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In this paper we consider the K-theory of smooth algebraic stacks, establish lambda and gamma operations, and show that the higher K-theory of such stacks is always a pre-lambda-ring, and is a lambda-ring if every coherent sheaf is the quotient of a vector bundle. As a consequence, we are able to define Adams operations and absolute cohomology for smooth algebraic stacks satisfying this hypothesis. We also obtain a comparison of the absolute cohomology with the equivariant higher Chow groups in certain special cases., Comment: To appear in the Annals of K-theory

In the present paper, we discuss applications of the derived completion theorems proven in our previous two papers. One of the main applications is to Riemann-Roch problems for forms of higher equivariant K-theory, which we are able to establish in great generality both for equivariant G-theory and equivariant homotopy K-theory with respect to actions of linear algebraic groups on normal quasi-projective schemes over a given field. We show such Riemann-Roch theorems apply to all toric and spherical varieties. We also obtain Lefschetz-Riemann-Roch theorems involving the fixed point schemes with respect to actions of diagonalizable group schemes. We also show the existence of certain spectral sequences that compute the homotopy groups of the derived completions of equivariant G-theory starting with equivariant Borel-Moore motivic cohomology.

In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known completion theorems for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem for equivariant Algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him. In an earlier work by the first two authors, we solved this conjecture by providing a derived completion theorem for equivariant G-theory. In the present paper, we provide a similar derived completion theorem for the homotopy Algebraic K-theory of equivariant perfect complexes, on schemes that need not be regular. Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any normal quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the Equivariant Homotopy Algebraic K-Theory of large classes of varieties like all toric varieties (for the action of a torus) and all spherical varieties (for the action of a reductive group). With finite coefficients invertible in the base fields, we are also able to obtain such derived completion theorems for equivariant algebraic K-theory but with respect to actions of diagonalizable group schemes. These enable us to obtain a wide range of applications, several of which are also explored., Comment: arXiv admin note: substantial text overlap with arXiv:1906.06827

In this paper, we settle an open conjecture regarding the assertion that the Euler-characteristic of $\rmG/\NT$ for a split reductive group scheme $\rmG$ and the normalizer of a split maximal torus $\NT$ over a field is $1$ in the Grothendieck-Witt ring with the characteristic exponent of the field inverted, under the assumption that the base field contains a $\sqrt -1$. Numerous applications of this to splittings in the motivic stable homotopy category and to Algebraic K-Theory are worked out in several related papers by Gunnar Carlsson and the authors., Comment: 16 pages. To appear in Advances in Mathematics. arXiv admin note: substantial text overlap with arXiv:2007.02249

We construct two functorial filtrations on the algebraic $K$-theory of schemes of finite type over a field $k$ that may admit arbitrary singularities and may be non-reduced, one called the coniveau filtration, and the other called the motivic coniveau filtration. Restricting to the subcategory of smooth $k$-schemes, our coniveau filtration coincides with the classical coniveau (also known as the topological) filtration on algebraic $K$-theory of D. Quillen, whereas our motivic coniveau filtration coincides with the homotopy coniveau filtration for algebraic $K$-theory of M. Levine., Comment: 47 pages

We show that the orthogonal spectral sequence introduced by the second author is strongly convergent in Voevodsky's triangulated category of motives DM over a field k. In the context of the Morel-Voevodsky motivic stable homotopy category we provide concrete examples where the spectral sequence is not strongly convergent, and give a criterion under which the strong convergence still holds. This criterion holds for Voevodsky's slices, and as a consequence we obtain a spectral sequence which converges strongly to the E1-term of Voevodsky's slice spectral sequence., Comment: 11 pages. arXiv admin note: text overlap with arXiv:1605.00717

The present paper is a continuation of earlier work by Gunnar Carlsson and the first author on a motivic variant of the classical Becker-Gottlieb transfer and an additivity theorem for such a transfer by the present authors. Here, we establish a motivic variant of the classical Segal-Becker theorem relating the classifying space of a 1-dimensional torus with the spectrum defining (algebraic) K-theory., Comment: To appear in the Annals of K-Theory

In this paper, which is a continuation of earlier work by the first author and Gunnar Carlsson, one of the first results we establish is the additivity of the motivic Becker-Gottlieb transfer, as well as their \'etale realizations. This extends the additivity results the authors already established for the corresponding traces. We then apply this to derive several important consequences: for example, in addition to obtaining the analogues of various double coset formulae known in the classical setting of algebraic topology, we also obtain applications to Brauer groups of homogeneous spaces associated to reductive groups over separably closed fields. We also consider the relationship between the transfer on schemes provided with a compatible action by a $1$-parameter subgroup and the transfer associated to the fixed point scheme of the $1$-parameter subgroup., Comment: In this updated version, the authors extend and expand on the additivity results obtained in the earlier version

We investigate vapor homogeneous nucleation in a superheated Lennard Jones liquid with computer simulations. Special simulation techniques are required to address this study since the nucleation of a critical vapor bubble --one that has equal chance to grow or shrink-- in a moderately superheated liquid is a rare event. We use the Seeding method, that combines Classical Nucleation Theory with computer simulations of a liquid containing a vapor bubble to provide bubble nucleation rates in a wide temperature range. Seeding has been successfully applied to investigate the nucleation of crystals in supercooled fluids and here we apply it for the fist time --to the best of our knowledge-- to the liquid-to-vapor transition. We find that Seeding provides nucleation rates that are consistent with independent calculations not based on the assumptions of Classical Nucleation Theory. Different criteria to determine the radius of the critical bubble give different rate values. The accuracy of each criterion depends of the degree of superheating. Moreover, Seeding simulations show that the surface tension depends on pressure for a given temperature. Therefore, using Classical Nucleation Theory with the coexistence surface tension does not provide good estimates of the nucleation rate., Comment: paper

Simulations are widely used to study nucleation in first order phase transitions due to the fact that they have access to the relevant length and time scales. However, simulations face the problem that nucleation is an activated process. Therefore, rare event simulation techniques are needed to promote the formation of the critical nucleus. The Seeding method, where the simulations are started with the nucleus already formed, has proven quite useful in efficiently providing estimates of the nucleation rate for a wide range of orders of magnitude. So far, Seeding has been employed in the NPT ensemble, where the nucleus either grows or redissolves. Thus, several trajectories have to be run in order to find the thermodynamic conditions that make the seeded nucleus critical. Moreover, the nucleus lifetime is short and the statistics for obtaining its properties is consequently poor. To deal with these shortcomings we extend the Seeding method to the NVT ensemble. We focus on the problem of bubble nucleation in a mestastable Lennard Jones fluid. We show that, in the NVT ensemble, it is possible to equilibrate and stabilise critical bubbles for a long time. The nucleation rate inferred from NVT-Seeding is fully consistent with that coming from NPT-Seeding. The former is quite suitable to obtain the nucleation rate along isotherms, whereas the latter is preferable if the dependence of the rate with temperature at constant pressure is required. Care should be taken with finite size effects when using NVT-Seeding. Further work is required to extend NVT seeding to other sorts of phase transitions., Comment: 10 pages, 9 figures

We construct a motivic spectral sequence for the relative homotopy invariant K-theory of a closed immersion of schemes $D \subset X$. The $E_2$-terms of this spectral sequence are the cdh-hypercohomology of a complex of equi-dimensional cycles. Using this spectral sequence, we obtain a cycle class map from the relative motivic cohomology group of 0-cycles to the relative homotopy invariant K-theory. For a smooth scheme $X$ and a divisor $D \subset X$, we construct a canonical homomorphism from the Chow groups with modulus $\CH^i(X|D)$ to the relative motivic cohomology groups $H^{2i}(X|D, \Z(i))$ appearing in the above spectral sequence. This map is shown to be an isomorphism when $X$ is affine and $i = \dim(X)$., Comment: 31 pages, Final version, To appear in Annali della Scuola Normale Superiore di Pisa

We examine the slice spectral sequence for the cohomology of singular schemes with respect to various motivic T-spectra, especially the motivic cobordism spectrum. When the base field k admits resolution of singularities and X is a scheme of finite type over k, we show that Voevodsky's slice filtration leads to a spectral sequence for MGL(X) whose terms are the motivic cohomology groups of X defined using the cdh-hypercohomology. As a consequence, we establish an isomorphism between certain geometric parts of the motivic cobordism and motivic cohomology of X. A similar spectral sequence for the connective K-theory leads to a cycle class map from the motivic cohomology to the homotopy invariant K-theory of X. We show that this cycle class map is injective for projective schemes. We also deduce applications to the torsion in the motivic cohomology of singular schemes., Comment: 37 pages. Final version. To appear in Annals of K-theory

We introduce a tower of localizing subcategories in Voevodsky's big (closed under infinite coproducts) triangulated category of motives. We show that the tower induces an interesting finite filtration on the motivic cohomology groups of smooth schemes over a perfect field. With rational coefficients, this finite filtration satisfies several of the properties of the still conjectural Bloch-Beilinson-Murre filtration., Comment: 18 pages; exposition simplified, removed unnecessary condition in former proposition 6.1.8

The main goal of this paper is to break up motivic cohomology into smaller pieces as suggested by the conjectural Bloch-Beilinson filtrations for the Chow groups., Comment: 28 pages

We verify a conjecture of Voevodsky, concerning the slices of co-operations in motivic $K$-theory., Comment: 20 pages

The main goal of this paper is to construct an analogue of Voevodsky's slice filtration in the motivic unstable homotopy category. The construction is done via birational invariants, this is motivated by the existence of an equivalence of categories between the orthogonal components for Voevodsky's slice filtration and the birational motivic stable homotopy categories constructed in \cite{Pelaez:2011fk}. Another advantage of this approach is that the slices appear naturally as homotopy fibres (and not as in the stable setting, where they are defined as homotopy cofibres) which behave much better in the unstable setting., Comment: 34 pages, added a comparison with the classical Postnikov tower. To appear in Transactions of the AMS

This paper is part of an endeavor to define an analogue of the slice filtration in the unstable motivic homotopy category. Our approach was inspired by the fact that the triangulated structures do not play a relevant role for the construction of birational homotopy categories as well as by the work of Kahn-Sujatha \cite{K-theory/0596} on birational motives, where the existence of a connection between the layers of the slice filtration and birational invariants is explicitly suggested. Our main result, shows that there is an equivalence of categories between the orthogonal components for the slice filtration and the birational motivic stable homotopy categories which are constructed in this paper. Relying on this equivalence, we are able to describe the slices for projective spaces (including $\mathbb P ^{\infty}$), Thom spaces and blow ups., Comment: Final version. To appear in Documenta Mathematica. arXiv admin note: text overlap with arXiv:1201.3657

Let $X$ be a Noetherian separated scheme of finite Krull dimension. We show that the layers of the slice filtration in the motivic stable homotopy category $\stablehomotopy$ are strict modules over Voevodsky's algebraic cobordism spectrum. We also show that the zero slice of any commutative ring spectrum in $\stablehomotopy$ is an oriented ring spectrum in the sense of Morel, and that its associated formal group law is additive. As a consequence, we get that with rational coefficients the slices are in fact motives in the sense of Cisinski-D{\'e}glise \cite{mixedmotives}, and have transfers if the base scheme is excellent. This proves a conjecture of Voevodsky \cite[conjecture 11]{MR1977582}., Comment: 9 pages, Following a remark of Denis-Charles Cisinski the condition on the base scheme being geometrically unibranch has been removed

Let $k$ be a field with resolution of singularities, and $X$ a separated $k$-scheme of finite type with structure map $g$. We show that the slice filtration in the motivic stable homotopy category commutes with pullback along $g$. Restricting the field further to the case of characteristic zero, we are able to compute the slices of Weibel's homotopy invariant $K$-theory extending the result of Levine, and also the zero slice of the sphere spectrum extending the result of Levine and Voevodsky. We also show that the zero slice of the sphere spectrum is a strict cofibrant ring spectrum $\mathbf{HZ}_{X}^{\slicefilt}$ which is stable under pullback and that all the slices have a canonical structure of strict modules over $\mathbf{HZ}_{X}^{\slicefilt}$. If we consider rational coefficents and assume that $X$ is geometrically unibranch then relying on the work of Cisinski and D{\'e}glise, we get that the zero slice of the sphere spectrum is given by Voevodsky's rational motivic cohomology spectrum $\mathbf{HZ}_{X}\otimes \mathbb Q$ and that the slices have transfers. This proves several conjectures of Voevodsky., Comment: Final version. To appear in J. K-Theory

We show that the slice filtration introduced by Voevodsky is compatible in a suitable sense with the symmetric monoidal structure in the category of motivic symmetric T-spectra constructed by Jardine. It follows from this compatibility that the zero slice of the sphere spectrum s_{0}(1) is a ring spectrum and that for every symmetric T-spectrum X, and every integer n; the n-slice s_{n}(X) is a module over s_{0}(1). In particular, if the base scheme is a field of characteristic zero, we have that all the slices s_{n}(X) are big motives in the sense of Voevodsky. We also get as a corollary that the smash product of symmetric T-spectra induces pairings in the motivic Atiyah-Hirzebruch spectral sequence., Comment: bibliography corrected, proof of former lemma 3.5.60 corrected