Related papers: A note on linear stacks
One fundamental consequence of a scheme $X$ being proper is that the functor classifying maps from $X$ to any other suitably nice scheme or algebraic stack is representable by an algebraic stack. This result has been generalized by…
Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of \v{S}penko and Van den Bergh, we construct equivalences between the derived categories of coherent sheaves…
In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable $\infty$-category…
Let X and Y be two smooth Deligne-Mumford stacks and consider a function f, resp. g, on X, resp. Y. Assume that there exists a complex F of sheaves on the fiber product of X and Y over A^1 (induced by f and g), such that the Fourier-Mukai…
A theory of dg schemes is developed so that it becomes a homotopy site, and the corresponding infinity category of stacks is equivalent to the infinity category of stacks, as constructed by Toen and Vezzosi, on the site of dg algebras whose…
We define derived versions of $F$-zips and associate a derived $F$-zip to any proper, smooth morphism of schemes in positive characteristic. We analyze the stack of derived $F$-zips and certain substacks. We make a connection to the…
This note is supposed to answer some questions on deformation theory in derived algebraic geometry. We show that derived algebraic geometry allows for a geometrical interpretation of the full cotangent complex and gives a natural setting…
Given a smooth projective toric variety $X_\Sigma$ of complex dimension $n$, Fang-Liu-Treumann-Zaslow \cite{FLTZ} showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves $Coh(X_\Sigma)$…
We show that every scheme/algebraic space/stack that is quasi-compact with quasi-finite diagonal can be approximated by a noetherian scheme/algebraic space/stack. More generally, we show that any stack which is etale-locally a global…
We study the interaction between geometric operations on stacks and algebraic operations on their categories of sheaves. We work in the general setting of derived algebraic geometry: our basic objects are derived stacks X and their…
By resolving an arbitrary perfect derived object over a Deligne-Mumford stack, we define its Euler class. We then apply it to define the Euler numbers for a smooth Calabi-Yau threefold in the 4-dimensional projective space. These numbers…
The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold $M$ with the Grothendieck group of constructible sheaves on $M$. When $M$ is a finite dimensional real vector space,…
We propose a simple mechanism by which scaling laws emerge from feature learning in multi-layer networks. We study a high-dimensional hierarchical target that is a globally high-degree function, but that can be represented by a combination…
We prove that the bounded derived category of coherent sheaves on a smooth projective complex variety reconstructs the isomorphism classes of fibrations onto smooth projective curves of genus $g\geq 2$. Moreover, in dimension at most four,…
In this paper we construct full support character sheaves for stably graded Lie algebras. Conjecturally these are precisely the cuspidal character sheaves. Irreducible representations of Hecke algebras associated to complex reflection…
We remove the global quotient presentation input in the theory of windows in derived categories of smooth Artin stacks of finite type. As an application, we use existing results on flipping of strata for wall-crossing of Gieseker…
Linear Logic refines Intuitionnistic Logic by taking into account the resources used during the proof and program computation. In the past decades, it has been extended to various frameworks. The most famous are indexed linear logics which…
We investigate conditions for a Fourier-Mukai transform between derived categories of coherent sheaves on smooth projective stacks endowed with actions by finite groups to lift to the associated equivariant derived categories. As an…
We prove the equivalence of two fundamental properties of algebraic stacks: being a quotient stack in a strong sense, and the resolution property, which says that every coherent sheaf is a quotient of some vector bundle. Moreover, we prove…
In this paper we determine the derived representation type of quadratic string algebras and we prove that every derived tame quadratic string algebra whose quiver has cycles is derived equivalent to some skewed-gentle algebra.