Irregular perverse sheaves

We introduce irregular constructible sheaves, which are $\mathbb{C}$-constructible with coefficients in a finite version of Novikov ring $\Lambda$ and special gradings. We show that the bounded derived category of cohomologically irregular constructible complexes is equivalent to the bounded derived category of holonomic $\mathcal{D}$-modules by a modification of D'Agnolo--Kashiwara's irregular Riemann--Hilbert correspondence. The bounded derived category of cohomologically irregular constructible complexes is equipped with the irregular perverse t-structure, which is a straightforward generalization of usual perverse t-structure and we see its heart is equivalent to the abelian category of holonomic $\mathcal{D}$-modules. We also develop the algebraic version of the theory. Furthermore, we discuss the reason of the appearance of Novikov ring by using a conjectural reformulation of Riemann--Hilbert correspondence in terms of certain Fukaya category.


Introduction
The regular Riemann-Hilbert correspondence (formulated and proved by Kashiwara [Kas84] and another proof by Mebkhout [Meb84]) states that the derived category of regular holonomic Dmodules is equivalent to the derived category of C-constructible sheaves. Under this equivalence, the abelian category of regular holonomic D-modules is mapped to the abelian category of perverse sheaves introduced by Beilinson-Bernstein-Deligne-Gabber [Kas75,BBD82,GM80].
After many efforts including understanding of formal and asymptotic structures [Maj84,Sab00,Moc11,Ked11], Stokes phenomena and Riemann-Hilbert correspondence for meromorphic connections [Mal83,Sib90,DMR07,Moc11,Sab13], sophistication of the regular Riemann-Hilbert correspondence [KS01], and developments of ind-sheaves and the discovery of its relation to asymptotic behavior [KS01,KS03], in a seminal paper [DK16b], D'Agnolo-Kashiwara formulated and proved the irregular Riemann-Hilbert correspondence for holonomic D-modules: where the left hand side is the derived category of cohomologically holonomic D-modules and the right hand side is the category of R-constructible C-valued enhanced ind-sheaves.
In the sequel [DK16a], they also introduced the notion of enhanced perverse t-structure on the right hand side of the embedding and proved that the embedding is t-exact in a slightly generalized sense. Moreover, Mochizuki [Moc16] proved that the image of the equivalence can be characterized by curve test.
In this paper, we modify the right hand side of the equivalence and make it more closer to the form of the regular Riemann-Hilbert correspondence.
As mentioned in their paper, D'Agnolo-Kashiwara's clever definition and use of enhanced sheaves are inspired from the construction of Tamarkin [Tam08]. Tamarkin's idea of adding one extra variable originally aimed to realize Novikov ring action in sheaf theory as in Fukaya category [FOOO09]. In this paper, we take a way which is more closer to this original idea instead of the use of enhanced sheaves. The replacement for the right hand side of (1.1) is expressed as graded modules (sheaves) over the "finite Novikov ring" Λ := k[R ≥0 ] where k ⊂ C is a field. An element of Λ is expressed as a finite sum a∈R ≥0 c a T a where T is the indeterminate. A priori, the hom-spaces Hom(V, W) of Λ-modules are defined over Λ. By taking the tensor product Hom(V, W) ⊗ Λ k where Λ → k is defined by T a → 1, we obtain a new category Mod I pre (Λ X ). We will further modify this category to obtain Mod I (Λ X ). Anyway, we can consider Mod I pre (Λ X ) as an approximate description of Mod I (Λ X ).
The category Mod I (Λ X ) is abelian and has enough injective and flat objects. We define an abelian subcategory of Mod I (Λ X ): the category of irregular constructible sheaves Mod ic (Λ X ). Then we set D b ic (Λ X ) as the full subcategory of the bounded derived category D b (Mod I (Λ X )) consisting of cohomologically irregular constructible sheaves. The meaning of irregular constructibility is as follows: As usual there exists a C-Whitney stratification and we have a sheaf which is locally constant as Λ-module over each stratum, but moreover with particular gradings coming from Sabbah-Mochizuki-Kedlaya's Hukuhara-Levelt-Turrittin theorem [Sab00, Moc11,Ked11]. Then we have the following: Theorem 1.2.
1. The category D b ic (Λ X ) has functors Hom, ⊗, f −1 , f ! for any morphism f and f ! for proper f .

If k = C, there exists an equivalence
In our formulation, the data of exponential factors of solutions of irregular differential equations are encoded in the grading of Λ-modules. We would like to apply the following trivial fact to our setting: For a graded ring R, the grading-forgetful functor from the abelian category of graded R-modules to the abelian category of R-modules is exact. Nevertheless our category Mod ic (Λ X ) has a bit exotic modification of hom-spaces, we still have the following: Theorem 1.3. There exists an exact functor F from Mod ic (Λ X ) to the abelian category of Cconstructible sheaves Mod c (k X ).
By using F, we can define the support of an irregular constructible sheaf V by supp V := supp F(V). By using this definition, we can define the irregular perverse t-structure by the same formula as in usual perverse sheaves: Let p D ≤0 ic (Λ X ) (resp. p D ≥0 ic (Λ X )) be the full subcategory of D b ic (Λ X ) spanned by objects satisfying dim supp H j (V) ≤ −j (resp. dim supp H j (DV) ≤ −j) for any j ∈ Z.
2. The heart of irregular perverse t-structure Ierv(C X ) over C is equivalent to the abelian category of holonomic D-modules under the equivalence (1.2).
We also prove the corresponding results in algebraic setting: Mostly, the statements are corollaries of analytic cases, although we also have f * and f ! for any morphism and can prove more stronger commutativity results for the Riemann-Hilbert functor (as in the case of algebraic regular Riemann-Hilbert correspondence).
As perverse sheaves have vast applications to mathematics including Hodge theory, topology, geometric representation theory, and etc, one can expect irregular perverse sheaves have such applications too, which are possible future works.
We also discuss a conjectural explanation of the appearance of Novikov ring using Fukaya category, which makes D'Agnolo-Kashiwara's approach closer to Tamarkin's one. Our main conjecture is the following (a slightly more precise form is presented in Section 11): Conjecture 1.5.
1. There exists a version of Fukaya category Fuk icnov (T * X) defined over finite Novikov ring Λ.
2. After taking derived category and reducing coefficients Λ to k, we denote the resulting category by D Fuk ic (T * X). Then we have an equivalence D Fuk ic (T * X) ≃ D b ic (Λ X ). In particular, over k = C, we have the Fukaya categorical Riemann-Hilbert correspondence If the conjecture is true, one can imagine K-theory classes of objects of D Fuk ic (T * X) as an irregular version of characteristic cycle. In the same vein, their supports can be considered as an irregular version of microsupports, which are no longer conic. Hence one can also imagine a generalization of microlocal analysis. Note that a version (real blowed-up version) of the equivalence (1.4) is already appeared if one fixes a formal type [STWZ15] (see also Remark 11.6). Also, another connection between Riemann-Hilbert correspondence and holomorphic Fukaya category is conjectured by Kontsevich [Kon], whose relation to our conjecture is also of interest.
The organization of this paper is as follows: In section 2, we define and discuss the preliminary version of the category of sheaves with coefficients in Λ. In section 3 and 4, we define the (derived) category of sheaves with coefficients in Λ over topological spaces with boundary and consider various (derived) functorial operations as in usual sheaf theory. In section 5, we define our main objects irregular constructible sheaves and again see various functorial operations. We also note that irregular constructible sheaves are actually sheaves. In section 6, we construct the functor F which relates irregular to usual sheaves. In section 7, we see the relationship between enhanced sheaves and our Λ-modules, which enables us to establish our version of Riemann-Hilbert correspondence using D'Agnolo-Kashiwara's theorem in section 8. We also prove some commutativity results for Riemann-Hilbert functor in section 8. In section 9, we define irregular perverse sheaves by using F and import results in the theory of perverse sheaves to irregular perverse sheaves. In section 10, we discuss algebraic version of the above story. In section 11, we give some discussions around Fukaya category and Riemann-Hilbert correspondence.

Acknowledgment
The author would like to thank Andrea D'Agnolo whose lectures three times (at Kashiwa, Berkeley, and Padova) gave him many insights about irregular Riemann-Hilbert correspondence. He also kindly pointed out some mistakes in an early draft. The author also would like to thank Takahiro Saito for having many discussions on many aspects of Riemann-Hilbert correspondence (at least once a week), and Takuro Mochizuki for kindly answering some questions. This work was supported by World Premier International Research Center Initiative (WPI), MEXT, Japan and JSPS KAKENHI Grant Number JP18K13405.

Λ X -modules
In this section, we introduce the "finite Novikov ring" Λ and its modules. We fix a field k ⊂ C once and for all.

The ring Λ
Let us see the set of non-negative real numbers R ≥0 as a semigroup by the addition. We denote the associated polynomial ring by Λ := Λ k := k[R ≥0 ]. For a ∈ R ≥0 , let us denote the corresponding indeterminate by T a . We set Gr a Λ := k · T a ⊂ Λ for a ≥ 0, which gives an R-grading on Λ.
Let Mod 0 (Λ) be the abelian category of R-graded Λ-modules with degree 0 morphisms. For an R-graded Λ-module V , let V a be the grading shift of M i.e., Gr b V a := Gr a+b V . We set for R-graded Λ-modules. The category Mod R (Λ) is consisting of R-graded modules with the hom-spaces defined by (2.1). We set Let X be a complex manifold. Let Λ X be the constant sheaf valued in Λ.
LetṼ be a sheaf of R-graded Λ X -modules. For an open subset U ⊂ X, we have an R-graded Λ-moduleṼ(U ). For an inclusion U ֒→ V , we have a mapṼ(V ) →Ṽ(U ) which respects the grading Gr aṼ (V ) → Gr aṼ (U ). Hence we have a sheaf of k-vector spaces Gr aṼ and an isomorphism V ∼ = a Gr aṼ as sheaves valued in k-vector spaces.
We denote the category of R-graded Λ X -modules by Mod 0 (Λ X ).
Notation.Ṽ a for a ∈ R is a-shift ofṼ as in the previous subsection. For f :Ṽ →W, f a means the shifted morphismṼ a →W a We set and Note that Hom Mod R (Λ X ) (Ṽ,W) is a Λ-module. We see k as a Λ-module by setting f · c := f | T =1 c for f ∈ Λ and c ∈ k. We set Definition 2.3. The category Mod I pre (Λ X ) is defined by the following data: the set of objects is the set of R-graded Λ X -modules. For an R-graded Λ X -moduleṼ, the corresponding object in which is the identity on objects and takes a morphism f to f ⊗ 1.
Definition 2.4. For an object V in Mod I pre (Λ X ), a lift is a pair of an objectṼ ∈ Mod 0 (Λ X ) and an isomorphism [Ṽ] ∼ = − → V. In the following, we usually do not write this isomorphism explicitly for simplicity.
Proposition 2.5. The category Mod I pre (Λ X ) is an abelian category. To prove this proposition, we prepare some lemmas.
Lemma 2.6. Let V be an R-graded Λ-module. Let s be a homogeneous element of V . If T a ·s = 0 for any a ∈ R ≥ 0, then s ⊗ 1 is nonzero in V ⊗ Λ k.
Proof. Note that l · s is nonzero for any l ∈ Λ\{0}. We have an inclusion Λ · s ֒→ V . Since the LHS is a free Λ-module, the tensoring (−) ⊗ Λ k preserves the inclusion. Hence s ⊗ 1 is nonzero in V ⊗ Λ k.
Proof. This is a case of Lemma 2.6 by setting V = Hom Mod R (Λ X ) (Ṽ,W) and s := f which is a lift of f .
Proof. Take a representative f = c f c ∈ c Hom c Mod R (Λ X ) (Ṽ,W). Since f c is zero except for finite c, we can take b to be a real number which is greater than or equal to the maximum of c for which f c is nonzero. Then we set Proof. By multiplying some T a 's we can assume that a 1 = a 2 . Since f 1 − f 2 represents 0 in Proof. By Lemma 2.9, it suffices to prove the objects defined for f ′ and T a f ′ are isomorphic.
We have morphisms f ′ :Ṽ →W a and T a f ′ :Ṽ →W a + b in Mod 0 (Λ X ). Note that ker f ′ ֒→ ker T a f ′ . Hence for anyP ∈ Mod 0 (Λ X ), we havẽ which induces a comparison morphism c : . It suffices to show that c is an isomorphism. For Mod R (Λ X ) (P, ker f ′ ). Hence T a g is in the image ofc. Since g and T a g represents the same morphism in Mod I pre (Λ X ), we have the surjectivity of c.
On the other hand, let g ∈ Hom . For a representative g ′ of g, we have T b g ′ = 0 for some b ∈ R ≥0 by Lemma 2.7. Hence . This gives the injectivity of c. Similar arguments prove the claims for im(f ′ ), coker(f ′ ), and coim(f ′ ).
Lemma 2.11. The objects defined in Lemma 2.10 actually give kernel, image, cokernel, and coimage in Mod I pre (Λ X ). Proof. Again, we will only prove for kernel and the others can be proved by similar arguments.
in Mod 0 (Λ X ). By replacingW withW c with sufficiently large c andf with T cf , we can take so thatf •g = 0 by Lemma 2.7. Then there exists a morhismP → kerf by the universality of the kernel. The commutative diagram descends to the commutative diagram We can lift h ′ toh ′ :P → kerf a for some a ∈ R ≥0 . Take b ∈ R ≥0 so that T bι a •h ′ = T a+bg is satisfied. Then we again get a commutative diagram.
On the other hand, we have the commutative diagram (2.14) By the universality of kerf b , we have T a+bh = T bh′ . Hence h = h ′ . This completes the proof.
Proof of Proposition 2.5. It remains to show that the isomorphism between im and coim. Let f be a morphism in Mod I pre (Λ X ) andf be a lift of f . As shown in Lemma 2.11, imf is given by [imf ] and coimf is given by [coimf ]. Since Mod 0 (Λ X ) is abelian, there exists a canonical isomorphism imf ∼ = coimf . This also induces an isomorphism between imf and coimf . This completes the proof.
Proof. This is obvious from Lemma 2.11.
It is useful to state a kind of the converse of the above corollary.
be an exact sequence of Mod I pre (Λ X ). Then there exists an exact sequence in Mod 0 (Λ X ) which is a lift of the above sequence.
Proof. Take a liftṼ ′f Then we have an exact sequence 0 →Ṽf − →Wg − →X → 0. (2.17) Heref andg are caonical morphisms. We have an associated morphismṼ ′ →Ṽ. In Mod I pre (Λ X ), this associates a morphism V → [Ṽ] = [kerg ′ ] = ker g. By the exactness of the given sequence, HenceṼ is a lift of V. In a similar way, one can see thatX is a lift of X . This completes the proof.
3 The category Mod I (Λ (X,D) ) In this section, we glue up Mod I pre (Λ X ) to obtain a modified category, especially for noncompact manifolds.

Topological space with boundary
In this paper, a topological space with boundary is a pair (X, D X ) of a good topological space X with a closed subset D X of X. We say D X is the boundary of (X, D X ) and X\D X is the interior of (X, D X ). A morphism between (X, D X ) and (Y , D Y ) is a continuous map f between X and Y preserving the interiors. We denote the interiors by X := X\D X and Y := Y \D Y . We also denote the induced map between interiors by f : X → Y by the abuse of notation. Let (X, D X ) be a topological space with boundary. The site Open (X,D X ) is defined by the following data: the underlying category is the category of open subsets of X\D X , a collection of open subsets {U i } i∈I in X\D X is said to define a cover of U if there exists a subset J of I such that the subcollection {U i } i∈J still defines an open covering of U and is locally finite over X. The following is clear:   Proof. Let {U i } i∈I be a cover of U in Open (Y ,D Y ) . Let J ⊂ I be as in the definition of the cover. Then {f −1 (U i )} is an open covering of f −1 (U ) in X. Let x ∈ X Take x ∈ X, then there exists a small neighborhood V of f (y) such that V only intersects with a finite subset of {U j } j∈J . Then f −1 (V ) also only intersects with a finite subset of {f −1 (U j )} j∈J . Hence {f −1 (U i )} i∈I is a cover of f −1 (U ) in Open (X,D X ) .

The category Mod
Let (X, D X ) be a topological space with boundary. We set X := X\D X . Let U ⊃ V be open subsets of X. Then we have a restriction functor Lemma 3.5. This restriction functor is exact.
Proof. A short exact sequence in Mod I pre (Λ U ) can be lifted to an short exact sequence in Mod 0 (Λ U ) by Corollary 2.12. Then we can restrict it to an exact sequence in Mod 0 (Λ V ). By Lemma 2.11, this also gives an exact sequence in Mod I pre (Λ V ).
These maps form a presheaf of categories over the site Open (X,D X ) . This is not always a stack (even a prestack) because the tensor product ⊗k on the hom-space breaks the sheaf property.
Take the stackfication (resp. prestackification) of this stack with respect to Open (X,D X ) . We denote it by Mod I (X,D X ) (resp. Mod I ps (X,D X ) ). See Appendix for a short exposition of stackification.
Definition 3.6. The global section category of Mod I (X,D X ) is denoted by Mod I (Λ (X,D X ) ). For a manifold X, we set Mod I (Λ X ) := Mod I (Λ (X,∅) ).
Proposition 3.7. The category Mod I (X,D X ) (U ) is an abelian category for any U ∈ Open (X,D X ) .
Proof. We will only consider about kernels. The similar argument holds for cokernels, images and coimages. Let f : V → W be a morphism in Mod I (X,D X ) (U ). Then there exists a covering {U i } of U such that we have a descent data f i : If it is necessary, we can replace the covering with a finer covering so that each f | U i is represented by a morphism f i : Then we have ker(f i ) since Mod I pre (Λ U i ) is an abelian category. Since the restriction functors are exact (Lemma 3.5), we have ker . This further gives a descend data and glues up to an object K ∈ Mod I (X,D X ) (U ). For a morphism g : X → V with f • g = 0, by taking a sufficiently fine cover {U i }, we can represent them in Mod I pre (Λ U i ). Then one get a unique factorizing morphism X | U i → ker f i . Again by taking a finer covering as in the previous part of the proof and the universality, the set of these factorizing morphisms gives a descent data and can be glued up into the unique factorizing X → K. This shows K is ker f . Let U be an open subset of X. Let α U : Mod I pre (Λ U ) → Mod I (X,D) (U ) be the canonical functor.
Lemma 3.8. The functor α U is an exact functor.
Proof. Since kernels, cokernels, images, and coimages are defined locally, the assertion is obvious.
Lemma 3.9. If U is compact, the functor α U is fully faithful.
Proof. We set D U := D X ∩ U . To show the claim, it is enough to prove Hom Mod I pre (Λ U ) (V, W) is a sheaf over the site Open (U ,D U ) . Since U is compact, any cover in Open (U ,D U ) has a finite subcover.
We first assume that there exists a finite cover {U i } of U such that the restriction of f ∈ Hom Mod I pre (Λ X ) (V, W) to each open subset is zero. Letf ∈ Hom Mod 0 (Λ X ) (Ṽ,W) be a representative. Then the restriction of f to each open subset U i is represented byf there exists big T a such that T af | U i = 0 by Lemma 2.7. Let A be the maximum of those a's. Then T Af = 0. Hence f = 0.
Let {f i } ∈ Hom Mod I pre (Λ U i ) (V| U i , W| U i ) satisfies the descent condition. Depending on i, we have a set of liftsf i :Ṽ| U i →W| U i a i in Mod 0 (Λ U i ). In our situation, we can take a i = a j for any i, j, since the indexes are finite. Then we resetW withW a i by Lemma 2.11. On U i ∩ U j ,f i andf j may not coincide, but f i = [f i ] coincides with f j = [f j ]. Hence there exists T a ij such that T a ijf i = T a ijf j . By taking the maximum among a ij , we resetf j by T af j then the set {T af i } satisfies the descent condition in Mod 0 (Λ U ). Hence we get a glued morphism.
Corollary 3.10. If X is compact, Mod I pre (Λ X ) is an abelian subcategory of Mod I (Λ X ).
Proof. The embedding is given by the proof of Lemma 3.9.
Proof. This is a consequence of Lemma 3.8 and Lemma 2.12.

Finite limits and finite colimits
Since Mod I (Λ (X,D X ) ) is an abelian category, it admits finite limits and finite colimits.
Lemma 3.12. Let F : U → Mod 0 (Λ X ) be a finite diagram without loops in Mod 0 (Λ X ). We have Proof. We will only prove the first one. The second one can be proved in a similar manner. It is enough to show that the left hand side satisfies the universality of the right hand side. Let V be an object which is over [F ]. Locally, we have a liftṼ → F . By the universality, we get a morphismṼ → lim ←− Again, from the discussion of Section 5, we can conclude This is a contradiction.

Operations
In this section, we will develop the six functors. As above, (X, D X ), (Y , D Y ) are topological spaces with boundaries. We set X := X\D X and Y := Y \D Y

Internal hom
LetṼ,W be objects in Mod 0 (Λ X ), the internal hom sheaf is defined by the assignment for an open subset U ⊂ X. This is canonically an R-graded Λ X -module. Let V, W be objects of Mod I (Λ (X,D X ) ). Then there exists an open covering Lemma 3.14. The set {[Hom(Ṽ i ,W i )]} satisfies the descent and gives an object of Mod I (Λ (X ,D X ), which we will denote by Hom(V, W). This is independent of the choice of local lifts.
We can take a liftf ijk :Ṽ i | U ijk →Ṽ j | U ijk a of this morphism and that of the inversẽ becomes 0 after multiplying T c for sufficiently big c by Lemma 2.7.
For simplicity, let us assume that W has a global lift i.e., W = [W] although we can do the same for general W. We have the following induced morphisms: . A similar argument as was done in Proposition 3.7 gives a gluing of these isomorphisms to give a global object in Mod I (Λ X ). The independence of the choice of local lifts is also clear.
for V ⊂ U . Then this assignment does not depend on the choice of the liftṼ U . Hence one can associate a sheaf over Open (X,D X ) . We write it V ⊗ k.
Lemma 3.15. For V, W ∈ Mod I (Λ (X,D X ) ), the sheaf Hom(V, W)⊗k over Open (X,D X ) is canonically isomorphic to Hom Mod I Proof. This is obvious from the construction.

Tensor product
First, for R-graded Λ-modules V and W , their tensor product is defined as follows: where the equivalence relations are generated by LetṼ,W be objects of Mod 0 (Λ X ). Then the tensor productṼ ⊗W is defined by the sheafification of the assignmentṼ } satisfies the descent and gives an object of Mod I (Λ (X,D X ) ), which we will denote by V ⊗ W. This is independent of the choice of local lifts.
Proof. We can prove in a similar manner as in the proof of Lemma 3.14.

Tensor-Hom adjunction
Proposition 3.17. For V, W, X ∈ Mod I (Λ (X,D X ) ), we have the following: (3.10) Proof. Let {U i } be a covering of X such that we have lifts of V, W, X over each open cover. Let us denote the liftings byṼ i ,W i ,X i . Then we have a canonical isomorphism (3.11) By tensoring k and taking sheafification over Open (U i ,U i ∩D X ) , we have an isomorphism as a sheaf over Open (U i ,U i ∩D X ) . The isomorphisms over U i 's are glued up and give a desired result.
Corollary 3.18. In the same setting as above, we have the following: Proof. This is clear from the above proposition.

Push-forward
We will define push-forwards for a class of morphisms.
In the following, we only consider the following class of maps: Definition 3.19. We say a morphism f : Remark 3.20. For a locally closed subset U ⊂ X, a canonical morphism (U, ∅) → (X, ∅) is not tame in general. However (U , U \U ) → (X, ∅) is tame. In this sense, we will consider the latter one as a standard inclusion morphism.
Let V be an object of Mod I (Λ (X ,D X ) ) and f be a tame map (X, In this case, we simply set (3.14) where push-forward of R-graded Λ X -moduleṼ is defined by Gr a f * Ṽ := f * Gr aṼ .
Lemma 3.21. This is well-defined.
Proof. LetṼ ′ be another representative. Take a covering {U i } of Open (X,D X ) such that we have By pushing forward these equations, we have (3.16) by Lemma 3.12. Combining with the first part of the proof, we get an isomorphism Moreover it does not depend on the choice of coordinates and lifts.
Proof. It can be proved by a similar argument as in Lemma 3.21.

Pull-back
Let V be an object of Mod I (Λ (X,D X ) ). There exists an open covering {U i } of X with R-graded we assign a sheaf [f −1Ṽ i ] and these can glue up together. We will denote the resulting object by f −1 V.

Push-Pull adjunction
Let V be an object of Mod I (Λ (X,D X ) ), W be an object of Lemma 3.23. We have the following natural isomorphism: Then By the definition of the push forward, we have This completes the proof.
Lemma 3.24. Assume the same setting as above. Then Proof. Taking ⊗k and the global sections (as in the paragraph above Lemma 3.15) of both sides of Lemma 3.23, the right hand side becomes Hom This completes the proof.

Proper push-forwards
Let V be an object of Mod I (Λ (X,D X ) ) and f be a map (X, D X ) → (Y , D Y ). We first assume that V hasṼ with [Ṽ] ∼ = V. In this case, we simply set Lemma 3.25. This is well-defined.
Proof. This can be proved in the same way as the proof of Lemma 3.21.
Again by the same construction as in the case of push-forwards, we can define f ! V in general under the assumption of the tameness.
Assumption 3.26. In the following, when we consider f * or f ! , we always assume the tameness of f .
In this section, we develop fundamentals about derived operations for Mod I (Λ (X ,D X ) ).

Injectives and flats
Proof. The first part is almost trivial. Let us prove the second part.
Let us take a locally finite covering We also get a liftW ih Then they are trivially glued up to give a desired lift of g.
Proof. Take V ∈ Mod I (Λ (X,D X ) ). Then there exists a locally finite covering {U i } of X with liftingṼ i . As usual, one can embedṼ i to an injective objectĨ i which is a product of skyscraper sheaves.
Hence we have the inclusion where the latter is a locally finite direct sum hence it exits. By Lemma 4.1, [ι i * Ĩi ] is also an injective object. This completes the proof.
The above proof also shows the following:

Flats
Lemma 4.4. LetF be a flat R-graded Λ-module. Then [F ] is a flat object.
Proof. Let V → W ∈ Mod I (Λ (X,D X ) ) be an injection. Let us take an open covering Here one can takef i as an injection by Lemma 2.13.  By a similar argument as in Corollary 4.3, we get the following:

Derived functors
Note that right and left exactness of various functors f * , f ! , f −1 , Hom, ⊗ are the same as in the case of k-modules, according to Lemma 2.7.

Derived functors
Proof. Since [·] is an exact functor, it suffices to show for an object Mod 0 (Λ X ) by a standard argument in homological algebra. ThenṼ has an injective resolutionĨ (4.1) This completes the proof.
Proof. This is clear from the definition of f −1 and its exactness on Mod 0 (Λ X ).
Proof. One can prove by the same argument as in Lemma 4.7 by using Corollary 4.6.

Derived adjuntions
Lemma 4.10. There exists the following isomorphism Proof. This can be proved by a standard argument. Let us take a flat resolution F of W and an injective resolution I of X . Then RHom(F, I) ≃ Hom(F, I) is again an injective object. Actually, we have Hom(−, Hom(F, I))) ∼ = Hom((−) ⊗ F, I) (4.3) by Lemma 3.17. Then both sides of the equality in the statement is quasi-isomorphic to Hom(V ⊗ F, I). This completes the proof.
Lemma 4.11. There exists the following isomorphism Proof. By Lemma 3.24 and the exactness of f −1 imply that push-forward of an injective is again injective. Also, pull-back of a flat object is again flat. Let F be a flat resolution of V and I be an injective resolution of W. By replacing with these resolutions, we can work with underived functors, then Lemma 3.23 completes the proof.

Upper shriek
To construct upper shriek, we follow the argument in [KS90]. Let f : Y → X be a map. Assume that f ! : Mod(Z X ) → Mod(Z Y ) has finite cohomological dimension. LetṼ be an object of Mod 0 (Λ X ) and K be a flat f -soft Z Y -module. We define a presheaf by ( This is actually a sheaf by [KS90, Lemma 3.1.3]. We set f ! KṼ := a∈R Gr a (f ! KṼ ) which is an object of Mod 0 (Λ Y ). Let us moreover supposeṼ be an injective object.
Lemma 4.12. Under the above assumption, we have the following: 2. For anyW ∈ Mod 0 (Λ Y ). we have a canonical isomorphism Proof. This is done by the same argument as in the proof of [KS90, Lemma 3.1.3].
Let us take K by the following. Let K + (Mod 0 (Λ X )) be the homotopy category of injective complexes bounded below of objects in Mod 0 (Λ X ). Then we have an equivalence D + (Mod 0 (Λ X )) ∼ = K + (Mod 0 (Λ X )). We set the composition (4.7) Lemma 4.14. The functor f ! is the right adjoint of Rf ! . We moreover have by the above lemma. Since f !W ⊗ K ≃ Rf !W , we complete the proof of the first assertion.
The second assertion can also be proved by the argument of the proof of [KS90, Proposition 3.1.10].
Let us now discuss the upper shriek in D b (Mod I (Λ (X ,D X ) )). Let V be an object of Mod I (Λ (X,D X ) ) and K be a Z Y -module.
Take a locally finite covering Shriek adjunction Proposition 4.19. There exists a functorial isomorphism: Proof. First, note that Rf ! W ≃ f ! (W ⊗ K) which is deduced from the local consideration. Let I be an injective resolution of V and C(Mod I (Λ (X ,D X ) )) be the category of bounded complexes of Mod I (Λ (X,D X ) ). Then the left hand side of the desired equality is (4.11) We also have a morphism coming from the morphism V → I. We would like to prove this is an isomorphism. Let us see locally on Y . From the construction in Lemma 4.2, the complex I is coming from an injective objectĨ locally. Hence we have an isomorphism Here we used the fact that [·] is exact and f !Ĩ is injective. Then Lemma 4.22 completes the proof.
Proposition 4.20. There exists a functorial isomorphism: (4.14) Proof. As usual sheaves, we have a canonical morphism We would like to see the composition is an isomorphism. By a local consideration, this can be deduced from the usual case.

Formulas
Lemma 4.21. Let δ : Proof. This is clear from the same formula for usual sheaves.
As in Lemma 3.15, we can relate R Hom and usual Hom as follows. Let V, W ∈ D b (Mod I (Λ (X,D X ) )), then from R Hom(V, W) ∈ Mod I (Λ (X ,D X ) ), we can construct a complex of sheaves R Hom(V, W)⊗ k as in the paragraph before Lemma 3.15. Actually ⊗k is exact, as we will see in the proof of Lemma 6.1. This completes the proof. D Y ) )), and a tame morphism f : (X, D X ) → (Y , D Y ), the followings hold: Hom(W, X )).
Lemma 4.24 (Projection formula). We have the following: Proof. We use Yoneda, Lemma 4.23, ad Proposition 4.20: ≃ Hom(f * V ⊗ L W, X ). (4.17) We omitted the subscripts to Hom-spaces to shorten the notation. This completes the proof.
Lemma 4.25. We have the following formula Proof. We use Yoneda, Lemma 4.23, Lemma 4.24: (4.19) We omitted the subscripts to Hom-spaces to shorten the notation. This completes the proof.
Lemma 4.26. We have the following formula for V, W ∈ D b (Mod I (Λ (X,D X ) )).
Proof. We use Yoneda, Lemma 3.23, Lemma 4.23: (4.21) We omitted the subscripts to Hom-spaces to shorten the notation. This completes the proof.

Irregular constructibility
In this section, we introduce the notion of C-constructibility for objects in Mod I (Λ (X,D X ) ). It is defined in the same way for stratification as usual constructible sheaves but with a strong assumption on gradings coming from Sabbah-Mochizuki-Kedlaya's Hukuhara-Levelt-Turritten theorem. In this section, we consider (X, D X ) = (X, ∅) with X is a complex manifold. We denote Mod I (Λ (X,D X ) ) by Mod I (Λ X ).

Formal structure
In this subsection, we recall as a motivation the theory of formal structures of meromorphic connections initiated by Sabbah [Sab00] and developed by Mochizuki (algebraic case) [Moc11] and Kedlaya (analytic case) [Ked11]. Let Z be a divisor in a complex manifold X and O x be the formal completion of O X at x ∈ X. Let M be a meromorphic connection over X with poles along Z. We set M Definition 5.1.
1. For φ ∈ O( * Z) x , we set E(φ) to be O( * Z) x as a O x -module with a connection ∇ over O x such that ∇s := ∂(φ) · s (5.1) for the generator s.
2. We assume that Z is a normal crossing divisor and take a local coordinate Definition 5.2. We continue the notations in Definition 5.1.2.

A good decomposition of M x is an isomorphism
where φ α ∈ O( * Z) x and each R α is regular with the conditions (a) Each φ α has the form u m j=1 x −i j i for some unit u ∈ O x and nonnegative integers i 1 , ..., i m .
for some unit u ∈ O x and nonnegative integers i 1 , ..., i m .
2. We say M admits a good decomposition at x ∈ Z if M x admits a good decomposition.
In general, meromorphic connections do not have good decompositions as explained in [Sab00]. Sabbah's conjecture says that they do after modifications, which is proved by Mochizuki and Kedlaya.  Let ̟ : X(Z) → X be the real blow-up of X along Z (with real analytic structure specified in [DK16b]). Let C ∞,temp X (Z) be the subsheaf of the sheaf of C ∞ -functions consisting of functions which are tempered at the exceptional divisor. Let further A X(Z) be the subsheaf of C ∞,temp X (Z) consisting of functions whose restrictions on X\Z are holomorphic. We set D A

X(Z)
Suppose that M has a good decomposition α∈I E(φ α ) ⊗ R α at x. For each φ α , by taking a representative locally around x, we set E(φ α ) to be a meromorphic connection (O( * Z), ∇) defined by ∇s := ∂(φ)s for the generator s. We also set R α to be a regular meromorphic connection defined locally around x corresponding to R α .
The following thoerem is proved in [Moc11] and explained in [Sab13].

Irregular constant sheaf Λ φ
In this subsection, we prepare some preliminary lemmas concerning a class of modules. Let (S, D S ) be a topological space with boundary. Let φ be a C-valued continuous function over S := S\D S . We set where k t≥Re φ is the constant sheaf supported on the set {(s, t) ∈ S × R | t ≥ Re φ(s)} and p : S × R → S is the projection.
Proof. Since the sheaf is globally presented as a direct sum, the restriction morphism preserves grading. The Λ-action is given as follows: For b ∈ R ≥0 , we have a canonical morphism This action gives the action of T b .
We would like to see the structure of Λ φ S a little bit closer.
Lemma 5.7. Let U be a connected open subset of S such that φ| U is bounded.
. This is the kernel of the restriction morphism Γ(U × R, k t≥Re φ ) → Γ(U × (−∞, −a), k t≥Re φ ). Since U is connected, the set defined by t ≤ Re φ is also connected. For given x ∈ S, let us set as follows: (5.5) These are R-graded Λ-modules with obvious gradings. Note that these are torsion-free Λ-modules and the ring Λ has a valuation. Hence these modules are flat. From this lemma, the following is clear. Proof. Let V → W be an injective morphism in Mod I (Λ (S,D S ) ). We would like to show the There exists a covering {U i } of S which is locally finite in S such that there exists representatives V i , W i of V and W over each U i . It is enough to prove the injectivity over each U i . By Lemma 2.13, one can assume the restriction V i → W i is still injective. Since the tensor product commutes with taking stalks, it reduces to show that V Corollary 5.8) is a torsion-free Λ-module, this completes the proof.
Lemma 5.11. Let φ 1 and φ 2 be C-valued continuous functions over connected S such that max{0, Re φ 1 − Re φ 2 } is bounded. Then there exists a canonical idenitification If moreover Re φ 1 − Re φ 2 is bounded, two objects are isomorphic.
Proof. Since max{0, Re φ 1 − Re φ 2 } is bounded, there exists a large c ∈ R such that Re φ 2 + c ≥ Re φ 1 . The restriction map k Re φ 1 ≥t → k Re φ 2 +c≥t induces a morphism Λ φ 1 S → Λ φ 2 S c of R-graded Λ S -modules. If max{0, Re φ 2 − Re φ 1 } is also bounded, in the same way, we also have a morphism Λ φ 2 S → Λ φ 1 S d for some d ≥ 0. The composition Λ φ 1 S → Λ φ 1 S c + d is given by T c+d . This is the identity of Λ φ (S,D S ) in Mod I (Λ X ). The same for the other direction. This completes the proof of the second part of the statement. We call the morphism Λ φ 1 S → Λ φ 2 S and its scalar multiples standard morphisms. In the below, we will see there are only standard morphisms.
Let f be a nonzero morphism in Hom Mod I (Λ (S,D S ) ) (Λ φ 1 (S,D S ) , Λ φ 2 (S,D S ) ). Let us take a representativef : Λ φ 1 S → Λ φ 2 +c S as a morphism of R-graded Λ-modules locally on U ⊂ S. We can take so that c + Re φ 2 > Re φ 1 and replace φ 2 with φ 2 + c We consider d ∈ R such that the grading d-part off is nonzero. To see this part more explicitly, let us prepare some notations. Let Then we have a commutative diagram Since S 1 and S 2 are connected, the hom-space between them is 1-dimensional. Hencef d is induced by a standard morphism. This completes the proof.
We prepare the following crucial lemma. The corresponding observation in the theory of enhanced ind-sheaves is a key to the formulation of irregular Riemann-Hilbert correspondence [DK16b]. Proof. For f ∈ Hom Mod I (Λ (S,D S ) ) (Λ φ 1 (S,D S ) , Λ φ 2 (S,D S ) ), let us take a representativef : Λ φ 1 (S,D S ) → Λ φ 2 +c (S,D S ) as a morphism between R-graded Λ S -modules. Since Re φ 2 −Re φ 1 is negatively divergent, there exists a neighborhood U of D S such that Re φ 2 + c − Re φ 1 is negative on U \D S . Hence over U \D S , the restriction off is zero there. By Lemma 5.11 and the connectedness of S, f is zero everywhere.
We also give the following.
Proof. We have Gr a Λ φ i S = k {x|Re φ(x)>−a} for i = 1, 2. Hence we have a map Gr a Λ φ 1 +φ 2 Hence the kernel and cokernel of m is killed by T a for any a ∈ R. Therefore the kernel and cokernel are zero in Mod I (Λ (S,D S ) ). This completes the proof.
Similarly, we have Proof. One can prove in a similar way as in the proof of Lemma 5.13.
The following will be repeatedly used later.

Definition
Let V be a neighborhood of 0 ∈ C n and consider a simple normal crossing D I = i∈I {z i = 0}∩V . For A := {a i } ∈ Z I , Φ A : C n → C n is defined by z a i i where a i = 0 for i ∈ I.
Definition 5.16. 1. A correspondence f : V \D I → C is a multi-valued meromorphic function if there exists A := {a i } ∈ Z I and a meromorphic function 2. A finite set of multi-valued meromorphic function is said to be good, if it satisfies the conditions in Definition 5.2 after taking the pull-backs along Φ A .
For a multi-valued meromorphic function φ and an open subset U on which φ is represented by a set of single-valued holomorphic functions {φ k } k∈K , we set Λ φ := k∈K Λ φ k .
For S a locally closed complex submanifold X, consider (S, D S := S\S) as a topological space with boundary.
Definition 5.17. Let V be an object of Mod I (Λ (S,D S ) ). We call V is a good irregular local system if the followings hold: 1. D S is normal crossing.
Here ι (U k ,U k ∩D S ) is the canonical map induced by the inclusion U k ֒→ S.
If the set of multi-valued functions is actually the set of meromorphic functions, we call it a unramified good irregular local system.
Remark 5.18. We believe the goodness assumption in 3.b in the above can be removed by a similar consideration done in [Moc16].
Proof. This is just from the definition of multi-valued meromorphic functions. We say an irregular local system is single-valued type (resp. multi-valued type) if the good irregular local system appeared in 2 is unramified (ramified).
Let V be an irregular local system on (S, D S ). Take a point x ∈ D S . Then by the definition of irregular local systems, there exists a relatively compact open neighborhood U of x with a modification p : U ′ → U . Then for any y ∈ p −1 (D S ) =: D ′ , there exists a finite cover {U k } k of U ′ \D ′ given in the definition of good irregular local systems. We have we get a finite covering U of U \D S such that V| U ,D S ∩U is isomorphic to a direct sum of irregular constant sheaves for each U ∈ U .
Definition 5.23. We call a finite covering U of U \D S given above a sectorial covering of V around x.
Lemma 5.24. For V, W ∈ Mod I (Λ (S,D S ) ) and x ∈ D S , there exists a neighborhood U of x with a modification (U ′ , D ′ ) → (U, D) such that p −1 (V| (U,U ∩D S ) ) and p −1 (W| (U,U ∩D S ) ) are irregular local systems. In particular, V and W have a common sectorial covering.
Proof. This is standard.
Then we would like to define one of the fundamental objects in this paper.
Definition 5.25. Let V be an object of Mod I (Λ X ). We say V is irregular constructible if the followings hold: There exists a C-analytic stratification S of X such that the restriction V| (S,D S :=S\S) to each stratum S ∈ S is an irregular local system as an object of Mod I (Λ (S,D S ) ).
Let us denote the full subcategory of Mod I (Λ X ) spanned by irregular constructible sheaves by Mod ic (Λ X ).
Proof. Since Mod I (Λ X ) is abelian, it suffices to show kernels, cokernels, images, and coimages of morphisms between irregular constructible sheaves are also irregular constructible sheaves. Let f : V → W be a morphism between irregular constructible sheaves. One can take a common C-Whitney stratification for V and W. Then it suffices to show Lemma 5.27 below.
Lemma 5.27. Kernels, cokernels, images, coimages of morphisms between irregular local systems are irregular local systems.
To prove Lemma 5.27, we prepare some notions and lemmas.
Definition 5.28. Let φ i (i = 1, 2) be meromorphic functions over U with poles in D. We say φ 1 and φ 2 are equivalent if there exists a bounded holomorphic function φ over U such that φ 1 = φ 2 + φ. We denote the set of meromorphic functions over (U, D) modulo this equivalence relation by M(U, D).
Recall that Λ φ 1 (U,D) and Λ φ 2 (U,D) are canonically for φ 1 = φ 2 ∈ M (U, D) by Lemma 5.11. Furthermore, we can pull-back more by a covering map Φ A to make p −1 V and p −1 W unramified irregular local systems. Then again, Φ −1 It suffices to show that this is an irregular local system. So we reset the notations. Let V and W be unramified good irregular local systems and f : V → W be a morphism. Then there exist sets of meromorphic functions Φ V and Φ W over (U, D) which are appeared in the definition of irregular local system. Take a point x ∈ D, a neighborhood U of x, and a sectorial covering U of U \D for V and W. On each U ∈ U , we have isomorphisms V| U ∼ = φ∈Φ V Λ φ (U ,U∩D) and W| U ∼ = ψ∈Φ W Λ ψ (U ,U∩D) . Suppose the following; there exists a sector U ∈ U such that the restriction of f to the is nonzero again. This implies max{Re φ − Re ψ} is bounded by Lemma 5.11. We can continue this procedure and we eventually will arrive a sector on which φ − ψ is negatively divergent since φ = ψ. This is a contradiction.
Hence we cannot have such a morphism. This means f | U is diagonal with respect to indices M (U, D) × M (U, D). Hence the morphism f | U is represented by a sum of c · T a : Λ φ (U ,U ∩D) → Λ φ (U ,U∩D) where c ∈ k by Lemma 5.11. The A(c · T a ) ia again of the form of a sum of Λ φ (U ,U ∩D) . This completes the proof.
We prepare the following lemma for the next subsection.
Lemma 5.29. The category Mod ic (Λ X ) is a thick subcategory of Mod I (Λ X ).
be an exact sequence in Mod I (Λ X ) with V, W ∈ Mod ic (Λ X ). Let S be a common stratification of V and W. Since pull-backs are exact, we can reduce to the case that V, W are irregular local systems on (S, D S ). For any point x ∈ D S , there exists a neighborhood U of x such that U \D S has a finite sectorial covering We have already seen that RHom(Λ ψ is free. This completes the proof.

Derived category and six operations
Definition 5.30. Cohomologically irregular constructible Λ (X,D X ) -module is an object of D b (Mod I (Λ (X,D X ) )) such that all the cohomologies are irregular constructible sheaves. We denote the full subcategory spanned by those objects by D b ic (Λ X ) Proposition 5.31. The category D b ic (Λ X ) is a triangulated category.
Proof. This is clear from Lemma 5.29.
We will now see Grothendieck six operations on this category.
Proof. This is obvious from Lemma 4.26 and Lemma 5.13.

Verdier duality
First, we prepare the following useful lemma: Let (S, D S ) be a topological space with boundary. Let U be an open subset of S and U be the closure inside S. Consider the map i : (U , D U := U \U ) → (S, D S ). We denote the closed complement of U in S by V . We denote the map j : (V, V ∩ D S ) → (X, D X ).
Lemma 5.33. There exists an exact triangle: (5.14) Proof. Note that i and j are tame maps. Then this is clear from the corresponding statement for usual sheaves and the commutativity results for [·] proved in 4.2.
For the constant map a X : X → * , we set ω Λ First note tha following: Proof. This is clear from Lemma 5.15.
Then we have: Proof. Let S be a stratification of V. Let U be the union of open subsets of S. By applying Lemma 5.33, we have an exact triangle Then we have (5.17) Since it is clear that irregular constructibility are preserved under i ! and j * , we can prove the desired result by the induction of the dimension of the strata and Lemma 5.34.
. Then a natural morphism V → DDV is an isomorphism.
Proof. It is also enough to show the statement for irregular local systems. Then the statement is clear from DDΛ φ X = Λ φ X .
Lemma 5.37. Let V, W ∈ D b ic (Λ X ). We have Proof. We have This completes the proof.
Corollary 5.38. The contravariant functor D : Proof. This is clear from Lemma 5.37 and Lemma 5.36.
Proposition 5.39. We have a natural isomorphism Proof. As usual, we can see that RHom(V, W) ≃ D(DW ⊗ L V). Then this is a corollary of the preceding results.

Pull-backs
Proof. For f −1 , this is clear from the definition. By using Lemma 5.36 below, we have Here the final form is in D b ic (Λ X ).

Proper push-forwards
Push-forwards are more difficult and we use irregular Riemann-Hilbert correspondence proved below.
Proposition 5.42. Let f : X → Y be a proper morphism. Then Proof. For V ∈ D b ic (Λ X ), take V ′ := V ⊗ k C. Then we get a holonomic D-module M := (Sol Λ ) −1 (V ′ ). Due to Malgrange [Mal04], there exists a good lattice on M. Hence the pushforward of M along f is again holonomic and we have f * V ′ ∈ D b ic ((Λ ⊗ k C) Y ) by [DK16b] and Lemma 8.12. Since the irregular constructibility is preserved under ⊗ k C. This completes the proof.

Global R-graded realization
The following proposition says "an object of D b ic (Λ X ) is actually a sheaf over X". It is logically not important, but conceptually makes us feel easy to irregular constructible sheaves. We use some results from the later sections to prove the following.
). This completes the proof.

Forgetting grading
In this section, we discuss the relationship between irregular constructible sheaves and constructible sheaves. For a topological space with boundary (X, D X ), we set X := X\D X .

Forgetting grading
Lemma 6.1. There exists an exact functor F : Mod I (Λ (X,D X ) ) → Mod(k X ). (6.1) Proof. For an object V, let us take a locally finite covering {U i } of X with representatives {Ṽ i } ⊂ Mod I pre (Λ X ). There exists an isomorphism f ij : . We can take a covering {U ijk } on which we have a descent data f ijk : Again, these morphisms can be glued up and give a k-module sheaf V ⊗ Λ k. By a similar argument, one can actually see this does not depend on the choice of lifts. For Then we get a set of morphisms {f i ⊗ Λ X k X }. One can see these are glued up to a morphism in Mod(k X ) depending only on f by a similar argument as above. The resulting morphism is denoted by F(f ). It is clear that this correspondence preserves the compositions. Hence F gives a functor. We would like to see the functor F is exact. Let be an exact sequence in Mod I (Λ (X,D X ) ). It is equivalent to that there exists a locally finite open covering {U i } of X such that we have an exact sequence over each U i . By Lemma 2.13, it can be lifted to an exact sequence of R-graded Λ X -modules Since tensor product is left exact, we get an exact sequencẽ It remains to showf i ⊗ id is injective. Let us take a homogeneous section of the kernel of f i ⊗ k. Since it is a subsheaf ofṼ i ⊗ Λ X k X , it is locally represented by the form s ⊗ 1. If s ⊗ 1 is nonzero, it means that T a · s = 0 inṼ i . Hence we have Λ U · s ֒→Ṽ i | U where U is the open set on which s is defined. Iff i (s) ⊗ 1 = 0, we have some T a such that T af i (s) = 0 by Lemma 2.6. Hence we have a sequence of morphisms over U of R-graded Λ-modules Proof. For an R-graded Λ X -module V, let us consider On the other hand, the sheaf f −1 • F(V) is a sheaf associated with the presheaf By the definition, (6.10) Hence they are the same.
Lemma 6.3. Let V ∈ Mod I (Λ (X,D X ) ) be an irregular local system. Then F(V) is a local system.

Proof.
There exists an open covering of U such that V is represented by a direct sum of irregular constant sheaves Λ φ . By the definition of F, it is enough to see Λ φ ⊗ Λ k is a constant sheaf over any enough small open subset. This is clear.
Proof. Let V ∈ Mod I (Λ (X,D X ) ) and take an injective resolution I • by using Proposition 4.2. Note that skyscraper sheaves Λ x used in this injective resolution are mapped to skyscraper sheaves k x . Combining with the exactness of F (Lemma 6.1), we can conclude that F(I • ) is an inejctive resolution of F(V). Hence we have Similarly, for a free R-graded Λ-module F, the module F(F U ) is a direct sum of k U , hence is flat. By Lemma 4.4, we can do a similar argument as above. This completes the proof.
Lemma 6.5. Let f be a proper map X → Y . We have an equality Proof. By Lemma 6.4, it is enough to show the underived version. For V ∈ Mod R (Λ X ) and an open subset U , both f ! •F(V) and F•f ! have V(f −1 (U ))⊗k over U . This completes the proof.
Lemma 6.6. Let i (X,D X ) : (X, D X ) → (X, ∅) be the canonical map and i X : X ֒→ X be the inclusion. We have an equality Proof. Again, we only prove the underived version. One can prove in a similar way to Lemma 6.5.

The case of irregular constructible sheaves
Proposition 6.7. The functor F is restricted to Mod ic (Λ X ) → Mod c (k X ).
Proof. For V ∈ Mod ic (Λ X ), let us take a stratification S of X. For each S ∈ S, let us denote the inclusions by i (S,D S ) : (S, D S ) ֒→ (X, ∅) and i S : S ֒→ X. Then we have i −1 S F(V) ∼ = F(i −1 (S,D S ) (V)) by Lemma 6.2. By Lemma 6.3, this is a local system. Hence F(V) is a constructible sheaf with respect to S.
We also denote the induced functor D b (Mod I (Λ X )) → D b (Mod(k X )) by F.
, since F is exact on the abelian categories (Lemma 6.1), we have ). By Proposition 6.7, we have F(H i (V • )) ∈ Mod c (k X ).
Lemma 6.9. If we have F(E) ≃ 0 for an irregular constructible sheaf E, we have E ≃ 0.
Proof. An irregular constructible sheaf is locally isomorphic to i∈I Λ φ i for some φ i 's. Since F( i Λ φ i ) ∼ = k |I| , F(E) ∼ = 0 is equivalent to |I| = 0. This means E ∼ = 0. This completes the proof.

R-constructible enhanced ind-sheaves
In this section, we recall the definition of R-constructible enhanced ind-sheaves. For more detailed accounts, we refer to the original [DK16b] and the survey [KS16]. Let X be a real analytic manifold. Let R be the two point compactification of R i.e. R ∼ = (0, 1) ֒→ [0, 1] = R. We define the category of enhanced ind-sheaves by two-steps: First, we set ). We set The category of enhanced ind-sheaves over X is defined by 3) The triangulated category E b (Ik M ) has monoidal operations + ⊗ and Ihom + . For a morphism M → N of real analytic manifolds, there are associated functors They form adjoint pairs Ef !! ⊣ Ef ! and Ef −1 ⊣ Ef * . We further set k E X := "lim" a→∞ k t≥a (7.6) as an object of E b (Ik M ). As usual, "lim" means Ind-colimit. 2. An enhanced R-constructible ind-sheaf E of E b (Ik M ) is said to be C-constructible if the following holds: There exists an open covering {U } of X and a C-stratification S U for each for each S ∈ S U are isomorphic to a direct sum of sheaves of the form k t≥φ(x) for some continuous function φ.
We denote the full subcategory spanned by R-constructible (resp. C-constructible) enhanced ind-sheaves by E b R-c (Ik M ) (resp. E b C-c (Ik M )). The category E b R-c (Ik M ) has a contravariant autoequivalence D, analogous to the Verdier dual.

From enhanced sheaves to Λ-modules
For a sheaf E on X × R, let us consider an object −a∈R p * Γ [−a,∞) E where p : X × R → X is the projection. This is equipped with the action of Λ as follows: The action of ). (7.8) Lemma 7.2. The correspondenceM is a left exact functor Mod(Ik X×R ) → Ind(Mod I (Λ X )).
Proof. SinceM is a functor obtained as the inditization of a left exact functor, which is again left exact [KS06].
We denote the right derived functor ofM by RM : Composing these with RM , we get ). (7.11) Lemma 7.3. Let E be an R-constructible sheaf over X × R. Then we have M (E Proof. By the definition of M and k E X , it is enough to show that the natural morphisms −a∈R are isomorphisms for any c ∈ R ≥0 . The cone is given by Since T c ∈ Λ is vanished on this object, this is zero. Hence the morphisms are isomorphisms. Lemma 7.4. The functor M is restricted to a functor E b R-c (Ik X ) → D b (Mod I (Λ X )), which is also denoted by M .
Proof. For an R-constructible enhanced ind-sheaf E, there exists a locally finite covering U of X such that we have E| U ×R ≃ E U + ⊗k E U and (n+2)-fold covers are empty. By the Cech construction, E is represented by a result of mapping cones of i ! (E U + ⊗k E U ). This impliesM (E) is obtained as a finite mapping cones ofM (i ! (E U + ⊗ k E U )). By Lemma 7.3, this means thatM (E) is in D b (Mod I (Λ X )). This completes the proof.
Let S be a locally closed subset in X and S be the closure of S in X and set D S := S\S. Let φ be a C-valued function on S and take Λ φ (S,D S ) . Since there exists a tame map i (S,D S ) : (S, D S ) → X, we get i (S,D S )! Λ φ (S,D S ) ∈ Mod I (Λ X ). We also set E φ := k Re φ≤t Proof. This is clear from the definitions and Lemma 7.3.
Lemma 7.6. There exists a canonical isomorphism (7.14) Proof. By Lemma 5.11 and Lemma 5.12, we have It is standard to see that the RHS also has the same formula. In the case that max{0, It is easy to see that the induced morphism only depends on the choice of f . By the proof of Lemma 5.11, this gives an isomorphism.

Irregular Riemann-Hilbert correspondence
In this section, we will prove our version of the irregular Rimann-Hilbert correspondence as a corollary of D'Agnolo-Kashiwara's one. In this section, we will work over C.

Notations for analytic D-modules
We refer the theory of analytic D-modules to [Kas03]. In this subsection, we simply recall the notations. For a complex manifold, D X is the sheaf of differential operators, Mod(D X ) is the category of left D-modules, and D b (D X ) is the bounded derived category of D-modules. We denote the full subcategory of D b (D X ) spanned by cohomologically holonomic D-modules by D b hol (D X ). The Verdier dual D is a contravariant autoequivalence of D b hol (D X ). For a morphism of complex manifolds f : X → Y , we can define the following functors: by using transfer D-modules D X←Y and D X→Y . The functor f † always preserves cohomologically holonomic modules. If f is proper, f also preserves cohomologically holonomic modules. For a proper f , the pair of functors form an adjoint pair f ⊣ f † . We also set f ⋆ :

Irregular Riemann-Hilbert correspondence using enhanced sheaves
We recall the irregular Riemann-Hilbert correspondence by D'Agnolo-Kashiwara: DK16b]). There exists a contravariant embedding Our convention is slightly different from the original one in [DK16b]: Let Sol E be the original one. We set Sol E := Sol E [dim X] We have Sol E := D•DR E where DR E is the same as the original one. We collect some properties of the irregular Riemann-Hilbert correspondence as follows: Proposition 8.2 ([DK16b, Theorem 9.4.8, Proposition 9.4.10]).
1. There exists a canonical isomorphism D • DR E ≃ DR E • D.

There exists an isomorphism Sol
We will also use the following fundamental result. Let Y be an analytic hypersurface of the complex manifold X. Take a meromorphic function φ with poles in Y ; φ ∈ O X ( * Y ). We set E φ := (D X · e φ )( * Y ).
Our convention for Sol E is shifted from D'Agnolo-Kashiwara's one to hold the following:

Irregular Riemann-Hilbert correspondence
Let us denote the essential image of Sol E by E b D (IC X ). Let π : Y ′ → Y be a result of normalization and a resolution of singularities of Y . Let E be the inverse image of the union of singularities of Y and M 1 . Then there exists a canonical morphism M 1 → M ′ 1 := π ⋆ (π ⋆ M)( * E) (8.6) Since (π ⋆ M)( * E) is a meromophic connection, M (Sol E (M ′ 1 )) is irregular constructible. Since the cone of this morphism is living on a divisor of Y , we can prove the desired statement by iterating these arguments.
Our version of irregular Riemann-Hilbert correspondence is the following: Theorem 8.5. The functor M is a contravariant exact equivalence: In particular, there exists a contravariant equivalence Let N 1 be the cone of a canonical morphism N → N ( * Y ). Then N 1 is supported on Y . By a similar procedure done in the proof in Lemma 8.4. We complete the proof of the fully faithfulness. Now we will prove the essential surjectivity. We only have to see that the functor hit each irregular local system by Lemma 5.33.
Let V be an irregular local system on (X, D X ). After a modification, we get a good irregular local system. This associates a good Stokes local system in the sense of [Sab13]. By the Riemann-Hilbert correspondence in [Sab13], this gives a meromorphic connection over the modification. By pushing forward to the original base, we get a desired object.
Remark 8.6. One can also prove Theorem 8.5 by using curve test criterion by Mochizuki [Moc16].
Corollary 8.7. There exists an exact equivalence

Functors
In this subsection, we prove the commutativity between Sol Λ and various functors. We assume that all the spaces are without boundary in this subsection.
In particular, (8.14) Proof. Since we know the functors are commutative wth Sol E , it is enought to see the commutativity with the functor M . Let us take an R-constructible sheaf E on X × R. Let f be the direct product of f : X → Y and id : R → R. Then we have . This proves the first line.
Hence it suffices to prove First, note that we have p * RΓ [a,∞) (E + ⊠ F) ≃ p * RΓ t 1 +t 2 ≥a (E ⊠ F). We also have a map By combining these, we get a map M (E) ⊠ M (F) → M (E + ⊠ F). It suffices to show that this map is locally an isomorphism. This is clear from Lemma 5.13 and an easy observation Proposition 8.10. We have Proof. Let δ : X → X × X be the diagonal embedding. Then M ⊗ N ∼ = δ † (M ⊠ N ). Then we have, where we used Lemma 4.21 and Lemma 8.9. The second claim follows form the adjunction. This completes the proof.
Proposition 8.11. Let f : X → Y . We have Proof. The first one is followed by the second one and Proposition 8.8. We have where we used Proposition 8.10 and the commutativity of Sol E with D ( [DK16b]). This completes the proof of the third line.

Irregular perverse sheaves
In this section, we define the irregular perverse t-structure on the category of irregular constructible complexes. Over C, the heart is equivalent to the category of holonomic D-modules.
We also prove t-exactness of various functors.
Definition 9.5. The heart of t-structre is called the category of irregular perverse sheaves and denoted by Ierv(k X ).
Theorem 9.6. The functor Sol Λ restrict to a contravariant equivalence Lemma 9.7. Let D i (i = 1, 2) be triangulated categories with t-structures (D ≤0 i , D ≥0 i ). Let F : D 1 → D 2 be a t-exact equivalence. Then F gives an equivalence between t-structures.
Proof. We have to show that F : D ≤0 1 → D ≤0 2 is essentially surjective. Let E be an object of D ≤0 2 . Then we have a standard triangle 1 . We can prove for the positive part in a similar manner. This completes the proof.
Proof of Theorem 9.6. By Lemma 9.7, it is enough to show that Sol Λ is t-exact. We only show the condition dim{supp H j (V)} ≤ −j. (9.9) The other case follows from the Verdier duality. Let M be a holonomic D-module. We will prove by the dimensional induction: Let us assume the assertion is true for any complex manifold with dim < dim X.
Let D be a divisor containing the singularities of M. Let p : X ′ → X be a resolution making D normal crossing. We denote the normal crossing divisor by D ′ . Then p † M is again a holonomic D X ′ -module (not a complex), since p is a submersion. If p † M satisfies the claim, we have a sequence of inequalities This means the claim also holds for M. Hence we can assume that D is normal crossing. Since the claim is local, we can further assume that D is a simple normal crossing: there is a set of coordinate hyperplanes {D i } such that D = i∈I D i . Consider the following exact triangle: Hence we also have Sol Λ (M( * D)) → Sol Λ (M) → Sol Λ (C) → (9.12) By Lemma 8.3, the complex Sol Λ (M( * D)) is concentrated in degree − dim X, it is enough to show the claim (9.9) for Sol Λ (C).
The holonomic complex C is supported on D and H j (C) = 0 for j = 0, 1. Hence the degree of DC is concentrated on −1, 0. Let ι i : D i ֒→ X be the closed embedding. Then by [HTT08, Proposition 1.5.16], the functor ι † i is a right derived functor. Hence ι † i DC is concentrated in −1, 0, 1. Hence ι ⋆ i C := Dι † i DC is also concentrated in degree −1, 0, 1. Let us write as Then the truncation has a natural map ι ⋆ i C → C i . By summing up the morphisms, we get a sequence of morphisms C → i∈I ι i ι ⋆ i C → i∈I ι i C i . Then we define the exact triangle (9.14) Then C 1 has degree 0, 1, 2. Since Sol Λ ( ι i C i ) = ι i * Sol Λ (C i ) and the assumption of the induction, the complex Sol Λ ( ι i C i ) satisfies the claim (9.9). Hence it is enough to show for C 1 . Since the morphism C → i∈I ι i C i is an isomorphism on the complement of intersections of D i 's, the complex C 1 is supported on intersections of D i 's. Set ι ij : D ij := D i ∩ D j ֒→ X for i < j. Again by a similar argument to C, we can see that ι ⋆ ij C 1 is concentrated on −2, ..., 2. By a similar truncation as above, we can define C ij with degree 0, 1, 2 and a map ι ⋆ ij C 1 → C ij . Again, we can define C 2 by the same argument and one can see that it is enough to show (9.9) for C 2 .
By proceeding this induction, we finally have a holonomic complex supported on i∈I D i with degree 0, ...., dim X − I. We can see that this holonomic complex again satisfies (9.9). This completes the proof.

Algebraic case
In this section, we deduce the algebraic version of the results.

Notations for algebraic D-modules
For the theory of algebraic D-modules, we refer to [HTT08]. For a smooth quasi-projective variety X, we denote the sheaf of algebraic differential operators by D X . We denote the category of left D X -modules by Mod(D X ), the bounded derived category by D b (D X ), and the full subcategory of cohomologically holonomic modules by D b hol (D X ). We denote the Verdier dual by D. For a morphism of algebraic varieties f : X → Y , we define f and f † by the same formula as in the analytic case. In algebraic case, both functors preserve holonomic objects without properness assumption. We set f ⋆ := D • f † • D and f ! := D • f •D. Then we have two adjoint pairs f ⋆ ⊣ f and f ! ⊣ f † .
Let X an be the complex manifold associated with X. The analytification functor is an exact functor (·) an : Mod(D X ) → Mod(D X an ). We also denote the induced functor on the derived categories by the same notation (·) an . It preserves the holonomicity. We note the following: 2. If f is proper, we have a canonical isomorphism f N an ≃ f an (N ) an for N ∈ D b hol (D X ).

Algebraic irregular constructible sheaves
Let X be a smooth quasi-projective variety. Let X be a smooth projective variety with a Zariski open embedding i X : X → X. We set j X : D X := X\X ֒→ X.
Definition 10.2. An object V ∈ Mod I (Λ (X ,D X ) ) is algebraic irregular constructible if the following holds: there exists an algebraic stratification S of X refining X = X ⊔ D X such that each restriction of V to S ∈ S is an irregular local system.
We denote the full subcategory of irregular constructible sheaves by Mod ic (Λ (X,D X ) ). Let i (X,D X ) : (X, D X ) → (X, ∅) be the canonical morphism, which is tame. We also denote the inclusion by i D X : (D X , ∅) → (X, ∅).
Lemma 10.3. The functors i (X,D X )! : Mod ic (Λ (X,D X ) ) → Mod I (Λ X ) is fully faithful embedding onto the full subcategory spanned by objects satisfying i −1 D X V ≃ 0. The functor i (X,D X ) * is also fully faithful. In both cases, the quasi-inverses are given by i −1 (X,D X ) .
Proof. This simply follows from Lemma 5.33 Lemma 10.4. The category Mod ic (Λ (X,D X ) ) does not depend on the choice of X.
Proof. We will prove the assertion in two steps, let us first assume that p : X ′ → X be a map between two projective compactifications of X extending id : X → X. Then it is clear that p * induces an desired equivalence of categories. Now let X ′ be an arbitrary projective compactification of X. Then there exists X ′′ with maps X ′′ → X and X ′′ → X ′ extending id : X → X. This can be done by taking a smooth blow-up replacement of the closure of the diagonal embedding X → X × X ′ . From the first part of this proof, we have done.
We will denote the category of algebraic irregular constructible sheaves by Mod ic (Λ X ).
Proof. One can prove by mimicking the proof of Lemma 5.29.
Let us denote the triangulated subcategory of D b (Mod I (Λ (X ,D X ) )) formed by cohomologically algebraic irregular constructible sheaves by D b ic (Λ X ). Let D b c (k X ) be the category of cohomologically algebraic constructible complexes.
Proposition 10.6. The functor F is restricted to a functor F : Proof. This is clear from the proof of Proposition 6.7.
It is also clear that what we proved in Section 5.4 also holds for D b (Λ X ). Addition to those, we have the following: Let f : X → Y be a morphism between algebraic varieties. Since we can always compactify this morphism as X → Y , we have a map of topological spaces with boundary f : (X, D X ) → (Y , D Y ). We set ). (10.1) Proposition 10.7. Let f : X → Y be a morphism between algebraic varieties. Then Rf * V, Rf ! V ∈ D b ic (Λ Y ) for D b ic (Λ X ).
Proof. This can be proved by the same argument used in Proposition 5.42 by using Proposition 10.11.

Algebraic Riemann-Hilbert correspondence
We first recall the following Malgrange's result.
Theorem 10.8 ( [Mal04]). If X is a smooth projective variety, analytic holonomic D X -modules are algebraic.
By using this, we have the following algebraic version of irregular Riemann-Hilbert correspondence.
Theorem 10.9. There exists an exact equivalence Proof. If X is projective, there is nothing to prove by Theorem 10.8. We suppose X is quasiprojective and X be a compactification of X. For M ∈ D b hol (D X ), we have i X M ∈ D b hol (D X ) where i X : X ֒→ X is the inclusion. We also set i (X,D X ) : (X, D X ) → (X, ∅) the canonical morphism. Then we get a functor 3) The middle equality is Chow's lemma. Note that the first three compositions are fully faithful.
Hence, to prove the fully faithfulness of Sol Λ X , it suffices to show that that image of Sol Λ 11 Fukaya categorical Riemann-Hilbert correspondence In this section, we speculate some constructions which explain the appearance of Λ-module more naturally by using Fukaya category. We hope more details or proofs will be discussed in future papers. Let us recall the following theorem: Let Z be a compact real analytic manifold and D b R-c (Z) be the bounded derived category of R-constructible sheaves.
Theorem 11.1 ( [NZ09,Nad09]). There exists a Fukaya-type A ∞ -category Fuk(T * Z) of T * Z and an equivalence D b R-c (Z) ≃ D Fuk(T * Z) (11.1) where the right hand side is the derived category of Fuk(T * Z).
For the definition of this kind of Fukaya category, see the original reference [NZ09]. We modify this story to explain irregular Riemann-Hilbert correspondence well. There are three ingredients.
The first ingredient is an observation that Nadler-Zalsow's Fukaya category is not enough sensitive for our purpose in the following sense.
Example 11.2. Let Z = R and consider functions f 1 = 1/x and f 2 = 1/x 2 defined over x > 0. For the sake of simplicity, we only consider locally around 0. Let us consider Λ f i . Then we have (11.2) Let L i be the graph of differentials in T * Z for each f i , which are Lagrangian submanifolds in T * Z.
We would like to consider Lagrangian intersection Floer theory CF (L 1 , L 2 ) and CF (L 2 , L 1 ) and compare them with hom-spaces in D b Mod I (Λ X ). Assume that they are well-defined. As usual, to deal with intersections at infinity, we consider Reeb perturbation for Lagrangians in the right of CF (·, ·). Then one can see that CF (L 1 , L 2 ) = k and CF (L 2 , L 1 ) = 0 (11.3) In the formalism of Nadler-Zaslow, this does not occur. Since both L i 's have the same asymptotic line, it should give the same object k (0,∞) in D b R-c (Z).
The first conjecture is the following.
Conjecture 11.3. There exists a version of Fukaya category Fuk m (T * Z) of T * Z which modifies Nadler-Zaslow's one to realize (11.3).
The second ingredient is R-graded realization of irregular constructible sheaf. For the below, we replace Z by X, which is a complex manifold. Let M be a meromorphic connection with poles in D. Then the corresponding object under the irregular Riemann-Hilbert correspondence is an irregular local system over (X, D). To simplify our explanation, we assume that the formal types of M is not ramified. We have a C ∞ function on X\D such that Λ f represents the irregular local system. The graph of the derivative of f gives a Lagrangian L M in T * X. Since the choice of f is only up to bounded modification, we can expect the following. One can also modify the above explanation to adopt to any irregular local system. Definition 11.9. A descent data for a cover {U i } i∈I is a set of objects {a i } i∈I (a i ∈ F(U i )) and a set of morphisms θ ij : a j | U i ∩U j → a i | U i ∩U j (i, j ∈ I) such that θ ii = 1, θ ij • θ jk = θ ik .
For an object a ∈ F( i∈I U i ), we can get a descent data by the restrictions.
Definition 11.10. A prestack is a stack if any descent data comes from the restrictions.
The existence of stackification (cf. [Moe02, Theorem 2.1]) tells us there exists a stackF associated to a given prestack. An object ofF(U ) for U ∈ Open (X,D X ) is represented by a cover {U i } i∈I and a descent data on this cover.