraic nature (induced sheaf on a subset of X, prolongation of a sheaf on a subspace, subsheaves, sheaf homomorphisms, factor sheaves, exact sequences of sheaves, tensor product of sheaves) or of a topological nature (cohomology of a space X in a sheaf F of additive groups on X). In defining this last notion, attention must be paid to the fact that X may fail to be a Hausdorff space (this will always happen in algebraic geometry). This fact entails certain difficulties; in particular, the exactness of the cohomology sequence associated to an exact sequence cannot be established in general. This fundamental theorem holds nevertheless for the cohomology of an algebraic variety V, provided only älgebraic coherent'' sheaves are considered. Let F be a sheaf of modules over a sheaf A of rings. Then F is said to be of finite type if, for every x∈X, there exist a finite number of sections si (1≤i≤p) of F over a suitable neighbourhood U of x such that, for all y∈U, Fy is generated as a module by the elements si(y). Let s1,⋯,sp be sections of F over an open set U. For each y∈U, let Ry be the set of elements (a1,⋯,ap) in Ayp such that ∑pi=1aisi(y)=0. Then the sets Ry form a sheaf R over U. If F is of finite type and R of finite type (for all choices of s1,⋯,sp, U), then F is called coherent. Coherent sheaves are discussed in relation to the general algebraic operations on sheaves. Chapter II contains an alternative definition of the notion of an (abstract) variety, due to A. Weil. An algebraically closed field K is given; each affine K-space Kr is endowed with the Zariski topology on this space. For any x∈Kr, let Qx be the local ring of x (the set of rational functions on Kr which are defined at x); the rings Ox form a sheaf of rings on Kr. A variety V is defined to be a space V together with a subsheaf OV of the sheaf of germs of K-valued functions on V, with two axioms of which the first says that X has an open finite covering (Ui) such that each Ui (with its induced sheaf) is isomorphic to a locally closed set in some Kr, with its sheaf of local rings, while the second expresses the condition of biregularity of the birational transformations between the Ui's which are defined by the fact that, if x∈Ui∩Uj, its local ring on Ui is the same as its local ring on Uj (both being the ring (OV)x). The notion introduced in this manner is slightly more general than the notion defined by Weil, inasmuch as it includes reducible varieties (unions of a finite number of Weil varieties). If OV is the sheaf which defines the structure of a variety V, then, for x∈V, (OV)x is called the local ring at x, and is denoted by Ox. By an algebraic sheaf on V is meant a sheaf of modules over OV. Every sheaf on a closed subvariety W of V may be extended to a sheaf on V which coincides with 0 outside W. This implies that, inasmuch as sheaf theory is concerned, the theory of a subvariety is not essentially more difficult than that of the ambient variety; in particular, sheaf theory on a projective variety may be reduced to sheaf theory on a projective space, in which specific homogeneous coordinates are used to the full. The author studies the theory of sheaves on affine varieties; there, it turns out that, for coherent sheaves, the cohomology is trivial in all dimensions >0; this seems to indicate that the natural domain of sheaf theory is the study of algebraico-geometric notions in which the completeness of the variety plays a crucial role. By using a finite affine covering of an abstract variety by affine varieties, it is shown that the cohomology sequence attached to an exact sequence A→B→C of sheaves is exact provided A is algebraic and coherent. Chapter III is concerned with the theory of sheaves on a projective variety. The projective space X of dimension r is covered in a natural manner by r+1 affine spaces; it is shown that this covering is already sufficient to compute the cohomology groups of a coherent algebraic sheaf; in particular, the cohomology in dimension >r is trivial. More precisely, if V is a closed projective variety of dimension n, then the cohomology of V (in any coherent algebraic sheaf) is trivial in any dimension >n. Let S be the ring of polynomials in r+1 variables. In order to study the cohomology of X in an algebraic sheaf F, the author associates to F a graded module M over S. Let x be a point of X. Introduce the ring On(x) generated by the elements P/Q where P and Q are homogeneous in S, degree of P=N+degree of Q and Q and Q(x)≠0 (n being any integer). The rings On(x) form a sheaf O(n); set F(n)=O(n)⊗F. Let (ΓF)n be the set of sections of F(n) over the whole of X, and ΓF the direct sum of the groups (ΓF)n, for all integers n. Then ΓF has an easily defined structure of graded module over the ring S. If F is coherent, then ΓF satisfies the following condition (TF): there exists a p such that ∑n>p(ΓF)n is finitely generated. Moreover, in that case, the sheaf F may be recovered from the module ΓF by a general procedure which associates a sheaf on X to every graded S-module. To every graded S-module M, the author attaches certain cohomology groups Hq(M) J. L. Koszul had introduced homology (not cohomology) groups for M Colloque de Topologie (espaces fibrés), Bruxelles, 1950, Thone, Liège, 1951, pp. 73–81; MR0042428 (13,109a); taking into account the self duality of an exterior algebra on a free module, Serre's groups are related in the classical manner to a generalization of Koszul's homology groups. If F is a coherent sheaf on X, then the cohomology groups of F are the same as those of the module ΓF. This allows one to obtain a certain number of properties of the cohomology groups Hq(X;F); in particular, these groups are finite-dimensional vector spaces over K, and, for q>0, Hq(X,F(n))=0 for n large. A new step is accomplished by comparing the cohomology groups Hq(M) of a graded S-module M to certain other groups, which are Ext-groups. Let M(n) be the module obtained from M by lowering the degree by n. The module Hq(M(n)) is a vector space over K; denote by (Tq(M))n the dual of the vector space Hq(M(−n)) and by Tq(M) the direct sum of the groups (Tq(M))n for all integers n; this is again a graded module over S. Let Ω be a free module of rank 1 over S, graded by the condition that the basic element be of degree r+1. Then it is proved that, for any q≠r, Tr−q(M) is isomorphic to Extq(M,Ω), while, for q=r, there is an exact sequence 0→Extr(M,Ω)→T0(M)→M∗→Extr+1(M,Ω)→0, where M∗ is the graded module whose elements of degree n are the linear forms on the vector space Mn over K of elements of degree n in M. The passing from −n to n which is involved in the definition of Tq(M), together with the fact that the dual of a vector space ≠0 is ≠0, allows one to infer from this results on the groups Hq(M(−n)) for n large, and therefore on the groups Hq(X,F(−n)), if F is a coherent sheaf on X. Let V be an irreducible variety of dimension p in X; let F be a coherent sheaf on V such that, for every x∈V, Fx is a free module over the local ring of x. Then, if V is free from singularity, Hq(V,F(−n))=0 for large n and 0≤q<p; if V is normal and p≥2, then H1(V;F(−n))=0 for large n. If F is a coherent sheaf on a projective variety V, it is possible to define the Euler-Poincaré characteristic χ(V,F)=∑q(−1)qhq(V,F), where hq(V,F) is the dimension of Hq(V,F) over K. If V is irreducible of dimension p, and F the sheaf of local rings, then χ(V,F)=1+(−1)qpa(V), where pa(F) is the arithmetic genus of V. If V=X, then, for any F, χ(X,F(n)) is a polynomial in n whose degree is the dimension of the carrier of F. Reviewed by C. Chevalley

@article{citeulike:11071789, abstract = {MR: This paper contains the basic material for the application of cohomology theory to algebraic geometry; the specific applications themselves are to appear in a forthcoming paper. Chapter I is concerned with the general theory of sheaves. A sheaf on a space X is a space F with a mapping p (the projection) of F into X which is a local homeomorphism. A continuous mapping s of an open set U of X into F such that p(s(x))=x for x∈U is called a section of F above U. In all algebraic applications, the sets Fx=p−1(x) have algebraic structures whose laws of composition are assumed to "vary continuously'' with x, in a sense easily made precise; the most interesting case is that of a sheaf of modules, where each Fx is a module over a ring Ax, the Ax's forming themselves a sheaf of rings. The notions introduced in sheaf theory are either of an algebraic nature (induced sheaf on a subset of X, prolongation of a sheaf on a subspace, subsheaves, sheaf homomorphisms, factor sheaves, exact sequences of sheaves, tensor product of sheaves) or of a topological nature (cohomology of a space X in a sheaf F of additive groups on X). In defining this last notion, attention must be paid to the fact that X may fail to be a Hausdorff space (this will always happen in algebraic geometry). This fact entails certain difficulties; in particular, the exactness of the cohomology sequence associated to an exact sequence cannot be established in general. This fundamental theorem holds nevertheless for the cohomology of an algebraic variety V, provided only "algebraic coherent'' sheaves are considered. Let F be a sheaf of modules over a sheaf A of rings. Then F is said to be of finite type if, for every x∈X, there exist a finite number of sections si (1≤i≤p) of F over a suitable neighbourhood U of x such that, for all y∈U, Fy is generated as a module by the elements si(y). Let s1,⋯,sp be sections of F over an open set U. For each y∈U, let Ry be the set of elements (a1,⋯,ap) in Ayp such that ∑pi=1aisi(y)=0. Then the sets Ry form a sheaf R over U. If F is of finite type and R of finite type (for all choices of s1,⋯,sp, U), then F is called coherent. Coherent sheaves are discussed in relation to the general algebraic operations on sheaves. Chapter II contains an alternative definition of the notion of an (abstract) variety, due to A. Weil. An algebraically closed field K is given; each affine K-space Kr is endowed with the Zariski topology on this space. For any x∈Kr, let Qx be the local ring of x (the set of rational functions on Kr which are defined at x); the rings Ox form a sheaf of rings on Kr. A variety V is defined to be a space V together with a subsheaf OV of the sheaf of germs of K-valued functions on V, with two axioms of which the first says that X has an open finite covering (Ui) such that each Ui (with its induced sheaf) is isomorphic to a locally closed set in some Kr, with its sheaf of local rings, while the second expresses the condition of biregularity of the birational transformations between the Ui's which are defined by the fact that, if x∈Ui∩Uj, its local ring on Ui is the same as its local ring on Uj (both being the ring (OV)x). The notion introduced in this manner is slightly more general than the notion defined by Weil, inasmuch as it includes reducible varieties (unions of a finite number of Weil varieties). If OV is the sheaf which defines the structure of a variety V, then, for x∈V, (OV)x is called the local ring at x, and is denoted by Ox. By an algebraic sheaf on V is meant a sheaf of modules over OV. Every sheaf on a closed subvariety W of V may be extended to a sheaf on V which coincides with 0 outside W. This implies that, inasmuch as sheaf theory is concerned, the theory of a subvariety is not essentially more difficult than that of the ambient variety; in particular, sheaf theory on a projective variety may be reduced to sheaf theory on a projective space, in which specific homogeneous coordinates are used to the full. The author studies the theory of sheaves on affine varieties; there, it turns out that, for coherent sheaves, the cohomology is trivial in all dimensions >0; this seems to indicate that the natural domain of sheaf theory is the study of algebraico-geometric notions in which the completeness of the variety plays a crucial role. By using a finite affine covering of an abstract variety by affine varieties, it is shown that the cohomology sequence attached to an exact sequence A→B→C of sheaves is exact provided A is algebraic and coherent. Chapter III is concerned with the theory of sheaves on a projective variety. The projective space X of dimension r is covered in a natural manner by r+1 affine spaces; it is shown that this covering is already sufficient to compute the cohomology groups of a coherent algebraic sheaf; in particular, the cohomology in dimension >r is trivial. More precisely, if V is a closed projective variety of dimension n, then the cohomology of V (in any coherent algebraic sheaf) is trivial in any dimension >n. Let S be the ring of polynomials in r+1 variables. In order to study the cohomology of X in an algebraic sheaf F, the author associates to F a graded module M over S. Let x be a point of X. Introduce the ring On(x) generated by the elements P/Q where P and Q are homogeneous in S, degree of P=N+degree of Q and Q and Q(x)≠0 (n being any integer). The rings On(x) form a sheaf O(n); set F(n)=O(n)⊗F. Let (ΓF)n be the set of sections of F(n) over the whole of X, and ΓF the direct sum of the groups (ΓF)n, for all integers n. Then ΓF has an easily defined structure of graded module over the ring S. If F is coherent, then ΓF satisfies the following condition (TF): there exists a p such that ∑n>p(ΓF)n is finitely generated. Moreover, in that case, the sheaf F may be recovered from the module ΓF by a general procedure which associates a sheaf on X to every graded S-module. To every graded S-module M, the author attaches certain cohomology groups Hq(M) [J. L. Koszul had introduced homology (not cohomology) groups for M [Colloque de Topologie (espaces fibr\'{e}s), Bruxelles, 1950, Thone, Li\`{e}ge, 1951, pp. 73–81; MR0042428 (13,109a)]; taking into account the self duality of an exterior algebra on a free module, Serre's groups are related in the classical manner to a generalization of Koszul's homology groups]. If F is a coherent sheaf on X, then the cohomology groups of F are the same as those of the module ΓF. This allows one to obtain a certain number of properties of the cohomology groups Hq(X;F); in particular, these groups are finite-dimensional vector spaces over K, and, for q>0, Hq(X,F(n))={0} for n large. A new step is accomplished by comparing the cohomology groups Hq(M) of a graded S-module M to certain other groups, which are Ext-groups. Let M(n) be the module obtained from M by lowering the degree by n. The module Hq(M(n)) is a vector space over K; denote by (Tq(M))n the dual of the vector space Hq(M(−n)) and by Tq(M) the direct sum of the groups (Tq(M))n for all integers n; this is again a graded module over S. Let Ω be a free module of rank 1 over S, graded by the condition that the basic element be of degree r+1. Then it is proved that, for any q≠r, Tr−q(M) is isomorphic to Extq(M,Ω), while, for q=r, there is an exact sequence 0→Extr(M,Ω)→T0(M)→M∗→Extr+1(M,Ω)→0, where M∗ is the graded module whose elements of degree n are the linear forms on the vector space Mn over K of elements of degree n in M. The passing from −n to n which is involved in the definition of Tq(M), together with the fact that the dual of a vector space ≠{0} is ≠{0}, allows one to infer from this results on the groups Hq(M(−n)) for n large, and therefore on the groups Hq(X,F(−n)), if F is a coherent sheaf on X. Let V be an irreducible variety of dimension p in X; let F be a coherent sheaf on V such that, for every x∈V, Fx is a free module over the local ring of x. Then, if V is free from singularity, Hq(V,F(−n))={0} for large n and 0≤q<p; if V is normal and p≥2, then H1(V;F(−n))={0} for large n. If F is a coherent sheaf on a projective variety V, it is possible to define the Euler-Poincar\'{e} characteristic χ(V,F)=∑q(−1)qhq(V,F), where hq(V,F) is the dimension of Hq(V,F) over K. If V is irreducible of dimension p, and F the sheaf of local rings, then χ(V,F)=1+(−1)qpa(V), where pa(F) is the arithmetic genus of V. If V=X, then, for any F, χ(X,F(n)) is a polynomial in n whose degree is the dimension of the carrier of F. Reviewed by C. Chevalley}, added-at = {2017-06-29T07:13:07.000+0200}, author = {Serre, Jean-Pierre}, biburl = {https://www.bibsonomy.org/bibtex/2cbb21160b0dbf3276b353a3608995e44/gdmcbain}, citeulike-article-id = {11071789}, citeulike-attachment-1 = {1969915.pdf; /pdf/user/gdmcbain/article/11071789/822710/1969915.pdf; 5752cd0f6f1dc29c6419b0254d029bbfa0dd376a}, citeulike-linkout-0 = {http://www.jstor.org/stable/1969915}, comment = {cited by Grothendieck (1960, Introduction): <<Le travail [14] (cit\'e FAC)) de J.-P. Serre peut aussi \^etre consid\'er\'e comme un expos\'e interm\'ediare entre le point de vue classique et le point de vue des sch\'emas en G\'eom\'etrie alg\'ebraique, et \`a ce titre, sa lecture peut constituer une excellente pr\'eparation \`a celle de nos \textit{\'El\'ements}.>>}, file = {1969915.pdf}, interhash = {612f512d136926629ff9fe76b1263e34}, intrahash = {cbb21160b0dbf3276b353a3608995e44}, journal = {Annals of Mathematics}, keywords = {13-02-commutative-algebra-research-exposition, 13d45-commutative-algebra-local-cohomology, 14-02-algebraic-geometry-research-exposition}, number = 2, pages = {197--278}, posted-at = {2012-08-16 13:13:48}, priority = {2}, timestamp = {2017-06-29T07:13:07.000+0200}, title = {{Faisceaux alg\'{e}briques coh\'{e}rents}}, url = {http://www.jstor.org/stable/1969915}, volume = 61, year = 1955 }