Koszul Resolution Research Articles

Algebraic theories of power operations

AbstractWe develop and exposit some general algebra useful for working with certain algebraic structures that arise in stable homotopy theory, such as those encoding well‐behaved theories of power operations for ring spectra. In particular, we consider Quillen cohomology in the context of algebras over algebraic theories, plethories, and Koszul resolutions for algebras over additive theories. By combining this general algebra with obstruction‐theoretic machinery, we obtain tools for computing with algebras over and over Lubin–Tate spectra. As an application, we demonstrate the existence of periodic complex orientations at heights .

Superconformal Algebras and Holomorphic Field Theories

We show that four-dimensional superconformal algebras admit an infinite-dimensional derived enhancement after performing a holomorphic twist. The type of higher symmetry algebras we find is closely related to algebras studied by Faonte–Hennion–Kapranov, Hennion–Kapranov, and the second author with Gwilliam in the context of holomorphic QFT. We show that these algebras are related to the two-dimensional chiral algebras extracted from four-dimensional superconformal theories by Beem and collaborators; further deforming by a superconformal element induces the Koszul resolution of a plane in mathbb {C}^2 cong {mathbb {R}}^4. The central charges at the level of chiral algebras arise from central extensions of the higher symmetry algebras.

Big principal series, -adic families and -invariants

In earlier work, the first named author generalized the construction of Darmon-style$\mathcal {L}$-invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic$\mathcal {L}$-invariants can be computed in terms of derivatives of Hecke eigenvalues in$p$-adic families. Our proof is novel even in the case of modular forms, which was established by Bertolini, Darmon and Iovita. The main new technical ingredient is the Koszul resolution of locally analytic principal series representations by Kohlhaase and Schraen. As an application of our results we settle a conjecture of Spieß: we show that automorphic$\mathcal {L}$-invariants of Hilbert modular forms of parallel weight$2$are independent of the sign character used to define them. Moreover, we show that they are invariant under Jacquet–Langlands transfer and, in fact, equal to the Fontaine–Mazur$\mathcal {L}$-invariant of the associated Galois representation. Under mild assumptions, we also prove the equality of automorphic and Fontaine–Mazur$\mathcal {L}$-invariants for representations of definite unitary groups of arbitrary rank. Finally, we study the case of Bianchi modular forms to show how our methods, given precise results on eigenvarieties, can also work in the absence of discrete series representations.

Twisting structures and morphisms up to strong homotopy

We define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the bar-cobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction. The associated Kleisli categories are the “strong homotopy” morphism categories. In an appendix, we study the co-ring associated to the canonical morphism of cooperads , which is exactly the two-sided Koszul resolution of the associative operad , also known as the Alexander-Whitney co-ring.

Endomorphisms of Koszul complexes: formality and application to deformation theory

We study the differential graded Lie algebra of endomorphisms of the Koszul resolution of a regular sequence on a unitary commutative $K$-algebra $R$ and we prove that it is homotopy abelian over $K$, while it is generally not formal over $R$. We apply this result to prove an annihilation theorem for obstructions of (derived) deformations of locally complete intersection ideal sheaves on projective schemes.

A note on the adjoint action of a semisimple group on the coordinate algebra of an infinitesimal subgroup

Let G be a connected, semisimple, simply connected, algebraic group over an algebraically closed field K of positive characteristic. The action of G on itself by conjugation induces on the coordinate algebra K[G], and hence on the coordinate algebra K[Gr] of the infinitesimal subgroup Gr, the structure of a rational G-module. In the recent paper, [3], by Bendel, Nakano, Pillen and Sobaje, the authors ask whether the tensor product of K[Gr] with the rth Steinberg module Str has a good filtration. Here we show that this is the case provided that the characteristic of K is good for G. We do so by relating K[Gr] to a certain quotient of the coordinate algebra K[g] of the Lie algebra g of G and using a Koszul resolution. We note that the argument may also be applied to certain polynomial modules occurring in the context of Doty's Conjecture, [17, 4.2.11].

Gerstenhaber brackets on Hochschild cohomology of quantum symmetric algebras and their group extensions

We construct chain maps between the bar and Koszul resolutions for a quantum symmetric algebra (skew polynomial ring). This construction uses a recursive technique involving explicit formulae for contracting homotopies. We use these chain maps to compute the Gerstenhaber bracket, obtaining a quantum version of the Schouten-Nijenhuis bracket on a symmetric algebra (polynomial ring). We compute brackets also in some cases for skew group algebras arising as group extensions of quantum symmetric algebras.

PBW deformations of Koszul algebras over a nonsemisimple ring

Let $B$ be a generalized Koszul algebra over a finite dimensional algebra $S$. We construct a bimodule Koszul resolution of $B$ when the projective dimension of $S_B$ equals 2. Using this we prove a Poincar\'e-Birkhoff-Witt (PBW) type theorem for a deformation of a generalized Koszul algebra. When the projective dimension of $S_B$ is greater than 2, we construct bimodule Koszul resolutions for generalized smash product algebras obtained from braidings between finite dimensional algebras and Koszul algebras, and then prove the PBW type theorem. The results obtained can be applied to standard Koszul Artin-Schelter Gorenstein algebras in the sense of Minamoto and Mori.

PBW Deformations of Skew Group Algebras in Positive Characteristic

We investigate deformations of a skew group algebra that arise from a finite group acting on a polynomial ring. When the characteristic of the underlying field divides the order of the group, a new type of deformation emerges that does not occur in characteristic zero. This analogue of Lusztig’s graded affine Hecke algebra for positive characteristic can not be forged from the template of symplectic reflection and related algebras as originally crafted by Drinfeld. By contrast, we show that in characteristic zero, for arbitrary finite groups, a Lusztig-type deformation is always isomorphic to a Drinfeld-type deformation. We fit all these deformations into a general theory, connecting Poincare-Birkhoff-Witt deformations and Hochschild cohomology when working over fields of arbitrary characteristic. We make this connection by way of a double complex adapted from Guccione, Guccione, and Valqui, formed from the Koszul resolution of a polynomial ring and the bar resolution of a group algebra.

Generalized Koszul Resolution

In this paper, we generalize a notion of Koszul resolutions and characterize modules which admits such resolutions. It turns out that for a noetherian ring $A$ and a coherent $A$-module $M$, $M$ has a two dimensional generalized Koszul resolution if and only if $M$ is a pure weight two module in the sense of [HM10]. As an application, we attack the Gersten conjecture for weight two case.

Homotopy Batalin–Vilkovisky algebras

This paper provides an explicit cofibrant resolution of the operad encoding Batalin–Vilkovisky algebras. Thus it defines the notion of homotopy Batalin–Vilkovisky algebras with the required homotopy properties. To define this resolution, we extend the theory of Koszul duality to operads and properads that are defined by quadratic and linear relations. The operad encoding Batalin–Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincaré–Birkhoff–Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal field theory carries a homotopy BV-algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad \mathcal{BV} . We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian–Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.

Quantum differentiation and chain maps of bimodule complexes

We consider a finite group acting on a vector space and the corresponding skew group algebra generated by the group and the symmetric algebra of the space. This skew group algebra illuminates the resulting orbifold and serves as a replacement for the ring of invariant polynomials, especially in the eyes of cohomology. One analyzes the Hochschild cohomology of the skew group algebra using isomorphisms which convert between resolutions. We present an explicit chain map from the bar resolution to the Koszul resolution of the symmetric algebra which induces various isomorphisms on Hochschild homology and cohomology, some of which have appeared in the literature before. This approach unifies previous results on homology and cohomology of both the symmetric algebra and skew group algebra. We determine induced combinatorial cochain maps which invoke quantum differentiation (expressed by Demazure-BGG operators).

Multiplicative Aspects of the Halperin–Carlsson Conjecture

We use the multiplicative structure of the Koszul resolution to give short and simple proofs of some known estimates for the total dimension of the cohomology of spaces which admit free torus actions and some analogous results for filtered differential modules over polynomial rings. We also point out the possibility of improving these results in the presence of a multiplicative structure on the so-called minimal Hirsch–Brown model for the equivariant cohomology of the space.

On the cooperation algebra of the connective Adams summand

The aim of this paper is to gain explicit information about the multiplicative structure of l* l, where l is the connective Adams summand at an odd prime p. Our approach differs from Kane's or Lellmann's because our main technical tool is the MU-based Kunneth spectral sequence. We prove that the algebra structure on l* l is inherited from the multiplication on a Koszul resolution of l* BP.

Quantization on Curves

Deformation quantization on varieties with singularities offers perspectives that are not found on manifolds. The Harrison component of Hochschild cohomology, vanishing on smooth manifolds, reflects information about singularities. The Harrison 2-cochains are symmetric and are interpreted in terms of abelian *-products. This paper begins a study of abelian quantization on plane curves over $$\mathbb{C}$$ , being algebraic varieties of the form $${\mathbb{C}}^2/R$$ , where R is a polynomial in two variables; that is, abelian deformations of the coordinate algebra $$\mathbb{C}[x,y]/(R$$ ). To understand the connection between the singularities of a variety and cohomology we determine the algebraic Hochschild (co)homology and its Barr–Gerstenhaber–Schack decomposition. Homology is the same for all plane curves $$\mathbb{C}[x,y]/R$$ , but the cohomology depends on the local algebra of the singularity of R at the origin. The Appendix, by Maxim Kontsevich, explains in modern mathematical language a way to calculate Hochschild and Harrison cohomology groups for algebras of functions on singular planar curves etc. based on Koszul resolutions.

Quantum Weyl reciprocity for cohomology

We study the relationship between the cohomology of $q$-Schur algebras and Hecke algebras in type $A$. A key tool, which has independent interest, involves a specific Hecke algebra resolution, introduced in a special case by Deodhar. This resolution is somewhat akin to the Koszul resolution for Lie algebra cohomology, in that it has an explicit description yet applies for all modules. An application of the properties of this resolution establishes a ?quantum Weyl reciprocity? connecting, in a range of degrees, the cohomology of the $q$-Schur algebra with that of the Hecke algebra. This work also provides, independently of the resolution, cohomological inequalities which apply in all degrees and which sometimes can be related to Kazhdan?Lusztig polynomials. Finally, we extend to cohomology a decomposition number method of Erdmann, and illustrate its usefulness in computations. This method enables us to exhibit some properties of symmetric group cohomology which, if true, would imply the $SL_n$-version of Lusztig's characteristic $p$ character conjecture.

HOMOLOGY OF POWERS OF REGULAR IDEALS

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. For a commutative ring $R$ with an ideal $I$, generated by a finite regular sequence, we construct differential graded algebras which provide $R$-free resolutions of $I^s$ and of $R/I^s$ for $s \geq 1$ and which generalise the Koszul resolution. We derive these from a certain multiplicative double complex ${\mathbf K}$. By means of a Cartan–Eilenberg spectral sequence we express ${\rm Tor}_*^R(R/I, R/I^s)$ and ${\rm Tor}_*^R(R/I, I^s)$ in terms of exact sequences and find that they are free as $R/I$-modules. Except for $R/I$, their product structure turns out to be trivial; instead, we consider an exterior product ${\rm Tor}_*^R(R/I, I^s)\,{\otimes_R}\,{\rm Tor}_*^R(R/I, I^t)\,{\to}\,{\rm Tor}_*^R(R/I, I^{s+t})$. This paper is based on ideas by Andrew Baker; it is written in view of applications to algebraic topology.

Torus Actions, Equivariant Moment-Angle Complexes, and Coordinate Subspace Arrangements

We show that the cohomology algebra of the complement of a coordinate subspace arrangement in the m-dimensional complex space is isomorphic to the cohomology algebra of the Stanley―Reisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) After that we calculate the latter cohomology algebra by means of the standard Koszul resolution of a polynomial ring. To prove these facts, we construct a homotopy equivalence (equivariant with respect to the torus action) between the complement of a coordinate subspace arrangement and the moment-angle complex defined by a simplicial complex. The moment-angle complex is a certain subset of the unit polydisk in the m-dimensional complex space invariant with respect to the action of the m-dimensional torus. This complex is a smooth manifold provided that the simplicial complex is a simplicial sphere; otherwise, the complex has a more complicated structure. Then we investigate the equivariant topology of the moment-angle complex and apply the Eilenberg―Moore spectral sequence. We also relate our results with well-known facts in the theory of toric varieties and symplectic geometry. Bibliography: 23 titles.

Koszulity for Nonquadratic Algebras

It is known that a Koszul algebra is defined as being a quadratic algebra with a “pure” resolution of the ground field. In this paper, we extend Koszulity to algebras whose relations are homogeneous of degree s>2. A cubic Artin–Schelter regular algebra has motivated our work. Generalized Koszulity is connected to lattice distributivity and to confluence. A generalized symmetric algebra is proved to be generalized Koszul, and the bimodule version of the generalized Koszul resolution is used for investigating its Hochschild homology.

Reduction number of links of irreducible varieties

The reductions of an ideal I give a natural pathway to the properties of I, with the advantage of having fewer generators. In this paper we primarily focus on a conjecture about the reduction exponent of links of a broad class of primary ideals. The existence of an algebra structure on the Koszul and Eagon-Northcott resolutions is the main tool for detailing the known cases of the conjecture. In the last section we relate the conjecture to a formula involving the length of the first Koszul homology modules of these ideals.