##
**The Grothendieck duality theorem via Bousfield’s techniques and Brown representability.**
*(English)*
Zbl 0864.14008

For a scheme \(X\), let \(D(qc/X)\) (resp. \(D^+(qc/X))\) denote the derived category of complexes of coherent sheaves (resp. bounded below complexes of coherent sheaves) on \(X\). The traditional Grothendieck duality theorem states that in many cases (say when the schemes involved are noetherian), for a proper map \(f:X\to Y\), the functor \(\mathbb{R} f_*:D^+ (qc/X) \to D^+(qc/Y)\) has a right adjoint \(f^!\). There is a short elegant proof of this due to Deligne [see the appendix to R. Hartshorne’s book: “Residues and duality” (1966; Zbl 0212.26101)]. If \(Y\) is the spectrum of a field \(k\), we only have to produce \(f^!k\), and Deligne’s proof is particularly transparent in this case as the reader can verify from what follows. For an open affine set \(U=\text{Spec} A\) of \(X\), consider the quasi-coherent sheaf \({\mathcal I}_U\) on \(X\) obtained by sheafifying the \(A\)-module \(\operatorname{Hom}_k(A,k)\) on \(U\), and then pushing it forward to all of \(X\). One checks that \({\mathcal I}_U\) is an injective \({\mathcal O}_X\)-module, and that for any quasi-coherent \({\mathcal O}_X\)-module \({\mathcal F}\), \(\operatorname{Hom}_{{\mathcal O}_X} ({\mathcal F}, {\mathcal I}_U)\) is dual, as a \(k\)-vector space, to \(\Gamma (U,{\mathcal F})\). Now let \({\mathcal U}= \{{\mathcal U}_i\}\) be an affine open cover of \(X\) and set \({\mathcal I}^{-p}= \bigoplus_{i_0< \cdots<i_p} {\mathcal I}_{U_{i_0} \cap \cdots \cap U_{i_p}}\) for \(p\geq 0\). The \({\mathcal I}^q\) string themselves together into a complex \({\mathcal I}^\bullet\) (the coboundary maps being “dual” to that obtained from the Čech construction) and one has \(\operatorname{Hom}^\bullet_{{\mathcal O}_X} ({\mathcal F}^\bullet, {\mathcal I}^\bullet) = \operatorname{Hom}_k^\bullet ({\mathcal C}^\bullet ({\mathcal U}, {\mathcal F}^\bullet),k)\), for all bounded below complexes \({\mathcal F}^\bullet\) of quasi-coherent sheaves, giving Grothendieck duality with \(f^!k={\mathcal I}^\bullet\). Digging deeper one finds that what is involved in the general case is a version of the Special Adjoint Functor theorem. The procedure gives right adjoints at the sheaf level which is transferred to the homotopy category via injective resolutions, and finally to the derived category. The procedure has been modified to work for maps of arbitrary quasi-compact quasi-separated schemes, for unbounded complexes (using Spaltenstein’s results), and recently for maps of noetherian formal schemes. The first two results are unpublished works of Lipman, and the last is a recent joint work of Alonso Tarrío, Jeremías Lopez and Lipman.

The approach of the paper under review is radically different. Ideas from homotopy theory are adapted (via work of Thomason on perfect complexes) to the problem on hand. Consequently, the adjoint to \(\mathbb{R} f_*\) is obtained directly at the derived category level, and the generalizations (with the exception of the result of formal schemes) mentioned above are obtained. The author gives a necessary and sufficient condition for the natural map \(\mathbb{L} f^*x \otimes^\mathbb{L} f^! {\mathcal O}_Y\to f'x\) to be an isomorphism, viz., that \(f^!\) commutes with co-products. The traditional sufficient condition is that \(f\) should be of “finite Tor dimension”. Further, in the bounded below situation, the author shows that \(f^!\) behaves well with base changes by open immersions, giving the sheafified version of Grothendieck duality – even in the non-noetherian situation. A counter-example for this is provided when one works with unbounded complexes.

The author’s techniques are general enough to give adjoints for the functor \(f_+\) between derived categories of complexes of \(D\)-modules. How does all this compare with Deligne’s approach (modified and extended by Lipman et al.)? Both approaches need a nice set of generators for the relevent category. Since Deligne first establishes adjointness at the sheaf level, coherent sheaves form generators. Neeman’s generators are perfect complexes, since he works entirely in the derived category. For this one needs the results of Thomason [see R. W. Thomason and T. Trobaugh in: The Grothendieck Festschrift, Vol. III, Prog. Math. 88, 247-435 (1990; Zbl 0731.14001)], a work that can overwhelm by its length. It should however be mentioned that the author has simplifications in a previous paper [see Ann. Sci. Écol. Norm. Supér., IV. Sér. 25, No. 5, 547-566 (1992)]. Deligne’s approach generalizes to give duality for maps of noetherian formal schemes. Since it is not clear that perfect complexes in this situation extend from open subsets, it is not clear that the author’s approach works here. On the other hand, the author has proved that the functor \(f_+\) mentioned above has a right adjoint.

The above comparison is not meant to diminish the very real achievements of the paper. The game is not for the shortest possible proof, but for conceptual clarification (or else, one would hold, with the Luddites, that Grothendieck’s proof of Stein factorization and Zariski’s main theorem are much too long, involving so much of homological algebra – a point of view which is surely untenable). This paper has brought fresh ideas into the game, and the reviewer has certainly learnt a lot.

The approach of the paper under review is radically different. Ideas from homotopy theory are adapted (via work of Thomason on perfect complexes) to the problem on hand. Consequently, the adjoint to \(\mathbb{R} f_*\) is obtained directly at the derived category level, and the generalizations (with the exception of the result of formal schemes) mentioned above are obtained. The author gives a necessary and sufficient condition for the natural map \(\mathbb{L} f^*x \otimes^\mathbb{L} f^! {\mathcal O}_Y\to f'x\) to be an isomorphism, viz., that \(f^!\) commutes with co-products. The traditional sufficient condition is that \(f\) should be of “finite Tor dimension”. Further, in the bounded below situation, the author shows that \(f^!\) behaves well with base changes by open immersions, giving the sheafified version of Grothendieck duality – even in the non-noetherian situation. A counter-example for this is provided when one works with unbounded complexes.

The author’s techniques are general enough to give adjoints for the functor \(f_+\) between derived categories of complexes of \(D\)-modules. How does all this compare with Deligne’s approach (modified and extended by Lipman et al.)? Both approaches need a nice set of generators for the relevent category. Since Deligne first establishes adjointness at the sheaf level, coherent sheaves form generators. Neeman’s generators are perfect complexes, since he works entirely in the derived category. For this one needs the results of Thomason [see R. W. Thomason and T. Trobaugh in: The Grothendieck Festschrift, Vol. III, Prog. Math. 88, 247-435 (1990; Zbl 0731.14001)], a work that can overwhelm by its length. It should however be mentioned that the author has simplifications in a previous paper [see Ann. Sci. Écol. Norm. Supér., IV. Sér. 25, No. 5, 547-566 (1992)]. Deligne’s approach generalizes to give duality for maps of noetherian formal schemes. Since it is not clear that perfect complexes in this situation extend from open subsets, it is not clear that the author’s approach works here. On the other hand, the author has proved that the functor \(f_+\) mentioned above has a right adjoint.

The above comparison is not meant to diminish the very real achievements of the paper. The game is not for the shortest possible proof, but for conceptual clarification (or else, one would hold, with the Luddites, that Grothendieck’s proof of Stein factorization and Zariski’s main theorem are much too long, involving so much of homological algebra – a point of view which is surely untenable). This paper has brought fresh ideas into the game, and the reviewer has certainly learnt a lot.

Reviewer: P.Sastry (Jhusi)

### MSC:

14F20 | Étale and other Grothendieck topologies and (co)homologies |

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

55P42 | Stable homotopy theory, spectra |

Full Text:
DOI

### References:

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