scholarly journals Gysin maps in Balmer–Witt theory

2007 ◽  
Vol 211 (1) ◽  
pp. 203-221 ◽  
Author(s):  
Alexander Nenashev
Keyword(s):  
Witt Theory ◽  
Gysin Maps

scholarly journals FRAMED CORRESPONDENCES AND THE MILNOR–WITT -THEORY

2016 ◽  
Vol 17 (4) ◽  
pp. 823-852 ◽  
Author(s):  
Alexander Neshitov
Keyword(s):  
Characteristic Zero ◽  
Triangulated Category ◽  
Algebraic Varieties ◽  
Base Field ◽  
Witt Theory

Given a field $k$ of characteristic zero and $n\geqslant 0$, we prove that $H_{0}(\mathbb{Z}F(\unicode[STIX]{x1D6E5}_{k}^{\bullet },\mathbb{G}_{m}^{\wedge n}))=K_{n}^{MW}(k)$, where $\mathbb{Z}F_{\ast }(k)$ is the category of linear framed correspondences of algebraic $k$-varieties, introduced by Garkusha and Panin [The triangulated category of linear framed motives $DM_{fr}^{eff}(k)$, in preparation] (see [Garkusha and Panin, Framed motives of algebraic varieties (after V. Voevodsky), Preprint, 2014, arXiv:1409.4372], § 7 as well), and $K_{\ast }^{MW}(k)$ is the Milnor–Witt $K$-theory of the base field $k$.

scholarly journals Gysin maps in oriented theories

2006 ◽  
Vol 302 (1) ◽  
pp. 200-213 ◽  
Author(s):  
Alexander Nenashev
Keyword(s):  
Gysin Maps

On the Witt groups of projective bundles and split quadrics: geometric reasoning

2008 ◽  
Vol 3 (3) ◽  
pp. 533-546 ◽  
Author(s):  
Alexander Nenashev
Keyword(s):  
Geometric Reasoning ◽  
Triangulated Categories ◽  
General Properties ◽  
Projective Bundles ◽  
Witt Theory ◽  
Witt Groups

AbstractFormulas for the derived Witt groups of projective bundles are obtained. We deduce them from general properties of Witt theory, with the help of twisted Thom isomorphisms, avoiding explicit use of triangulated categories. Witt groups of completely split quadrics are also considered.

scholarly journals Localizing virtual structure sheaves for almost perfect obstruction theories

2020 ◽  
Vol 8 ◽  
Author(s):  
Young-Hoon Kiem ◽  
Michail Savvas
Keyword(s):  
Obstruction Theory ◽  
Euler Characteristics ◽  
Moduli Stacks ◽  
Wall Crossing ◽  
Wall Crossing Formula ◽  
Gysin Maps

Abstract Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and K-theoretic invariants for many moduli stacks of interest, including K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the K-theory and Gysin maps of sheaf stacks. In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory. To this end, we further investigate the K-theory of sheaf stacks and its functoriality properties. As applications of the localization formulas, we establish a K-theoretic wall-crossing formula for simple $\mathbb{C} ^\ast $ -wall crossings and define K-theoretic invariants refining the Jiang-Thomas virtual signed Euler characteristics.

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