We construct the logarithmic and tropical Picard groups of a family of logarithmic curves and realize the latter as the quotient of the former by the algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of logarithmic abelian varieties over the moduli space of Deligne–Mumford stable curves, but does not possess an underlying algebraic stack. However, the logarithmic Picard group does have logarithmic modifications that are representable by logarithmic schemes, all of which are obtained by pullback from subdivisions of the tropical Picard group.

In this paper we study the subgroup of the Picard group of Voevodsky’s category of geometric motives $\operatorname{DM}_{\text{gm}}(k;\mathbb{Z}/2)$ generated by the reduced motives of affine quadrics. Our main tools here are the functors of Bachmann [On the invertibility of motives of affine quadrics, Doc. Math. 22 (2017), 363–395], but we also provide an alternative method. We show that the group in question can be described in terms of indecomposable direct summands in the motives of projective quadrics over $k$. In particular, we describe all the relations among the reduced motives of affine quadrics. We also extend the criterion of motivic equivalence of projective quadrics.

We suggest an analog of the Bass–Quillen conjecture for smooth affinoid algebras over a complete non-archimedean field. We prove this in the rank-1 case, i.e. for the Picard group. For complete discretely valued fields and regular affinoid algebras that admit a regular model (automatic if the residue characteristic is zero) we prove a similar statement for the Grothendieck group of vector bundles $K_{0}$.

We present a new proof of Anderson's result that the real K-theory spectrum is Anderson self-dual up to a fourfold suspension shift; more strongly, we show that the Anderson dual of the complex K-theory spectrum KU is C2-equivariantly equivalent to Σ4KU, where C2 acts by complex conjugation. We give an algebro-geometric interpretation of this result in spectrally derived algebraic geometry and apply the result to calculate 2-primary Gross-Hopkins duality at height 1. From the latter we obtain a new computation of the group of exotic elements of the K(1)-local Picard group.

We show that for a field k, algebraic over ℚ, and F a field of characteristic zero, the triangular spectra of mixed Tate motives (resp. mixed Artin-Tate motives), denoted by DMT(k)F (resp. DMAT(k)F), are both isomorphic to spec(F). We further compute the Picard groups of DMT(k)F and DMAT(k)ℚ.

We introduce the fundamental group $\mathcal{F}\left( A \right)$ of a simple $\sigma $-unital ${{C}^{*}}$–algebra $A$ with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of Fundamental Group of Simple${{C}^{*}}$-algebras with Unique Trace I and II by Nawata and Watatani. Our definition in this paper makes sense for stably projectionless ${{C}^{*}}$-algebras. We show that there exist separable stably projectionless ${{C}^{*}}$-algebras such that their fundamental groups are equal to $\mathbb{R}_{+}^{\times }$ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian.

Let $A$ be a commutative comodule algebra over a commutative bialgebra $H$. The group of invertible relative Hopf modules maps to the Picard group of $A$, and the kernel is described as a quotient group of the group of invertible group-like elements of the coring $A\otimes H$, or as a Harrison cohomology group. Our methods are based on elementary $K$-theory. The Hilbert 90 theorem follows as a corollary. The part of the Picard group of the coinvariants that becomes trivial after base extension embeds in the Harrison cohomology group, and the image is contained in a well-defined subgroup $E$. It equals $E$ if $H$ is a cosemisimple Hopf algebra over a field.

In this paper, we study the family of algebraic K3 surfaces generated by the smooth intersection of a (1, 1) form and a (2, 2) form in ${\open P}^2\times{\open P}^2$ defined over ${\open C}$ and with Picard number 3. We describe the group of automorphisms ${\cal A}={\rm Aut}(V/{\open C})$ on V. For an ample divisor D and an arbitrary curve C0 on V, we investigate the asymptotic behavior of the quantity $N_{{\cal A}(C_0)}(t)=\#\{C \in {\cal A}(C_0): C\cdot D<t\}$. We show that the limit $\lim_{t\to \infty } {\log N_{{\cal A}(C\,)}(t)\over \log t}=\alpha\fleqno{}$ exists, does not depend on the choice of curve C or ample divisor D, and that .6515<α<.6538.

We derive lower bounds for the rank of Picard groups of modular varieties associated with natural congruence subgroups of the orthogonal group of an even lattice of signature (2, l). As an example we consider the Siegel modular group of genus 2. The analytic part of this paper also leads to certain class number identities.

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPick(A) is the group of triangle auto-equivalences of Db(mod A) induced by two-sided tilting complexes. We study the group DPick(A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPick, as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPick(A) on a certain infinite quiver Γirr. This representation is faithful when the quiver Δ of A is a tree, and then DPick(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPick(A). When A is hereditary, DPick(A) coincides with the full group of k-linear triangle auto-equivalences of Db(mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to Db(mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.

The local class group of a Krull domain A is the quotient group G(A) = CI(A)/Pic(A). A Krull domain A is locally factorial if and only if G(A) = 0. In this paper, we characterize the Krull domains for which G(A) is a torsion group. We evaluate the local class group of several examples and finally, we explain why every abelian group is the local class group of a Krull domain.