Pedro Tamaroff
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View article: Tangent complexes and the Diamond Lemma
Tangent complexes and the Diamond Lemma Open
The celebrated Diamond Lemma of Bergman gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras. We revisit that result in the context of deformation theory and homotopical algebra…
View article: Generalized Cohomological Field Theories in the Higher Order Formalism
Generalized Cohomological Field Theories in the Higher Order Formalism Open
In the classical Batalin—Vilkovisky formalism, the BV operator is a differential operator of order two with respect to a commutative product; in the differential graded setting, it is known that if the BV operator is homotopically trivial,…
View article: Resolutions of operads via Koszul (bi)algebras
Resolutions of operads via Koszul (bi)algebras Open
We introduce a construction that produces from each bialgebra H an operad $$\mathsf {Ass}_H$$ controlling associative algebras in the monoidal category of H -modules or, briefly, H -algebras. When the underlying algebra of this bialgeb…
View article: The cohomology of coalgebras in species
The cohomology of coalgebras in species Open
Aguiar and Mahajan introduced a cohomology theory for the twisted coalgebras of Joyal, with particular interest in the computation of their second cohomology group, which gives rise to their deformations. We use the Koszul duality theory b…
View article: Generalized cohomological field theories in the higher order formalism
Generalized cohomological field theories in the higher order formalism Open
In the classical Batalin--Vilkovisky formalism, the BV operator $Δ$ is a differential operator of order two with respect to the commutative product. In the differential graded setting, it is known that if the BV operator is homotopically t…
View article: Homotopical and effective methods for associative algebras
Homotopical and effective methods for associative algebras Open
This thesis contains four main chapters based on four different papers. In the third chapter, we solve the problem of computing the minimal model of an arbitrary associative monomial algebra. Our methods are combinatorial and depend on a d…
View article: The Tamarkin–Tsygan calculus of an associative algebra<i> à la</i> Stasheff
The Tamarkin–Tsygan calculus of an associative algebra<i> à la</i> Stasheff Open
We show how to compute the Tamarkin-Tsygan calculus of an associative algebra by providing, for a given cofibrant replacement of it, a 'small' Calc ∞ -model of its calculus, which we make somewhat explicit at the level of Calc-algebras.To …
View article: Minimal models for monomial algebras
Minimal models for monomial algebras Open
Using combinatorics of chains going back to works of Anick, Green, Happel and Zacharia, we give, for any monomial algebra $A$, an explicit description of its minimal model. This also provides us with formulas for a canonical $A_\infty$-str…
View article: Tangent complexes and the Diamond Lemma
Tangent complexes and the Diamond Lemma Open
The celebrated Diamond Lemma of Bergman gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras. We present a new way to interpret and prove this result from the viewpoint of homot…
View article: Differential forms on smooth operadic algebras
Differential forms on smooth operadic algebras Open
The classical Hochschild--Kostant--Rosenberg (HKR) theorem computes the Hochschild homology and cohomology of smooth commutative algebras. In this paper, we generalise this result to other kinds of algebraic structures. Our main insight is…
View article: A spectral sequence for tangent cohomology of algebraic operads
A spectral sequence for tangent cohomology of algebraic operads Open
Operadic tangent cohomology generalizes the existing theories of Harrison cohomology, Chevalley--Eilenberg cohomology and Hochschild cohomology. These are usually non-trivial to compute. We complement the existing computational techniques …
View article: A spectral sequence for Andr\'e--Quillen cohomology of algebraic operads
A spectral sequence for Andr\'e--Quillen cohomology of algebraic operads Open
Operadic cohomology generalizes the existing theories of Harrison cohomology, Chevalley--Eilenberg cohomology and Hochschild cohomology. These are usually non-trivial to compute. We complement the existing computational techniques by produ…
View article: The cohomology of twisted coalgebras
The cohomology of twisted coalgebras Open
In this paper, which is based on the author's MSc thesis, we study in detail the cohomology theory for twisted coalgebras introduced in Monoidal Functors, Species and Hopf Algebras by M. Aguiar and S. Mahajan. We compute it completely in v…
View article: Derived Poincaré-Birkhoff-Witt theorems (with an appendix by Vladimir Dotsenko)
Derived Poincaré-Birkhoff-Witt theorems (with an appendix by Vladimir Dotsenko) Open
We define derived Poincaré--Birkhoff--Witt maps of dg operads or derived PBW maps, for short, which extend the definition of PBW maps between operads of V.~Dotsenko and the second author in 1804.06485, with the purpose of studying the univ…
View article: Derived Poincar\\'e-Birkhoff-Witt theorems (with an appendix by Vladimir\n Dotsenko)
Derived Poincar\\'e-Birkhoff-Witt theorems (with an appendix by Vladimir\n Dotsenko) Open
We define derived Poincar\\'e--Birkhoff--Witt maps of dg operads or derived\nPBW maps, for short, which extend the definition of PBW maps between operads of\nV.~Dotsenko and the second author in 1804.06485, with the purpose of studying\nth…
View article: Finite generation for Hochschild cohomology of Gorenstein monomial algebras
Finite generation for Hochschild cohomology of Gorenstein monomial algebras Open
We show that a finite dimensional monomial algebra satisfies the finite generation conditions of Snashall-Solberg for Hochschild cohomology if and only if it is Gorenstein. This gives, in the case of monomial algebras, the converse to a th…
View article: The Tamarkin--Tsygan calculus of an algebra a la Stasheff
The Tamarkin--Tsygan calculus of an algebra a la Stasheff Open
We show how to compute the Tamarkin-Tsygan calculus of an associative algebra by providing, for a given cofibrant replacement of it, a `small' $\mathsf{Calc}_\infty$-model of its calculus, which we make somewhat explicit at the level of $\…