3,050 research outputs found
The negative side of cohomology for Calabi-Yau categories
We study integer-graded cohomology rings defined over Calabi-Yau categories.
We show that the cohomology in negative degree is a trivial extension of the
cohomology ring in non-negative degree, provided the latter admits a regular
sequence of central elements of length two. In particular, the product of
elements of negative degrees are zero. As corollaries we apply this to
Tate-Hochschild cohomology rings of symmetric algebras, and to Tate cohomology
rings over group algebras. We also prove similar results for Tate cohomology
rings over commutative local Gorenstein rings.Comment: 14 page
Gorenstein homological algebra and universal coefficient theorems
We study criteria for a ring—or more generally, for a small category—to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to develop machinery for proving new ones. Among the universal coefficient theorems covered by our methods we find, besides all the classic examples, several exotic examples arising from the KK-theory of C*-algebras and also Neeman’s Brown–Adams representability theorem for compactly generated categories
Gorenstein homological algebra and universal coefficient theorems
We study criteria for a ring - or more generally, for a small category - to
be Gorenstein and for a module over it to be of finite projective dimension.
The goal is to unify the universal coefficient theorems found in the literature
and to develop a machinery for proving new ones.
Among the universal coefficient theorems covered by our methods we find,
besides all the classic examples, several exotic examples arising from the
KK-theory of C*-algebras and also Neeman's Brown-Adams representability theorem
for compactly generated categories.Comment: 43 page
Invertible modules for commutative -algebras with residue fields
The aim of this note is to understand under which conditions invertible
modules over a commutative S-algebra in the sense of Elmendorf, Kriz, Mandell
and May give rise to elements in the algebraic Picard group of invertible
graded modules over the coefficient ring by taking homotopy groups. If a
connective commutative S-algebra R has coherent localizations (R_*)_m for every
maximal ideal m in R_*, then for every invertible R-module U, U_* is an
invertible graded R_*-module. In some non-connective cases we can carry the
result over under the additional assumption that the commutative S-algebra has
`residue fields' for all maximal ideals m in R_* if the global dimension of R_*
is small or if R is 2-periodic with underlying Noetherian complete local
regular ring R_0.Comment: Revised version. One serious flaw correcte
Witt's theorem for noncommutative conics
Let k be a field. We show that all homogeneous noncommutative curves of genus
zero over k are noncommutative P^1-bundles over a (possibly) noncommutative
base. Using this result, we compute complete isomorphism invariants of
homogeneous noncommutative curves of genus zero, allowing us to generalize a
theorem of Witt.Comment: Section two generalize
Matrix factorizations for quantum complete intersections
We introduce twisted matrix factorizations for quantum complete intersections
of codimension two. For such an algebra, we show that in a given dimension,
almost all the indecomposable modules with bounded minimal projective
resolutions correspond to such matrix factorizations.Comment: 13 page
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