Periodic Report Summary 1 - HOMOQCSSTACKS (Homotopy theory of quasi-coherent sheaves on stacks)
In the late 1960's Deligne and Mumford invented the notion of an stack and used it to prove the irreducibility of the moduli space of smooth algebraic curves of given genus, over an arbitrary base field. They introduced a class of stacks, which they called algebraic, over which many of the constructions and results of algebraic geometry are applicable. These stacks have the capacity to capture the geometry of a moduli problem better than traditional algebraic varieties or schemes. Later Artin identified a more general class of algebraic stacks and gave a general criterion for a given moduli problem to correspond faithfully to such a stack. In the more than 30 years that have passed since these groundbreaking works appeared, stacks have had many important applications to a wide variety of fields, for instance to the foundations of Gromov-Witten theory, the study of Brauer groups, Interstection theory, Deformation theory, and Non-abelian Hodge theory. Stacks are now an indispensable tool to the modern algebraic geometer, and are important in related fields such as Differential Geometry, where they are known as orbifolds. Stacks play a key role in geometric approaches to Representation Theory, especially in the Geometric Langlands Correspondence. Work of Witten and others have established connections of this area with Mathematical Physics, including Quantum Cohomology, Mirror Symmetry, and D-brane charges in String Theory. Recently, generalizations of the notion of algebraic stacks have begun to emerge. The importance of notions and basic properties of higher stacks and derived stacks are now being exposed by Lurie and others.
The deep connection between stable homotopy theory and the theory of stacks, finds its origin in the work of Quillen who proved that the complex cobordism ring is isomorphic to the Lazard ring classifying 1-dimensional formal group laws. Together with work of Landweber it implies that the cohomology theory of complex cobordism takes values in an algebraic category associated to the theory of formal group laws. Quillen's results predate the theory of stacks and so were not stated in these terms. What they imply though, is that the complex cobordism of a space or spectrum X is a quasi-coherent sheaf of the moduli stack of one dimensional commutative formal groups. These results led to detailed analysis of the algebra of formal group laws and the next wave of major results in stable homotopy theory during the 1970's and 1980's. In this way stacks have come to play an extremely important role in algebraic topology and the connections with algebraic geometry and number theory have become a central area of activity.
One of the challenges facing this subject is to develop a comprehensive theory of stacks that can handle the needs of stable homotopy theory. One difficulty is that the theory of stacks was developed mostly under Noetherian hypotheses which are natural in classical algebraic geometry. However, these conditions do not hold for the problems that appear in stable homotopy theory and other branches of geometry. I have developed a new categorical foundation for the study of stacks. Among the strengths of this theory is that it applies equally well to classical algebraic geometric problems and those which arise in stable homotopy theory. The new perspective embodied in the homotopy theory of stacks also suggests innovative approaches to fundamental problems about stacks which are of interest even in the classical algebreo-geometric setting. The proposed research aims at answering some of these questions.
A crucial aspect in the theory of stacks is the study of quasi-coherent sheaves on a stack. The overall objectives of the proposed research are to prove a number of theorems for these categories of quasi-coherent sheaves. In particular, I proposed to investigate various types of decompositions of the category of quasi-coherent sheaves on a stack which are in various ways determined by geometric decompositions of the stack. Some of these are motivated by specific applications to stable homotopy theory, while others address foundational questions which are of general interest. I proposed to generalize fundamental statements which are true for well behaved schemes but which turn out to be much more complicated for stacks, even those of Deligne and Mumford or Artin. Our approach to these theorems is very different than the classical, being more categorical in nature, and lends easily to generalization and applications to other geometric and abstract contexts. In the case where the stack that we are looking at is the moduli stack of formal groups, the proposed theorems could be used to generalize both important and difficult theorems in stable homotopy theory, lend greater conceptual elegance to the narrative, and point at deeper structure.
In a previous work, I introduce the notion of infinity-algebraic stack generalizing both the stacks of Deligne and Mumford and those of Artin. In this context 1-algebraic corresponds to algebraic in the sense of Deligne and Mumford, while 2-algebraic that of Artin, and infinity-algebraic being the most general in a hierarchy algebraicity. Furthermore infinity-algebraic can be considered as a relative condition with respect to a Grothendieck site.
I have proposed two main objectives for the first year of the project:
1. Prove that for any map of infinity-algebraic stacks there is an adjoint pair.
2. Prove that this adjoint pair commutes with flat maps.
Both of these objectives were fully achieved. Using the theorems summarized above, I have obtained theorems about stacks which directly imply the Miller-Ravenel and Morava change of rings theorems and also algebraic chromatic convergence. I have also investigated how this new approach to algebraic chromatic convergence yields new approach to topological chromatic convergence, both generalizing and strengthening the known theorems.
For the second year I have proposed the following main objectives:
3. Prove that the category QC(M) is complete and co-complete.
4. Given an infinity-algebraic stack M and a closed substack C, construct a completion along C endofunctor on QC(M).
Both of these objectives were fully achieved. I also proposed to prove in the second year:
5. Given an infinity-algebraic stack M and an open substack U, show that for each quasi-coherent sheaf F there is a derived pullback square, writing F as a homotopy push-out of its completion along the complement of U and its pull-back to U.
This objective is still a work in progress. In addition, I have made progress on problems that had originally been only planned for the last year of the project, and have obtained partial results.
With the growing body of work relating stable homotopy theory to a broader collection of stacks (beyond the moduli stack of formal groups), I expect that the abstract results which I intend to prove will find a variety of applications leading to further advances both in conceptual understanding and in computation power in stable homotopy theory. A proper formulation of the basic categorical constructions for quasi-coherent sheaves which appear in this project should also yield the right language in which to formulate the algebraic results needed as input for new results in stable homotopy theory. This perspective broadens one's view and adds greater flexibility to the set of tools one uses for handling stacks. At the same time the greater generality of the framework has led to both more general theorems and easier proofs of many of the algebraic theorems in chromatic stable homotopy theory. In this framework we intend to develop both the language and abstract machinery to tackle this vast range of problems.
I have ongoing fruitful collaborations with top researchers in the U.S. and Europe which are mutually beneficial. My aim is not only to nurture these existing collaborations but also to invest in their further development with the objective generating new knowledge and transferring it to my host institute and to the European scientific community at large. By partly funding the formation of a new research group in Homotopy theory at BGU, which is expected to collaborate with other leading research groups in the EC, this project will no doubt contribute to the enhancement of European competitiveness in my field. This will be achieved through personal collaborations with European colleagues, attending conferences and seminars in the EU region, and by attracting excellent graduate and post-graduate students to BGU.
The deep connection between stable homotopy theory and the theory of stacks, finds its origin in the work of Quillen who proved that the complex cobordism ring is isomorphic to the Lazard ring classifying 1-dimensional formal group laws. Together with work of Landweber it implies that the cohomology theory of complex cobordism takes values in an algebraic category associated to the theory of formal group laws. Quillen's results predate the theory of stacks and so were not stated in these terms. What they imply though, is that the complex cobordism of a space or spectrum X is a quasi-coherent sheaf of the moduli stack of one dimensional commutative formal groups. These results led to detailed analysis of the algebra of formal group laws and the next wave of major results in stable homotopy theory during the 1970's and 1980's. In this way stacks have come to play an extremely important role in algebraic topology and the connections with algebraic geometry and number theory have become a central area of activity.
One of the challenges facing this subject is to develop a comprehensive theory of stacks that can handle the needs of stable homotopy theory. One difficulty is that the theory of stacks was developed mostly under Noetherian hypotheses which are natural in classical algebraic geometry. However, these conditions do not hold for the problems that appear in stable homotopy theory and other branches of geometry. I have developed a new categorical foundation for the study of stacks. Among the strengths of this theory is that it applies equally well to classical algebraic geometric problems and those which arise in stable homotopy theory. The new perspective embodied in the homotopy theory of stacks also suggests innovative approaches to fundamental problems about stacks which are of interest even in the classical algebreo-geometric setting. The proposed research aims at answering some of these questions.
A crucial aspect in the theory of stacks is the study of quasi-coherent sheaves on a stack. The overall objectives of the proposed research are to prove a number of theorems for these categories of quasi-coherent sheaves. In particular, I proposed to investigate various types of decompositions of the category of quasi-coherent sheaves on a stack which are in various ways determined by geometric decompositions of the stack. Some of these are motivated by specific applications to stable homotopy theory, while others address foundational questions which are of general interest. I proposed to generalize fundamental statements which are true for well behaved schemes but which turn out to be much more complicated for stacks, even those of Deligne and Mumford or Artin. Our approach to these theorems is very different than the classical, being more categorical in nature, and lends easily to generalization and applications to other geometric and abstract contexts. In the case where the stack that we are looking at is the moduli stack of formal groups, the proposed theorems could be used to generalize both important and difficult theorems in stable homotopy theory, lend greater conceptual elegance to the narrative, and point at deeper structure.
In a previous work, I introduce the notion of infinity-algebraic stack generalizing both the stacks of Deligne and Mumford and those of Artin. In this context 1-algebraic corresponds to algebraic in the sense of Deligne and Mumford, while 2-algebraic that of Artin, and infinity-algebraic being the most general in a hierarchy algebraicity. Furthermore infinity-algebraic can be considered as a relative condition with respect to a Grothendieck site.
I have proposed two main objectives for the first year of the project:
1. Prove that for any map of infinity-algebraic stacks there is an adjoint pair.
2. Prove that this adjoint pair commutes with flat maps.
Both of these objectives were fully achieved. Using the theorems summarized above, I have obtained theorems about stacks which directly imply the Miller-Ravenel and Morava change of rings theorems and also algebraic chromatic convergence. I have also investigated how this new approach to algebraic chromatic convergence yields new approach to topological chromatic convergence, both generalizing and strengthening the known theorems.
For the second year I have proposed the following main objectives:
3. Prove that the category QC(M) is complete and co-complete.
4. Given an infinity-algebraic stack M and a closed substack C, construct a completion along C endofunctor on QC(M).
Both of these objectives were fully achieved. I also proposed to prove in the second year:
5. Given an infinity-algebraic stack M and an open substack U, show that for each quasi-coherent sheaf F there is a derived pullback square, writing F as a homotopy push-out of its completion along the complement of U and its pull-back to U.
This objective is still a work in progress. In addition, I have made progress on problems that had originally been only planned for the last year of the project, and have obtained partial results.
With the growing body of work relating stable homotopy theory to a broader collection of stacks (beyond the moduli stack of formal groups), I expect that the abstract results which I intend to prove will find a variety of applications leading to further advances both in conceptual understanding and in computation power in stable homotopy theory. A proper formulation of the basic categorical constructions for quasi-coherent sheaves which appear in this project should also yield the right language in which to formulate the algebraic results needed as input for new results in stable homotopy theory. This perspective broadens one's view and adds greater flexibility to the set of tools one uses for handling stacks. At the same time the greater generality of the framework has led to both more general theorems and easier proofs of many of the algebraic theorems in chromatic stable homotopy theory. In this framework we intend to develop both the language and abstract machinery to tackle this vast range of problems.
I have ongoing fruitful collaborations with top researchers in the U.S. and Europe which are mutually beneficial. My aim is not only to nurture these existing collaborations but also to invest in their further development with the objective generating new knowledge and transferring it to my host institute and to the European scientific community at large. By partly funding the formation of a new research group in Homotopy theory at BGU, which is expected to collaborate with other leading research groups in the EC, this project will no doubt contribute to the enhancement of European competitiveness in my field. This will be achieved through personal collaborations with European colleagues, attending conferences and seminars in the EU region, and by attracting excellent graduate and post-graduate students to BGU.