ar X iv : m at h / 06 03 73 3 v 3 [ m at h . A C ] 1 8 D ec 2 00 6 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 SqB/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 ρ : M ≃ −→ SqB/A M then the pair (M,ρ) 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♭(M, ρ) (resp. f♯(M, ρ)) along a finite (resp. essentially smooth) ring homomorphism f : B → C. In the subsequent paper [YZ4] we consider rigid dualizing complexes over commutative rings, building on the results of the present paper. The project culminates in [Ye4], 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 [VdB]. 0. Introduction Rigid dualizing complexes were invented by Van den Bergh [VdB] in the context of noncommutative algebraic geometry. Since then the theory of rigid dualizing complexes was developed further by several people, with many applications in the areas of noncommutative algebra and noncommutative algebraic geometry. See the papers [YZ1, YZ2, YZ3] and their references. The present paper is the first in a series of three papers in which we apply the rigidity technique to the areas of commutative algebra and algebraic geometry. Of course the commutative setup is “contained in”, and is always much “easier” than the corresponding noncommutative setup. The point is that we want to get stronger and more general results when we restrict attention to commutative rings and schemes. Let us remind the reader that in the papers cited above the noncommutative algebras in question were always assumed to be over some base field K. The first generalization that comes to mind when considering a commutative theory of rigid dualizing complexes is to be able to work over any commutative base ring K, or at least a sufficiently nice base ring, such as K = Z for instance. This seemingly innocent generalization turns out to be quite hard. The present paper is devoted to Date: 9 December 2006.