Related papers: A proof of Grothendieck's base change theorem

We establish an analogue of the Grothendieck inequality where the rectangular matrix is replaced by a symmetric/Hermitian matrix and the bilinear form by a quadratic form. We call this the symmetric Grothendieck inequality; despite its…

In this article, we develop a theory of Grothendieck's six operations for derived categories in \'etale cohomology of Artin stacks, for both torsion and adic coefficients. We prove several desired properties of the operations, including the…

In this paper we study flatness of the restriction on some special subgerms (e.g. the reduction and the unmixed part) of the total space of a flat morphism over a smooth base space. We give a relationship between reducedness of the total…

This note records that in the setting of complex varieties, the cohomological consequence of Ehresmann's fibration theorem holds without the smooth assumption on the base or the total space.

We derive a permutability theorem for the Christoffel, Goursat and Darboux transformations of isothermic surfaces. As a consequence we obtain a simple proof of a relation between Darboux pairs of minimal surfaces in Euclidean space, curved…

We provide an explicit characterization of the covariant isotropy group of any Grothendieck topos, i.e. the group of (extended) inner automorphisms of any sheaf over a small site. As a consequence, we obtain an explicit characterization of…

We prove that, for smooth quasi-projective varieties over a field, the $K$-theory $K(X)$ of vector bundles is the universal cohomology theory where $c_1(L\otimes \bar L)=c_1(L)+c_1(\bar L)-c_1(L)c_1(\bar L)$. Then, we show that…

In this paper, it is proved, that for varieties with (m-1)-Du Bois singularities, the natural morphism from the Grothendieck dual of the m-th graded Du Bois complex to the Grothendieck dual of its zero-th cohomology sheaf is injective on…

Motivated by questions in the study of relative trace formulae, we construct a generalization of Grothendieck's simultaneous resolution over the regular locus of certain symmetric pairs. We use this space to prove a relative version of…

The purpose of this note is to define sheaves for diffeological spaces and give a construction of their \v{C}ech cohomology. As an application, we prove that the first degree \v{C}ech cohomology classes for the sheaf of smooth functions to…

In this article we study cohomology of a group with coefficients in representations on Banach spaces and its stability under deformations. We show that small, metric deformations of the representation preserve vanishing of cohomology. As…

In this paper, we prove a sharp Mei's Lemma with assuming the bases of the underlying general dyadic grids are different. As a byproduct, we specify all the possible cases of adjacent general dyadic systems with different bases. The proofs…

We prove a trace formula in stable motivic homotopy theory over a general base scheme, equating the trace of an endomorphism of a smooth proper scheme with the "Euler characteristic integral" of a certain cohomotopy class over its scheme of…

The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.

We state a conjecture relating de Rham cohomology of a smooth rigid analytic variety to its compactly supported pro-\'etale cohomology. We prove the conjecture in the cases where the variety is a Stein curve of dimension one or a Stein…

In this paper we prove the existence of an algebraic model for quasi-coherent sheaves on certain non-connective geometric stacks arising in stable homotopy theory and spectral algebraic geometry using the machinery of adapted homology…

In this article, we prove the Hodge conjecture for a desingularization of the moduli space of rank 2, semi-stable, torsion-free sheaves with fixed odd degree determinant over a very general irreducible nodal curve of genus at least 2. We…

We provide a short proof on the change-of-basis coefficients from the Specht basis to the Kazhdan-Lusztig basis, using Kazhdan-Lusztig theory for parabolic Hecke algebra.

We look at homotopy-coherent diagrams of spaces (after Segal, Leitch, Vogt, Mather, Cordier) over a Grothendieck site; we call these ``flexible presheaves''. After some preliminary materiel, we define the ``flexible sheaf'' condition. This…

A theorem of Grothendieck asserts that over a perfect field k of cohomological dimension one, all non-abelian H^2-cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of…