Related papers: Grothendieck duality under Spec Z

We develop the theory of module categories over a Grothendieck-Verdier category, i.e. a monoidal category with a dualizing object and hence a duality structure more general than rigidity. Such a category C comes with two monoidal structures…

For any ring $A$ and a small, preadditive, Hom-finite, and locally bounded category $Q$ that has a Serre functor and satisfies the (strong) retraction property, we show that the category of additive functors from $Q$ to the category of…

We show that for quasi-compact quasi-separated schemes of finite dimension, the constructibility condition in real \'etale cohomology agrees with a notion of constructibility arising naturally from topology. As application we prove that the…

We give a new characterization of silting subcategories in the stable category of a Frobenius extriangulated category, generalizing the result of Di et al. (J. Algebra 525 (2019) 42-63) about the Auslander-Reiten type correspondence for…

Toen has interpreted the schematization problem as originally imagined by Grothendieck in "Pursuing Stacks" in such a way that solution(s) to this problem could be given. As he pointed out, there are many solutions available, and he gave…

We give an alternative treatment of the foundations of parametrized spectra, with an eye toward applications in fixed-point theory. We cover most of the central results from the book of May and Sigurdsson, sometimes with weaker hypotheses,…

New homotopy invariant finiteness conditions on modules over commutative rings are introduced, and their properties are studied systematically. A number of finiteness results for classical homological invariants like flat dimension,…

For a locally presentable abelian category $\mathsf B$ with a projective generator, we construct the projective derived and contraderived model structures on the category of complexes, proving in particular the existence of enough homotopy…

We show that every sheaf on the site of smooth manifolds with values in a stable (infinity,1)-category (like spectra or chain complexes) gives rise to a differential cohomology diagram and a homotopy formula, which are common features of…

Let (X;OX) be a locally noetherian scheme with a dualizing complex D. We prove that DOX - : K(PinfX)----> K(InjX) is an equivalence of triangulated categories where K(InjX) is the homotopy category of injective quasi-coherent OX- modules…

We construct spectral sequences in the framework of Baues-Wirsching cohomology and homology for functors between small categories and analyze particular cases including Grothendieck fibrations. We also give applications to more classical…

In this paper, we introduce the notion of Grothendieck enriched categories for categories enriched over a sufficiently nice Grothendieck monoidal category $\mathcal{V}$, generalizing the classical notion of Grothendieck categories. Then we…

In this paper the Serre spectral sequence of Moerdijk and Svensson is extended from Bredon cohomology to RO(G)-graded cohomology for finite groups G. Special attention is paid to the case G=Z/2 where the spectral sequence is used to compute…

We give a detailed proof of the following fundamental result: the singularity category of a ring is triangle equivalent to the stabilization of its stable module category. The result yields singular equivalences between rings of different…

In this paper, we investigate the split Grothendieck group $K^{\rm sp}_{0}(\mathcal{M})$ of a $d$-rigid subcategory $\mathcal{M}$ in an extriangulated category $\mathscr{C}$. As applications, we prove the following results: (1) If…

Gillam proved that the category of locally ringed spaces admits a fully faithful embedding into a certain category, which has a right adjoint that maps some simple objects to the spectra of rings. In this paper, we use condensed mathematics…

A parametrized spectrum E is a family of spectra E_x continuously parametrized by the points x of a topological space X. We take the point of view that a parametrized spectrum is a bundle-theoretic geometric object. When R is a ring…

We define a natural 2-categorical structure on the base category of a large class of Grothendieck fibrations. Given any model category $\mathbf{C}$, we apply this construction to a fibration whose fibers are the homotopy categories of the…

Making use of topological periodic cyclic homology, we extend Grothendieck's standard conjectures of type C and D (with respect to crystalline cohomology theory) from smooth projective schemes to smooth proper dg categories in the sense of…

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes. A quasi-coherent sheaf on X gives rise, by taking sections over the covering sets, to a diagram of modules over the various coordinate rings. The resulting…