English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this brief postscript to our paper "Integral transforms and Drinfeldcenters in derived algebraic geometry", we describe a Morita equivalence forderived, categorified matrix algebras implied by theory developed since itsappearance. We work in the setting of perfect stacks X and their stableinfinity-categories Q(X) of quasicoherent sheaves. Perfect stacks include allvarieties and common stacks in characteristic zero, and their stableinfinity-categories of sheaves are well behaved refinements of theirquasicoherent derived categories, satisfying natural analogues of commonproperties of function spaces.  To a morphism of perfect stacks pi:X-->Y, we associate the categorifiedmatrix algebra Q(X x_Y X) of sheaves on the derived fiber product equipped withits monoidal convolution product. We show that for pi faithfully flat (as acorollary of the 1-affineness theorem of Gaisgory) or for pi proper andsurjective and X,Y smooth (as an application of proper descent, cf. Gaitsgoryand Preygel), there is a Morita equivalence between Q(X x_Y X) and Q(Y), thatis, an equivalence of their infinity-categories of stable moduleinfinity-categories. In particular, this immediately implies an identificationof their Drinfeld centers (as previously established in 0805.0157), and moregenerally, an identification of their associated topological field theories.Another consequence is that for an affine algebraic group G in characteristiczero and an algebraic subgroup K, passing to K-invariants induces anequivalence from stable infinity-categories with algebraic G-action to modulesfor the Hecke category Q(K\G/K).

Morita equivalence for convolution categories: Appendix to arXiv:0805.0157