{'fraction_non_alphanumeric': 0.06735576178338568, 'fraction_numerical': 0.017349424737012575, 'mean_word_length': 3.5950562961045684, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 29, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 28, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Tilting classes over commutative rings

We classify all tilting classes over an arbitrary commutative ring via certain sequences of Thomason subsets of the spectrum, generalizing the classification for noetherian commutative rings by Angeleri-Posp\'i\v{s}il-\v{S}\v{t}ov\'i\v{c}ek-Trlifaj. We show that the n-tilting classes can be equivalently expressed as classes of all modules vanishing in the first n degrees of one of the following homology theories arising from a finitely generated ideal: Tor_*(R/I,-), Koszul homology, \v{C}ech homology, or local homology (even though in general none of those theories coincide). Cofinite-type n-cotilting classes are described by vanishing of the corresponding cohomology theories. For any cotilting class of cofinite type, we also construct a corresponding cotilting module, generalizing the construction of \v{S}\v{t}ov\'i\v{c}ek-Trlifaj-Herbera. Finally, we characterize cotilting classes of cofinite type amongst the general ones, and construct new examples of n-cotilting classes not of cofinite type, which are in a sense hard to tell apart from those of cofinite type.

Infinitely generated tilting and cotilting modules were introduced in [CT95, AHC01] about two decades ago as a formal generalization of Miyashita tilting modules [Miy86]. The main motivation for studying Miyashita tilting modules is that they represent equivalences of derived categories. In the last few years, it became clear that infinitely generated modules represent derived equivalences as well (see [Baz10,BMT11,Š14,FMS15,PŠ17]), but also that there is more than that.
In the realm of commutative noetherian rings, it was shown [AHPŠT14] that tilting modules have a very close relation to the underlying geometry of the corresponding affine schemes. In fact, a full classification of tilting modules up to additive equivalence in terms of geometric data was obtained there. From a wider perspective, this was explained by Angeleri and Saorín [AHS14] who exhibited a direct relation of the dual cotilting modules to t-structures of the derived category, and the resulting t-structures were known to have a similar classification from [ATJLS10]. The outcome was that in the commutative noetherian setting, cotilting modules represent a class of t-structures (namely those, which are compactly generated and induce a derived equivalence to the heart).
The motivating question of the present work is, what remains true for commutative rings which are not necessarily noetherian. A posteriori, the general setting forced us to look for more conceptual proofs and gave more insight in the problem even for noetherian rings. Here is a short overview of highlights of the paper: (1) For a commutative ring R, we give a full classification of tilting R-modules up to additive equivalence in terms of filtrations of Thomason subsets of Spec R (Theorem 6.2).
(2) We obtain more insight in R-modules which are perfect in the derived category as well as in resolving subcategories formed by such modules, in that every such resolving subcategory is generated by syzygies of Koszul complexes (this follows from Theorem 6.2(iv) combined with Proposition 5.12).
(3) We observe that tilting modules up to additive equivalence are also classified by vanishing of the derived completion functors as well as by vanishing of theČech homology (Theorem 7.7). (4) We give a construction of dual cotilting modules to tilting modules (Theorem 8.8) and present some intriguing examples in the last section.
A starting point for the classification in this paper (as well as for its predecessor [Hrb16] which focused on (co)tilting modules of homological dimensions at most one) is the 1997 paper of Thomason [Tho97]. There he generalized Neeman-Hopkins's classification of thick subcategories of the derived category of perfect complexes over a commutative noetherian ring to a general commutative ring. The parametrizing family for this classification is the set of all sets of the spectrum of the ring, which are open with respect to the Hochster dual of the usual Zariski topology. If the ring is noetherian, these Thomason sets coincide precisely with the specialization closed subsets (that is, upper subsets of Spec R with respect to set-theoretic inclusion). It turns out that the main classification theorem for tilting modules and classes induced by them in [AHPŠT14] (but not its proof) remains valid in the non-noetherian setting provided that we simply change 'specialization closed set' to 'Thomason set' in the statement.
As was the case in the noetherian case, it is convenient to carry out the main steps of the proof in the dual setting of cotilting modules and cotilting classes induced by them first, and then transfer the results via elementary duality. The reason why the dual setting is more graspable seems to lie in the fact that, over a commutative ring, the cotilting classes dual to a tilting class are closed under injective envelopes (Proposition 5.5). In homological dimension one, this already leads to the wellunderstood theory of hereditary torsion pairs, as explained in [Hrb16]. We do not know of any analogous closure property for tilting classes.
In other respects, our approach differs considerably from [AHPŠT14]. For example, we cannot use the Matlis theory of injectives, the classical theory of associated primes or the Auslander-Bridger transpose anymore. Instead, we use the classification of hereditary torsion pairs of finite type from [Hrb16] and prove directly that a cotilting class is described cosyzygy-wise by a sequence of such torsion pairs (Lemma 5.8).
Another problem we have to tackle is that tilting classes bijectively correspond to resolving subcategories of modules having finite projective resolutions by finitely generated projectives (in the sense of Theorem 4.2). The description of such resolving subcategories in the noetherian case was obtained, independently of [AHPŠT14], by Dao and Takahashi [DT15]. It turns out that a sufficiently rich supply of Rmodules with such a finite resolution, for any commutative ring R, comes from tails of classical Koszul complexes associated with finite sets of elements of R. The key point here is to understand when exactly such tails are exact, i.e. when high enough Koszul homologies of R vanish.
Apart from the homological description of tilting classes which was obtained for noetherian rings in [AHPŠT14], i.e. in terms of vanishing of certain degrees of Tor * (R/I, −), we also obtain another one in terms of vanishing of the Koszul homology. We proceed further here to show that two other homology theories also fit in this place-the local and theČech homology associated to a finitely generated ideal I. The interesting point is that, unless the ring is noetherian, these two homologies need not be isomorphic. However, the vanishing of the first n homologies of a module is always equivalent for all of the four homology theories in play. The analogous result holds for the dual setting too for more classical local andČech cohomologies.
The paper is organized as follows. Section 2 gathers the required results about hereditary torsion pairs of commutative rings, focusing on those of finite type. In Section 3 we prove the key Proposition 3.9 which shows that vanishing of Ext i R (R/I, −) for i = 0, 1, . . . , n − 1 is equivalent to vanishing of the corresponding Koszul cohomology for any commutative ring. We link this vanishing property with the notion of vaguely associated prime ideal, yielding results which are analogous to characterizations of the grade of a finitely generated module over a noetherian ring. The fourth section recalls the fundamentals of the theory of large tilting modules over an arbitrary ring. These three sections prepare the ground for the core Section 5, in which we classify the n-cotilting classes of cofinite type over a commutative ring. These results are then reformulated and translated for the tilting side of the story in Section 6. The aforementioned connection with the derived functors of torsion and completion, as well as theČech (co)homology, is explained in Section 7. In the following Section 8, we show how a cotilting module associated to any cotilting class of cofinite type over a commutative ring can be constructed, building on the idea from [ŠTH14]. In the final Section 9, we characterize the cofinite-type cotilting classes amongst the general ones, and we construct new examples of ncotilting classes which are not of cofinite type, but which are in a sense difficult to tell apart from cofinite type cotilting classes.

Torsion pairs over commutative rings
In this section we give a classification of hereditary torsion pairs of finite type over commutative rings by Thomason subsets of the Zariski spectrum. This will be a key tool further in the paper. The material is probably known or not difficult to see for experts and almost all the fragments of the classification are present in the literature, but we have not been able to find a convenient reference.
Regarding our notation, R will always stand for an associative and unital ring and Mod-R for the category of right R-modules. We will always assume that R is also commutative unless stated otherwise. If I ⊆ R is an ideal, we denote by the corresponding Zariski closed set. We begin our discussion with so-called Hochster duality of spectral topological spaces and Thomason sets.
2.1. Spectral spaces, Hochster duality, and Thomason sets. The set of Thomason subsets of Spec R was used as an index set in Thomason's classification [Tho97] of thick subcategories of the category of perfect complexes. Although Thomason sets already appear in the work of Hochster [Hoc69], their name is customary nowadays since they were used in connection with other classification problems (see e.g. [KP17] and the references therein) and, in particular, with tilting theory in [Hrb16]. Let us start the discussion with a definition, which is relatively simple: Definition 2.1. Given a commutative ring R, a subset X ⊆ Spec R is a Thomason set if it can be expressed as a union of complements of quasi-compact Zariski open sets. That is, there exists a collection G of finitely generated ideals of R such that To put this into the right context, one should note that Thomason  In fact, Hochster [Hoc69] proved that spectral spaces are up to homeomorphism precisely the ones of the form Spec R for a commutative ring R. If X is a spectral topological space, the collection of quasi-compact open sets with the operations of the set-theoretic union and intersection is a distributive lattice. In fact, X is fully determined by this distributive lattice and every distributive lattice arises like that. Recall that an ideal I in a lattice L is a prime ideal if x ∧ y ∈ I implies x ∈ I or y ∈ I.  Given a Thomason set X ⊆ Spec R, we assign to it the filter of quasi-compact open sets {U | Spec R \ U ⊆ X}. Conversely, given a filter F , we assign to it X = U∈F (Spec R \ U ).

2.2.
Hereditary torsion pairs of finite type. A torsion pair in Mod-R is a pair of classes (T , F ) of modules such that Hom R (T, F ) = 0 for each T ∈ T and F ∈ F and for each X ∈ Mod-R there exists a short exact sequence 0 → T → X → F → 0. This short exact sequence is unique up to a unique isomorphism. The class T is called a torsion class and torsion classes of a torsion pair T are characterized by the closure properties-they are closed under coproducts, extensions and factor modules. Dually, torsion-free classes F are characterized as those being closed under products, extensions and submodules.
A torsion pair is hereditary if T is closed under submodules. Equivalently, F is closed under injective envelopes, [Ste75,§VI.3].
We will be mostly interested in the case when the torsion class is generated by a set of finitely presented modules. That is, F = Ker Hom R (T 0 , −) for a set of finitely presented modules T 0 . Although all the concepts can be defined over any ring, commutative or not, the following is special in the commutative case.
Proposition 2.6. Let R be a commutative ring and (T , F ) be a torsion pair such that F = Ker Hom R (T 0 , −) for a set of finitely presented modules T 0 . Then (T , F ) is hereditary.
A hereditary torsion pair is determined by the class of cyclic torsion modules, or equivalently by the set of ideals I for which R/I is torsion, [Ste75, Proposition VI.3.6]. Such a set of ideals corresponding to a hereditary torsion pair is called a Gabriel topology. We refer to [Ste75,§VI.5] for the general definition of a Gabriel topology, while a much simpler special case will be discussed below in Lemma 2.9.
Proposition 2.7. Let R be a commutative ring. There is a bijective correspondence between hereditary torsion pairs (T , F ) in Mod-R and Gabriel topologies G on R. Moreover, the following are equivalent for a hereditary torsion pair (T , F ): (1) There is a set T 0 of finitely presented modules with F = Ker Hom R (T 0 , −).
(2) F is closed under direct limits.
(3) The corresponding Gabriel topology G has a basis of finitely generated ideals (i.e. each ideal in G contains a finitely generated ideal in G).
Definition 2.8. The torsion pairs satisfying the conditions (1)-(3) above will be called hereditary torsion pairs of finite type.
If G is a Gabriel topology on R, we denote by G f the collection of all finitely generated ideals in G. It is always closed under products of ideals (i.e. if I 1 , I 2 ∈ G, then I 1 ·I 2 ∈ G) and finitely generated overideals. If Propositions 2.7(3) holds, G f completely determines G and the two latter closure properties in fact completely characterize such Gabriel topologies.
Lemma 2.9 ([Hrb16, Lemma 2.3]). Let R be a commutative ring. Then a filter G of ideals of R with a basis of finitely generated ideals is a Gabriel topology if and only if it is closed under products of ideals.
The following correspondence is a consequence of standard commutative algebra.
if and only if I n ⊆ I ′ for some n ≥ 1. Since also U I1·I2 = U I1 ∩ U I2 , the correspondence which assigns to a Gabriel topology G as in (1) the filter {U I | I ∈ G} of the lattice of quasi-compact open sets is a bijective.
If we combine the discussion above with Corollary 2.5, we obtain the following parametrization of hereditary torsion pairs of finite type (see [AHH16,Hrb16] for closely related results and compare also to [Ste75, Proposition VI.6.15]). Here, if M ∈ Mod-R, Supp(M ) = {p ∈ Spec R | M p = 0} denotes as usual the support of M .
Proposition 2.11. Given a commutative ring R, there is a bijective correspondence between (1) hereditary torsion pairs (T , F ) of finite type, and (2) Thomason subsets of Spec R. Given a torsion pair (T , F ), we assign to it the subset X = T ∈T Supp(T ). Conversely, if X is a Thomason set, we put Proof. The fact that there is a bijective correspondence between (1) and (2) follows by combining Proposition 2.7, Lemma 2.10 and Corollary 2.5. The particular correspondence given by the three statements is rather explicit. To (T , F ) we assign the Thomason set X of all prime ideals p ∈ Spec R for which there exists a finitely generated ideal I ⊆ p with R/I ∈ T . Conversely, to a Thomason set X we assign the unique hereditary torsion pair of finite type such that, given a finitely generated ideal I, R/I is torsion precisely when V (I) ⊆ X.
Suppose now that (T , F ) is of finite type and X is the corresponding Thomason set. Since every object in T is an epimorphic image of a coproduct of cyclic modules contained in T , we have X = R/I∈T Supp(R/I) = T ∈T Supp(T ), as required. Next let T ′ = {T ∈ Mod-R | Supp(T ) ⊆ X}. This is clearly a hereditary torsion class and, given a finitely generated ideal I ⊆ R, we have R/I ∈ T ′ if and only if Supp R/I ⊆ X. As Supp R/I = V (I) in this case, T ′ is also sent to X under the bijective correspondence, and hence T = T ′ .
We conclude the discussion with a description of the torsion-free classes under the correspondence from Proposition 2.11. This was important in the classification of tilting classes in the noetherian case [AHPŠT14] and the current version comes from [Hrb16].  Proof. This has been proved in [Hrb16]. Namely, given p ∈ Spec R and a finitely

Generalized grade of a module
A very important concept in homological algebra for modules over commutative noetherian rings is the one of a regular sequence and of the grade of a module. The maximal length of a regular sequence in a given ideal has various homological characterizations. Appropriate forms of these characterizations still remain equivalent over non-noetherian commutative rings which will be useful for us. We shall give details in this section.
We shall use the following notation here. Given M ∈ Mod-R and i ≥ 0, we denote by Ω i (M ) the i-th syzygy of M (uniquely determined only up to projective equivalence). If i < 0, we let Ω i M stand for the minimal |i|-th cosyzygy of M (determined uniquely as a module).
3.1. Derived categories and truncation of complexes. In this section it will also be useful to argue using the derived category D(Mod-R) of Mod-R; see for instance [KS06,Chapter 13]. The suspension functor will be denoted by and we will typically use the homological indexing of components of complexes: In this context, we will also use the homological truncation functors with respect to the (suspensions of the) canonical t-structure on D(Mod-R) (see [BBD82, Examples 1.3.22] or [KS06, §12.3]). We shall slightly adapt our notation for it to be compatible with our homological indexing. Thus, given a complex X ∈ D(Mod-R) and n ∈ Z, we shall denote by X ≥n the complex The subcomplex inclusion i : X ≥n : → X (when we consider the complexes in the usual category of complexes C(Mod-R)) induces an isomorphism on the k-th homology for each k ≥ n and H k (X ≥n ) = 0 for all k < n. Dually, the projection morphism p : X → X/X ≥n induces an isomorphism on the k-th homology for all k < n and H k (X/X ≥n ) = 0 whenever k ≥ n.
In fact, (X ≥n | n ∈ Z) yields a filtration of X in C(Mod-R). Since the homologies of X ≥n /X ≥n+1 in degrees different from n vanish, we have for each n ∈ Z a distinguished triangle in D(Mod-R). This observation, which formalizes how a complex can be built by extension from its homology modules will be especially useful in §5.

3.2.
Koszul complexes and Koszul (co)homology. A particularly useful class of complexes here will be the so-called Koszul complexes. Here R will stand for a commutative ring.
. Given x ∈ R, we define the Koszul complex with respect to x, denoted K • (x), as follows by which we mean a complex concentrated in degree 1 and 0, with the only non-zero map R → R being the multiplication by x. Here we use the homological indexing (i.e. the differential has degree −1). More generally, given a finite sequence x = (x 1 , x 2 , . . . , x n ) of elements of R, we define the complex K • (x) as the tensor product In particular, K • (x) is a complex of finitely generated free R-modules concentrated in degrees 0 to n. Given an arbitrary module, we can define the corresponding Koszul homology and cohomology.
Definition 3.2. Given a finite sequence x = (x 1 , x 2 , . . . , x n ) of elements of R, a module M ∈ Mod-R, and i ∈ Z, the i-th Koszul homology of x with coefficients in M is defined as Remark 3.3. The Koszul cohomology has a particularly easy interpretation in the derived category D(Mod-R). Namely, we have Typically we will start with a finitely generated ideal I = (x 1 , . . . , x n ) of R and we will consider the Koszul complex or homology or cohomology of x = (x 1 , x 2 , . . . , x n ). These notions are not invariant under change of the generating set of I-see [BH93, Proposition 1.6.21] for a precise discussion on how the complex changes. However, the following consequence of this discussion will be crucial for us and will make the particular choice of a finite generating set of I unimportant. Proof. This is an immediate consequence of [BH93, Corollary 1.6.22] and [BH93, Corollary 1.6.10(d)].
Hence, we introduce the following slightly abused notation.
Notation 3.5. Given a commutative ring R and a finitely generated ideal I, we always fix once and for ever a system of generators x = (x 1 , x 2 , . . . , x n ). We then say that K • (x) is the Koszul complex of the ideal I and denote it by K • (I).
Similarly for M ∈ Mod-R and i ∈ Z we define the i-th Koszul homology of I with coefficients in The following two well-known properties of Koszul complexes which will be important in our application.  In fact, the latter lemma allows us to relate the Koszul cohomology of I to ordinary Ext-groups. Namely, we have a map of complexes . This map is natural in M and compares the Ext-group to the Koszul cohomology with coefficients in M (see also [BH93, Proposition 1.6.9]). It is easy to see that q 0 M is always an isomorphism, but for i > 0 the relation is more complicated and will be studied in the next subsection. To get a quick impression of the potential difficulties, note that H i (I; M ) always commutes with direct limits, while Ext i R (R/I, −) need not if R is not a coherent ring.
3.3. More on the relation between the Koszul cohomology and Ext groups. Our strategy will be to try to quantify the difference between K • (I) and R/I in the derived category. We start with an easy lemma.
Proof. Since N is an R/I-module, there is an exact sequence The lemma is then proved by applying Hom R (−, M ) on this sequence and a straightforward induction on k, using the fact that K is also an R/I-module.
The last result extends in a straightforward way to the derived category.
Lemma 3.8. Let I be a finitely generated ideal and suppose that M is an R-module Proof. Using the notation from §3.1 for truncations of complexes, we first recall the well-known fact that Hom D(Mod-R) (X ≥1 , M ) = 0; see [KS06, Propositions 3.1.8 and 3.1.10]. Since by our assumption The latter lemma implies that the comparison map in (3) in §3.2 between the i-th Koszul cohomology of I and Ext i R (R/I, M ) is an isomorphism under certain assumptions.
Proposition 3.9. Let R be a commutative ring and I be a finitely generated ideal of R. Suppose that M is a module such that M ∈ Proof. Since the two rightmost terms F 1 is always an isomorphism. This proves the lemma for n = 0.
If n > 0, we consider the triangle in D(Mod-R) induced by the short exact sequence of complexes in the first three terms (see also (2) in §3.2). If we apply Hom D(Mod-R) (−, Σ n M ), we obtain an exact sequence of abelian groups Since all the homologies of K • (I) ≥1 are R/I-modules, the leftmost and the rightmost term vanish by Lemma 3.8 provided that M ∈ n−1 i=0 Ker Ext i R (R/I, −). Hence q n M is an isomorphism in such a case.
An immediate but particularly useful consequence is the following identification.
Corollary 3.10. If R is a commutative ring, then as subcategories of Mod-R for each finitely generated ideal I and n ≥ 0.
Remark 3.11. The dual versions of results from §3.3 also hold, and will be used in §7. Explicitly, for each finitely generated ideal I and n ≥ 0 we have Ker H i (I; −), and for any module M belonging to this class, there is a natural isomorphism q M n : H n (I; M ) → Tor R n (R/I, M ), obtained by applying H n (− ⊗ L R M ) onto (2). This can be proven either directly using similar arguments as in this section, or it follows by using the elementary duality (−) + (see §4 for the definition). Indeed, using the Hom-⊗-adjunction and exactness of the elementary duality, we have for any M ∈ It is straightforward to check that the same properties ensure that (q M n ) + is naturally equivalent to q n M + , which is an isomorphism by Proposition 3.9. Since (−) + is exact and faithful, we conclude that q M n is an isomorphism.
3.4. Vaguely associated primes revisited. Now we give another way to express the very same class as in Corollary 3.10 by giving a homological generalization of Proposition 2.13. We start with an easy observation Lemma 3.12. Let R be a commutative ring, I be a finitely generated ideal. Then F = Ker Hom R (R/I, −) is a hereditary torsion-free class of finite type whose corresponding Thomason set (in the sense of Proposition 2.11) is V (I).
Proof. By Lemma 2.9, the smallest Gabriel topology G containing I consists of the ideals J such that I m ⊆ J for some m ≥ 1. The corresponding cyclic module R/J then admits a filtration by R/I-modules of length m. Hence the smallest torsion class T containing R/I coincides with the smallest torsion class containing G. In particular, (T , F ) is a hereditary torsion pair. The fact that it corresponds to the Thomason set V (I) follows from the proof of Proposition 2.11. Now can we state and prove the promised result.
Proposition 3.13. Let R be a commutative ring, I be a finitely generated ideal, M be an R-module and n ≥ 1. Then the following are equivalent: Proof. We prove the proposition by induction on n. Let F = Ker Hom R (R/I, −). The statement for n = 1 is precisely Proposition 2.13. Suppose now that n > 1, and consider the exact sequence where E is the injective envelope of Ω −(n−2) (M ). An application of Hom R (R/I, −) yields an exact sequence so Ω −(n−2) (M ) ∈ F by Proposition 2.13. Since F is a hereditary torsion-free class, also E ∈ F and the leftmost term in (4) vanishes. As also Ext n−1 R (R/I, M ) = 0 by the assumption, we have Hom R (R/I, Ω −(n−1) (M )) = 0 and VAss Ω −(n−1) (M ) ∩ V (I) = ∅ as required.
3.5. Characterizations of grade. Now we are in a position to state and prove the main result of the section. As the definition of the grade of a module is specific to the noetherian situation and will not be so important for the rest of the paper, we only refer to [BH93, § §I.1.1-I.1.2] for the corresponding standard definitions. The equivalences below are well-known under additional finiteness conditions (R noetherian, M finitely generated)-see for instance [BH93, Theorems I.2.5 and I.6.17].
Theorem 3.14. Let R be a commutative ring, M be an R-module and n ≥ 1. Then the following are equivalent: If, moreover, R is noetherian and M finitely generated, the statements are further equivalent to: (4) The grade of I on M is at least n.

Infinitely generated tilting theory
At this point, we quickly recall basic terminology and facts about module approximations and cotorsion pairs, two essential tools of the forthcoming sections. We also remind the reader of the notion of (not necessarily finitely generated) ntilting and n-cotilting module, as defined by [CT95] and [AHC01], and the duality between those two. 4.1. Module approximations. We briefly recall the definitions of (pre)covers and (pre)envelopes of modules. Let C be a class of right R-modules, and M ∈ Mod-R. We say that a map f : C → M is a C-precover of M provided that C ∈ C, and for any C ′ ∈ C the map Hom R (C ′ , f ) is surjective. Furthermore, if any map g ∈ End R (C) such that f = f g is necessarily an automorphism, we say that f is a C-cover. Finally, a surjective map f : C → M is called a special C-precover if C ∈ C and Ker(f ) ∈ ⊥1 C. It is easy to see that any special C-precover is a C-precover. Also, by the Wakamatsu Lemma ([GT12, Lemma 5.13]), any surjective C-cover is a special C-precover. Finally, we say that a class C is (special) (pre)covering, if any module M ∈ Mod-R admits a (special) C-(pre)cover.
The notions of C-preenvelope, C-envelope, and special C-preenvelope are defined dually.
4.2. Cotorsion pairs. Given a class C ⊆ Mod-R, we fix the following notation: 4.3. Tilting and cotilting modules and classes. Let T be a right R-module and n ≥ 0. We say that T is n-tilting if the following three conditions hold: (T1) pd T ≤ n, (T2) Ext i R (T, T (X) ) = 0 for all i > 0 and all sets X, (T3) there is an exact sequence 0 → R → T 0 → · · · → T n → 0, where T i is a direct summand of a direct sum of copies of T for all i = 0, 1, . . . , n.
A module T is tilting if it is n-tilting for some n ≥ 0. An n-tilting module T induces a hereditary and complete cotorsion pair (A, T ) = ( ⊥1 (T ⊥ ), T ⊥ ). The class T is called an (n-)tilting class. Two tilting modules T, T ′ are equivalent if they induce the same tilting class. Even though the tilting modules in our setting are in general big (indeed, over a commutative ring, any finitely generated tilting module is projective), the tilting classes arise from small modules in the following sense. Let mod-R denote the full subcategory of Mod-R consisting of strongly finitely presented modules, that is, modules having finite projective resolution consisting of finitely generated projectives. A full subcategory S of mod-R is called resolving if it contains all finitely generated projectives, is closed under extensions and direct summands, and A ∈ S whenever there is an exact sequence There is a bijective correspondence between n-tilting classes T and resolving subcategories S of mod-R consisting of R-modules of projective dimension ≤ n. The correspondence is given by mutually The cotilting modules have a formally dual definition -a left R-module C is (n-)cotilting if the following conditions hold: a direct summand of a direct product of copies of T for all i = 0, 1, . . . , n, and W is an injective cogenerator of R-Mod. As for the tilting modules, a cotilting module C induces a cotilting class C = ⊥ C, and two cotilting modules C, C ′ are equivalent if they induce the same cotilting class. There is also an explicit duality between tilting and cotilting modules. If R is a k-algebra over a commutative ring k (e.g. k = Z), we denote by (−) + = Hom k (−, E) the duality with respect to an injective cogenerator E of k -Mod. Then for any right tilting R-module T , the dual T + is a left cotilting R-module. We say that a tilting class T in Mod-R and cotilting class C in R-Mod are associated if there is a tilting module T inducing T such that T + induces C.
It is not true that every cotilting class is associated to a tilting class. We say that a class C is of cofinite type provided that there is a set of strongly finitely presented modules S of projective dimension bounded by n such that C = S ⊺ . An example of a cotilting class not of cofinite type was first exhibited in [Baz07]. In §9, we show a more general construction of such classes.
All tilting and all cotilting classes are definable (i.e., closed under pure submodules, direct products, and direct limits). Furthermore, a pair of associated tilting class and cotilting class is dual definable in the following sense: Lemma 4.4. Let R be a ring, T a tilting class in Mod-R, and C the cotilting class of cofinite type in R-Mod associated to T . Then for any M ∈ Mod-R, and N ∈ R-Mod: Proof. The proof is the same as that of [AHH16, Lemma 3.3]. As T is of finite type, there is a set S ⊆ mod-R such that T = S ⊥ . By [GT12,Theorem 15.19, and the paragraph following it], also C = S ⊺ . Using the Hom-⊗ adjunction, exactness of (−) + , and [EJ00, Theorem 3.2.10], we have Then we can apply (ii) tho obtain (i).
Finally, as in [AHPŠT14], we fix the following notation for cotilting classes of lower dimensions induced by a cotilting class, which will be useful for arguing by induction on the dimension.
Notation 4.5. Given an n-cotilting class C induced by a cotilting module C, we let

Cotilting classes of cofinite type
In this section we classify cotilting classes of cofinite type over a commutative ring. The parametrizing set for this classification will consist of sequences of torsion-free classes of hereditary torsion-free pairs of finite type satisfying some extra conditions. Using results from Section 2, these classes are in bijective correspondence with certain Thomason sets, and thus generalize in a direct way the parametrizing sets used in the noetherian case in [AHPŠT14].
Definition 5.1. Let R be a commutative ring and n ≥ 0. We say that a sequence of torsion-free classes S = (F 0 , F 1 , . . . , F n−1 ) is a characteristic sequence (of length n) if (i) F i is hereditary and of finite type for each i = 0, 1, . . . , n − 1, Notation 5.2. Given a characteristic sequence S = (F 0 , F 1 , . . . , F n−1 ) and i = 0, 1, . . . , n − 1 we put F i (S) = F i , and we define a class Our goal in this section is to prove the following: Theorem 5.3. Let R be a commutative ring and n ≥ 0. Then the assignment is a bijection between the set of all characteristic sequences of length n and all n-cotilting classes in Mod-R of cofinite type.
The proof of Theorem 5.3 will be done in several steps throughout this section, by proving subsequently that Ψ is surjective, well-defined, and injective.

Ψ is surjective.
Lemma 5.4. Let R be a commutative ring, M ∈ Mod-R, and P a finitely generated projective R-module.
Proof. Let P ′ ∈ Mod-R be a finitely generated projective module such that P ⊕P ′ ≃ R n for some n ∈ ω. As E(M ) n is injective, the map R n ⊗ R ι = ι n is the injective envelope of M n by [AF92, Proposition 6.16(2)]. As Proposition 5.5. Let R be a commutative ring and C be a cotilting class in Mod-R. If C is of cofinite type, then it is closed under injective envelopes.
Proof. Fix a module M ∈ C, and let us show that E(M ) ∈ C, provided that C is of cofinite type. Under this assumption, there is a set S of strongly finitely presented modules of projective dimension bounded by n such that C = S ⊺ . We proceed by induction on n. If n = 0 the claim is clear as C = Mod-R; henceforth assume that we proved the claim for all k < n. Pick S ∈ S and fix its projective resolution consisting of finitely generated projectives. Tensoring the complex P • with the injective envelope ι : M → E(M ) yields a commutative diagram in Mod-R (this is where we use the commutativity of R): (In the case of n = 1 we put P 2 = 0.) Since P • consists of projective modules, the columns are exact. We have there is nothing to prove. Otherwise, since P 1 ⊗ R ι is the injective envelope of P 1 ⊗ R M by Lemma 5.4, P 1 ⊗ R ι is an essential monomorphism. Therefore, the module is non-zero, and thus essential in Ker(σ 1 ⊗ R E(M )). It follows that Im(σ 2 ⊗ R E(M )) is essential in Ker(σ 1 ⊗ R E(M )). Now we use the induction hypothesis, which implies that Tor R k (S, E(M )) ≃ Tor R 1 (Ω k−1 S, E(M )) = 0 for all k > 1. This means that H k (P • ⊗ R E(M )) = 0 for all k > 1, and as this complex consists of injectives and is left-bounded, the map (σ 2 ⊗ R E(M )) is a split monomorphism. The only case for which this is not a contradiction is when Im(σ 2 ⊗ R E(M )) = Ker(σ 1 ⊗ R E(M )), showing that 0 = H 1 (P • ⊗ R E(M )) = Tor R 1 (S, E(M )), proving finally that E(M ) ∈ C.
Given a fixed module C, we can assign to any module X the canonical map η X : X → C HomR(X,C) = C X . This map is in fact (covariantly) functorial, as we recall in the following Lemma.
Lemma 5.6. The map η X is functorial in X. That is, given any map Proof. For any β ∈ Hom R (Y, C), let π β : C Y → C be the projection onto the β-th coordinate. We define η f by the following rule: For any c = (c α ) α∈HomR(X,C) ∈ C X , and any β ∈ Hom R (Y, C), we let It is easy to see that η f is an R-module homomorphism. Also, for any β ∈ Finally, the equality η g η f = η gf can be checked by direct computation from (5).
Lemma 5.7. Let R be a ring and let C be an cotilting class in R-Mod closed under injective envelopes. Then there is a hereditary faithful torsion-free class of finite type F such that Proof. Let C be the cotilting module associated to C. Applying Hom R (−, C) to the exact sequence 0 → M → E(M ) → Ω −1 M → 0, and using that C is closed under injective envelopes by Proposition 5.5, we infer that M ∈ C if and only if E(M ) ∈ C and Ω −1 M ∈ C (1) .
We let F be the closure of C under submodules. As C is closed under injective envelopes, injectives of C and F coincide. From this it is easy to infer that C = {M ∈ Mod-R | M ∈ F & Ω −1 M ∈ C (1) }. We are left to show that F is a hereditary faithful torsion-free class of finite type. It is easy to check that R ∈ F , and that F is closed under submodules, injective envelopes, extensions, and products.
It remains to show that F is closed under direct limits. Note that F = Cogen(C). Let (X i ) i∈I be a directed system of modules from F . As X i is cogenerated by C, the canonical map η Xi : X i → C Xi is monic. Using the functoriality proved in Lemma 5.6, we actually have a directed system (X i → C Xi ) i∈I of monic maps. Taking the direct limit yields a monic map lim − →i∈I X i → lim − →i∈I C Xi . As C is definable, the latter direct limit is in C, proving that lim − →i∈I X i is indeed in F .
Lemma 5.8. Let R be a commutative ring and C be an n-cotilting class in Mod-R of cofinite type. Then there is a characteristic sequence (F 0 , F 1 , . . . , F n−1 ) such that C = {M ∈ Mod-R | Ω −i M ∈ F i for each i = 0, 1, . . . , n − 1}. In particular, map Ψ is surjective.
Definition 5.9. Let S be a characteristic sequence of length n. We let G i (S) denote the Gabriel topology associated to the torsion-free class F i (S) in the sense of Proposition 2.7 for each i = 0, 1, . . . , n − 1.
In particular, G i (S) is a Gabriel topology of finite type, and Ker Hom R (R/I, −).
Lemma 5.10. Let S be a characteristic sequence of length n. Then Proof. Let S = (F 0 , F 1 , . . . , F n−1 ). We prove by induction on 0 < k ≤ n that In what follows, we denote by (−) * the regular module duality Hom R (−, R).
Definition 5.11. Let I be a finitely generated ideal, and let us denote the Koszul complex K • (I) as follows where F k is in degree k for all k = 0, 1, 2, . . . , n + 1. We denote by S I,k the cokernel of the map d * k . That is, S I,k is (up to stable equivalence) the Auslander-Bridger transpose of Coker(d k ). Proof. Let us adopt the notation for K • (I) as in Definition 5.11. Let 0 < k ≤ n. Applying (−) * to K • (I) we obtain complex which is exact by our assumption and Proposition 3.9. This proves (i).
Denote by C • the complex (6) with S I,k deleted, and F * k in the degree zero. Then C • is a projective resolution of S I,k , and thus H 1 (C • ⊗ R M ) ≃ Tor R 1 (S I,k , M ). But as F j is finitely generated projective for all j = 0, 1, . . . , n, we have by [AF92,Proposition 20.10] a natural isomorphism F * j ⊗ R M ≃ Hom R (F j , M ). Hence, using Proposition 3.9 we obtain Ker Ext i R (R/I, −). Now we prove (ii). Note first that S I,k is an (n − k)-th syzygy of S I,n . Hence, Tor R k (S I,n , M ) ≃ Tor R 1 (S I,n−k+1 , M ). Then (S I,n ) ⊺ = n k=1 Ker Tor R 1 (S I,k , −). A straightforward induction on k = 1, 2, . . . , n together with (7) proves that the latter class is equal to n k=1 Ker Ext k−1 R (R/I, −) as desired. Finally, (iii) follows directly from (ii) by Theorem 4.3.
Lemma 5.13. Let S be a characteristic sequence of length n, then C(S) is a ncotilting class of cofinite type. In particular, map Ψ is well-defined.
Proof. We have the following chain of equalities: The first equality is Lemma 5.10, the second one follows easily from G f i ⊇ G f i+1 for each i = 0, 1, . . . , n − 2, and the last one is an application of Proposition 5.12(ii). Then Theorem 4.3 yields the result.
Since cotilting classes contain all projectives modules, this long exact sequence yields Hom R (R/I, ΩM ) = 0, and Ext j R (R/I, ΩM ) ≃ Ext j−1 R (R/I, M ) for all j = 1, 2, . . . , i. Therefore, This concludes (i) by Lemma 5.10. (ii) It follows directly from the definition that Then (ii) is proved by (i) and a straightforward induction.
Lemma 5.15. Let S and S ′ be two characteristic sequences. If S = S ′ , then C(S) = C(S ′ ). In particular, map Ψ is injective.
, but E(M ) ∈ F 0 (S) proving the statement for i = 0. Let now i = 0, 1, . . . , n − 1 and suppose towards contradiction that C(S) = C(S ′ ). Then also C(S) (i) = C(S ′ ) (i) , but this is a contradiction using the case i = 0 and Lemma 5.14(ii). 5.4. The result. Now we are ready to prove our classification theorem.
Proof of Theorem 5.3. The assignment Ψ : S → S(S)) is a well-defined map from the set of all characteristic sequences of length n to n-cotilting classes of cofinite type by Lemma 5.13. This map is injective by Lemma 5.15 and surjective by Lemma 5.8.

Main classification results
In this section we rephrase Theorem 5.3 in terms of Thomason sets, and state our characterization of tilting classes over commutative rings.
Proof. Start with a cofinite-type cotilting class C, and let S = (F 0 , F 1 , . . . , F n−1 ) be the characteristic sequence corresponding to C in the sense of Theorem 5.3. Note that Ass(C (j) ) = Ass(F j ) for each j = 0, 1, . . . , n − 1. Indeed, one inclusion follows trivially, as C (j) ⊆ F j by Lemma 5.14, while the second inclusion follows from C (j) and F j being both closed under injective envelopes and having the same injectives.
The rest of the proof is a combination of Theorem 5.3, Proposition 2.11, and Proposition 2.13.
(ii) n-cotilting classes C in Mod-R of cofinite type, (iii) n-tilting classes T in Mod-R, (iv) resolving subcategories S of mod-R consisting of modules of projective dimension ≤ n. The correspondences are given as follows: Finally, by Theorem 4.2, resolving subcategories S as in (iv) correspond bijectively to tilting classes T via the assignment T → ( ⊥ T ) ∩ mod-R. Whenever T = S ⊥ for some set S ⊆ mod-R, we have that ⊥ T equals to all direct summands of all {R} ∪ S-filtered modules by Theorem 4.1, and thus ⊥ T ∩ mod-R equals to direct summands of all finitely {R} ∪ S-filtered modules. By (i) → (iii), we can chose S = {S I,i+1 | I ∈ G f i , i = 0, 1, . . . , n − 1}, establishing (i) → (iv). Remark 6.3. If the ring R is coherent, we can use a projective resolution of R/I in mod-R instead of the Koszul complex. Therefore, in this case S I,i+1 can be replaced by module Tr(Ω i R/I) for each i = 0, 1, . . . , n−1, where Tr is the Auslander-Bridger transpose (cf. [AHPŠT14]).
Example 6.4. Let R be a commutative perfect ring. Then the only tilting class in Mod-R is the trivial one, that is the whole Mod-R. By Theorem 6.2, it is enough to show that if I is a finitely generated ideal such that Hom R (R/I, R) = 0, then I = R. Indeed, since I is finitely generated, the descending chain (I n | n ∈ ω) stabilizes at some n (see [Lam13,Theorem 23.20, p. 345]). Then either I is nilpotent, and thus Hom R (R/I, R) = 0 unless I = R, or I n is idempotent. As I is finitely generated, the latter case implies that I n is a direct summand of R, and thus again Hom R (R/I, R) = 0 unless I = R.

Derived functors of torsion and completion andČech (co)homology
In Section 6, we characterized cofinite type n-cotilting classes over a commutative ring as classes of all modules which vanish in certain degrees of Koszul cohomologies, arising from a set of finitely generated ideals. In this section, we show that we can replace Koszul complexes by two kinds of more canonically defined cohomology theories associated to an ideal-Čech cohomology, and local cohomology. Our results are valid for a general commutative ring, even though the latter two cohomology theories do not in general coincide for non-noetherian rings. An analogous description of n-tilting classes viaČech homology and local homology will also be accomplished, after dealing with a few extra technical difficulties. The main sources we use in this section are [GM92,Pos16,PSY14,Sch03].
Throughout this section, let R be a commutative ring. A module M is said to be I-torsion if Γ I (M ) = M , and we denote the full subcategory of all I-torsion modules by T I . Then T I is an abelian category with exact direct sums, and the embedding T I ⊆ Mod-R is exact and clearly admits Γ I as is its right adjoint.
In particular, Γ I is left exact, and we can form the right derived functor RΓ I , called the local cohomology functor of I. The situation is a bit more tricky in the case of completion functors. Following Positselski [Pos16], we say that a module M is an I-contramodule provided that

L∆I
However, note that in general not even in the derived picture we can swap L∆ I for LΛ I (see [Pos16,Example 2.6]). This will be further discussed in §7.4. 7.2.Čech (co)homology. As discussed in §3.2, the Koszul complex K • (I) for a finitely generated ideal I is not a well-defined object even in the derived category, as the homology can change when passing from one generating system of I to another. This can be mended by stepping outside of the realm of perfect complexes and using generators of I to form aČech cochain complex (also called a stable Koszul complex ).
Definition 7.1. Let x be an element of R. TheČech complex with respect to x, denoted byČ • (x), which is defined as where R x = R[x −1 ], ι is the natural morphism, and the cochain complex is concentrated in (cohomological) degrees 0 and 1. Given a sequence x = (x 1 , x 2 , . . . , x n ) of elements of R, we defineČ • (x) as the tensor product n i=1Č • (x i ). Lemma 7.2 legitimizes the following notation: Given a finitely generated ideal I with a finite sequence of generators x, we denoteČ • (I) =Č • (x). The cochain complexČ • (I) is then well-defined as an object of the derived category.
Similar to Koszul complexes, we can computeČech cohomology and homology. First we address the former, defined as follows: is a bounded complex of flat modules, we can drop the left derivation symbol L from the formula. ThisČech cohomology can also be viewed as a limit version of Koszul cohomology in the following way. Given x ∈ R, the Koszul chain complexes (K • (x j ) | j > 0) form an inverse system with the following connecting maps: Dualizing this with respect to R, we obtain a direct system (K • (x j ) * | j > 0) of cochain complexes, and it is easy to check that its limit is preciselyČ • (x). As direct limit commutes with tensor product, we getČ • (I) = lim − →j>0 K • (x j ) * , where we fix the notation x j = (x j 1 , x j 2 , . . . , x j m ) for a set of generators x = x 1 of I. Given a module M , we infer from exactness of the direct limit functor the following isomorphism Using this representation, we can already show that theČech complexes classify the cofinite type cotilting classes (see also Theorem 7.7 below). In the following proofs, let always I j denote the ideal generated by the sequence x j (not to be confused with I j ).
By the same argument using Lemma 3.7 as above, we have Ext n−1 R (I j /I j+1 , M ) = 0. It follows that the map Ext n R (π, M ) is a monomorphism. From the construction and naturality of q n M , we infer that the there is a commutative diagram (14) where φ j is the j-th map from the direct system (H n (x j ; M ) | j > 0). Since the horizontal maps of (14) are isomorphisms, we finally infer that this direct system consists of monomorphisms, as desired.
Now we treat theČech homology, which we define, following [Sch03], in this way: Because this functor a priori inhabitates strictly the derived category, we would like to replaceČ • (I) by its projective resolution, in a way that respects the limit construction ofČ • (I). To this end, we recall the telescope cochain complex (here we follow [PSY14]). For any subset X of ω, let F [X] be the free R-module with basis {δ j | j ∈ X}. Given an element x ∈ R we let be the cochain complex concentrated in (cohomological) degrees 0 and 1, where the differential d is defined on the above basis as follows For any j > 0, we let be the subcomplex of Tel(x), so that Tel(x) = j>0 Tel j (x). More generally, given a sequence of elements x = (x 1 , x 2 , . . . , x n ) of R, we let Tel(x i ).
Note that again Tel(x) = j>0 Tel j (x). It follows from ([PSY14, Lemma 5.7]) that there are natural homotopy equivalences w x,j : Tel j (x) → K • (x j ) * , such that their limit map is a quasi-isomorphism. If I is the ideal generated by x, we can now represent thě Cech homology as follows: ). Of course, in general, taking homology does not commute with inverse limits. On the other hand, the inverse system (Tel j (x) * ⊗ R M | j > 0) consists of epimorphisms, and thus satisfies the Mittag-Leffler condition (see [GT12, Definition 3.5 and Lemma 3.6]). Using [Wei94, Theorem 3.5.8], we have for each i ≥ 0 the following exact sequence: Here, the symbol lim ← − 1 j>0 stands for the first right derived functor of the inverse limit functor lim ← −j>0 . Furthermore, because w * x,j : K • (x j ) → Tel j (x) * is also a natural homotopy equivalence of complexes, we can rewrite this sequence as: We are ready to prove thatČech complexes allow to classify tilting classes (see also Theorem 7.7 below).
Lemma 7.4. Let R be a commutative ring, I a finitely generated ideal, and n > 0. Proof. We proceed again by induction on n ≥ 0, and fix throughout the proof a module M ∈ n−1 i=0 Ker H i (I; −) = n−1 i=0 KerȞ i (I; −). Suppose first that H n (I; M ) = 0. By Remark 3.11, H i (x j ; M ) is naturally isomorphic to Tor R i (R/I j , M ) for each j > 0 and i = 0, 1, . . . , n+1. An argumentation analogous to the one in the proof of Lemma 7.3 then yields that H n (x j , M ) = 0 for each j > 0, and that the inverse system (H n+1 (x j ; M ) | j > 0) consists of epimorphisms. In particular, this system is Mittag-Leffler, and thus [GT12, Lemma 3.6] implies that lim ← − 7.3. Main theorem revisited. In this section, we show that instead of Ext/Tor or Koszul (co)homology, we can use either local (co)homology, orČech (co)homology, in the formulation of Theorem 6.2. We prove the remaining parts in the following Lemmas, and then state the alternative classification Theorem.
Lemma 7.5. Let R be a commutative ring and S = (F 0 , F 1 , . . . , F n−1 ) a characteristic sequence of length n. Then Proof. This follows directly from the definition of C(S) (see Notation 5.2). Indeed, if I ∈ G f m and 0 ≤ m < n, then a module M ∈ C(S) must be in the class Lemma 7.6. Let R be a commutative ring, I a finitely generated ideal, and n > 0. Then: Proof. The shape of the proof is the same as that of Lemma 7.4, using the exact sequence [GM92, Proposition 1.1] instead of (15). For the convenience of the reader, we provide details here. We proceed by induction on n ≥ 0 (the case of n = 0 is a vacuous statement). For the induction step, we will assume that M ∈ n−1 i=0 Ker L i Λ I . By [GM92, Proposition 1.1], there is an exact sequence: Tor R n (R/I j , M ) → 0.
If L n Λ I (M ) = 0, then rightmost term also vanishes. By the induction hypothesis we have M ∈ n−1 i=0 Ker Tor R i (R/I, −). Applying − ⊗ R M to the exact sequence 0 → I j /I j+1 → R/I j+1 → R/I j → 0, and noting that Tor R n−1 (I j /I j+1 , M ) = 0, we infer that the inverse system (Tor R n (R/I j , M ) | j > 0) consists of epimorphisms. This shows that Tor R n (R/I, M ) = 0, proving one inclusion. To prove the other inclusion, suppose now that Tor R n (R/I, M ) = 0. It follows easily that Tor R n (R/I j , M ) = 0 for all j > 0, and thus the rightmost term of (16) is zero. By repeating the same argument as above for a homological degree shifted by 1, we obtain that the inverse system (Tor R n+1 (R/I j , M ) | j > 0) consists of epimorphisms, and thus is Mittag-Leffler. Hence, the leftmost term of (16) also vanishes by [GT12, Lemma 3.6], and thus L n Λ I (M ) = 0, finishing the proof.
7.4. The big picture. As the four homological and four cohomological theories used in the classification Theorems 6.2 and 7.7 may feel a little overwhelming, we devote this and the next subsection to a short explanation of the relations between these. In particular we show that theČech (co)homology, analogously to the local (co)homology, also induces a pair of adjoint functors between derived categories, and that there are always comparison functors between theČech and local (co)homologies which are equivalences under a technical condition. Although in this material is not really new, it requires some effort to extract it from the existing literature [ATJLL97,GM92,PSY14,Sch03]. Here we especially rely on a recent and original presentation in [Pos16,Theorem 3.4]. In fact, all we want to say is essentially a reformulation of [Pos16, Theorem 3.4] and its proof, and we refer the reader to [Pos16] for a more comprehensive treatment. We will freely use the theory of localization of the derived category, as well as recollements and their translation to TTF triples. For these we refer to [BBD82], [Kra10], or [NS09]. Using the machinery of localization theory of triangulated categories (see [Kra10], namely [Kra10,4.9.1, 4.13.1, 5.3.1, 5.4.1, 5.5.1]) there is a recollement (we adopt the convention that going up amounts to taking left adjoints) In particular it is a localizing subcategory, and hence a tensor ideal by [KP17, Lemma 1.1.8]. Consider the triangle induced by the identity map R → R in degree 0. Let M be a complex and apply − ⊗ L R M to (19) in order to obtain a triangle R xi,xj → · · · → R x1,x2,...,xn .
Since Y is thick and a tensor ideal, it follows from 18 that Cone There is a technical condition on I, so-called weak proregularity of I, which ensures (and in fact is equivalent to) that both adjunctions on the left in (21) and (22) are in fact equivalences (see [Pos16, Theorem 1.3, Corollary 2.10]). We will discuss this in §7.5 Here we conclude by noting that in such a case, the local (co)homology coincides with theČech (co)homology. Furthermore, weak proregularity also implies that L∆ I is naturally equivalent to LΛ I ([Pos16, Lemma 2.5]) and, since D TI (Mod-R) and D CI (Mod-R) are always equivalent (cf. the recollement (17) or [Pos16, Theorem 3.4]), weak proregularity also implies that: The latter statement is known as the Matlis-Greenlees-May duality, [DG02,PSY14,Pos16]. 7.5. Weak proregularity. A classical result ( [Gro63]) says that, over a commutative noetherian ring, the local cohomology coincides with theČech cohomology. The dual result for the left derived completion functor andČech homology ([GM92, PSY14, Sch03, ATJLL97]) is a much more recent development. However, over a general commutative ring, the local andČech (co)homologies need not be the same, despite the fact that we can use either of them in Theorem 7.7 to classify (co)tilting classes. Here we gather relevant results to understand the issue. (1) An inverse system (M i , f ji | j ≥ i) of modules is pro-zero if for every i there is j ≥ i such that f ji is zero.
(2) Let R be x = (x 1 , x 2 , . . . , x n ) be a sequence of elements of a commutative ring R. We say that x is weakly proregular if the inverse system (H i (x j ; R) | j > 0) (see §7.1) is pro-zero for each i > 0. This legitimizes us to define a finitely generated ideal I to be weakly proregular, if any of its finite generating sequences is weakly proregular.
If R is noetherian, then any ideal is weakly proregular ([PSY14, Theorem 4.34]). On the other hand, over a general commutative rings, there can easily be nonweakly proregular finitely generated ideals ([GM92, Example 1.4]). It turns out that this property characterizes precisely when the local (co)homology of an ideal coincides with theČech (co)homology. The result on the side of cohomology is [Sch03, Theorem 3.2]. In the same paper, the homology analog is proved under the extra assumption that each element of the generator sequence forms itself a oneelement weakly proregular sequence ( [Sch03,Theorem 4.5]). This extra assumption was removed in [PSY14]. Therefore, any example of a non-weakly proregular ideal I (such as [GM92, Example 1.4]) gives a situation where both the local cohomology and homology are not computed via theČech complex. Indeed, not only the functorial map from Theorem 7.10 fails to be a quasi-isomorphism, but inspecting the proofs in [Sch03], some flat (injective) module has a non-zero higherČech homology (cohomology), but the local homology (cohomology) will vanish, respectively.

Construction of the corresponding cotilting modules
In this section we construct to each cotilting class of cofinite type over a commutative ring a cotilting module inducing it. The construction generalizes ideas from [ŠTH14].
Lemma 8.1. Let R be a ring and C a n-cotilting class in R-Mod. Suppose that C is a left R-module satisfying ⊥ C = C, C ∈ C, and C ⊆ Cogen(C). Then C is a cotilting module.
Proof. We prove that Cogen n (C) = C, which implies that C is a n-cotilting module by [Baz04,Theorem 3.11]. Since C is a n-cotilting class and C ∈ C, we have inclusion Cogen n (C) ⊆ C. To show the other inclusion, let M ∈ C, put I = Hom R (M, C), and let ϕ : M → C I be the coevaluation map. Since M ∈ Cogen(C), we have that the map ϕ is injective. Applying Hom R (−, C) onto the exact sequence 0 → M ϕ − → C I → X → 0 yields is clearly surjective, we have that Ext 1 R (X, C) = 0. Using the fact that C I , M ∈ C = ⊥ C, we infer that X ∈ C = ⊥ C. Repeating this argument shows that indeed M ∈ Cogen n (C).
Corollary 8.2. Let R be a commutative ring and C a cotilting module such that the induced cotilting class C = ⊥ C is of cofinite type. Let S be the characteristic sequence such that C = C(S). Let W j be an injective module such that Cogen(W j ) = F j (S). Then Ω −j (C) ⊕ W j is a cotilting module inducing the cotilting class C (j) .
In the rest of the section let R be a commutative ring and let S = (F 0 , F 1 , . . . , F n−1 ) be a characteristic sequence. Our goal is to construct a cotilting module C(S) such that ⊥ C(S) = C(S).
Construction 8.3. We aim to construct a (co)complex of injective modules where E i is in cohomological degree i, satisfying the following properties: (i) the cohomology of the complex vanishes with the exception of degree 0, (ii) for each i = 0, 1, . . . , n, the kernel C i of ψ i is a (n − i)-cotilting module such that ⊥ C i = C (i) . We construct the complex by backwards induction on i = n, n − 1, . . . , 1, 0. For the step i = n, we let E n be an injective cogenerator of Mod-R, and ψ n be the zero map.
Suppose that we have already constructed the complex down to degree k + 1 for some 0 ≤ k < n so that it is exact in degrees > k + 1 and satisfies (ii). By [GT12,Theorem 15.9], there is a C (k) -cover f : Proof. Because F k ∈ C (k) , we have by Lemma 5.14(ii) that Ω −1 F k ∈ C (k+1) . By the inductive premise, C k+1 is a cotilting module such that ⊥ C k+1 = C (k+1) , and therefore C k+1 ∈ C ⊥ (k+1) . It follows that the cover f : F k → C k+1 can be extended to a map f ′ : E(F k ) → C k+1 . As E(F k ) ∈ C (k) by Proposition 5.5, it can be easily seen that f ′ is an C (k) -precover of C k+1 . Therefore, F k is a direct summand of E(F k ), proving that F k is injective. Now we let E k = F k ⊕W k , where W k is an injective module such that Cogen(W k ) = F k (e.g. W k = {E(R/J) | R/J ∈ F k }). We define ψ k : E k → E k+1 by setting ψ k ↾ F k = f and ψ k ↾ W k = 0. Since W k is injective and belongs to F k , it is in C (k) , and then it easily follows that ψ k : E k → C k+1 is a C (k) -precover of C k+1 = Ker(ψ k+1 ) = Im(ψ k ).
Lemma 8.5. We have ⊥ C k = C (k) and C k ∈ C (k) .
Proof. By the inductive premise of the construction, we know ⊥ Ω −1 C k = C (k+1) , and thus j>1 Ker Ext j R (−, C k ) = C (k+1) . Let M ∈ C (k+1) . Then Ext 1 R (M, C k ) = 0 if and only if any map g : M → C k+1 can be factorized through ψ k : E k → C k+1 . If M ∈ C (k) , then this factorization is always possible, because ψ k is an C (k) -precover. This proves that C (k) ⊆ ⊥ C k .
For the converse inclusion, suppose that M ∈ C (k) . Consider first the case where even M ∈ F k and let T be the torsion part of M with respect to the torsion pair (T k , F k ). By construction, C k+1 has an injective direct summand W k+1 which cogenerates the torsion-free class F k+1 . Since M and also T belong to F k+1 , there is a non-zero map T → W k+1 , which extends to a map g : M → W k+1 ⊆ C k+1 . Such map does not factor through ψ k since Hom R (T, E k ) = 0. The remaining case is where M ∈ F k , but Ω −1 M ∈ C (k+1) . Consider the long exact sequence obtained by applying Hom R (−, C k ) to Since M ∈ F k , we have E(M ) ∈ C (k) , and thus we obtain isomorphisms Ext i Finally, that C k ∈ C (k) follows from Lemma 5.14 and from Ω −j C k being a direct summand of C k+j for each j = 1, 2, . . . , n − k − 1.
Lemma 8.6. The module C k is an (n − k)-cotilting module.
Proof. Since Cogen(C k ) = F k ⊇ C (k) , and by Lemma 8.5, the module C k and the (n − k)-cotilting class C (k) satisfy the hypothesis of Lemma 8.1. Therefore, C k is (n − k)-cotilting by that lemma.
This concludes the inductive step, and hence also the construction.
Notation 8.7. We put C(S) = Ker(ψ 0 ), and conclude:  Proof. If C is of cofinite type, then C (i) is easily seen to be of cofinite type too for any i = 0, 1, . . . , n − 1. Then C (i) is closed under injective envelopes by Proposition 5.5.
The other implication follows from Lemma 5.7, the proof of Lemma 5.8, and Lemma 5.13.
In the rest of the section, we exhibit for each n ≥ 2 a concrete example of an ncotilting class which is not of cofinite type, but for which C (i) is closed under injective envelopes for all i = 0, 1, . . . , n − 2. To this end, we first recall a characterization of cotilting classes which is valid for any (even non-commutative) ring: Proposition 9.2. ([AHPŠT14, Proposition 3.14]) Let R be a ring, n ≥ 0, and C a class of left R-modules. Then C is n-cotilting if and only if the following conditions hold: (i) C is definable, (ii) R ∈ C and C is closed under extensions and syzygies, (iii) each n-th syzygy module is in C, In particular, a class of left R-modules is 1-cotilting precisely when it is a definable torsion-free class containing R.
In order to construct the aforementioned examples, we need a suitable family of examples of non-cofinite type 1-cotilting classes to start with. The following is a generalization of [Baz07, Proposition 4.5]: Example 9.3. Let R be a local commutative ring admitting a non-trivial idempotent ideal J. Let G be a Gabriel topology of finite type such that J ∈ G, and such that G is faithful, i.e. Hom R (R/I, R) = 0 for all I ∈ G. Let F = I∈G Ker Hom R (R/I, −) be the torsion-free class of the associated hereditary torsion pair of finite type. Given a module M , let Soc J (M ) = {m ∈ M | Jm = 0}. We define a class C as follows: Alternatively, where t G is the torsion radical associated to the torsion pair (T , F ).
We will show that C is a 1-cotilting class, but not of cofinite type. First, we claim that C = I∈G Ker Hom R ((J + I)/I, −). Let M be a module such that there is a non-zero map f : (J + I)/I → M for some I ∈ G. Since (J + I)/I is J-divisible (i.e. Observe that if JX = 0, we would have a non-zero morphism R/I → JX which would similarly as above restrict to a non-zero morphism (J + I)/I → JX. This is, however, not possible since we assume that Hom R ((J +I)/I, M ) = 0. Thus JX = 0 and X ⊆ Soc J (M ). But thenX = 0, contradicting thatf is a non-zero map. This establishes the other inclusion of the claim. In particular, we have proved that C is a torsion-free class.
As a torsion-free class, C is closed under products and (pure) submodules. To prove that C is definable, it remains to treat direct limits. Since (T , F ) is of finite type, the torsion radical t G commutes with direct limits. Given a directed system M i , i ∈ I with M i ∈ C, we compute: The latter is a direct limit of R/J-modules, proving that Jt G (lim − →i M i ) = 0 as desired. Finally, since G is faithful, we have that R ∈ F ⊆ C. Using Proposition 9.2, we infer that C is a 1-cotilting class.
Finally, we prove that C is not of cofinite type. Indeed, if it was of cofinite type, Theorem 6.2 would provide us with a Gabriel topology H of finite type such that C = K∈H Ker Hom R (R/K, −). Since Mod-R/J ⊆ C, this implies that J + K = R for all K ∈ H. Since R is local, the only possibility is H = {R}, which forces C = Mod-R. Recall that we assumed J ∈ G, and thus there is a finitely generated ideal I ∈ G with I ⊆ J. If R/I ∈ C, then necessarily R/I = Soc J (R/I), which implies I = J. But J is a non-trivial idempotent ideal in a local commutative ring, so it cannot be finitely generated, a contradiction.
As a next step, we would like to extend the example to n-cotilting classes for n ≥ 1. The strategy is to reverse (under suitable assumptions) the process of Lemma 5.7. That is, we would like to combine a hereditary faithful torsion-free class F and an n-cotilting class, which is not necessarily of cofinite type, to an (n + 1)-cotilting class. Note that in Lemma 5.7 we have Inj-R ∩ F ⊆ C, where Inj-R ⊆ Mod-R is the class of injective modules. We will adopt this assumption for the next auxiliary result which can be viewed as an analog of [AHPŠT14, Lemma 3.10]. be an injective envelope and let w : M → E(L) be a morphism such that wu = v, obtained by the injectivity of E(L). If f : L → C is any homomorphism, it extends tof : E(L) → C since L ∈ D. In particular f =f v =f wu, showing that the leftmost morphism in the following exact sequence is surjective: Hom R (M, C) → Hom R (L, C) → Ext 1 R (N, C) → Ext 1 R (M, C) = 0 Thus Ext 1 R (N, C) = 0 and N ∈ C. Conversely suppose that N ∈ C. Then L ∈ F ∩ C by the above observations. To see that L ∈ D, it remains to prove that each f : L → C extends tof : E(L) → C. Consider the following commutative square with injective envelopes in rows, where the right vertical map is completed using the injectivity of E(M): Then Hom R (u, C) is surjective since Ext 1 R (N, C) = 0 and Hom R (z, C) is surjective since M ∈ D. It follows that Hom R (v, C) is surjective, as required.
Now we prove a result which, under more restrictive assumptions, allows us to combine a hereditary torsion-free class with an n-cotilting class.
Proposition 9.5. Let R be a ring and n ≥ 1. Suppose that all of the following conditions are satisfied: (1) C is an n-cotilting class in Mod-R and Ω −1 (R) ∈ C.
(2) (T , F ) is a faithful hereditary torsion pair of finite type in Mod-R.
Proof. Let C ∈ Mod-R be a cotilting module such that C = ⊥ C and let us denote the class from condition (4) by I. If G is the Gabriel topology corresponding to (T , F ), we have Indeed, the first equality follows by an argument analogous to the one for Lemma 3.7 while the second equality follows from Corollary 3.10. Since all the H i (I; −) commute with direct products and direct limits, the class I is definable by [CB98, § §2.1-2.3].
Next we claim that Inj-R ∩ F ⊆ I ⊆ D. The first inclusion is trivial and to see the second one, let M ∈ I and consider the short exact sequence 0 → M → E(M ) → Ω −1 (M ) → 0. We must show that M ∈ D. Clearly M ∈ F by assumption, and we also have E(M ) ∈ F . If T ∈ T is torsion, it follows that Now we prove that D is (n + 1)-cotilting by checking the assumptions of Proposition 9.2. If M ∈ D and N ⊆ M is a pure submodule, then M/N ∈ C since M ∈ C and definable classes are closed under pure quotients; [Pre09, Theorem 3.4.8]. Hence N ∈ D by Lemma 9.4. If (M i ) i∈I is a directed system in D, we fix an injective module F ∈ F which cogenerates F (see [Ste75,Proposition VI.3.7]) and consider the directed system of maps (M i → F Mi = F Hom R (Mi,F ) ) i∈I given by Lemma 5.6. All the maps in the system are monomorphisms by the choice of F , and there is an exact sequence As F Mi ∈ I for each i ∈ I, we have lim − →i∈I F Mi ∈ I ⊆ D. Since F Mi ∈ D, we have F Mi /M i ∈ C for each i ∈ I, and so lim − →i∈I F Mi /M i ∈ C. It follows from Lemma 9.4 that lim − →i∈I M i ∈ D. Finally, since D is closed under products by its very definition, we have verified Proposition 9.2(i).
Conditions (1) and (2) imply that R ∈ D. Since D is definable, it must contain all projective modules as well. Hence given any M ∈ Mod-R, we have Ω(M ) ∈ D if and only if M ∈ C by Lemma 9.4. In particular D is closed under taking syzygies, and that D is closed under extensions follows from the Horseshoe Lemma. Thus Proposition 9.2(ii) holds for D. Finally, if M ∈ Mod-R, then Ω n (M ) ∈ C since C is assumed to be n-cotilting, so Ω n+1 (M ) ∈ D by what we have just shown. Hence D is an (n + 1)-cotilting class by Proposition 9.2 and we have also proved that The closure of D under injective envelopes is obvious from the definition. Now we formulate an easier way to apply this result for constructing n-cotilting classes not of cofinite type. The constructed classes are almost identical to what we obtained for classes of cofinite type in Lemma 5.8. The only difference is that the last torsion-free class in the sequence need not be of cofinite type (viewed as a 1-cotilting class).
Then D = {M ∈ Mod-R | Ω −i M ∈ F i for each i = 0, 1, . . . , n − 1} is an n-cotilting class such that D (i) is closed under injective envelopes for each i = 0, 1, . . . , n − 2. In particular, D is of cofinite type if and only if the definable torsion-free class F n−1 is hereditary.
Proof. We will inductively apply Proposition 9.5 and prove the result along with the following equality: where for each i, T i is the torsion class corresponding to F i .
We need to check conditions (1)-(3) of Proposition 9.5 for F 0 and C and formula (23) for D. However, (1) and (2)  We conclude by an explicit construction of an n-cotilting class not of cofinite type, by combining Corollary 9.6 and Example 9.3.
Example 9.7. Let n > 0 and R be a local commutative ring satisfying: (i) there is a non-trivial idempotent ideal J in R, (ii) there is a finitely generated ideal I ⊆ J satisfying Ext i R (R/I, R) = 0 for all i = 0, 1, . . . , n − 1.
First we provide a concrete example of such a ring R. Let k be a field, and let R be the ring of Puiseux series in n variables. That is, is the ring of formal power series in n variables with exponents which are rational, but for each particular series the denominators are bounded. This ring has a unique maximal ideal J consisting of all series with zero coefficient in degree 0, and J is easily seen to be idempotent. Also, put I = Span(x 1 , x 2 , . . . , x n ) and note that the elements (x 1 , x 2 , . . . , x n ) form a regular sequence. Then K • (I) is a projective resolution of R/I, implying that H i (I; R) = 0 for all i = 0, 1, . . . , n − 1. Therefore, we have Ext i R (R/I, R) = 0 for all i = 0, 1, . . . , n − 1 by Corollary 3.10. Put F n−1 = {M ∈ Mod-R | M/ Soc J (M ) ∈ Ker Hom R (R/I, −)}. Then F n−1 fits the construction of Example 9.3 (the Gabriel topology G is generated by the ideal I, and the choice of non-trivial idempotent ideal is J). This shows that F n−1 is a 1-cotilting class not of cofinite type, i.e. a definable torsion-free class containing R which is not hereditary. It also follows from the assumption on I that Ker Hom R (R/I, −) ⊆ F n−1 contains Ω −i R for all i = 0, 1, . . . , n − 1.
We put F k = Ker Hom R (R/I, −) for all k = 0, 1, . . . , n − 2, and note that those are hereditary torsion-free classes of finite type. Then it is straightforward to check that the sequence (F 0 , F 1 , . . . , F n−1 ) satisfies conditions of Corollary 9.6. We conclude that D = {M ∈ Mod-R | Ω −i M ∈ F i for all i = 0, 1, . . . , n − 1} is an n-cotilting class such that D (i) is closed under injective envelopes for all i = 0, 1, . . . , n − 2, but not for i = n − 1. In particular, D is not of cofinite type.