DG structure on the length 4 big from small construction

The big from small construction was introduced by Kustin and Miller in [A. Kustin and M. Miller, Constructing big Gorenstein ideals from small ones, J. Algebra 85 (1983) 303–322] and can be used to construct resolutions of tightly double linked Gorenstein ideals. In this paper, we expand on the DG-algebra techniques introduced in [A. Kustin, Use DG methods to build a matrix factorization, preprint (2019), arXiv:1905.11435 ] and construct a DG [Formula: see text]-algebra structure on the length [Formula: see text] big from small construction. The techniques employed involve the construction of a morphism from a Tate-like complex to an acyclic DG [Formula: see text]-algebra exhibiting Poincaré duality. This induces homomorphisms which, after suitable modifications, satisfy a list of identities that end up perfectly encapsulating the required associativity and DG axioms of the desired product structure for the big from small construction.

Cohomology of generalized configuration spaces

Let $X$ be a topological space. We consider certain generalized configuration spaces of points on $X$, obtained from the cartesian product $X^{n}$ by removing some intersections of diagonals. We give a systematic framework for studying the cohomology of such spaces using what we call ‘twisted commutative dg algebra models’ for the cochains on $X$. Suppose that $X$ is a ‘nice’ topological space, $R$ is any commutative ring, $H_{c}^{\bullet }(X,R)\rightarrow H^{\bullet }(X,R)$ is the zero map, and that $H_{c}^{\bullet }(X,R)$ is a projective $R$-module. We prove that the compact support cohomology of any generalized configuration space of points on $X$ depends only on the graded $R$-module $H_{c}^{\bullet }(X,R)$. This generalizes a theorem of Arabia.

Decorated Marked Surfaces III: The Derived Category of a Decorated Marked Surface

We study the Ginzburg dg algebra $\Gamma _{\mathbf {T}}$ associated with the quiver with potential arising from a triangulation $\mathbf {T}$ of a decorated marked surface ${\mathbf {S}}_\bigtriangleup$, in the sense of [22]. We show that there is a canonical way to identify all finite-dimensional derived categories $\operatorname {\mathcal {D}}_{fd}(\Gamma _{\mathbf {T}})$, denoted by $\operatorname {\mathcal {D}}_{fd}({\mathbf {S}}_\bigtriangleup )$. As an application, we show that the spherical twist group $\operatorname {ST}({\mathbf {S}}_\bigtriangleup )$ associated with $\operatorname {\mathcal {D}}_{fd}({\mathbf {S}}_\bigtriangleup )$ acts faithfully on its space of stability conditions.

Residual intersections are Koszul–Fitting ideals

We describe generators of disguised residual intersections in any commutative Noetherian ring. It is shown that, over Cohen–Macaulay rings, the disguised residual intersections and algebraic residual intersections are the same, for ideals with sliding depth. This coincidence provides structural results for algebraic residual intersections in a quite general setting. It is shown how the DG-algebra structure of Koszul homologies affects the determination of generators of residual intersections. It is shown that the Buchsbaum–Eisenbud family of complexes can be derived from the Koszul–Čech spectral sequence. This interpretation of Buchsbaum–Eisenbud families has a crucial rule to establish the above results.

Holomorphic field theories and Calabi–Yau algebras

We consider the holomorphic twist of the worldvolume theory of flat D[Formula: see text]-branes transversely probing a Calabi–Yau manifold. A chain complex, constructed using the BV formalism, computes the local observables in the holomorphically twisted theory. Generalizing earlier work in the case [Formula: see text], we find that this complex can be identified with the Ginzburg dg algebra associated to the Calabi–Yau. However, the identification is subtle; the complex is the space of fields contributing to the holomorphic twist of the free theory, and its differential arises from interactions. For [Formula: see text], this holomorphically twisted theory is related to the elliptic genus. We give a general description for D1-branes probing a Calabi–Yau fourfold singularity, and for [Formula: see text] quiver gauge theories. In addition, we propose a relation between the equivariant Hirzebruch [Formula: see text] genus of large-[Formula: see text] symmetric products and cyclic homology.

Formality conjecture for K3 surfaces

We give a proof of the formality conjecture of Kaledin and Lehn: on a complex projective K3 surface, the differential graded (DG) algebra$\operatorname{RHom}^{\bullet }(F,F)$is formal for any sheaf$F$polystable with respect to an ample line bundle. Our main tool is the uniqueness of the DG enhancement of the bounded derived category of coherent sheaves. We also extend the formality result to derived objects that are polystable with respect to a generic Bridgeland stability condition.

THE CURVED A∞-COALGEBRA OF THE KOSZUL CODUAL OF A FILTERED DG ALGEBRA

The goal of this article is to study the coaugmented curved A∞-coalgebra structure of the Koszul codual of a filtered dg algebra over a field k. More precisely, we first extend one result of B. Keller that allowed to compute the A∞-coalgebra structure of the Koszul codual of a nonnegatively graded connected algebra to the case of any unitary dg algebra provided with a nonnegative increasing filtration whose zeroth term is k. We then show how to compute the coaugmented curved A∞-coalgebra structure of the Koszul codual of a Poincaré-Birkhoff-Witt (PBW) deformation of an N-Koszul algebra.