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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
##### Keywords:
finite projective dimension
Full Text:
##### References:
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