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Dimension projective finie et cohomologie locale. Applications à la demonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck. (French) Zbl 0268.13008
This beautiful and important paper contains proofs that in characteristic \(p>0\) and for certain local rings containing the rationals, M. Auslander’s zero-divisor conjecture holds and Bass’ question (if a local ring \(R\) possesses a nonzero module of finite type and finite injective dimension, is \(R\) Cohen-Macaulay?) has an affirmative answer. The class of local rings essentially of finite type over a field is included.
The authors also show that if \(X\) is a closed subscheme of projective space \(P =\mathbb P^n_K\) over a field \(K\) of characteristic \(p>0\), \(d\) is the smallest of the dimensions of the irreducible components of \(X\), and \(i\leq d\) is an integer such that \(X\) satisfies the condition \(S_i\) of Serre (i. e. for every \(x\in X\), \[ \mathrm{prof}\, {\mathcal O}_{X,x}\geq \inf (i, \dim{\mathcal O}_{X,x})), \] then for every coherent sheaf \(\mathcal O\) on \(P-X\), \(H^s(P-X, \mathcal O)\) is a finite-dimensional vector space for \(s\geq n-i\) and, moreover, \(H^s(P-X, \mathcal O(r))=0\) for all sufficiently large positive \(r\). (This was conjectured by A. Grothendieck for the case where \(X\) is locally a complete intersection.) [Cf. the recent paper of A. Ogus [Ann. Math. (2) 98, 327–365 (1973; Zbl 0308.14003); e.g. theorem 2.7, for results in the case \(\mathrm{char}\, X = 0\).]
In fact, the authors establish the essentially stronger result (theorem 4.9) that if \(R\) is a regular local ring of characteristic \(p>0\), \(I\) an ideal of \(R\) and \(i\) an integer such that (1) for every irreducible component \(Y\) of \(\mathrm{Spec}(R/I)\), \(i < \dim Y\); and (2) if \(U\) is the complementary open set to the closed point of \(\mathrm{Spec}(R)\), then \(R/I\) restricted to \(U\) satisfies \(S_i\) (condition of Serre), then for every \(R\)-module \(M\) and each \(s\geq \dim R-i\), the local cohomology modules \(H^s_I(M)\) are artinian \(R\)-modules.
The proofs of the zero-divisor conjecture and Bass’ question depend on showing that both follow from the following conjecture of the authors (intersection theorem): if \((R,P)\) is local, and \(M,N\) are \(R\)-modules of finite type such that \(\mathrm{Supp}(M\otimes_K N) = \{P\}\), then \(\dim N\leq \mathrm{pd}_RM\). For characteristic \(p\) the authors prove this (and theorem 4.9) by ingenious use of the interplay between local cohomology, the Frobenius functor, and modules of finite projective dimension. They then obtain their characteristic 0 results by a very clever application of M. Artin’s approximation theorem.
[The reviewer has since obtained the intersection theorem, hence, the zero-divisor conjecture, Bass’ question, etc. for all local rings \(R\) such that \(R_{\text{red}}\) contains a field, by a different method: see M. Hochster, Deep local rings (Aarhus University Preprint Series, No. 8 (1973/74). See also Bull. Am. Math. Soc. 80, 683–686 (1974; Zbl 0289.13007).]
Finally, it should be mentioned that the paper under review contains very many other important and interesting results: too many to permit even a listing here.

MSC:
13C10 Projective and free modules and ideals in commutative rings
13D05 Homological dimension and commutative rings
14A05 Relevant commutative algebra
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References:
[1] M. Artin, Algebraic approximation of structures over complete local rings,Publ. Math. I.H.E.S.,36, 1969. · Zbl 0181.48802
[2] M. Auslander, Modules over unramified regular local rings,Ill. J. of Math.,5 (1961), 631–645. · Zbl 0104.26202
[3] M. Auslander, Modules over unramified regular local rings,Proc. Intern. Congress of math., 1962, 230–233.
[4] M. Auslander, On the purity of branch locus,Amer. J. of Math.,84 (1962), 116–125. · Zbl 0112.13101 · doi:10.2307/2372807
[5] M. Auslander etD. Buchsbaum, Homological codimension and multiplicity,Ann. of Math.,68 (1958), 626–657. · Zbl 0092.03902 · doi:10.2307/1970159
[6] H. Bass, On the ubiquity of Gorenstein rings,Math. Zeitschr.,82 (1963), 8–28. · Zbl 0112.26604 · doi:10.1007/BF01112819
[7] N. Bourbaki,Algèbre commutative, Hermann, Paris, 1961–65.
[8] H. Cartan etS. Eilenberg,Homological algebra, Princeton University Press, 1956.
[9] Gabriel, Objects injectifs dans les catégories abéliennes,Séminaire Dubreil-Pisot, Algèbre et théorie des nombres, 12e année, 1958–1959, no 17, 32 p.
[10] A. Grothendieck,Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), North Holland, Amsterdam, 1968. · Zbl 0197.47202
[11] A. Grothendieck, Local cohomology,Lectures notes on mathematics, no 41, Springer Verlag, 1967. · Zbl 0185.49202
[12] A. Grothendieck, Éléments de géométrie algébrique,Publ. Math. I.H.E.S., nos 4, 8, 11, 17, 20, 24, 32.
[13] R. Hartshorne, Residues and duality,Lectures notes on mathematics, no 20, Springer Verlag, 1966. · Zbl 0212.26101
[14] R. Hartshorne, Cohomological dimension of algebraic varieties,Ann. of Math.,88 (1968), 403–450. · Zbl 0169.23302 · doi:10.2307/1970720
[15] R. Hartshorne, Ample subvarieties of algebraic varieties,Lectures notes in mathematics, no 156, Springer Verlag, 1970. · Zbl 0208.48901
[16] Horrocks, Vector bundles on the punctured spectrum of a local ring,Proc. Lond. Math. Soc. (3),14 (1964), 689–713. · Zbl 0126.16801 · doi:10.1112/plms/s3-14.4.689
[17] S. Kleimann, On the vanishing of H n (X,J) for ann-dimensional variety,Proc. Amer. Math. Soc.,18 (1967), 940–944. · Zbl 0165.24001
[18] E. Kunz, Characterisations of regular local rings of characteristicp, Amer. J. of Math.,41 (1969), 772–784. · Zbl 0188.33702 · doi:10.2307/2373351
[19] S. Lichtenbaum, On the vanishing of Tor in regular local rings,Ill. J. of Math.,10 (1966), 220–226. · Zbl 0139.26601
[20] Macaulay,Algebraic theory of modular systems, Cambridge tracts, no 19, 1916. · JFM 46.0167.01
[21] D. Rees, The grade of an ideal or module,Proc. Camb. phil. soc.,53 (1957), 28–42. · Zbl 0079.26602 · doi:10.1017/S0305004100031960
[22] P. Samuel,Séminaire d’algèbre commutative, Anneaux de Gorenstein et torsion en algèbre commutative. Secrétariat mathématique, 11, rue Pierre Curie, Paris (5e), 1967.
[23] J.-P. Serre, Algèbre locale et multiplicité,Lectures notes in mathematics, no 11, Springer Verlag, 1965.
[24] D. A. Buschbaum, Complexes associated with the minors of a matrix,Symposia Matematica, vol. IV, 1970, Bologna.
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