scholarly journals A Spectral Sequence for the Hochschild Cohomology of a Coconnective DGA

2013 ◽  
Vol 112 (2) ◽  
pp. 182 ◽  
Author(s):  
Shoham Shamir
Keyword(s):  
Spectral Sequence ◽  
Loop Space ◽  
Hochschild Cohomology ◽  
Derived Category ◽  
Orientable Manifold ◽  
Support Varieties ◽  
Mod P

A spectral sequence for the computation of the Hochschild cohomology of a coconnective dga over a field is presented. This spectral sequence has a similar flavour to the spectral sequence presented in [7] for the computation of the loop homology of a closed orientable manifold. Using this spectral sequence we identify a class of spaces for which the Hochschild cohomology of their mod-$p$ cochain algebra is Noetherian. This implies, among other things, that for such a space the derived category of mod-$p$ chains on its loop-space carries a theory of support varieties.

scholarly journals On vector bundles for a Morse decomposition of $L\mathbb{C}\mathrm{P}^n$

2017 ◽  
Vol 121 (2) ◽  
pp. 186
Author(s):  
Iver Ottosen
Keyword(s):  
Loop Space ◽  
Vector Bundles ◽  
Morse Decomposition ◽  
Energy Integral ◽  
Chern Classes ◽  
Free Loop Space ◽  
Stiefel Manifolds ◽  
Mod P

We give a description of the negative bundles for the energy integral on the free loop space $L\mathbb{C}\mathrm{P}^n$ in terms of circle vector bundles over projective Stiefel manifolds. We compute the mod $p$ Chern classes of the associated homotopy orbit bundles.

scholarly journals THE HOCHSCHILD COHOMOLOGY RING OF A CLASS OF SPECIAL BISERIAL ALGEBRAS

2010 ◽  
Vol 09 (01) ◽  
pp. 73-122 ◽  
Author(s):  
NICOLE SNASHALL ◽  
RACHEL TAILLEFER
Keyword(s):  
London Math ◽  
Hochschild Cohomology ◽  
Cohomology Ring ◽  
Krull Dimension ◽  
Finitely Generated ◽  
Cohomology Rings ◽  
Hochschild Cohomology Ring ◽  
Special Biserial Algebras ◽  
Support Varieties

We consider a class of self-injective special biserial algebras ΛN over a field K and show that the Hochschild cohomology ring of dΛN is a finitely generated K-algebra. Moreover, the Hochschild cohomology ring of ΛN modulo nilpotence is a finitely generated commutative K-algebra of Krull dimension two. As a consequence the conjecture of [N. Snashall and Ø. Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc.88 (2004) 705–732], concerning the Hochschild cohomology ring modulo nilpotence, holds for this class of algebras.

Derived Category and Canonical Bundle --II

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Hochschild Cohomology ◽  
Derived Categories ◽  
Kodaira Dimension ◽  
Derived Category ◽  
Canonical Bundle ◽  
Derived Equivalence ◽  
One Step ◽  
Canonical Ring ◽  
Special Case

Based on the work of Orlov, Kawamata, and others, this chapter shows that the (numerical) Kodaira dimension and the canonical ring are preserved under derived equivalence. The same techniques can be used to derive the invariance of Hochschild cohomology under derived equivalence. Going one step further, it is shown that the nefness of the canonical bundle is detected by the derived category. The chapter also studies the relation between derived and birational (or rather K-) equivalence. The special case of a central conjecture predicts that two birational Calabi-Yau varieties have equivalent derived categories.

scholarly journals INVARIANCE OF THE GERSTENHABER ALGEBRA STRUCTURE ON TATE-HOCHSCHILD COHOMOLOGY

2019 ◽  
pp. 1-36
Author(s):  
Zhengfang Wang
Keyword(s):  
Hochschild Cohomology ◽  
Derived Category ◽  
Algebra Structure ◽  
Gerstenhaber Algebra ◽  
Morita Type

Keller proved in 1999 that the Gerstenhaber algebra structure on the Hochschild cohomology of an algebra is an invariant of the derived category. In this paper, we adapt his approach to show that the Gerstenhaber algebra structure on the Tate–Hochschild cohomology of an algebra is preserved under singular equivalences of Morita type with level, a notion introduced by the author in previous work.

Correction to ‘Mod p cohomology theories and the Bockstein spectral sequence’

1967 ◽  
Vol 63 (4) ◽  
pp. 935-935 ◽  
Author(s):  
C. R. F. Maunder
Keyword(s):  
Spectral Sequence ◽  
Cohomology Theory ◽  
Bockstein Spectral Sequence ◽  
Mod P

In (1), Theorem 4·1·4, and hence Theorem 4·3·1, are incorrect as stated, unless h* denotes ordinary cohomology theory. However, if h* denotes complex K-theory, corresponding results still hold, provided everything is graded over Z2 instead of Z.

scholarly journals HIGHER STRING TOPOLOGY ON GENERAL SPACES

2006 ◽  
Vol 93 (2) ◽  
pp. 515-544 ◽  
Author(s):  
PO HU
Keyword(s):  
Simplicial Complex ◽  
Hochschild Cohomology ◽  
Chain Complex ◽  
Derived Category ◽  
Cochain Complex ◽  
String Topology ◽  
Sphere Bundle ◽  
Chain Complexes ◽  
Natural Notion ◽  
Finite Simplicial Complex

In this paper, I give a generalized analogue of the string topology results of Chas and Sullivan, and of Cohen and Jones. For a finite simplicial complex $X$ and $k \geq 1$, I construct a spectrum $Maps(S^k, X)^{S(X)}$, which is obtained by taking a generalization of the Spivak bundle on $X$ (which however is not a stable sphere bundle unless $X$ is a Poincaré space), pulling back to $Maps(S^k, X)$ and quotienting out the section at infinity. I show that the corresponding chain complex is naturally homotopy equivalent to an algebra over the $(k + 1)$-dimensional unframed little disk operad $\mathcal{C}_{k + 1}$. I also prove a conjecture of Kontsevich, which states that the Quillen cohomology of a based $\mathcal{C}_k$-algebra (in the category of chain complexes) is equivalent to a shift of its Hochschild cohomology, as well as prove that the operad $C_{\ast}\mathcal{C}_k$ is Koszul-dual to itself up to a shift in the derived category. This gives one a natural notion of (derived) Koszul dual $C_{\ast}\mathcal{C}_k$-algebras. I show that the cochain complex of $X$ and the chain complex of $\Omega^k X$ are Koszul dual to each other as $C_{\ast}\mathcal{C}_k$-algebras, and that the chain complex of $Maps(S^k, X)^{S(X)}$ is naturally equivalent to their (equivalent) Hochschild cohomology in the category of $C_{\ast}\mathcal{C}_k$-algebras.

Spherical and Exceptional Objects

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Projective Space ◽  
Spectral Sequence ◽  
Mirror Symmetry ◽  
Final Section ◽  
Derived Category ◽  
Braid Groups ◽  
Spherical Object ◽  
Geometric Context ◽  
Orthogonal Decompositions ◽  
Exceptional Sequences

Spherical objects — motivated by considerations in the context of mirror symmetry — are used to construct special autoequivalences. Their action on cohomology can be described precisely, considering more than one spherical object often leads to complicated (braid) groups acting on the derived category. The results related to Beilinson are almost classical. Section 3 of this chapter gives an account of the Beilinson spectral sequence and how it is used to deduce a complete description of the derived category of the projective space. This will use the language of exceptional sequences and semi-orthogonal decompositions encountered here. The final section gives a simplified account of the work of Horja, which extends the theory of spherical objects and their associated twists to a broader geometric context.

Sign in / Sign up

Export Citation Format

Share Document