Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHomotopy coherent nerve in Deformation theory
368
0
0.0
(
0
)
Authors
V. Hinich
Ask ChatGPT about the research
No Arabic abstract
In this note we explain that homotopy coherent simplicial nerve has to used intead of the standard definition in the authors papers on formal deformation theory. A convenient version of the notion of fibered category is presented which is useful once one works with simplicial categories.
rate research
Read More
We construct a version of Dijkgraaf-Witten theory based on a compact abelian Lie group within the formalism of Turaevs homotopy quantum field theory. As an application we show that the 2+1-dimensional theory based on U(1) classifies lens spaces up to homotopy type.
In this paper, we develop the deformation theory controlled by pre-Lie algebras; the main tool is a new integration theory for pre-Lie algebras. The main field of application lies in homotopy algebra structures over a Koszul operad; in this case, we provide a homotopical description of the associated Deligne groupoid. This permits us to give a conceptual proof, with complete formulae, of the Homotopy Transfer Theorem by means of gauge action. We provide a clear explanation of this latter ubiquitous result: there are two gauge elements whose action on the original structure restrict its inputs and respectively its output to the homotopy equivalent space. This implies that a homotopy algebra structure transfers uniformly to a trivial structure on its underlying homology if and only if it is gauge trivial; this is the ultimate generalization of the $dd^c$-lemma.
We construct geometric models for classifying spaces of linear algebraic groups in G-equivariant motivic homotopy theory, where G is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the homotopy K-theory of G-schemes (which we construct as an E-infinity-ring) is stable under arbitrary base change, and we deduce that homotopy K-theory of G-schemes satisfies cdh descent.
Let $G=Sp_{2n}(mathbb{C})$, and $mathfrak{N}$ be Katos exotic nilpotent cone. Following techniques used by Bezrukavnikov in [5] to establish a bijection between $Lambda^+$, the dominant weights for a simple algebraic group $H$, and $textbf{O}$, the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of the isotropy group of the orbit, we prove an analogous statement for the exotic nilpotent cone. First we prove that dominant line bundles on the exotic Springer resolution $widetilde{mathfrak{N}}$ have vanishing higher cohomology, and compute their global sections using techniques of Broer. This allows to show that the direct images of these dominant line bundles constitute a quasi-exceptional set generating the category $D^b(Coh^G(mathfrak{N}))$, and deduce that the resulting $t$-structure on $D^b(Coh^G(mathfrak{N}))$ coincides with the perverse coherent $t$-structure. The desired result now follows from the bijection between costandard objects and simple objects in the heart of this $t$-structure on $D^b(Coh^G(mathfrak{N}))$.
For a finite Galois extension of fields L/k with Galois group G, we study a functor from the G-equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical Galois correspondence. We show that after completing at a prime and eta (the motivic Hopf map) this results in a full and faithful embedding whenever k is real closed and L = k[i]. It is a full and faithful embedding after eta-completion if a motivic version of Serres finiteness theorem is valid. We produce strong necessary conditions on the field extension L/k for this functor to be full and faithful. Along the way, we produce several results on the stable C_2-equivariant Betti realization functor and prove convergence theorems for the p-primary C_2-equivariant Adams spectral sequence.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
