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Register a new userTransfer of A-infinity structures to projective resolutions
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Authors
Jesse Burke
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We show that an A-infinity algebra structure can be transferred to a projective resolution of the complex underlying any A-infinity algebra. Under certain connectedness assumptions, this transferred structure is unique up to homotopy. In contrast to the classical results on transfer of A-infinity structures along homotopy equivalences, our result is of interest when the ground ring is not a field. We prove an analog for A-infinity module structures, and both transfer results preserve strict units.
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This paper is dedicated to triangulated categories endowed with weight structures (a new notion; D. Pauksztello has independently introduced them as co-t-structures). This axiomatizes the properties of stupid truncations of complexes in $K(B)$. We also construct weight structures for Voevodskys categories of motives and for various categories of spectra. A weight structure $w$ defines Postnikov towers of objects; these towers are canonical and functorial up to morphisms that are zero on cohomology. For $Hw$ being the heart of $w$ (in $DM_{gm}$ we have $Hw=Chow$) we define a canonical conservative weakly exact functor $t$ from our $C$ to a certain weak category of complexes $K_w(Hw)$. For any (co)homological functor $H:Cto A$ for an abelian $A$ we construct a weight spectral sequence $T:H(X^i[j])implies H(X[i+j])$ where $(X^i)=t(X)$; it is canonical and functorial starting from $E_2$. This spectral sequences specializes to the usual (Delignes) weight spectral sequences for classical realizations of motives and to Atiyah-Hirzebruch spectral sequences for spectra. Under certain restrictions, we prove that $K_0(C)cong K_0(Hw)$ and $K_0(End C)cong K_0(End Hw)$. The definition of a weight structure is almost dual to those of a t-structure; yet several properties differ. One can often construct a certain $t$-structure which is adjacent to $w$ and vice versa. This is the case for the Voevodskys $DM^{eff}_-$ (one obtains certain new Chow weight and t-structures for it; the heart of the latter is dual to $Chow^{eff}$) and for the stable homotopy category. The Chow t-structure is closely related to unramified cohomology.
Given a graded module over a commutative ring, we define a dg-Lie algebra whose Maurer-Cartan elements are the strictly unital A-infinity algebra structures on that module. We use this to generalize Positselskis result that a curvature term on the bar construction compensates for a lack of augmentation, from a field to arbitrary commutative base ring. We also use this to show that the reduced Hochschild cochains control the strictly unital deformation functor. We motivate these results by giving a full development of the deformation theory of a nonunital A-infinity algebra.
Let A be a connected graded algebra and let E denote its Ext-algebra. There is a natural A-infinity algebra structure on E, and we prove that this structure is mainly determined by the relations of A. In particular, the coefficients of the A-infinity products m_n restricted to the tensor powers of Ext^1 give the coefficients of the relations of A. We also relate the m_ns to Massey products.
Given a Lagrangian V cong CP^n in a symplectic manifold (M,omega), there is an associated symplectomorphism phi_V of M. We define the notion of a CP^n-object in an A-infinity-category A and use this to construct algebraically an A-infinity-functor Phi_V and prove that it induces an autoequivalence of the derived category DA. We conjecture that Phi_V corresponds to the action of phi_V and prove this in the lowest dimension n=1. The construction is designed to be mirror to a construction of Huybrechts and Thomas.
Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the old homological algebra (of derived functors between abelian categories) was established. This new homological algebra, of derived categories and derived functors between them, provides a significantly richer and more flexible machinery than the old homological algebra. For instance, the important concepts of dualizing complex and tilting complex do not exist in the old homological algebra. This paper is an edited version of the notes for a two-lecture minicourse given at MSRI in January 2013. Sections 1-5 are about the general theory of derived categories, and the material is taken from my manuscript A Course on Derived Categories (available online). Sections 6-9 are on more specialized topics, leaning towards noncommutative algebraic geometry.
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