On the fundamental class of an essentially smooth scheme-map

Joseph Lipman; Amnon Neeman

doi:10.14231/AG-2018-005

On the fundamental class of an essentially smooth scheme-map

Joseph Lipman, Amnon Neeman

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

Let f: X → Z be a separated, essentially finite-type, flat map of noetherian schemes and δ: X → X×_ZX the diagonal map. The fundamental class C_f (globalizing residues) is a map from the relative Hochschild functor Lδ*δ_*f* to the relative dualizing functor f^!. We show a compatibility between C_f and the derived tensor product. The main result is that, in a suitable sense, C_f generalizes Verdier's classical isomorphism for smooth f with fibers of dimension d, an isomorphism that binds f^! to relative d-forms.

Original language	English
Pages (from-to)	131-159
Number of pages	29
Journal	Algebraic Geometry
Volume	5
Issue number	2
DOIs	https://doi.org/10.14231/AG-2018-005
Publication status	Published - 1 Mar 2018

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title = "On the fundamental class of an essentially smooth scheme-map",

abstract = "Let f: X → Z be a separated, essentially finite-type, flat map of noetherian schemes and δ: X → X×ZX the diagonal map. The fundamental class Cf (globalizing residues) is a map from the relative Hochschild functor Lδ*δ*f* to the relative dualizing functor f!. We show a compatibility between Cf and the derived tensor product. The main result is that, in a suitable sense, Cf generalizes Verdier's classical isomorphism for smooth f with fibers of dimension d, an isomorphism that binds f! to relative d-forms.",

keywords = "Fundamental class, Grothendieck duality, Relative dualizing complex",

author = "Joseph Lipman and Amnon Neeman",

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year = "2018",

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T1 - On the fundamental class of an essentially smooth scheme-map

AU - Lipman, Joseph

AU - Neeman, Amnon

N1 - Publisher Copyright: © Foundation Compositio Mathematica 2018.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - Let f: X → Z be a separated, essentially finite-type, flat map of noetherian schemes and δ: X → X×ZX the diagonal map. The fundamental class Cf (globalizing residues) is a map from the relative Hochschild functor Lδ*δ*f* to the relative dualizing functor f!. We show a compatibility between Cf and the derived tensor product. The main result is that, in a suitable sense, Cf generalizes Verdier's classical isomorphism for smooth f with fibers of dimension d, an isomorphism that binds f! to relative d-forms.

AB - Let f: X → Z be a separated, essentially finite-type, flat map of noetherian schemes and δ: X → X×ZX the diagonal map. The fundamental class Cf (globalizing residues) is a map from the relative Hochschild functor Lδ*δ*f* to the relative dualizing functor f!. We show a compatibility between Cf and the derived tensor product. The main result is that, in a suitable sense, Cf generalizes Verdier's classical isomorphism for smooth f with fibers of dimension d, an isomorphism that binds f! to relative d-forms.

KW - Fundamental class

KW - Grothendieck duality

KW - Relative dualizing complex

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DO - 10.14231/AG-2018-005

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VL - 5

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EP - 159

JO - Algebraic Geometry

JF - Algebraic Geometry

IS - 2

ER -

On the fundamental class of an essentially smooth scheme-map

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