Descent for coherent sheaves along formal/open coverings

For a regular noetherian scheme $X$ with a divisor with strict normal crossings $D$ we prove that coherent sheaves satisfy descent w.r.t. the 'covering' consisting of the open parts in the various completions of $X$ along the components of $D$ and their intersections.


Introduction
Let X be a regular noetherian scheme and D ⊂ X a divisor with strict normal crossings (cf.Definition 7.1).In applications it is useful to have a descent theory for coherent sheaves on X relative to a 'covering' constisting of the open parts in the various completions of X along the components of D and their intersections.For example when X is the (integral model of a) toroidal compactification of a Shimura variety, these completions can be described as completions of relative torus embeddings on mixed Shimura varieties.In the case that D consists of one component, these questions have been treated in the literature, see e.g.[2,3,6], [7,Tag 0BNI].In this short article, we generalize these results to arbitrary divisors with normal crossings, sticking to the noetherian case, however.The result is as follows: Let {Y } be the coarsest stratification of X into locally closed subschemes such that every component of D is the closure Y of a stratum Y .For each stratum we define a sheaf of O X -algebras on X: where U is an open subset of X, and The definition of R Y does not depend on the choice of U ′ for R Y may also be described as the push-forward where C Y X is the formal completion of X along Y , and C Y X Y is the open formal subscheme with underlying topological space equal to Y and ι is the composed morphism of formal schemes C Y X Y → X.
For any chain of disjoint strata Y 1 , Y 2 , . . ., Y n such that Y i ⊂ Y i−1 for i = 2, . . ., n, we define inductively a sheaf R Y 1 ,...,Yn of O X -algebras on X which coincides with (1) for n = 1.For n > 1 we set Here ⊗ is the usual tensor product (not completed!) and Again this definition does not depend on U ′ .Let [ ( S, R)-coh-cocart ] be the category of the following descent data (it will be defined in a different way in the article, which will explain also the notation): For each stratum Y a coherent sheaf M Y of R Y -modules together with isomorphisms for any Y, Z with Z ⊂ Y , which are compatible w.r.t.any triple Y, Z, W of strata with Z ⊂ Y and W ⊂ Z in the obvious way.
be the category of coherent sheaves on X.
Then we have Main Theorem 7.6.The natural functor is an equivalence of categories.
1 Generalities 1.1.Let D → S be a bifibered category such that the fibers have all limits and all colimits.We will be interested mainly in the following two cases where [ ring ] is the category of commutative rings with 1 and [ mod ] is the bifibered category of modules over such rings.Furthermore let X be a topological space.Then we consider where [ X-ring ] is the category of ring sheaves on X which are coherent over themselves in the sense of ringed spaces and [ X-mod ] is the bifibered category of sheaves of modules over such ring sheaves.Back in the general case, for a morphism f ∈ Mor(S) we denote by f • and f • the corresponding push-forward and pull-back functors.By definition of a bifibered category f • is always left adjoint to f • .Those functors are only defined up to a (unique) natural isomorphism.Whenever we write f • or f • we assume that a choice has been made.For example, for a morphism f ∶ S → T of rings, f • is the forgetful map that considers a T -module as an S-module via f (hence there is a canonical choice in this case) and f • is its left-adjoint, the functor − ⊗ S T (which is also only defined up to a unique natural isomorphism).
1.2.Pairs (I, S) consisting of a diagram I (i.e. a small category) and a functor S ∶ I → S form a 2-category Dia op (S), called the category of diagrams in S (cf. also [5]).A morphism of S-diagrams (α, µ) 1.3.For each pair (I, S), we define the category D(I, S) of (I, S)-modules, whose objects are lifts of the functor S D I S / / M 8 8 q q q q q q q q q q q q q S to the given bifibered category, and whose morphisms are natural transformations between those.An object is called coCartesian if all the morphisms M (ρ) for ρ ∶ i → j are coCartesian.This defines a full subcategory D(I, S) cocart of D(I, S).If p ∶ I → E is a functor between small categories, we also define the subcategory of E-coCartesian objects as those for which the morphisms M (ρ) are coCartesian for all ρ such that p(ρ) is an identity.

1.4.
We need a refinement of the categories defined above.Suppose we are given a full subcategory D f of D whose objects shall be called finite.We assume that D f → S is still opfibered (i.e.pushforward preserves finiteness) but not necessarily fibered.We define the full subcategories D(I, S) f and D(I, S) f,cocart requiring point-wise finiteness.
Definition 1.5.In the example lying over the sheaf of rings R is finite, if it is coherent in the sense of ringed spaces, i.e. locally on X, we have an exact sequence 1.6.For each morphism (α, µ) ∶ (I, S) → (J, T ), we have a corresponding pull-back (α, µ) * given by (α, µ) * has a right adjoint (α, µ) * given by Kan's formula where an object in the slice category j × J I is denoted by a pair (i, ν), with i ∈ I and ν ∶ j → α(i).
If (α, µ) ∶ (I, S) → (J, T ) is purely of diagram type, i.e. if µ ∶ T ○ α → S is the identity, then (α, µ) * does have also a left adjoint (α, µ) !given by where an object in the slice category I × J j is denoted by a pair (i, ν), with i ∈ I and ν ∶ α(i) → j.
Example 1.7.For the fibered category [ mod ] → [ ring ] and for each morphism (α, µ) ∶ (I, S) → (J, T ), the pull-back is given by where each M i is considered an T j -module via the composition 2 Morphisms of (finite) descent We keep the notations introduced in the previous section.
Definition 2.1.We say that a morphism (α, µ) in Dia op (S) is of descent if is an equivalence of categories.We say that a morphism (α, µ) in Dia op (S) is of finite descent if is an equivalence of categories.
If α ∶ I → J is a Grothendieck fibration then the statement holds with the slice category j × J I replaced by the fibered product j × J I.
4. Let α ∶ I → J be a morphism of diagrams and let (α, id) ∶ (I, α * S) → (J, S) be a morphism of diagrams in S of pure diagram type.If (I × J j, S k ) → (⋅, S k ) is of descent (resp. of finite descent) for all j, k ∈ J then (α, id) is of descent (resp. of finite descent).is equivalent to the small category whose set of objects is π 0 (I) and whose morphism sets are In particular (I, S) → (⋅, S) is of descent (resp. of finite descent) if π 0 (I) = π 1 (I) = {⋅}.
6. Let ∆ be the simplex category, and let ∆ ○ be the injective simplex category.Let (∆, S • ), resp.(∆ ○ , S • ) be a cosimplicial, resp.a cosemisimplicial object in S. Then the category D(∆, S • ) cocart , resp.D(∆ ○ , S • ) cocart , is equivalent to the category of classical descent data, whose objects are an object M in D(S 0 ) together with an isomorphism in D(S 1 ) such that the following equality of morphisms in D(S 2 ) holds true: Here δ i k is the strictly increasing map {0, . . ., k − 1} → {0, . . ., k} omitting i.
We start by showing that both unit and counit are isomorphisms when restricted to the subcategories of coCartesian objects.Let j ∶ (⋅, T j ) → (J, T ) be the embedding.We have to check that is an isomorphism for all j.
Consider the 2-commutative diagram: (j × J I, ι * j S) By the explicit point-wise formula for (α, µ) * , the morphism (3) is the same as The morphism induced by the 2-morphism in the diagram ι * j (α, µ) * E → p * j j * E is an isomorphism on coCartesian objects by definition.Since the unit id → p j, * p * j is an isomorphism by assumption, we are done.We now show that (α, µ) * preserves coCartesian objects.Let ρ ∶ j 1 → j 2 be a morphism in J.It induces a map of fibers (purely is an isomorphism.This can be checked after pull-back along p 2 ∶ (j 2 × J I, pr * 2 S) → (j 2 , T j 2 ) because this induces an equivalence of the categories of coCartesian objects by assumption.Since S(ρ)p 2 = p 1 ρ we get the morphism which is the same as To see that the counit is an isomorphism on coCartesian objects, we have to see that is an isomorphism for all j.Since (α, µ) * preserves coCartesian objects, this is again the morphism and hence the morphism induced by the counit This is an isomorphism on coCartesian objects by assumption.Proof of the additional statement: If α is a Grothendieck fibration, we have an adjunction with κ j ι j = id and such that there is a 2-morphism ι j κ j ⇒ id.Hence by 2., these morphisms are of descent (resp. of descent for finitely generated modules).Hence we may replace j × J I by j × J I in the statement.4. We will show again that both unit and counit are isomorphisms when restricted to the subcategories of coCartesian objects.Let j ∶ (⋅, T j ) → (J, T ) be the embedding.We have to see that is an isomorphism for all j.
Consider the 2-commutative diagram: By the explicit point-wise formula for (α, µ) !, the morphism ( 5) is the same as The morphism induced by the 2-morphism in the diagram ι * (α, µ) * E → p * j j * E is an isomorphism on coCartesian objects by definition.Since the counit p j,! p * j → id is an isomorphism by assumption, we are done.We now show that (α, µ) !preserves coCartesian objects.Let ρ ∶ j 1 → j 2 be a morphism in J.We have to show that is an isomorphism.After inserting the point-wise formula and denoting This is the morphism induced by the counit ρ! ρ * for the morphism ρ ∶ (I × J j 1 , T j 2 ) → (I × J j 2 , T j 2 ) (composition with ρ).Now observe that p ′ 2,! , resp.p 2,! , by assumption, can be computed on coCartesian elements just by evaluation at any element of I × J j 1 resp.I × J j 2 .Therefore ( 6) is an isomorphism.5. is obvious.6.We have to show that the inclusion ι ∶ (∆ ○ ≤3 , S • ) → (∆ ○ , S • ) is of descent (resp. of finite descent), the category D(∆ ○ ≤3 , S • ) cocart being clearly just the category of classical descent data.By 4. and 5. this amounts to showing that ∆ is trivial.This is well-known.Note that it is essential to take 3 terms of ∆ ○ here.For example The same holds with ∆ ○ replaced by ∆ because there is an adjunction and the morphism is of descent (resp. of finite descent).To prove the latter assertion, by 4. it suffices to show that ∆ ○ × ∆ ∆ n is contractible.To see this, consider the projection p ∶ ∆ ○ × ∆ ∆ n → ⋅.It has a section s given by mapping ⋅ to id ∆n .We construct a morphism and mapping an injective morphism We have p ○ s = id {⋅} and here are obvious 2-morphisms showing that ∆ ○ × ∆ ∆ n is contractible, or, what matters here, that (∆ ○ × ∆ ∆ n , S) → (⋅, S) is of descent (resp.finite descent) for any S ∈ S.
We need a refinement of Proposition 2.2 3./4.which also is specific to the situation of fibered categories and will not hold in any context of cohomological descent.Call an object j ∈ J initial if no morphism j ′ → j exists with j = j ′ .
Lemma 2.3.Let again D → S be a bifibered category with choice of a full subcategory of finite objects D f as above.Let (α, µ) ∶ (I, S) → (J, T ) be a morphism of diagrams in S. Assume that for any object j there is a morphism k → j from an initial object k.
Then (α, µ) is of descent (resp. of finite descent) if p j ∶ (j × J I, pr * 2 S) → (j, T j ) is such that p * j is fully-faithful for any j ∈ J and such that p j is of descent (resp. of finite descent) for any initial object j.If α is a Grothendieck fibration then the same holds with j × J I replaced by j × J I.
Proof.By the proof of Proposition 2.2, 3. (α, µ) * is fully-faithful because all p * j are fully-faithful.We show by direct construction that (α, µ) * is essentially surjective.Let M be an (I, S)-module and j ∈ J an object.Choose a morphism α j ∶ k → j such that k is initial, which is the identity if j is already initial.Define ).Note that (p k ) * is an inverse to the equivalence (p k ) * .For a morphism ν ∶ j 1 → j 2 we must define a morphism N (j 1 ) ⊗ T (j 1 ) T (j 2 ) → N (j 2 ) or, in other words (⋅, T (ν)) * N (j 1 ) → N (j 2 ).We have the standard 2-commutative diagram Denote µ ν the morphism induced by ν: (j 2 × J I, pr * 2 S) → (j 1 × J I, pr * 2 S).We have We give the morphism ).Because of fully-faithfulness we may do so after pulling back via p * j 2 and hence define p * j 2 applied to it as the following composition using that p * k 1 and p k 1 , * define an equivalence.One checks that this association is functorial.
3 Descent for modules Lemma 3.1.Let R → R ′ be a ring homomorphism.For a morphism (I, S) → (J, T ) of diagrams of R-algebras the property of being of descent for arbitrary modules implies that (I, S ⊗ R R ′ ) → (J, T ⊗ R R ′ ) is of descent for arbitrary modules.
If R → R ′ is finite then for a morphism (I, F ) → (J, G) of diagrams of R-algebras the property of being of descent for finitely generated modules implies that (I, S ⊗ R R ′ ) → (J, T ⊗ R R ′ ) is of descent for finitely generated modules.
Proof.The category of (I, S ⊗ R R ′ )-modules is equivalent to the category of (I, S)-modules with R ′ -action, i.e. to the category whose objects are pairs consisting of an object X ∈ [ (I, S)-mod ] and of a homomorphism of R-algebras ρ ∶ R ′ → End R (X).
Lemma 3.2.Let (α, µ) ∶ (I, S) → (J, T ) be a morphism of diagrams of rings such that I × J j is a finite diagram for all j.If (α, µ) * is faithful then (α, µ) * M finitely generated implies M finitely generated.In particular "of descent" implies "of descent for finitely generated modules".
Proof.This is similar to the statement that a module is finitely generated if it becomes finitely generated after a faithfully flat ring extension.Let M be a coCartesian module over (J, T ) such that (α, µ) * M is finitely generated.Let j ∈ J.For each (i, ρ ∶ α(i) → j) ∈ I × J j we know that M (α(i)) ⊗ T α(i) S i is a finitely generated S i -module.Let {ξ i,ρ k } k be images in M (j) of the (finitely many) M (α(i))-components of those generators.We claim that the union over those finite sets for all objects in I × J j generates M (j).For let N (j) be the submodule generated by them, and assume that N (j) is different from M (j).The non-zero morphism j * M → M (j) N (j) induces a non-zero morphism M → j * (M (j) N (j)) and therefore a non-zero morphism (α, µ) * M → (α, µ) * j * (M (j) N (j)) because (α, µ) * is faithful.For any i consider the morphism i This is the tensor product with S(i) of the map induced by the canonical ones M (α(i)) → M (j).By construction of N (j) this map is zero, a contradiction.

Descent for modules on ringed spaces
Lemma 4.1.For the bifibered category we have that (I, S) → (J, T ) is of descent (resp. of finite descent) if for any open set U ⊂ X there is a cover U = ⋃ i U i such that (I, S U i ) → (J, T U i ) is of descent (resp. of finite descent) for the bifibered category Proof.This is an obvious glueing argument.Alternatively one could construct a commutative square of diagram-morphisms where the vertical morphisms consist point-wise in I (resp.J) of the restrictions of S i (resp.T j ) to a hypercovering of X such that the top horizontal morphism consists point-wise in ∆ ○ of a morphism of descent (resp. of finite descent).By explicit construction one sees that the upper horizontal morphism is also of descent (resp. of finite descent).The vertical morphisms are then of descent by the definition of sheaf.This shows that also the lower horizontal morphism is of descent (resp. of finite descent).

Descent and projective systems
Let S be a noetherian ring, a an ideal of S and consider the diagram (N op , S • ) where S n = S a n S for every n ∈ N.
Lemma 5.1.For an object M • ∈ [ (N op , S • )-mod ] the following assertions are equivalent 2. M 1 is finitely generated and for each for each k ≤ l, the sequence Proof.The exact sequence 0 / / a k S / / S / / S a k S / / 0 tensored with M l yields the sequence Hence coCartesianity of the diagram is equivalent to the exactness of the sequence above.It suffices to show that for a coCartesian diagram the condition that M 1 is finitely generated implies that M k is finitely generated.Consider the sequence of S a l S-modules Since a is nilpotent in S a l S, this implies that M l is finitely generated by Nakayama's lemma.
Lemma 5.2.Let R be a noetherian ring, a an ideal and consider a diagram (I, F ) of a-adically complete and separated noetherian R-algebras.Let (I × N op , F • ) to be the diagram with value in which the upper horizontal morphism is of descent for finitely generated modules and the vertical ones are of descent for finitely generated modules by Lemma 5.2.Hence so is the lower horizontal one.

Descent along basic formal/open coverings
is of descent for arbitrary modules (resp.for finitely generated modules).
2. For any sequence of ideals I i and elements f i such that the functor p * is fully-faithful.Here I f , for an element f ∈ R, denotes the functor Proof.We will actually need only the following axioms on the diagram of rings D: (c) For each object (i.e.descent datum Let us verify that the axioms (a-c) hold in the situation of the lemma.(a) Flatness of R f is clear.R is flat, because R is noetherian.The tensor property holds by construction.
(b) That the last map is surjective is clear.Hence the statement boils down to the Cartesianity of the diagram of R-modules R for any n.This diagram is Cartesian because f is not a zero divisor in R (and hence neither in R) and we have an isomorphism . Let m f be any element of M f .We have for all q ∈ Rf : where p j ∈ R are independent of q.Hence this element is of the form m ⊗ 1 if q is sufficiently divisible by f .Therefore, writing any given m f ⊗ p, where p ∈ Rf as m ′ f ⊗ 1 + m f ⊗ q where q is sufficiently divisible by f , we see that the map M ⊕ M f → M f is surjective.Now assume that the axioms (a-c) hold.First observe that also Rf is flat over R (base change of flatness).Let M be an R-module.Tensoring the sequence (7) with M over R yields the exact sequence 0 , and M ⊗ R Rf .This shows that M can be reconstructed as the limit over the diagram Consequently the unit id → p * p * of the adjunction is an isomorphism.For the second assertion of the proposition observe that application of the functors C I i , and Therefore also after applying those functors to D, and R, respectively, we have that the unit id → p * p * of the new adjunction is an isomorphism.Now let ( M , M f , M f ) be a descent datum.We form the exact sequence (cf.axiom (c) for the surjectivety of the map to M f ): This shows that the natural map N ⊗ R R → M is an isomorphism.That also N ⊗ R R f → M f is an isomorphism follows because the axioms (a-c) are completely symmetric in R and R f .Actually, if we have, as in the formulation of the proposition, that R → R f is an epimorphism of R-algebras, i.e. that R f = R f ⊗ R R f , then this is even easier.Finally the statement of descent for finitely generated modules follows from Lemma 3.2.
7 Descent along completions w.r.t. a divisor with normal crossings This definition differs slightly from [1, Exposé I, 3.1.5,p. 24] to the extent that there the existence of f 1 , . . ., f m is claimed to exist globally.Note that it follows from the definition that all V (f i ) (defined locally) are regular themselves.
7.2.Let X be a regular noetherian scheme and let D ⊂ X be a divisor with strict normal crossings.Let {Y } be the coarsest stratification of X into locally closed subvarieties such that every component of D is the closure of a stratum Y .For each stratum we define a sheaf of O X -algebras on X: Recall that for a sheaf of O X -algebras R as above, a coherent sheaf M of R-modules is a sheaf of R-modules such that on a covering {U i } of X, we have an exact sequence for every i.Proof.This is shown as the analogous property for coherent sheaves of O X -modules on a noetherian formal scheme X .
Coming back to the stratification {Y } of X, we define the following semi-simplicial set S.
is equivalent to the category of the following descent data: For each stratum Y a coherent sheaf M Y of R Y -modules together with isomorphisms for any Y, Z with Z ⊂ Y , which are compatible w.r.t.any triple Y, Z, W of strata with Z ⊂ Y and W ⊂ Z in the obvious way.
Main Theorem 7.6.The morphism ( ∫ S, R) → (⋅, O X ) is of descent for coherent sheaves.In other words the natural functor is an equivalence of categories.
Proof.By Lemma 4.1 and Lemma 7.4 we are reduced to a local, affine situation of the following kind: We may assume that D is defined by an equation f 1 ⋯f n = 0 such that f 1 , . . ., f n are part of a minimal sequence of generators of the maximal ideal of a point p ∈ D. We may also assume (by possibly shrinking the affine cover) that all V (f i 1 , . . ., f i k ) are irreducible for each subset {i 1 , . . ., i n } ⊂ {1, . . ., n}.The strata Y are then all of the form V We similarly get a diagram ( ∫ S, R) in Dia op ([ ring ]) and have to show that is an equivalence.Note that our semi-simplicial set S here is equal to the (semi-simplicial) nerve of the partially ordered set P({1, . . ., n}) (power set).We show by induction on n that (9) is an equivalence as follows.Denote the diagram defined before by ( ∫ S (n) , R (n) ) where D has equation f 1 ⋯f n .Decreasing n by 1 means forgetting the last element f n .For n = 1, S (1) is the nerve of the partially ordered set {} < {1} hence ∫ S (1) is the diagram ⌜ and we are precisely in the situation of Proposition 6.1, 1. Therefore ( ∫ S (1) , R (1) ) → (⋅, R) is of descent.For n > 1 consider the stratification {Y } for f 1 , . . ., f n−1 .Then cut each stratum Y in the two pieces Y ′ = Y ∩ V (f n ) and Y ′′ = Y ∖ Y ′ .This yields a chain of stratifications and refinements like considered in the following Lemma.
Lemma 7.7.We consider arbitrary stratifications of U such that each stratum is locally closed and a union of the ones considered before.They are all of the form V (f i 1 , . . ., f i k ) ∖ V (f i k+1 , . . ., f i l ), where however not necessarily {i 1 , . . ., i l } = {1, . . ., n}.The members of such a stratification form a partially ordered set P as before and we denote by S the corresponding (semi-simplicial) nerve.Let ν ∶ P ′ → P be a morphism of such partially ordered sets which comes from a refinement of stratifications such that precisely two strata Y ′ , Y ′′ get mapped to one stratum Y and we have It induces a map of semi-simplicial nerves, denoted ν by abuse of notation, as follows: 1.If Y does not occur in the list, it consists of ξ itself, considered as an element of ∫ S ′ .
2. If Y occurs in the list, the fiber consists of the diagram For a morphism in ∫ S ′ , say there is an obvious pull-back functor between these fibers which establishes ν as a Grothendieck fibration.
Using the claim and Proposition 2.2, 3. we would have to show that the fibers of (S ′ ξ , R ′ ξ ) → (ξ, R(ξ)) are of descent for any ξ = (∆ k , Y 1 < ⋯ < Y k ) ∈ ∫ S ′ .Actually we have the refinement Lemma 2.3 which reduces us to prove that for all Z the fiber above (∆ 0 , Z) (these are the initial objects of ∫ S in the sense of Lemma 2.3) is of descent, and that for all other p ξ ∶ (S ′ ξ , R ′ ξ ) → (ξ, R(ξ)) the pull-back p * ξ is fully faithful.In other words, we have to see that 1. (only Z = Y is non-trivial) is of descent for finitely generated modules.This is Proposition 6.

5 .
Let S ∈ S be an object and I a diagram.Denote by (I, S) the corresponding constant diagram.There is an equivalence of categories D(I, S) cocart → D(I[Mor(I) −1 ], S) cocart (resp.decorated with f) where I[Mor(I) −1 ] is the universal groupoid to which I maps.I[Mor(I) −1 ] of descent for finitely generated modules.Therefore by Proposition 2.2, 4. the morphism (I × N op , F • ) → (J × N op , G • ) is of descent for finitely generated modules.Now we have the commutative diagram of diagrams of rings

Proposition 6 . 1 .
Let R be a noetherian ring and f a non-zero divisor of R. Denote R the completion of R w.r.t.(f )-adic topology and let R f , and Rf , the rings R[f −1 ], and R[f −1 ], respectively.Then 1.The morphism of diagrams

Definition 7 . 1 .
A subscheme D on a regular scheme S is called a divisor with strict normal crossings if it is the zero-locus of a Cartier divisor which is Zariski locally around any p ∈ D of the form f 1 ⋯f m , where f 1 , . . ., f m are part of a sequence of minimal generators of the maximal ideal at p.
clear that the definition of R Y does not depend on the choice of U ′ for R Y may also be described asι * O C Y X Y , where C Y X is the formal completion of X along Y ,and C Y X Y is the open formal subscheme with underlying topological space equal to Y and ι is the composed morphism of formal schemes C Y X Y → X.For any chain of disjoint strata Y 1 , Y 2 , . . ., Y n such that Y i ⊂ Y i−1 , we define inductively a sheaf R Y 1 ,...,Yn of O X -algebras on X which coincides with the previous one for n = 1.For n > 1 we set and this definition does not depend on the choice of U ′ .Example 7.3.Consider X = A 1 k and D = {0}.Then the strata are D and Y ∶= X ∖ D. We have

Lemma 7 . 4 .
If U ⊂ X is affine then for any of the sheaves R = R Y 1 ,...,Yn of O X -modules defined above, we have an equivalence of categories of coherent sheaves of R U -modules and finitely generated R(U )-modules.
The set S n consists of chains of disjoint strata [Y 1 , . . ., Y n ] such that Y i ⊂ Y i−1 (we write also Y i < Y i−1 ) for i = 2 . . .n with the obvious face maps.Alternatively consider the stratification as a partially ordered set where Z ≤ Y if Z ⊂ Y .S is then the (semi-simplicial) nerve of this partially ordered set.We define also the category ∫ S whose objects are pairs (n, ξ), where n ∈ N and ξ ∈ S n , and whose morphisms µ ∶ (n, ξ) → (m, ξ ′ ) are morphisms µ ∶ ∆ n → ∆ m in ∆ ○ such that S(µ)(ξ ′ ) = ξ.This is the fibered category associated with the functor (∆ ○ ) op → [ set ] ⊂ [ cat ] via the Grothendieck construction.