On the derived functors of \(\varprojlim\). Applications. (Sur les foncteurs dérivés de \(\varprojlim\). Applications.) (French) Zbl 0102.02501

Roos, Jan-Erik

C. R. Acad. Sci., Paris 252, 3702-3704 (1961).

Let \(\mathcal A\) be an abelian category with Grothendieck’s Axiom \(AB4^*\) and let \(I\) be a partially ordered set. The author constructs a complex which yields the satellites \(\varprojlim^p\) of the inverse limit functor \(\mathcal A^I\to\mathcal A\) where \(\mathcal A^I\) denotes the category of projective systems parameterized by \(I\). The construction can be dualized so that if \(\mathcal A\) has \(AB 5\) and enough projectives then, for an inductive system \(A\), there exists a spectral sequence converging to \(\mathrm{Ext}^n(\lim A, B)\) such that \(E_2^{p,q}=\varprojlim^p\mathrm{Ext}^q (A_{\alpha}, B)\).
Further, if \(\mathcal A\) is the category of right \(\Lambda\)-modules over the noetherian ring \(\Lambda\) and if \(M\) is a left \(\Lambda\)-module of finite type then there are two spectral sequences converging to the same limit with first terms \(E_2^{p,q}= \mathrm{Tor}_{-p}(\varprojlim^{(q)}(A, M))\) and \(E_2^{p,q}=\varprojlim^{(p)}(\mathrm{Tor}_{-q}((A_{\alpha}, M))\). Finally the author indicates a modification of Čech homology which satisfies the exactness axioms.
Editorial remark (2021): A. Neeman [Invent. Math. 148, No. 2, 397–420 (2002; Zbl 1025.18007)] gave a counterexample which showed that this result is not true in general; the author gave himself a corrected version in [J. Lond. Math. Soc., II. Ser. 73, No. 1, 65–83 (2006; Zbl 1089.18007)].

Reviewer: G. F. Leger

MSC:

18E10	Abelian categories, Grothendieck categories
18A30	Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
18E10	Abelian categories, Grothendieck categories
13D45	Local cohomology and commutative rings
55P65	Homotopy functors in algebraic topology
18G10	Resolutions; derived functors (category-theoretic aspects)

Keywords:

category theory; projective limit

Citations:

Zbl 1025.18007; Zbl 1089.18007