Punctual gluing of $t$-structures and weight structures

TL;DR

This paper introduces the concept of punctual gluing for $t$-structures and weight structures, applying it to motivic categories to recover and extend key constructions in algebraic geometry.

Contribution

It formulates a new notion of punctual gluing and applies it to establish the restriction of relative $t$-structures and construct motivic weight structures.

Findings

Relative $t$-structure restricts to compact motives.

Constructs weight structure on motivic sheaves over any base.

Builds relative Artin and Picard motives for any variety.

Abstract

We formulate a notion of "punctual gluing" of $t$-structures and weight structures. As our main application we show that the relative version of Ayoub's $1$-motivic $t$-structure restricts to compact motives. We also demonstrate the utility of punctual gluing by recovering several constructions in literature. In particular we construct the weight structure on the category of motivic sheaves over any base $X$ and we also construct the relative Artin motive and the relative Picard motive of any variety $Y/X$.