Titlebook: Determinants, Gr?bner Bases and Cohomology; Winfried Bruns,Aldo Conca,Matteo Varbaro Book 2022 Springer Nature Switzerland AG 2022 Determi

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Abbreviate

Algebras Defined by Minors,rated by minors through their initial algebras. Since the initial algebras are normal monoid domains, toric algebra can be applied to them. Since normal monoid domains are very well understood, we can draw strong consequences for the algebras defined by minors.

reflection

-singularities of Determinantal Rings,lop them far enough to prove that determinantal rings are strongly F-regular. For deformation arguments one needs F-rationality. It is closely related to the rationality of singularities in characteristic 0, so that we can at least briefly discuss this property for determinantal rings. F-rationality

痛得哭了

,Castelnuovo–Mumford Regularity,ratch, but in a more general version for standard graded algebras over Noetherian base rings. As in the classical case, regularity can be computed from local cohomology, minimal free resolutions and Koszul homology. In the given generality we prove the theorems on the regularity of powers and produc

Ostrich

Grassmannians, Flag Varieties, Schur Functors and Cohomology,hasis on vanishing statements. We develop the theory in the relative setting, which offers significant flexibility for the inductive arguments that we employ. We also introduce Schur functors and explain their relationship to direct images of line bundles associated to dominant weights. We end with

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https://doi.org/10.1007/978-3-319-77568-5The second chapter pushes the discussion of the Gr?bner deformation forward in several directions: finer control of minimal and associated primes, connectedness, tight connections between an ideal and its squarefree initial ideals, should the latter exist. Some of these results were very recently established by Conca and Varbaro.

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Generations of Women HistoriansIn this chapter gives a short introduction to standard bitableaux and the straightening law. This powerful technique is the key to structural properties of determinantal rings. But it is also of central importance for the computation of Gr?bner and Sagbi bases on the one hand and for the representation theoretic approach on the other.