Sheaves of -infinity algebras and applications to algebraic varieties and singular spaces
We describe the -infinity algebra structure on the complex of singular cochains of a topological space, in the context of sheaf theory. As a first application, for any algebraic variety we define a weight filtration compatible with its -infinity structure. This naturally extends the theory of mixed Hodge structures in rational homotopy to -adic homotopy theory. The spectral sequence associated to the weight filtration gives a new family of multiplicative algebraic invariants of the varieties for any coefficient ring, carrying Steenrod operations. As a second application, we promote Deligne’s intersection complex computing intersection cohomology, to a sheaf carrying E-infinity structures. This allows for a natural interpretation of the Steenrod operations defined on the intersection cohomology of any topological pseudomanifold.
Key words and phrases:-infinity algebras, cochain theory, Godement resolution, Steenrod operations, -adic homotopy theory, weight filtration, intersection cohomology.
2010 Mathematics Subject Classification:
The singular cochain complex of every topological space , with coefficients in a commutative ring , has the natural structure of an -algebra. This structure encodes the commutativity of the cup product at the cochain level up to higher coherent homotopies and turns out to be extremely powerful in order to classify homotopy types: as shown by Mandell [Mandell], when and for sufficiently nice spaces, the -structure captures the -adic homotopy theory of the space. This result can be understood as an algebraization of -adic homotopy theory, analogous to Sullivan’s approach to rational homotopy via commutative dg-algebras. Moreover, finite type nilpotent spaces are weakly equivalent if and only if their singular cochains with integer coefficients are quasi-isomorphic as -algebras [Mandell2].
In this paper, we describe the -algebra structure on via sheaf theory. The study of sheaves of -algebras is not new and some related constructions appear in [HiSc], [ManCochains], [MaySheaf] and [RR0], [RR]. In particular, Mandell used the theory of homotopy limits of operadic algebras developed by Hinich and Schechtman to define a cosimplicial normalization functor in the category of -algebras over . The same construction was used by May to outline a theory of -algebras and C̆ech cochains as an approach to sheaf cohomology. However, these studies are not sufficient for our purposes. In this paper, we combine the above cosimplicial normalization functor with Godement’s cosimplicial resolution functor to define derived direct image and derived global image functors for sheaves of -algebras on topological spaces. This idea goes back to Godement, who already claimed in [Godement] that the good behavior with respect to products, of his canonical cosimplical resolutions by flasque sheaves, should allow one to study Steenrod operations in the context of sheaf theory.
We show that the assignment , where denotes the constant sheaf of -modules on , defines a functor from the category of Hausdorff, paracompact and locally contractible topological spaces, to the category of -algebras, which is naturally quasi-isomorphic to the functor defined by the complex of singular cochains with its -algebra structure (Theorem LABEL:quistocochains_thm). In particular, the above functor is a cochain theory in the sense of Mandell [ManCochains]. Such a cochain theory is a cochain-level refinement of the Eilenberg-Steenrod axioms, defined as a contravariant functor from spaces to cochain complexes, satisfying homotopy, excision, product and dimension axioms analogous to the usual axioms for cohomology. If such a cochain theory lifts to the category of -algebras, then this lift is uniquely quasi-isomorphic to the usual functor of singular cochains.
When , the above recover results of Navarro-Aznar [Na], who used the Thom-Whitney simple functor to give a solution to the commutative cochains problem over the field of rational numbers via sheaf theory.
As already pointed out by May in [MaySheaf], the study of sheaves of -algebras
has many potential applications to algebraic geometry. This is where the weight of this paper lies. We develop two main applications: to the weight filtration for algebraic varieties and to the intersection cohomology for topological pseudomanifolds.
Both of these theories owe much to Deligne and are based on sheaf-theoretic considerations.
The weight filtration. Deligne [DeHII], [DeHIII] introduced a functorial increasing filtration on the rational cohomology of every complex algebraic variety , called the weight filtration. This filtration relates the cohomology of the variety with cohomologies of smooth projective varieties. The successive quotients of this filtration become pure Hodge structures of different weights, leading to the notion of mixed Hodge structure.
The study of the rational homotopy of complex algebraic varieties using Deligne’s mixed Hodge theory and in particular, the weight filtration, has proven to be very fruitful. The initial findings are due to Morgan [Mo], who refined Deligne’s construction to endow the commutative dg-algebra models of smooth varieties with mixed Hodge structures. By introducing the Thom-Whitney simple functor and using the theory of cohomological descent, Navarro-Aznar [Na] extended Morgan’s theory to possibly singular varieties. These constructions have led to numerous topological consequences, mostly related to the formality of algebraic varieties in the sense of rational homotopy.
The weight filtration has also been defined on the cohomology with coefficients in an arbitrary commutative ring (see [GilletSoule], [GN]). Also, Totaro observed in his ICM address [To], that the weight filtration could also be defined on the cohomology with compact supports of any complex or real analytic space endowed with an equivalence class of compactifications. Totaro’s remarks have been made precise in [MCPI], [MCPII] for real algebraic varieties and in [CG3] for complex and real analytic spaces. Also, in [LiPr], cap and cup products are studied for the weight filtration on compactly supported -cohomology of real algebraic varieties.
In view of the above results, it is natural to ask for a weight filtration defined at the cochain level, with coefficients in an arbitrary commutative ring, and compatible with the -structure, thus generalizing all of the above constructions.
We obtain this multiplicative weight filtration using the extension criterion of functors of [GN]. This is based on the assumption that the target category is a cohomological descent category which, essentially, is a category endowed with a saturated class of weak equivalences, and a simple functor sending every cubical codiagram of to an object of and satisfying certain axioms analogous to those of the total complex of a double complex. In Theorem LABEL:descent1, we use the Mandell’s cosimplicial normalization functor to obtain a cohomological descent structure on the category of -algebras and prove a filtered version of this result in Theorem LABEL:cohdesccats.
The basic idea of the extension criterion of functors is that, given a functor compatible with smooth blow-ups from the category of smooth schemes to a cohomological descent category, there exists an extension to all schemes. Such an extension is essentially unique and is compatible with general blow-ups. Here, we use a relative version of this result, which extends functors from the category of pairs where is a smooth compactification of a smooth open variety . Given such a pair, we define the weight filtration by taking the canonical filtration on , promoting Deligne’s approach to the multiplicative setting. After extension, this gives a functor from the category of complex schemes to a certain homotopy category of filtered -algebras (Theorem LABEL:mainteoweight). The multiplicative weight spectral sequence of is defined as the spectral sequence associated to the filtered -algebra .
Over the rationals, the weight spectral sequence always degenerates at the second stage. However, this is not the case when working with arbitrary coefficients. Nevertheless, it converges to the cohomology of and from the second stage onwards, the th-term is a new and well-defined algebraic invariant of the variety, which carries a commutative bigraded structure as well as Steenrod operations when working over . In [To], Totaro speculates that the mixed motive of a real analytic space should not involve much more information than the weight spectral sequence over with an action of a Steenrod algebra. While the verification of this idea is beyond the scope of the present work, we do extend our constructions to real and complex analytic spaces showing, in Theorem LABEL:E2Steenrod, that the -term of the weight spectral sequence carries Steenrod operations
which are well-defined invariants for these spaces. We develop some examples illustrating that this is a new non-trivial invariant.
Intersection cohomology. Intersection cohomology is a Poincaré duality cohomology theory for singular spaces, introduced by Goresky and MacPherson. It is defined for any topological pseudomanifold and depends on the choice of a multi-index called perversity, measuring how far cycles are allowed to deviate from transversality.
Goresky and MacPherson, first defined intersection homology in [GMP1] as the homology of a subcomplex of the ordinary chains of , given by those cycles which meet the singular locus of with a controlled defect of transversality with respect to the chosen perversity and stratification. Subsequently, Deligne proposed a sheaf-theoretic approach, which was developed in [GMP2]. In this case, intersection cohomology is defined as the hypercohomology a complex of sheaves defined by starting with the constant sheaf on the regular part of and iteratively extending it to bigger open sets determined by the stratification and truncating it in the derived category with respect to the chosen perversity. The intersection complex is uniquely characterized by a set of axioms in the derived category of sheaves.
Intersection cohomology does not define an algebra, but it has a product compatible with perversities. In order to promote Deligne’s additive sheaf to a sheaf carrying an -algebra structure, we adapt Hovey’s [Hov2] formalism of perverse algebras to define the notion of perverse -algebra as an -algebra internal to the category of functors from the poset of perversities to cochain complexes. We then construct a sheaf of perverse -algebras whose underlying sheaf of complexes is Deligne’s intersection complex and show, in Theorem LABEL:uniquenessIC, that this is uniquely characterized by a set of axioms in the homotopy category of perverse -algebras. Different constructions related to the presence of multiplicative structures in intersection cohomology already existed in the literature, starting by Goresky and MacPherson’s study of cup products in intersection cohomology. There is also an ad. hoc. construction of Steenrod operations in intersection cohomology due to Goresky [Goresky] which is further studied in [CSTGor]. Also, recent work of the first author together with Saralegi and Tanré ([CSTSheaf],[CSTBl],[CST]) studies questions related to intersection rational homotopy. Our uniqueness result ensures that the intersection complex of perverse -algebras defined in this paper, recovers all of the above. In particular, when working over the field of rational numbers, we obtain a sheaf of commutative dg-algebras, which gives a solution to the problem of commutative cochains in the context of intersection cohomology.
The techniques developed in this paper have other potential applications. For instance, one could also endow the functors of nearby and vanishing cycles with -structures, as done in the last part of [Na] for the rational case. Also, Banagl’s [Banagl] theory of intersection space complexes and a recent generalization of Banagl’s theory due to Agustín and Fernández de Bobadilla [AgBo] should also allow for a treatment in the multiplicative setting.
The paper is organized as follows. In Section 2 we develop the necessary tools from sheaf theory of -algebras. In Section LABEL:SecFil we study filtered -algebras. We define filtered versions of the cosimplicial normalization functor and obtain cohomological descent structures on filtered -algebras. These results are applied in Section LABEL:SecWeight, where we define the weight filtration at the cochain level. Lastly, Section LABEL:SecInter is devoted to intersection cohomology. This last section can be read independently of Sections LABEL:SecFil and LABEL:SecWeight.
2. Sheaves of -algebras
In this section, we combine Mandell’s [ManCochains] cosimplicial normalization functor for -algebras together with Godement’s [Godement] cosimplicial resolution functor for sheaves of complexes to study sheaves of -algebras and their cohomology. We define derived direct image and global sections functors for sheaves of -algebras and use them to describe the -structure on the complex of singular cochains of a topological space, in terms of a resolution of its constant sheaf.
2.1. Preliminaries on -algebras
Throughout this paper, will be a commutative ring. All operads considered will be in the category of cochain complexes of -modules.
Let denote the operad of commutative algebras, given in each arity , by concentrated in degree 0, with acting by the identity.
An operad is called acyclic if there is a map of operads such that in each arity , the map is a quasi-isomorphism.
An operad is called -free if for each arity , the underlying graded -module of is free in each degree.
An -operad is a -free acyclic operad such that each graded module is concentrated in non-positive degrees.
Throughout this paper we will let be a cofibrant -operad. Recall that being cofibrant, means that has the lifting property with respect to maps of operads that are surjective quasi-isomorphisms (see for instance [Hinich]).
Denote by the category of -algebras over . An object of is given by a cochain complex together with an action of the -operad . We will mostly restrict to -algebras whose underlying cochain complex is non-negatively graded.
The category is complete and cocomplete. Limits and filtered colimits commute with the forgetful functor to cochain complexes. We will denote by the class of quasi-isomorphisms of -algebras (those morphisms of -algebras inducing quasi-isomorphisms at the cochain level) and by
the homotopy category defined by formally inverting quasi-isomorphisms.
2.2. Cosimplicial normalization functor for -algebras
We recall Mandell’s construction of the cosimplicial normalization functor for -algebras [ManCochains].
Consider the functor of normalized cochains
This is given by the total complex of the double complex obtained by normalizing degree-wise. When dealing with unbounded complexes, the total complex is constructed using the cartesian product, rather than the direct sum. The following description of the normalized cochains functor will be useful. Denote by
the normalized chain complex of -modules of the standard -simplex simplicial set, where
denotes the free -module functor, right adjoint to the forgetful. We will often consider as a non-positively graded cochain complex. Denote by
the internal hom of cochain complexes. The following Proposition is well-known:
If is a cosimplicial cochain complex, then may be identified with the end
of the functor given by .
A key ingredient to promote the cosimplicial normalization functor from cochain complexes to -algebras is the Eilenberg-Zilber operad, which we recall next (see [HiSc], see also [MaySheaf]).
The Eilenberg-Zilber operad is the endomorphism operad of the functor
In each arity , is the normalization of the cosimplicial (non-positively graded) cochain complex . We may write:
In particular, we have . The structure maps
of are given by the generalized Alexander-Whitney maps
where denotes the degree of the map .
The main theorem of [HiSc] (see also Theorem 5.5 of [ManCochains]) asserts that given a cosimplicial -algebra , where is an arbitrary operad, then is a -algebra, that is natural in maps of the operad and of the cosimplicial -algebra .
Since is an acyclic operad, by the lifting property of the cofibrant operad , we may choose a quasi-isomorphism of operads . Furthermore, there is a map of operads in such a way that both compositions
are the identity (see [ManCochains, Lemma 5.7]. This gives:
The cosimplicial normalization functor
is defined by composed with the reciprocal image functor of the composition
The functor satisfies the following properties:
By forgetting the algebra structures we recover the normalized cochains functor .
For a constant cosimplicial -algebra (in which all face and degeneracy maps are identities), the isomorphism of cochain complexes is a morphism of -algebras.
We refer to [ManCochains] for proofs of the above facts.
2.3. Derived direct image and global sections functors
Denote by the category of sheaves of cochain complexes of -modules on a topological space . It is a symmetric monoidal category, with the product of two sheaves and given by the sheafification of the presheaf . The unit is the constant sheaf .
The cosimplicial Godement resolution of a sheaf is an augmented cosimplicial sheaf satisfying:
the map is a quasi-isomorphism, and
is a complex of flasque sheaves on ,
where denotes the total simple functor, defined by the total complex of the double complex obtained without normalizing.
The functor is naturally homotopy equivalent to the cosimplicial normalization functor (see for instance [MacLane, Theorem 6.1]). We refer to [Godement, Appendix] for the construction and properties of . The assignment defines a lax symmetric monoidal functor (see for instance [RR, Proposition 3.12]). In particular, it induces a cosimplicial Godement resolution functor
on sheaves of -algebras.
Let be a continuous map of topological spaces. Recall:
The direct image functor
is defined by
The inverse image functor
is defined by taking to be the sheafification of the presehaf
for any sheaf on .
The direct image functor is left exact, but not right exact in general. The inverse image functor is exact and left adjoint to . In particular, there are natural adjunction morphisms and .
The functor is lax symmetric monoidal.
Define a map via the adjunction morphism
This makes compatible with the unit. We next define a natural transformation
Denote by the presheaf defined by . Consider the solid diagram of presheaves