Holmstrom Cohomology of diagrams

Cohomology of diagrams

http://hopf.math.purdue.edu/cgi-bin/generate?/Blanc-Johnson-Turner/lgss

Title: Local-to-global spectral sequences for the cohomology of diagrams

Authors: David Blanc, Mark W. Johnson, and James M. Turner

Address: Department of Mathematics, University of Haifa, 31905 Haifa, Israel Department of Mathematics, Penn State Altoona, Altoona, PA 16601, USA Department of Mathematics, Calvin College, Grand Rapids, MI 49546, USA

The cohomology of diagrams arises in various areas of mathematics, such as deformation theory, classifying diagrams of groups, and in homotopy theory, in the context of the rectification of homotopy-commutative diagrams, and thus in the study of higher homotopy and cohomology operations.

For this purpose we construct “cal-to-global’‘ spectral sequences for the cohomology of a diagram, which can be used to compute the cohomology of the full diagram in terms of smaller pieces. We also explain why such a local-to-global approach is relevant to higher operations.

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Cohomology of diagrams

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Cohomology of diagrams

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Cohomology of diagrams

arXiv:1101.2976 The cohomology of lambda-rings and Psi-rings from arXiv Front: math.KT by Michael Robinson In this thesis we develop the cohomology of diagrams of algebras and then apply this to the cases of the $\lambda$-rings and the $\Psi$-rings. A diagram of algebras is a functor from a small category to some category of algebras. For an appropriate category of algebras we get a diagram of groups, a diagram of Lie algebras, a diagram of commutative rings, etc

We define the cohomology of diagrams of algebras using comonads. The cohomology of diagrams of algebras classifies extensions in the category of functors. Our main result is that there is a spectral sequence connecting the cohomology of the diagram of algebras to the cohomology of the members of the diagram

$\Psi$-rings can be thought of as functors from the category with one object associated to the multiplicative monoid of the natural numbers to the category of commutative rings. So we can apply the theory we developed for the diagrams of algebras to the case of $\Psi$-rings. Our main result tells us that there is a spectral sequence connecting the cohomology of the $\Psi$-ring to the André-Quillen cohomology of the underlying commutative ring

The main example of a $\lambda$-ring or a $\Psi$-ring is the $K$-theory of a topological space. We look at the example of the $K$-theory of spheres and use its cohomology to give a proof of the classical result of Adams. We show that there are natural transformations connecting the cohomology of the $K$-theory of spheres to the homotopy groups of spheres. There is a very close connection between the cohomology of the $K$-theory of the $4n$-dimensional spheres and the homotopy groups of the $(4n-1)$-dimensional spheres.

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