A classical theorem of Hochster provides purely topological characterization of prime spectra of commutative rings. In this paper, we first prove an analogous statement for idempotent semirings, showing that for a spectral space X, we can construct an idempotent semiring A in such a way that the saturated prime spectrum of A is homeomorphic to X. We further provide examples of spectral spaces arising from sets of congruence relations of semirings. In particular, we prove that the space of valuations and the space of prime congruences on an idempotent semiring A are spectral, and there is a natural bijection of sets between two. We then develop several aspects of commutative algebra of semirings. We mainly focus on the notion of closure operations for semirings, and provide several examples. In particular, we introduce an integral closure operation and a Frobenius closure operation for idempotent semirings.

In this paper, we study modules over quotient spaces of certain categorified fiber bundles. These are understood as modules over entwining structures involving a small K-linear category D and a K-coalgebra C. We obtain Frobenius and separability conditions for functors on entwined modules. We also introduce the notion of a C-Galois extension E ⊆ D of categories. Under suitable conditions, we show that entwined modules over a C-Galois extension may be described as modules over the subcategory E of C-coinvariants of D.

Let H be a Hopf algebra. We consider H-equivariant modules over a Hopf module category C as modules over the smash extension C#H. We construct Grothendieck spectral sequences for the cohomologies as well as the H-locally finite cohomologies of these objects. We also introduce relative (D, H)-Hopf modules over a Hopf comodule category D. These generalize relative (A, H)-Hopf modules over an H-comodule algebra A. We construct Grothendieck spectral sequences for their cohomologies by using their rational Hom objects and higher derived functors of coinvariants. MSC(2010) Subject Classification: 16S40, 16T05, 18E05

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We present several naturally occurring classes of spectral spaces using commutative algebra on pointed monoids. For this purpose, our main tools are finite type closure operations and continuous valuations on monoids which we introduce in this work. In the process, we make a detailed study of different closure operations on monoids. We prove that the collection of continuous valuations on a topological monoid with topology determined by any finitely generated ideal is a spectral space.Recently, Finocchiaro [8] developed a new criterion involving ultrafilters to characterize spectral spaces. This helped them to show that certain familiar objects appearing in commutative algebra can be realized as the spectrum of a ring (see, for instance, [9], [10]). For example, Finocchiaro, Fontana and Spirito proved that the collection of submodules of a module over a ring has the structure of a spectral space [10]. They further used closure operations from classical commutative ring theory to bring more such spectral spaces in light. Moreover, the methods of [10] were used in [2], [3] to uncover many spectral spaces coming from objects in an abelian category and from modules over tensor triangulated categories. In this paper, we use both Hochster's characterization and Finocchiaro's criterion to present natural classes of spectral spaces involving pointed monoids.Our interest in monoids began by looking at the paper [12] by Weibel and Flores which studies certain geometric structures involving monoids. This interest in the geometry over monoids lies in its natural association to toric geometry, which was pointed out in the work of Cortias, Haesemeyer, Walker and Weibel [6]. A detailed work on the commutative and homological algebra on monoids was presented by Flores in [11]. As such, in this work we look into the topological aspects of monoids via spectral spaces. We begin by showing that the collection of all prime ideals of a monoid or, in other words, the spectrum of a monoid, endowed with the Zariski topology is homeomorphic to the spectrum of a ring, i.e., it is a spectral space. We further prove that the collection of all ideals as well as the collection of all proper ideals of a monoid are also spectral spaces (Corollary 3.4). As in [11], the notion of A-sets over a monoid A is the analogue of the notion of modules over a ring. We introduce closure operations on monoids and obtain natural classes of spectral spaces using finite type closure operations on A-sets. In the process, different notions of closure operations like integral, saturation, Frobenius and tight closures are introduced for monoids inspired by the corresponding closure operations on rings from classical commutative algebra. We discuss their persistence and localization properties in detail. Other than rings, valuation has also been studied on ring-like objects like semirings (see, for instance [18], [19]). Valuation monoids were studied in [6]. In this work, we prove that the collection of all valuation monoids having the same group completion forms a s...