146 research outputs found
A Hochschild-Kostant-Rosenberg theorem for cyclic homology
Let be a commutative algebra over the field . We show that there is a natural algebra homomorphism which is an isomorphism when is a smooth algebra. Thus, the
functor can be viewed as an approximation of negative cyclic homology
and ordinary cyclic homology is a natural -module. In
general, there is a spectral sequence . We find associated approximation functors and for
ordinary cyclic homology and periodic cyclic homology, and set up their
spectral sequences. Finally, we discuss universality of the approximations.Comment: To appear in J. Pure Appl. Algebr
Cobordism obstructions to independent vector fields
We define an invariant for the existence of r pointwise linearly independent
sections in the tangent bundle of a closed manifold. For low values of r,
explicit computations of the homotopy groups of certain Thom spectra combined
with classical obstruction theory identifies this invariant as the top
obstruction to the existence of the desired sections. In particular, this shows
that the top obstruction is an invariant of the underlying manifold in these
cases, which is not true in general. The invariant is related to cobordism
theory and this gives rise to an identification of the invariant in terms of
well-known invariants. As a corollary to the computations, we can also compute
low-dimensional homotopy groups of the Thom spectra studied by Galatius,
Tillmann, Madsen, and Weiss.Comment: 46 page
A geometric interpretation of the homotopy groups of the cobordism category
The classifying space of the embedded cobordism category has been identified
in by Galatius, Tillmann, Madsen, and Weiss as the infinite loop space of a
certain Thom spectrum. This identifies the set of path components with the
classical cobordism group. In this paper, we give a geometric interpretation of
the higher homotopy groups as certain cobordism groups where all manifolds are
now equipped with a set of orthonormal sections in the tangent bundle. We also
give a description of the fundamental group as a free group with a set of
geometrically intuitive relations.Comment: 23 page
Base change for semiorthogonal decompositions
Consider an algebraic variety over a base scheme and a faithful base
change . Given an admissible subcategory \CA in the bounded derived
category of coherent sheaves on , we construct an admissible subcategory in
the bounded derived category of coherent sheaves on the fiber product
, called the base change of \CA, in such a way that the
following base change theorem holds: if a semiorthogonal decomposition of the
bounded derived category of is given then the base changes of its
components form a semiorthogonal decomposition of the bounded derived category
of the fiber product. As an intermediate step we construct a compatible system
of semiorthogonal decompositions of the unbounded derived category of
quasicoherent sheaves on and of the category of perfect complexes on .
As an application we prove that the projection functors of a semiorthogonal
decomposition are kernel functors.Comment: 24 pages; derived category of countably-coherent sheaves which
appeared in the first version for technical reasons is replaced by the usual
quasicoherent categor
On cyclic fixed points of spectra
For a finite p-group G and a bounded below G-spectrum X of finite type mod p,
the G-equivariant Segal conjecture for X asserts that the canonical map X^G -->
X^{hG} is a p-adic equivalence. Let C_{p^n} be the cyclic group of order p^n.
We show that if the C_p Segal conjecture holds for a C_{p^n} spectrum X, as
well as for each of its C_{p^e} geometric fixed points for 0 < e < n, then then
C_{p^n} Segal conjecture holds for X. Similar results hold for weaker forms of
the Segal conjecture, asking only that the canonical map induces an equivalence
in sufficiently high degrees, on homotopy groups with suitable finite
coefficients
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