Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes

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For any scheme Z, Zqc resp. In a short while we will give a quick summary of the Deligne- Verdier approach DV approach for short to Duality. Deligne's and Verdier's results apply to finite-type, separated maps between schemes of fi- nite KruU dimension. This is generalized to arbitrary schemes by Alonso Tarrio, Jeremias Lopez and Lipman in Q in fact their results are far more general than we need in this paper. They work with formal schemes. Since our interest is not Date: February 1, The key results in the DV approach to Duality are a the existence of a right adjoint to the derived direct image functor for a proper map — the twisted inverse image functor in Verdier's terminology pp.

We should point out that Neeman has an intriguingly different approach to the above results see Q. Here then is promised summary of the key points of the DV-approach. We say more about this in Remark 1. Recently there have been other proofs of Nagata's result by Liitkebohmert l6j and independently Conrad Remark 1.

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Here is how the local nature of "upper shriek" is proved using flat base change. The Problem: To explain the problem, we will first consider a simpler situa- tion, in which we have more hypotheses than we really need. With Conrad consider first a proper map f : X Y oi finite type between Noetherian schemes, which is Cohen- Macaulay of relative dimension r. It is well-known that in this situation for some coherent sheaf the relative dualizing sheaf on X p.

It is further proved in loc. It should be pointed out that the statement in loc. Conrad's main results, when Y and Y' carry dualizing complexes, are 1.

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But, perhaps we are not being imaginative enough. That then is the problem. In this paper, we answer all three questions afhrma- tively. Our techniques are such that we do not need dualizing complexes or their Cousin versions — residual complexes. The author confesses to having a soft corner for the DV approach. Here, for what it is worth, is our idea of the first mile of the royal road. In later work with S. Remark 1. We have quoted Lemma 1, p. The same Lemma also asserts that the ojf is well behaved with respect to base change, but this assertion is not completely proved there.

The proof given in loc. The Main Results 2.

Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes resources

Theorem 3 p. The theorem depends only upon his flat base change theorem [loc. Moreover, from the proof of loc.

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See Remark 1. Remark 2. In view of the remarks made towards the end of subsection 2. However, its — r th cohomology does make sense. Over triple intersections, these objects formally satisfy cocycle rules. But that is not enough to glue them together as objects in the reason why Hartshorne upgrades his constructions to Cousin complexes. However, the — rth cohomology do es g lue together since we are now in the category of sheaves!

Studies In Duality On Noetherian Formal Schemes And Non Noetherian Ordinary Schemes

This is the slick way of understanding ld, p. Now suppose we have a cartesian diagram 2. Explanation: Item a needs slight elaboration. Xu is equidimensional and proper over Y. Therefore Theo- rem 2. Theorem 2. Let he a cartesian square of schemes, with f Cohen-Macaulay of relative dimension r. UJf, commutes, where the vertical isomorphisms arise from uniqueness up to unique isomorphism of dualizing pairs.

The residue map is a formal analogue of the integral One shows that this residue map for special Z we call such Z's good has a local duality property and is well behaved with respect to base change. By taking the closure of the diagonal embedding of X in Xi Xy X2 if necessary, we may assume that we have a commutative diagram with the square being cartesian.

Our interest is in "good" immersions, which we now define: Definition 3. Three key propositions: For proving Theorem 2. Th e pr oofs of these key propositions will be given later, af ter we show in subsection 3. For any coherent sheaf defined in an open neighborhood of Z in X, let Tz denote the completion of J- along Z. We then have: Proposition 3. Pro of of Theorem 2.

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Then as consequence of Proposition 3. This is where the Cohen- Macaulay property helps. In the flat topology on X we have a plentiful supply of good immersions, and then faithful flat descent gives the rest. We bring the above ideas down to earth as follows: Let X G X he a. If Z is the closed subscheme of U defined by the t's, then Z must be finite over T, for T is the spectrum of a complete local ring.

In view of 2. Parts a and b of Theorem 2. Remark 3.

The fundamental local isomorphism; adjunction 4. Sign convention for complexes: We follow the following standard sign conventions. If P is a finitely generated projective module, we identify P with its double dual in the standard way. Then one checks using the conventions above that: 1. Note the order in which the tensor product is taken. Now, is a free rank 1 P-module with ti A. A U a generator. There is, from comments in the previous subsection a complex of free i? We re- call first the explicit description of duality for the map j. Note also that the isomorphism 4.

We now come to the main point of all these seemingly meaningless exercises Proposition 4. Part a is proved in the exactly the same way in which Proposition 3. Since h is finite, it then follows that h is Cohen-Macaulay of relative dimension. Now set in a suggestive notation and define in another suggestive notation h by the composition Here O r.

Our intent clearly! Since B is flat and finite over A i. We will relate this pairing to the pairing stated in the Proposition to reach the desired conclusion. My profile My library Metrics Alerts. Sign in. Universidade de Santiago de Compostela. Verified email at usc. Articles Cited by Co-authors. Annales scientifiques de l'Ecole normale superieure 30 1 , , Canadian Journal of Mathematics 52 2 , , Transactions of the American Mathematical Society 6 , , Proceedings of the American Mathematical Society 2 , , Journal of Pure and Applied Algebra 7 , , Applied Categorical Structures 19 6 , , Journal of Pure and Applied Algebra 4 , , Articles 1—20 Show more.

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