## Recursos de colección

1. #### Deformation bicomplex of module algebras

Yau, Donald
The deformation bicomplex of a module algebra over a bialgebra is constructed. It is then applied to study algebraic deformations in which both the module structure and the algebra structure are deformed. The cases of module coalgebras, comodule (co)algebras, and (co)module bialgebras are also considered.

2. #### On affine morphisms of Hopf algebroids

Powell, Geoffrey M.L.
The paper considers the FPQC stacks which are associated to affine groupoid schemes. Using a formulation of a descent datum in terms of morphisms of affine groupoid schemes, explicit arguments are given which avoid appeal to the general principle of faithfully flat descent. This theory is applied to consider the notion of affine morphism.

3. #### The Euler characteristic of a category as the sum of a divergent series

Berger, Clemens; Leinster, Tom
The Euler characteristic of a cell complex is often thought of as the alternating sum of the number of cells of each dimension. When the complex is infinite, the sum diverges. Nevertheless, it can sometimes be evaluated; in particular, this is possible when the complex is the nerve of a finite category. This provides an alternative definition of the Euler characteristic of a category, which is in many cases equivalent to the original one.

4. #### Excision for $K$-theory of connective ring spectra

Dundas, Bjørn Ian; Kittang, Harald Øyen
We extend Geisser and Hesselholt’s result on “bi-relative $K$-theory” from discrete rings to connective ring spectra. That is, if $\mathcal{A}$ is a homotopy cartesian n-cube of ring spectra (satisfying connectivity hypotheses), then the $(n + 1)$-cube induced by the cyclotomic trace $$K(\mathcal{A}) \to TC(\mathcal{A})$$ is homotopy cartesian after profinite completion. In other words, the fiber of the profinitely completed cyclotomic trace satisfies excision.

5. #### Splittings in the Burnside ring and in $SF_G$

French, Christopher P.
Let $G$ be a finite $p$-group, $p \neq 2$. We construct a map from the space $J_G$, defined as the fiber of $\psi^k-1: B_G O \to B_G Spin$, to the space $(SF_G)_p$, defined as the 1-component of the zeroth space of the equivariant $p$-complete sphere spectrum. Our map produces the same splitting of the $G$-connected cover of $(SF_G)_p$ as we have described in previous work, but it also induces a natural splitting of the $p$-completions of the component groups of fixed point subspaces.

6. #### Hopf-Hochschild (co)homology of module algebras

Kaygun, Atabey
We define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called module algebras. We show that this (co)homology, called Hopf-Hochschild (co)homology, can also be defined as a derived functor on the category of representations of an equivariant analogue of the enveloping algebra of a crossed product algebra. We investigate the relationship of our theory with Hopf cyclic cohomology and also prove Morita invariance of the Hopf-Hochschild (co)homology.

7. #### Homotopy types of truncated projective resolutions

Mannan, W. H.
We work over an arbitrary ring $R$. Given two truncated projective resolutions of equal length for the same module, we consider their underlying chain complexes. We show they may be stabilized by projective modules to obtain a pair of complexes of the same homotopy type.

8. #### Cofibrations in the category of Frölicher spaces: Part I

Dugmore, Brett; Ntumba, Patrice Pungu
Cofibrations are defined in the category of Frölicher spaces by weakening the analog of the classical definition to enable smooth homotopy extensions to be more easily constructed, using flattened unit intervals. We later relate smooth cofibrations to smooth neighborhood deformation retracts. The notion of smooth neighborhood deformation retract gives rise to an analogous result that a closed Frölicher subspace A of the Frölicher space $X$ is a smooth neighborhood deformation retract of $X$ if and only if the inclusion $i : A \hookrightarrow X$ comes from a certain subclass of cofibrations. As an application we construct the right Puppe sequence.

9. #### Model structure on operads in orthogonal spectra

Kro, Tore August
We generalize Berger and Moerdijk's results on axiomatic homotopy theory for operads to the setting of enriched symmetric monoidal model categories, and show how this theory applies to orthogonal spectra. In particular, we provide a symmetric fibrant replacement functor for the positive stable model structure.

10. #### Beyond the hit problem: Minimal presentations of odd-primary Steenrod modules, with application to CP($\infty) and BU Pengelley, David J.; Williams, Frank We describe a minimal unstable module presentation over the Steenrod algebra for the odd-primary cohomology of infinite-dimensional complex projective space and apply it to obtain a minimal algebra presentation for the cohomology of the classifying space of the infinite unitary group. We also show that there is a unique Steenrod module structure on any unstable cyclic module that has dimension one in each complex degree (half the topological degree) with a fixed alpha-number (sum of 'digits') and is zero in other degrees. 11. #### A statistical approach to persistent homology Bubenik, Peter; Kim, Peter T. Assume that a finite set of points is randomly sampled from a subspace of a metric space. Recent advances in computational topology have provided several approaches to recovering the geometric and topological properties of the underlying space. In this paper we take a statistical approach to this problem. We assume that the data is randomly sampled from an unknown probability distribution. We define two filtered complexes with which we can calculate the persistent homology of a probability distribution. Using statistical estimators for samples from certain families of distributions, we show that we can recover the persistent homology of the underlying distribution. 12. #### A cohomological interpretation of Brion's formula Hüttermann, Thomas A subset$P$of$mathbb{R}^n$gives rise to a formal Laurent series with monomials corresponding to lattice points in$P$. Under suitable hypotheses, this series represents a rational function$R(P)$; this happens, for example, when$P$is bounded in which case$R(P)$is a Laurent polynomial. Michel Brion [2] has discovered a surprising formula relating the Laurent polynomial$R(P)$of a lattice polytope$P$to the sum of rational functions corresponding to the supporting cones subtended at the vertices of$P$. The result is re-phrased and generalised in the language of cohomology of line bundles on complete toric varieties. Brion's formula is the special case of an ample... 13. #### Heller triangulated categories Künzer, Matthias Let$\mathcal{E}$be a Frobenius category. Let$\underset {=} {\mathcal{E}}$denote its stable category. The shift functor on$\underline {E}$induces, by pointwise application, an inner shift functor on the category of acyclic complexes with entries in$\underset {=} {\mathcal{E}}$. Shifting a complex by 3 positions yields an outer shift functor on this category. Passing to quotient modulo split acyclic complexes, Heller remarked that inner and outer shift become isomorphic, via an isomorphism satisfying yet a further compatibility. Moreover, Heller remarked that a choice of such an isomorphism determines a Verdier triangulation on$\underset {=} {\mathcal{E}}$, except for the octahedral axiom. We generalise the notion of acyclic complexes such that... 14. #### A chain coalgebra model for the James map Hess, Kathryn; Parent, Paul-Eugène; Scott, Jonathan Let$EK$be the simplicial suspension of a pointed simplicial set$K$. We construct a chain model of the James map,$\alpha_K : CK \to \Omega CEK$. We compute the cobar diagonal on$\Omega CEK$, not assuming that$EK $is 1-reduced, and show that$\alpha_K$is comultiplicative. As a result, the natural isomorphism of chain algebras$TCK \cong \Omega CK$preserves diagonals. In an appendix, we show that the Milgram map,$\Omega (A \otimes B) \to \Omega A \otimes \Omega B$, where$A$and$B$are coaugmented coalgebras, forms part of a strong deformation retract of chain complexes. Therefore, it is a chain equivalence even when$A$and$B$are... 15. ####$DG$-models of projective modules and Nakajima quiver varieties Eshmatov, Farkhod Associated to each finite subgroup$\Gamma$of${\tt SL}_2(\mathbb{C})$there is a family of noncommutative algebras$O^\tau(\Gamma)$, which is a deformation of the coordinate ring of the Kleinian singularity$\mathbb{C}^2/\Gamma$. We study finitely generated projective modules over these algebras. Our main result is a bijective correspondence between the set of isomorphism classes of rank one projective modules over$O^\tau$and a certain class of quiver varieties associated to$\Gamma$s. We show that this bijection is naturally equivariant under the action of a “large” Dixmier-type automorphism group$G$. Our construction leads to a completely explicit description of ideals of the algebras$O^\tau$. 16. #### Thick subcategories of the derived category of a hereditary algebra Brüning, Kristian We classify thick subcategories of the bounded derived category of a hereditary abelian category A in terms of subcategories of A. The proof can be applied to characterize the localizing subcategories of the full derived category of A. As an application we prove an algebraic analog of the telescope conjecture for the derived category of a representation finite hereditary artin algebra. 17. #### Erratum to `Category of$A_\infty$-categories' Lyubashenko, Volodymyr The erroneous statement (HHA 5 (2003), no. 1, 1–48) that the collection of unital$A_\infty$-categories, all$A_\infty$-functors, and all$A_\infty$-transformations (resp. equivalence classes of natural$A_\infty$- transformations) form a$\mathcal{K}-2$-category$\mathcal{K}^u A_\infty$(resp. ordinary 2-category$^u A_\infty$) is corrected as follows. All 2-category axioms are satisfied, except that$1_e \cdot f$does not necessarily equal$1_{ef}$for all composable 1-morphisms$e, f$. The axiom$e \cdot 1_f = 1_{ef}$does hold. The mistake does not affect results on invertible 2-morphisms and quasi-invertible 1-morphisms in$^u A_\infty$. 18. #### A few localisation theorems Kahn, Bruno; Sujatha, R. Given a functor$T : \mathcal{C} \to \mathcal{D}$carrying a class of morphisms$S \subset \mathcal C$into a class$S^\prime \subset \mathcal{D}$, we give sufficient conditions for$T$to induce an equivalence on the localised categories. These conditions are in the spirit of Quillen’s Theorem A. We give some applications in algebraic and birational geometry. 19. #### From loop groups to 2-groups Baez, John C.; Stevenson, Danny; Crans, Alissa S.; Schreiber, Urs We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String($n$). A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the 'Jacobiator.' Similarly, a Lie 2-group is a categorified version of a Lie group. If$G$is a simply-connected compact simple Lie group, there is a 1-parameter family of Lie 2-algebras$\mathfrak{g}_k$each having$\mathfrak{g}$as its Lie algebra of objects, but with a Jacobiator built from the canonical 3-form on$G$. There appears to be no Lie 2-group having$\mathfrak{g}_\mathcal{k}$as its Lie 2-algebra, except when$k = 0$.... 20. #### On higher nil groups of group rings Juan-Pineda, Daniel Let$G$be a finite group and$mathbb{Z}[G]$its integral group ring. We prove that the nil groups$N^j K_2 \mathbb{Z}[G])$do not vanish for all$j \geq 1$and for a large class of finite groups. We obtain from this that the iterated nil groups$N^j K_i (\mathbb{Z}[G])$are also nonzero for all$i \geq 2, j \geq i - 1\$

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