θ-Invariants of Infinite-Dimensional Hermitian Vector Bundles: Definitions and First Properties

2020 ◽  
pp. 137-154
Author(s):  
Jean-Benoît Bost
Keyword(s):  
Vector Bundles ◽  
Infinite Dimensional ◽  
Hermitian Vector Bundles

scholarly journals QUILLEN EQUIVALENT MODELS FOR THE DERIVED CATEGORY OF FLATS AND THE RESOLUTION PROPERTY

2020 ◽  
pp. 1-19
Author(s):  
SERGIO ESTRADA ◽  
ALEXANDER SLÁVIK
Keyword(s):  
Vector Bundles ◽  
Derived Categories ◽  
Derived Category ◽  
Projective Modules ◽  
Dimensional Vector ◽  
Infinite Dimensional ◽  
Equivalent Models ◽  
Flat Modules ◽  
Quillen Equivalent ◽  
Resolution Property

We investigate the assumptions under which a subclass of flat quasicoherent sheaves on a quasicompact and semiseparated scheme allows us to ‘mock’ the homotopy category of projective modules. Our methods are based on module-theoretic properties of the subclass of flat modules involved as well as their behaviour with respect to Zariski localizations. As a consequence we get that, for such schemes, the derived category of flat quasicoherent sheaves is equivalent to the derived category of very flat quasicoherent sheaves. If, in addition, the scheme satisfies the resolution property then both derived categories are equivalent to the derived category of infinite-dimensional vector bundles. The equivalences are inferred from a Quillen equivalence between the corresponding models.

SUPER KRICHEVER FUNCTOR

1991 ◽  
Vol 02 (06) ◽  
pp. 741-760 ◽  
Author(s):  
MOTOHICO MULASE ◽  
JEFFREY M. RABIN
Keyword(s):  
Line Bundle ◽  
Vector Bundles ◽  
Vector Spaces ◽  
Contravariant Functor ◽  
Infinite Dimensional ◽  
Arbitrary Rank ◽  
Geometric Data

A supersymmetric generalization of the Krichever map is proposed. This map assigns injectively a point of an infinite dimensional super Grassmannian to a set of geometric data consisting of an arbitrary algebraic super manifold of dimension 1|1 defined over a field of any characteristic and a line bundle on it. The naturality of this map comes from the fact that it is obtained as the restriction of a contravariant functor to certain special objects. This functor gives an antiequivalence between the category of infinite dimensional super vector spaces satisfying the Fredholm condition together with their stabilizers, and the category of algebraic super curves and certain sheaves on them including all even vector bundles of arbitrary rank.

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