Rigid complexes via DG algebras

Let $A$ be a commutative ring, $B$ a commutative $A$-algebra and $M$ a complex of $B$-modules. We begin by constructing the square $\operatorname{Sq}_{B / A} M$, which is also a complex of $B$-modules. The squaring operation is a quadratic functor, and its construction requires differential graded (DG) algebras. If there exists an isomorphism $\rho : M \xrightarrow{\simeq} \operatorname{Sq}_{B / A} M$, then the pair $(M, \rho)$ is called a rigid complex over $B$ relative to $A$ (there are some finiteness conditions). There is an obvious notion of rigid morphism between rigid complexes.

We establish several properties of rigid complexes, including their uniqueness, existence (under some extra hypothesis), and formation of pullbacks $f^{\flat}(M, \rho)$ (resp. $f^{\sharp}(M, \rho)$) along a finite (resp. essentially smooth) ring homomorphism $f^* : B \to C$.

In the subsequent paper, Rigid Dualizing Complexes over Commutative Rings, we consider rigid dualizing complexes over commutative rings, building on the results of the present paper. The project culminates in our paper Rigid Dualizing Complexes and Perverse Sheaves on Schemes, where we give a comprehensive version of Grothendieck duality for schemes.

The idea of rigid complexes originates in noncommutative algebraic geometry, and is due to Van den Bergh (1997).