arXiv subjects

Alessio Cipriani

Publications and source records attributed to Alessio Cipriani.

6 recordsLinked to original sources

arXiv2026

Indecomposable extensions of perverse sheaves over a closed stratum

Given a topologically stratified space $X$, we develop a categorical framework for extensions of perverse sheaves over a closed stratum $S$. We introduce the notion of $S$-small extensions and extension pairs relative to a fixed perverse sheaf on $X\smallsetminus S$. By replacing the homotopical assumptions $\pi_1(S)=\pi_2(S)=0$ in the work of MacPherson and Vilonen (Invent. Math., 84(2):403-435, 1986) with the categorical condition that the category of local systems on $S$ is semisimple, we construct an equivalence of additive categories between $S$-small extensions and extension pairs. This extends the MacPherson--Vilonen description to a more general setting and yields a maximal extension functor, generalising Beilinson's construction. As a consequence, we obtain a structural classification of indecomposable $p$-perverse sheaves on $X$: every such object is either an extension by zero of an indecomposable local system on $S$, or arises from an extension pair whose canonical morphism does not decompose as a direct sum.

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arXiv2026

Faithful perversities

We show that the faithful highest weight hearts in an algebraic triangulated category are the serially faithful glued hearts, equivalently the hearts containing a dual pair of full exceptional collections in the sense of Bodzenta--Bondal (arXiv:2601.22004). We then characterise faithful highest weight categories of perverse sheaves on topologically stratified spaces algebraically, in terms of the exactness of certain functors, and topologically, in terms of the vanishing of certain cohomology groups of pairwise links. We prove that the global dimension of a faithful category of perverse sheaves on a topologically stratified space $X$ with finitely many strata is bounded by the dimension of $X$. Finally, we show that in this setting the hypercohomology of a perverse sheaf can be computed from a projective resolution of the constant sheaf, and conversely that the multiplicities of the terms in a minimal projective resolution of the constant sheaf can be computed as intersection cohomology groups.

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arXiv2026

Highest weight categories and stability conditions

Highest weight categories are an abstraction of the representation theory of semisimple Lie algebras introduced by Cline, Parshall and Scott in the late 1980s. There are by now many characterisations of when an abelian category is highest weight, but most are hard to verify in practice. We present two new criteria - one numerical in terms of the Grothendieck group, and one in terms of Bridegland stability conditions - which are easier to verify. The stability criterion naturally generalises to a characterisation of properly stratified categories. The numerical criterion implies a criterion of Green and Schroll for when modules over a monomial algebra are highest weight.

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arXiv2025

Serre functor and $\mathbb{P}$-objects for perverse sheaves on $\mathbb{P}^n$

We show that the inverse Serre functor for the constructible derived category $\mathbf{D}^\mathrm{b}_\mathrm{c}(\mathbb{P}^n)$ is given by the $\mathbb{P}$-twist at the simple perverse sheaf corresponding to the open stratum. Moreover, we show that all indecomposable perverse sheaves on $\mathbb{P}^n$ are $\mathbb{P}$-like objects, and explicitly construct morphisms spanning their total endomorphism spaces.

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arXiv2023

Wall And Chamber Structure For A Special Biserial Algebra Coming From Perverse Sheaves on $\mathbb{P}^n$

We describe the wall and chamber structure of a special biserial algebra whose module category is equivalent to the category of (middle) perverse sheaves on the complex projective space $\mathbb{P}^n$. In particular, by the well known classification of indecomposable modules for special biserial algebras, we deduce that the algebra of interest is of finite representation type and we provide an explicit description of the walls of the structure. By a result of Bridgeland this wall and chamber structure coincides with the chamber structure in an open subset of the space of stability conditions on the bounded derived category of constructible sheaves on $\mathbb{P}^n$.

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arXiv2020

Perverse sheaves and finite-dimensional algebras

Let $X$ be a topologically stratified space, $p$ be any perversity on $X$, and $k$ be a field. We show that the category of $p$-perverse sheaves on $X$, constructible with respect to the stratification and with coefficients in $k$, is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra if and only if $X$ has finitely many strata and the same holds for the category of local systems on each of these. The main component in the proof is a construction of projective covers for simple perverse sheaves.

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