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Results tagged with homological-algebra
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user 91245
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
7
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Definitions of Hochschild Cohomology $HH^{\bullet}(A)$
Let $A$ be an associative unital $k$-algebra, and let $M$ be a bimodule of $A$. The Hochschild cohomology of $A$ with coefficients in $M$ can be defined as
$$HH^{n}(A,\,M)=\mathrm{Ext}^{n}_{A^{e}}(A,\ …
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6
votes
2
answers
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Path algebras are formally smooth
In Ginzburg's notes Lectures on Noncommutative Geometry, he claim that the path algebra of a quiver is formally smooth (See Section 19.2).
I have two questions. First, how to show this claim and whi …
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4
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0
answers
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Universal enveloping algebra functor preserves quasi-isomorphism
Let $k$ be a field of characteristic 0. Let $\mathtt{DGA}_{k}$ denote the category of DG algebras and $\mathtt{DGLA}_{k}$ denote the category of DG Lie algebras. It is well known that there are model …
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2
votes
0
answers
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Gerstanharber bracket and derived Hom
Let $A$ be a honest algebra or more generally, a DG algebra. It is known that the Hochschild cochain complex is quasi-isomorphic to the derived Hom complex, i.e. one has
$$\mathrm{HH}^{\bullet}(A,\,A …
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2
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1
answer
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An exercise from Loday and Vallette about Koszul morphism
I tried to solve the following exercise from Loday and Vallette's Algebraic Operad. The first three parts are straightforward, however I have no idea how to solve the last part. I can't find any state …
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