arXiv subjects

Ana Jeremías

Publications and source records attributed to Ana Jeremías.

5 recordsLinked to original sources

arXiv2026

Homotopy limits of complexes

We propose a notion of homotopy limit in the category of complexes over an abelian category with products by totalizing the classic construction of the Roos' complex that computes derived inverse limits. For complexes of modules over a non-necessarily commutative ring, we show that our construction of homotopy limits computes the derived limit complex, and under an acyclicity hypothesis on the inverse system, we prove that it is quasi-isomorphic to the limit. We further show that, in general, the construction is appropriately dual of the previous construction of homotopy colimits of complexes from [Alonso, Jeremías and Souto: Localization in categories of complexes and unbounded resolutions. \textit{Canad. J. Math.} (2000)], and that there also is a dual behavior between derived limits and colimits in derived categories of modules. Finally, we show that colocalizing subcategories are stable for homotopy limits.

math.CT↗

arXiv2026

Localizing and colocalizing subcategories on schemes

A full triangulated subcategory $\mathsf{L} \subset \mathsf{T}$ of triangulated category $\mathsf{T}$ is localizing if it is stable for coproducts. If, further, $\mathsf{T}$ is $\otimes$-triangulated, we say that $\mathsf{H}$ is $\otimes$-ideal if $F \otimes G \in \mathsf{L}$ for all $G \in \mathsf{L}$ and all $F \in \mathsf{T}$. Analogously, a full triangulated subcategory $\mathsf{C} \subset \mathsf{T}$ is colocalizing if it is stable for products. If, further, $\mathsf{T}$ is closed, i.e. $\otimes$-triangulated with internal homs (denoted $[-,-]$), we say that $\mathsf{C}$ is $\mathcal{H}$-coideal if $[F, G] \in \mathsf{C}$ for all $G \in \mathsf{C}$ and all $F \in \mathsf{T}$. For a point generated concentrated scheme $X$, we prove that all $\otimes$-ideal localizing subcategories of $\mathbf{D}_{qc}(X)$ are classified by the subsets of $X$. As a consequence, we prove that for $\mathcal{H}$-coideal colocalizing subcategories of $\mathbf{D}_{qc}(X)$ the same holds. Moreover, every such colocalizing subcategory $\mathsf{C}$ is of the form $\mathsf{C}= \mathsf{L}^\perp$, where $\mathsf{L}$ is a $\otimes$-ideal localizing subcategory of $\mathbf{D}_{qc}(X)$.

math.AG↗

arXiv2026

Colocalizing subcategories on differentially graded algebras

Let $A$ be a bounded non positive commutative differential graded algebra $A$. Let $\mathbf{D}(A)$ its derived category of DG-modules. If $\mathbf{D}(A)$ is generated by the DG-modules corresponding to the residue fields of the ordinary ring $H^0(A)$ then its localizing subcategories and its colocalizing subcategories are in bijection with the subsets of $\textrm{Spec}(H^0(A))$. These results generalize well-known theorems by A. Neeman (from 1992 and 2011, respectively), because any Noetherian ring satisfies this condition.

math.CT↗

arXiv2023

Unbounded Algebraic Derivators

We show that the unbounded derived category of a Grothendieck category with enough projective objects is the base category of a derivator whose category of diagrams is the full 2-category of small categories. With this structure, we give a description of the localization functor associated to a specialization closed subset of the spectrum of a commutative noetherian ring. In addition, using the derivator of modules, we prove some basic theorems of group cohomology for complexes of representations over an arbitrary base ring.

math.CT↗

arXiv1995

Local Homology and Cohomology on Schemes

We prove a sheaf-theoretic derived-category generalization of Greenlees-May duality (a far-reaching generalization of Grothendieck's local duality theorem): for a quasi-compact separated scheme X and a "proregular" subscheme Z---for example, any separated noetherian scheme and any closed subscheme---there is a sort of sheafified adjointness between local cohomology supported in Z and left-derived completion along Z. In particular, the i-th left-derived completion functor is the "local homology" sheaf $Ext^i(\RΓ_ZØ_X, -)$. Sheafified generalizations of a number of duality theorems scattered about the literature result, e.g., the Peskine-Szpiro duality sequence (generalizing local duality), the Warwick Duality theorem of Greenlees, the Affine Duality theorem of Hartshorne. Using Grothendieck Duality, we also get a generalization of a Formal Duality theorem of Hartshorne, and of a related local-global duality theorem. In a sequel we will develop the latter results further, to study Grothendieck duality and residues on formal schemes.

alg-geom↗