Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Grothendieck duality via homotopy theory   Grothendieck proved that if $f:X\longrightarrow Y$ is a proper morphism of
nice schemes, then $Rf_*$ has a right adjoint, which is given as tensor product
with the relative canonical bundle. The original proof was by patching local
data. Deligne proved the existence of the adjoint by a global argument, and
Verdier showed that this global adjoint may be computed locally.
  In this article we show that the existence of the adjoint is an immediate
consequence of Brown's representability theorem. It follows almost as
immediately, by ``smashing'' arguments, that the adjoint is given by tensor
product with a dualising complex. Verdier's base change theorem is an immediate
consequence.

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The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: cond-mat
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: hep-th
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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: econ
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The categories of this paper are: physics
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The categories of this paper are: solv-int