Čech cohomology of semiring schemes

Jun, Jaiung

doi:10.1016/j.jalgebra.2017.04.001

Cited by 9 publications

(17 citation statements)

References 18 publications

(12 reference statements)

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“…In this section, we briefly recall basic definitions and properties of monoid schemes and semiring schemes (also tropical schemes as a special case) which play a key role in the later sections. Most of the material in this section can be found in [GG16], [Jun17a], [JMT19], [JRT20].…”

Section: Preliminariesmentioning

confidence: 99%

Vector bundles on tropical schemes

Jun,

Mincheva,

Tolliver

2020

Preprint

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We define vector bundles for tropical schemes, and explore their properties. The paper largely consists of three parts; (1) we study free modules over zero-sum free semirings, which provide the necessary algebraic background for the theory (2) we relate vector bundles on tropical schemes to topological vector bundles and vector bundles on monoid schemes, and finally (3) we show that all line bundles on a tropical scheme can be lifted to line bundles on a usual scheme in the affine case.

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Section: Preliminariesmentioning

confidence: 99%

Vector bundles on tropical schemes

Jun,

Mincheva,

Tolliver

2020

Preprint

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“…if and only if f and g are same as functions on T n . It is well-known (see for example [25] or [23]) that in this case B is cancellative. Now let X = Spec B.…”

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“…It follows from Proposition 4.11 that Pic(X ) = CaCl(X ). But we know from [25,Corollary 4.23.] that Pic(X ) is the trivial group and hence so is CaCl(X ).…”

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confidence: 99%

Picard groups for tropical toric schemes

2018

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From any monoid scheme X (also known as an F 1 -scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) X S by scalar extension to an idempotent semifield S. We prove that for a given irreducible monoid scheme X (satisfying some mild conditions) and an idempotent semifield S, the Picard group Pic(X) of X is stable under scalar extension to S (and in fact to any field K). In other words, we show that the groups Pic(X) and Pic(X S ) (and Pic(X K )) are isomorphic. In particular, if X C is a toric variety, then Pic(X) is the same as the Picard group of the associated tropical scheme. The Picard groups can be computed by considering the correct sheaf cohomology groups. We also define the group CaCl(X S ) of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme X S and prove that CaCl(X S ) is isomorphic to Pic(X S ).

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“…In general, a semiring scheme is defined to be a locally semiringed space which is locally isomorphic to Spec A for some semiring A. For details, we refer the readers to [Jun17].…”

Section: Preliminariesmentioning

confidence: 99%

Lattices, Spectral Spaces, and Closure Operations on Idempotent Semirings

Jun¹,

Ray²,

Tolliver³

2020

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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.

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Čech cohomology of semiring schemes

Cited by 9 publications

References 18 publications

Vector bundles on tropical schemes

Vector bundles on tropical schemes

Picard groups for tropical toric schemes

Lattices, Spectral Spaces, and Closure Operations on Idempotent Semirings

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