Anne-Marie Simon

Preliminaries and Auxiliary Results

Though this first part is mainly devoted to modules, we first fix the complex terminology and notations to be used when some derived functors come into play, recalling some useful well-known facts. Then we present some basic facts on Matlis duality, and on inverse and direct limits. Most readers will not be tempted to look at this first chapter but may prefer to refer back to it for notations and definitions when needed.

Koszul Complexes, Depth and Codepth

The notions of Ext-depth and Tor-codepth with respect to an ideal \(\mathfrak {a}\) of a commutative ring R were introduced and investigated by Strooker for R-modules. He also proved (see Strooker (Homological questions in local algebra, Cambridge University Press, Cambridge, 1990)[1, 6.1.6, 6.1.7]) that these can be computed by the use of a Koszul complex when the ideal \(\mathfrak {a}\) is finitely generated.

Completion, Čech and Local Homology and Cohomology: Interactions Between Them

Part I: Modules,- 1. Preliminaries and auxiliary results.- 2. Adic topology and completion.- 3. Ext-Tor vanishing and completeness criteria.- PartII: Complexes.- 4. Homological Preliminaries.- 5. Koszul complexes, depth and codepth.- 6. Cech complexes, Cech homology and cohomology.- 7. Local cohomology and local homology.- 8. The formal power series Koszul complex.- 9. Complements and Applications.- Part III: Duality.- 10. Cech and local duality.- 11. Dualizing complexes.- 12. Local duality with dualizing complexes and other dualities.- Appendix.- References.- Notation.- Subject Index.

Local Cohomology and Local Homology

Let R denote a commutative ring and \(\mathfrak {a}\) an ideal of R. The \(\mathfrak {a}\)-torsion functor and the \(\mathfrak {a}\)-completion functor, defined respectively by \(\varGamma _{\mathfrak {a}}(M):= \{m\in M \mid \mathfrak {a}^t m=0 \text { for some } t\in \mathbb {N}\}\) and \(\varLambda ^{\mathfrak a}(M):= \varprojlim (R/\mathfrak {a}^t \otimes _R M)\) for any R-module M, extends naturally to complexes. In this chapter we first recall that \(\mathrm{L} \varLambda ^{\mathfrak a}\) and \(\mathrm{R} \varGamma _{\mathfrak a}\) are well defined in the derived category and fix some notations. Then we investigate when \(\mathrm{L} \varLambda ^{\mathfrak a}(X)\) and \(\mathrm{R} \varGamma _{\mathfrak a}(X)\) vanish. For complexes homologically-bounded on the good size we obtain more precise results, previously known when the ring is Noetherian, possibly new in the present generality. We also provide a description of \(\mathrm{L} \varLambda ^{\mathfrak a}(X)\) and \(\mathrm{R} \varGamma _{\mathfrak a}(X)\) in terms of microscope and telescope. These descriptions do not refer to the resolutions of X, which could be an advantage.

Čech and Local Duality

In this chapter we provide some duality formulas for the Cech cohomology of an unbounded complex, which involve the general Matlis dual \({\check{C}}_{\underline{x}}^{\vee }\) of the Cech complex. When the sequence \(\underline{x}\) is a system of parameters of a Noetherian local ring our formulas provide a version of the Grothendieck Local Duality for Cohen–Macaulay or Gorenstein local rings. As a byproduct we obtain new characterizations of Gorenstein local rings in terms of local homology. As another byproduct there are some characterizations of \(\mathfrak {m}\)-torsion and \(\mathfrak {m}\)-pseudo complete modules over a Gorenstein local ring. When the sequence \(\underline{x}\) is a system of parameters of a complete Noetherian local ring, it turns out that the complex \({\check{C}}_{\underline{x}}^{\vee }\) is a bounded complex of injective modules with finitely generated cohomology. For that reason we start the chapter with an investigation of such complexes.

Local Duality with Dualizing Complexes and Other Dualities

In this chapter we present a version of Grothendieck local duality for a Noetherian local ring admitting a dualizing complex. We derive it from a more general result involving the local cohomology with respect to an arbitrary ideal of a Noetherian ring admitting a dualizing complex, originally proved by Hartshorne for a regular ring of finite Krull dimension. We also extend Hartshorne’s affine duality stated for regular rings of finite Krull dimension to any Noetherian ring with a dualizing complex and provide a counterpart in local homology. Among other things we provide duality results involving both local homology and local cohomology, a recurrent theme in this monograph. We also investigate the local homology of a complex with Artinian homology, more generally with mini-max homology. We end the chapter with a short approach to Greenlees’ Warwick duality.

Adic Topology and Completion

In the first three sections of this chapter we summarize the basic results on the subject, include results for which there is no reference in the literature and complete the picture with some new observations. A few of these facts hold in full generality; but most of them require that the adic topology is taken with respect to a finitely generated ideal. The case when the ring is Noetherian is easier to handle, though far from being obvious when dealing with infinitely generated modules. In Sect. 2.4 we provide an extension of Bartijn’s result stating that the adic completion of a flat module over a Noetherian ring is flat. Section 2.5 contains the first information on the left-derived functors of the adic completion functor with respect to an ideal \(\mathfrak {a}\). Writing them, we have been astonished by how much can be said assuming only the hypothesis that the ideal \(\mathfrak {a}\) is finitely generated. As usual there are finer results when the ring is Noetherian, and it is helpful in Part II to see that some of these hold in greater generality and in a more general setting. In Sect. 2.6 we introduce a notion of relative flatness, which is helpful for the study of local homology. Section 2.7 contains some remarks on torsion modules and relatively injective modules.

The Formal Power Series Koszul Complex

In his book Matsumura proved the following Theorem (see Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 320 p., 1986 [58, Theorem 8.12]).

Complements and Applications

Here we will add some extensions and applications to the previous chapters. We present various aspects of local homology and cohomology, some more properties of the completion and torsion functors. We start with the composites of the derived functors of the completion and the torsion. Their interplay with the Hom-functors and the tensor product is also considered. In the second section there are results about duality and adjointness linking local homology and local cohomology with respect to an ideal generated by a weakly pro-regular sequence. It also provides a first version of local duality. In the third section we prove results about the endomorphism complex of \({\check{C}}_{\underline{x}}\) and related objects. Then the Mayer–Vietoris sequences for local homology and local cohomology are proved. We also consider Mayer–Vietoris sequences for Cech homology and local cohomology. There are also applications of the classes \(\mathcal {C}_{\mathfrak {a}}\) and \(\mathcal {B}_{\mathfrak {a}}\) to unbounded complexes. Homologically complete and cohomologically torsion complexes are studied in Sect. 9.6. This is completed in Sect. 9.7 by studying the cosupport. In the final section there are change of rings theorems.

Čech Complexes, Čech Homology and Cohomology

Cech complexes are important tools in various fields of Mathematics, in particular in Algebraic Geometry and Commutative Algebra. In Commutative Algebra the Cech complex is known for its relation to local cohomology in the case when the underlying ring is Noetherian. Here we start with a general investigation of the construction of the Cech complex with respect to a sequence of elements \(\underline{x}= x_1,\ldots , x_k\) of a commutative ring R. We investigate Cech homology and cohomology and prove the Ext-depth Tor-codepth sensitivity of the Cech complex as well as some inequalities. One of the new features here is the general assumption of a finite set of elements in a commutative ring R and unbounded R-complexes X.