Picard groups and class groups of monoid schemes

Flores, Jaret; Weibel, Charles A.

doi:10.1016/j.jalgebra.2014.06.002

Cited by 18 publications

(22 citation statements)

References 10 publications

(11 reference statements)

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“…This way we obtain that H 1 (X , O × X ) = H 1 (X C , O × X C ). This gives us a different proof of Theorem 6.6 of [17]. As observed in [17] the Cartier divisors on X lift to torus-invariant Cartier divisors on X C and this way we recover Fulton's result in [19] Section 3.4 that the Picard group of a toric variety is generated by torus invariant divisors.…”

Section: Picard Groups Of Tropical Toric Schemessupporting

confidence: 56%

“…Next, we briefly recall the definition of invertible sheaves on a monoid scheme X . We refer the readers to [6] and [17] for details. (1) By an M-set, we mean a set with an M-action.…”

Section: Picard Groups For Monoid Schemesmentioning

confidence: 99%

“…Therefore, Pic(X ) is a group. One can use the classical argument to prove that Pic(X ) ≃ H 1 (X , O × X ) (for instance, see [17,Lemma 5.3.]). Recall that for any topological space X and a sheaf F of abelian groups on X , sheaf cohomology H i (X , F ) andČech cohomologyȞ i (X , F ) agree for i = 0, 1.…”

Section: Picard Groups For Monoid Schemesmentioning

confidence: 99%

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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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Section: Picard Groups Of Tropical Toric Schemessupporting

confidence: 56%

“…Next, we briefly recall the definition of invertible sheaves on a monoid scheme X . We refer the readers to [6] and [17] for details. (1) By an M-set, we mean a set with an M-action.…”

Section: Picard Groups For Monoid Schemesmentioning

confidence: 99%

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Picard groups for tropical toric schemes

2018

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“…From the point of view of algebraic geometry over fields, monoid schemes can be seen as a direct generalization of toric geometry and Kato fans of logarithmic schemes; see [3,6,8,9,27] among others. The central position of monoid schemes within F 1 -geometry is confirmed by their numerous links to other areas of mathematics, such as Weyl groups as algebraic groups over F 1 [29,44], computational methods for toric geometry [7,8,14], a framework for tropical scheme theory [15], applications to representation theory [20,[40][41][42]51] and, last but not least, stable homotopy theory as K-theory over F 1 [3,10], a theme on which we dwell in this paper.…”

Section: Introductionmentioning

confidence: 99%

Algebraic $$K\!$$-theory and Grothendieck–Witt theory of monoid schemes

2022

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We study the algebraic $$K\!$$ K -theory and Grothendieck–Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $$K\!$$ K -theory space of an integral monoid scheme X in terms of its Picard group $${{\,\mathrm{Pic}\,}}(X)$$ Pic ( X ) and pointed monoid of regular functions $$\Gamma (X, {\mathcal {O}}_X)$$ Γ ( X , O X ) and a complete description of the Grothendieck–Witt space of X in terms of an additional involution on $${{\,\mathrm{Pic}\,}}(X)$$ Pic ( X ) . We also prove space-level projective bundle formulae in both settings.

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“…We will asume that M is finitely generated, commutative, torsion free and cancellative. See [Böt15] for basic properties of this framework, for related concepts see [Flo15], [FW14], [LPL11], [Lor12]. A central question in approaching the local Picard group of K[M ] is whether it can be computed purely combinatorially and to what extent it depends on the base field K. On the combinatorial side, we have the finite combinatorial spectrum Spec M , and its punctured variant Spec It is known that in the (normal) toric setting this is an isomorphism, see [DFM93], but it is not true for other combinatorial structures represented by a binoid.…”

Section: Introductionmentioning

confidence: 99%

Local Picard group of pointed monoids and their algebras

Alberelli

Brenner

2019

Communications in Algebra

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The main goal of this paper is to give an explicit formula forin terms of simplicial and reduced simplicial cohomology, where K[△] is the Stanley-Reisner ring of the simplicial complex △. In particular we compute the local Picard group of K [△]. To achieve this we study the corresponding purely combinatorial problem on the punctured spectrum of the pointed monoid defined by △. The cohomology of the sheaf of units on Spec • K[△] is then the direct sum of this combinatorial cohomology, which has a decomposition along the vertices, and another part depending on the field K.Mathematical Subject Classification (2010): 13F55, 13C20, 14C22. graded normal ring the Picard group is trivial, Carlo Traverso in 1970 [Tra70], Richard G. Swan in 1980 [Swa80] and David F. Anderson in 1981 [And81] covered the seminormal case, showing that Pic(A) = Pic(A[X 1 , . . . , X n ]) in this case. We will prove that for Stanley-Reisner rings and their localizations, the cohomology of the sheaf of units vanishes. The next task is to determine the units of D(x i ), where the combinatorial units and the base field have to be considered. Already the affine line shows that there is not a direct splitting of the sheaf of units into combinatorial units and field units. However, in many favorable situations there is such a splitting on the combinatorial topology, the topology generated by the D(x i )s, and so the two aspects can be studied separately. In such situations, the first part is determined completely by the combinatorial situation, whereas the second part depends on the constant sheaf given by the units of the field. In the nonintegral case, this part contributes to the local Picard group. Main resultsWe give an overview of our main results, in particular for binoids M △ and their algebra K[△] that arise from a simplicial complex △. In Lemma 2.6 we observe that the intersection pattern of the open subsets D(x i ) of Spec M is given directly by the faces of the simplicial complex, thus leading us to prove, in Theorem 2.10, that the cohomology of a constant sheaf can be computed entirely in terms of simplicial cohomology. In Theorem 2.15, we show that the localization of a simplicial binoid at a face is isomorphic to the smash product of the simplicial binoid of the link of that face and a free group on that face. This opens the door to Theorem 2.21, where we show that we can rewrite the sheaf O * M △ as a direct sum of smaller sheaves, indexed by the vertices. These sheaves are actually defined as extensions by zeros of the constant sheaf Z on D(x i ), which is homeomorphic to the spectrum of the link of the corresponding vertex. This brings us to Theorem 2.25, which shows that we can compute sheaf cohomology (and thus the local Picard group) by means of reduced simplicial cohomology, via the formulaWe then use these results to understand the sheaf of units on the binoid algebra and their cohomology. In order to do so, we introduce in Definition 3.17 the combinatorial topology on Spec K[M ], which builds a bridge between the topology on ...

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Picard groups and class groups of monoid schemes

Cited by 18 publications

References 10 publications

Picard groups for tropical toric schemes

Picard groups for tropical toric schemes

Algebraic $$K\!$$-theory and Grothendieck–Witt theory of monoid schemes

Local Picard group of pointed monoids and their algebras

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