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Results tagged with homological-algebra
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user 123731
(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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Is there an adjoint to the inclusion of I-adically complete modules to all modules?
A module $M$ over a ring $R$ is $I$-adically complete with respect to the ideal $I$, if the canonical map $M \to \lim M/I^nM$ is an isomorphism. There exists a completion functor: $M \mapsto \lim M/I^ …
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3
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0
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On homotopically inverting maps in DG categories
In a DG category $\mathcal{C}$, call a (closed degree 0) morphism $f: X \to Y$ a homotopy equivalence, if there exists $g: Y \to X$ such that $gf-1_X$ and $fg-1_Y$ are exact.
For a full DG subcatego …
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7
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143
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DG functors along which contractions can be lifted
For an object $X$ in a DG category, its contraction is $r \in Hom^{-1}(X,X)$ such that $d(r)=1_X$. Let us say that contractions lift along a DG functor $F: \mathcal{C} \to \mathcal{D}$, if for a contr …
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3
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Applying a Hochschild cocycle to a Maurer-Cartan element: how one should think of this?
Let $C^{\bullet}(A,M)$ be the Hochschild cochain complex of a DG-algebra $A$ with coefficients in a DG-bimodule $M$. Let $\zeta \in C^0(A,M)$ be a cocycle. Let $a \in A$ be a Maurer-Cartan element, $d …
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