# Tagged Questions

(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.

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### N-periodic derived categories

I have some seemingly basic questions about $N$-periodic derived categories to which I have not found answers in any of the usual places. Let $R$ be a ring, and let $D(R)_{\mathbb Z/N\mathbb Z}$ ...
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### Lie algebra (co)homology of the Lie algebra of differential operators

Let $X$ be an algebraic variety (or a complex manifold) over $\mathbb C$. Let $D(X)$ be its algebra of differential operators. Mostly I am interested in algebraic differential operators, but the case ...
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### Could Partial Tiltings be studied as Almost Complete Tiltings?

The first part of what follows is a brief recap of the definitions, setting and motivations for my questions. Experts can find the questions at the end. Here $k$ denotes an algebraically closed field,...
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### On the ordered set of real numbers, does sheaf+cosheaf imply constant?

I have a technical question about unbounded chain complexes. I couldn't think of a descriptive title for it. Let $P$ be a chain complex of contravariant functors on $\mathbf{R}$ (the real numbers ...
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### Is there an obstruction which classifies “quasi-isomorphism but not chain equivalence”?

Fix a ring $R$ and let $C_\bullet$, $D_\bullet$ be (possibly unbounded) chain complexes of $R$-modules. Assume that $f_\bullet:C_\bullet \to D_\bullet$ is a quasi-isomorphism: that is to say, $f$ is a ...
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### $E_{\infty}$ algebra in characteristic zero

I asked this question on MSE(http://math.stackexchange.com/q/1579026/239218). Let $A^{\bullet}$ be a cosimplicial commutative algebra over a field $\Bbbk$. Denote with $N(A)^{\bullet}$ the ...
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### Is the sheaf of smooth functions flat?

Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Is the sheaf of smooth functions on $X$ flat as an $\mathcal{O}_X$ module?
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### Classification of representation-finite algebras up to stable equivalence of Morita type

Assume $K$ is an algebraically closed field. I wanted to ask if there is a classification of the representation-finite $K$-algebras up to stable equivalence of Morita type (at least for some small ...
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### Information and intuition packed in the Chern character for coherent sheaves

even after quite some time learning it, I still get somehow puzzled by the Chern character. Let me recall some stuff to get notation and setting. Let us consider a smooth projective algebraic variety ...
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### How do we get the quotient $Ext^1(N,M)/Hom(N,M)$?

If $0\longrightarrow M\longrightarrow E\longrightarrow N\longrightarrow 0$ is a short exact sequence of torsion free coherent sheaves on a surface. Here $M$ is a line bundle, $E$ a vector bundle of ...
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### Second homology of mapping class group via universal coefficient theorem

Let $S_g$ be the closed, orientable surface of genus $g\ge 2$ and $Mod(S_{g,1})$ the mapping class group of homeomorphisms fixing a base point $x_0\in S_g$ modulo isotopies fixing $x_0$. It is known ...
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### dg-resolution of the polynomial algebra

I am intersted in constructing a cofibrant resolution of the commutative polynomial algebra in some number of variables in the category of dg-algebras(not necceserily commutative). The resolutions ...