A Grothendieck-Lefschetz theorem for equivariant Picard groups

We prove a Grothendieck-Lefschetz theorem for equivariant Picard groups of non-singular varieties with finite group actions.Comment: 7 pages; to appear in JPA

Algebraic K-theory of quotient stacks

We prove some fundamental results like localization, excision, Nisnevich descent and the Mayer-Vietoris property for equivariant regular blow-up for the equivariant K-theory of schemes with an affine group scheme action. We also show that the equivariant K-theory with finite coefficients is invariant under equivariant vector bundle morphisms. We show that the equivariant homotopy K-theory is invariant under equivariant vector bundle morphisms and satisfies all the above properties, including nil-invariance.Comment: 21 pages, Title changed, Final version, to appear in Annals of K-Theor

Rigidity for equivariant pseudo pretheories

We prove versions of the Suslin and Gabber rigidity theorems in the setting of equivariant pseudo pretheories of smooth schemes over a field with an action of a finite group. Examples include equivariant algebraic $K$-theory, presheaves with equivariant transfers, equivariant Suslin homology, and Bredon motivic cohomology.Comment: 19 pages; to appear in Journal of Algebr

Equivariant vector bundles, their derived category and KK-theory on affine schemes

Let $G$ be an affine group scheme over a noetherian commutative ring $R$. We show that every $G$-equivariant vector bundle on an affine toric scheme over $R$ with $G$-action is extended from \Spec(R) for several cases of $R$ and $G$. We show that given two affine schemes with group scheme actions, an equivalence of the equivariant derived categories implies isomorphism of the equivariant $K$-theories as well as equivariant $K'$-theories.Comment: 34 pages, minor corrections, final version, to appear in Annals of K-Theor

Virtual equivariant Grothendieck-Riemann-Roch formula

For a $G$-scheme $X$ with a given equivariant perfect obstruction theory, we prove a virtual equivariant Grothendieck-Riemann-Roch formula, this is an extension of a result of Fantechi-G\"ottsche to the equivariant context. We also prove a virtual non-abelian localization theorem for schemes over $\mathbb{C}$ with proper actions.Comment: 28 pages. Comments are welcome

Localization theorems for algebraic stacks

In this paper we consider three types of localization theorems for algebraic stacks: i) Concentration, or cohomological localization. Given an algebraic group acting on a scheme or stack, we give a sufficient criterion for its localized equivariant Borel-Moore homology to be concentrated in a given closed substack. We deduce this from a new kind of stacky concentration theorem. ii) Atiyah-Bott localization, or localization of (virtual) fundamental classes to fixed loci of torus actions. In particular, this gives a conceptual new proof of the Graber-Pandharipande formula without global embedding or global resolution hypotheses. iii) Cosection localization, or localization of virtual fundamental classes to degeneracy loci of cosections (of the "obstruction sheaf"). We recast this in terms of a notion of "cohomological reductions" of (-1)-shifted 1-forms on derived stacks. These results also apply to oriented Borel-Moore homology theories, such as (higher) Chow homology and algebraic bordism, and hold over arbitrary fields and even in mixed characteristic. In an appendix, we study various types of fixed loci for algebraic group actions on algebraic stacks.Comment: 78 pages, comments welcom

Molecular pathogenesis and diagnostic imaging of metastatic jaw tumors

Metastasis is the spread of malignant cells from a primary tumor to distant sites through lymphatics or blood vessels. Malignant lesions metastasizing to the oral and perioral region are a rarity indeed. Malignant lesions could metastasize to both soft tissue of oral cavity and the hard tissues of the jaws and recent meta-analysis showed that metastasis is more common in the jaws than oral soft tissues because of rich vascular supply. The incidence is very low when compared to the incidence of primary oral cancers; nevertheless, one has to include in the diagnostic workup, metastatic malignant lesions, when an irregular ill-defined radiolucency or radiodensity with ragged edges in noted. It could be a challenging task for a diagnostician, in cases with the presence and location of the primary tumor is unknown. Advanced oral imaging technologies and biochemical markers play a vital role in diagnosing such lesions