Related papers: A Frobenius theorem in algebraic geometry
Deligne's conjecture that $\ell$-adic sheaves on normal schemes over a finite field admit $\ell'$-companions was proved by L. Lafforgue in the case of curves and by Drinfeld in the case of smooth schemes. In this paper, we extend Drinfeld's…
In this paper, we define $\Lambda$-quot-functors on Deligne-Mumford stacks. We prove that the $\Lambda$-quot-functor is representable by an algebraic space. Then, we construct the moduli space of $\Lambda$-modules on a projective…
In this article, we derive many properties of \'etale stacks in various contexts, and prove that \'etale stacks may be characterized categorically as those stacks that arise as prolongations of stacks on a site of spaces and local…
The notion of a Frobenius manifold appears in relation to various topics in algebraic and analytic geometry, such and quantum cohomology, deformation of meromorphic connections, unfolding of singularities and others. In the local setting…
Lurie's theorem states that there exists a sheaf of ring spectra on the site of formally \'etale Deligne--Mumford stacks over the moduli stack of $p$-divisible groups of height $n$, which agrees with the classical Landweber exact functor…
Deformation theory is treated for locally notherian formal schemes (non necessarily smooth). The cotangent complex is defined in the derived category through the homology localization functor. The basic properties and results of a…
Based on the methods used by the author to prove the Riemann-Roch formula for algebraic stacks, this paper contains a description of the rationnal G-theory of Deligne-Mumford stacks over general bases. We will use these results to study…
We extend the definition of the Chen-Ruan cohomology ring to smooth, proper, tame, Deligne-Mumford stacks over fields of positive characteristic and prove that a modified version of the Frobenius action preserves the product.
Although contemporary model theory has been called "algebraic geometry minus fields", the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of logical schemes,…
We prove that the dg category of perfect complexes on a smooth, proper Deligne-Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated…
In view of applications to the construction of moduli spaces of objects in algebraic supergeometry, we start a systematic study of stacks in that context. After defining a superstack as a stack over the \'etale site of superschemes, we…
In this paper, we introduce variants of formal nearby cycles for a locally noetherian formal scheme over a complete discrete valuation ring. If the formal scheme is locally algebraizable, then our nearby cycle gives a generalization of…
We develop an anabelian framework for general Deligne-Mumford curves, showing that their stack and orbifold structures are encoded in the group-theoretic properties of their \'etale fundamental groups. After establishing the required…
We give a simple proof of the Riemann-Roch theorem for Deligne-Mumford stacks using the equivariant Riemann-Roch theorem and the localization theorem in equivariant K-theory together with some basic commutative algebra of Artin rings.
This is the first paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In this paper, we lay the foundations for this study by introducing the…
Using the notion of Levi form of a smooth distribution, we discuss the local and the global problem of existence of one horizontal section of a smooth vector bundle endowed with a horizontal distribution. The analysis will lead to the…
The representability theorem for stacks, due to Artin in the underived setting and Lurie in the derived setting, gives conditions under which a stack is representable by an $n$-geometric stack. In recent work of Ben-Bassat, Kelly, and…
We give several related versions of global Grothendieck Duality for unbounded complexes on noetherian formal schemes. The proofs, based on a non-trivial adaptation of Deligne's method for the special case of ordinary schemes, are reasonably…
To a topological groupoid endowed with an involution, we associate a topological groupoid of fixed points, generalizing the fixed-point subspace of a topological space with involution. We prove that when the topological groupoid with…
This is the companion article to the Bourbaki talk of the same name given in March 2009. The main theme of the talk and the article is to explain the interplay between homotopy theory and algebraic geometry through the Hopkins-Miller-Lurie…