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Recollements in stable ∞-categories · arXivDesk

1507.03913Jul 14, 2015The unexpected (and actually undue) symmetric behavior of stable recollements (Lemma 4.3 v1) turned out to be the far reaching consequence of a typo in one of the commutative diagrams on page 9. This has now been corrected (i.e., Lemma 4.3 and his corollaries have been removed). Luckily, this was only minimally affecting the remaining part of the article, which has now been revised accordingly

Recollements in stable $\infty$ -categories

Domenico Fiorenza, Fosco Loregian

Category Theory Algebraic Geometry

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Abstract

We develop the theory of recollements in a stable $\infty$ -categorical setting. In the axiomatization of Beilinson, Bernstein and Deligne, recollement situations provide a generalization of Grothendieck's "six functors" between derived categories. The adjointness relations between functors in a recollement $\mathbf{D}^0\leftrightarrow \mathbf{D} \leftrightarrow \mathbf{D}^1$ induce a "recollée" $t$

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-structure

\mathfrak{t}_0\uplus\mathfrak{t}1

\mathbf{D}

, given

t

-structures

\mathfrak{t}_0,\mathfrak{t}_1

\mathbf{D}^0, \mathbf{D}^1

. Such a classical result, well-known in the setting of triangulated categories, is recasted in the setting of stable

\infty

-categories and the properties of the associated (

\infty

-categorical) factorization systems are investigated. In the geometric case of a stratified space, various recollements arise, which "interact well" with the combinatorics of the intersections of strata to give a well-defined, associative

\uplus

operation. From this we deduce a generalized associative property for

n

-fold gluing

\mathfrak{t}_0\uplus\cdots\uplus \mathfrak{t}_n

, valid in any stable

\infty

-category.

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