14

The answer to the question posed in the title of your post is yes, the tensor product of chain complexes is a Day convolution product. The important thing to note is that, to define a Day convolution monoidal structure on the $\mathcal{V}$-enriched functor category $[\mathcal{C},\mathcal{V}]$ (where $\mathcal{V}$ is a complete and cocomplete symmetric ...

10

The answer is yes. Given any projective module $P$ over $S^{-1}A$, where $A=\mathbb{Z}[t]$ (and works for many other rings too), it is the localization $S^{-1}M$ of a projective module over $A$. The reason is, you can always find such a finitely generated module $M$ with $S^{-1}M=P$, but you may replace $M$ with its double dual without affecting the ...

8

Spectral sequences are obtained by applying the snake lemma ad nauseam, essentially. You are doing that (finitely many times) in your spectral sequence argument.
Let me illustrate. Consider the case you have a diagram that has three exact rows and four exact columns, and start with the top right corner space $C_{1,4}$, and pick $x$ there.
If $d_v(x)=0$, ...

8

Robert Thomason was the first person to draw attention to this question, before derived schemes and infinity categories. I believe that he proved that for a quasi-compact and quasi-separated scheme that $D_{qc}=\textrm{Ind}(\textrm{Perf})$. For example, see Thomason-Trobaugh section 2.3, though at first glance it appears that only proves the weaker statement ...

ag.algebraic-geometry homotopy-theory homological-algebra stable-homotopy derived-algebraic-geometry

6

The boundary conditions of the B-model, ie, the D-branes, are the objects in $\mathcal{D}^b(X)$. A little bit of playing with pictures gives that the space of closed string states must be in the center of the algebra of open strings for any given boundary condition. For a category $\mathcal{C}$, this gives a map to $\mathcal{Nat}(id_\mathcal{C},id_\mathcal{C}...

ag.algebraic-geometry homological-algebra topological-quantum-field-theory mirror-symmetry hochschild-cohomology

6

The rings satisfying your condition (for right modules) are the right pure semisimple rings. There are many equivalent conditions. You can find a lot of information in Section 4.5 of the book
Prest, Mike, Purity, spectra and localisation., Encyclopedia of Mathematics and its Applications 121. Cambridge: Cambridge University Press (ISBN 978-0-521-87308-6/hbk)...

5

I'll take your question as license to advertise a relatively recent paper in a slightly more specialized but concretely calculational direction: http://nyjm.albany.edu/j/2014/20-53p.pdf. Its title is
Six model structures for DG-modules over DGAs: Model category theory in homological action. The theme is how different model structures can illuminate ...

5

If you use the category $C$ to represent chain complexes and you mean day convolution using a functor $C \otimes C\to C$ it is not possible. This boils down to whether you can obtain the totalization functor from bi-complexes to chain complexes, as a left adjoint to restriction for some functor $m: C \otimes C \to C$.
You cannot do this because the left ...

5

On question 1, for what rings all MCM ideals are principal, we can say quite a bit more if one knows that $R$ is parafactorial (that is, the Picard group of the punctured spectrum $Spec^o(R):=Spec(R)-\{m\}$ is trivial). For instance:
If R is a local complete intersection which is locally a UFD in codimension $3$, then any MCM ideal $I$ is free.
Proof: ...

4

You can always reduce an arbitrary "zigzag" as in the first definition, to one of length two, as in the second definition, by applying the following trick:
Whenever you encounter morphisms $E_{j-1}\to E_j\to E_{j+1}$ or $E_{j-1}\leftarrow E_j\leftarrow E_{j+1}$ that go in the same direction, take their composite.
Whenever you encounter morphisms of the form ...

3

Yes, there are zillions of model structures on Ch(A), corresponding to whatever class of projectives you choose to use for your homological algebra. This is all spelled out in the paper "Quillen model structures for relative homological algebra" by Christensen and Hovey.
More generally, the theory of cotorsion pairs builds model structures in many ...

3

Reiner's "Maximal Orders", Theorem 3.20, for instance.

3

Remark. Exact sequence $0 \to L \to E \to M \to 0$ corresponds to $Ext^1(M,L)$, not to $Ext^1(L,M)$.
Q1. $a \in k^\times$ acts on $Ext^1(L,M)$ via pullback along $a:L \to L$ or via pushout along $a: M \to M$.
Q2. There are two options: either one can check that the split sequence is the neutral element for the addition, or that in the long exact sequence
$$...

3

I believe these questions were studied and answered in the this 1984 paper:
HIROSHIMA MATH. J. 14 (1984), 359-369
On the set of free homotopy classes and Brown's construction
Takao MATUMOTO, Norihiko MINAMI and Masahiro SUGAWARA
They also have counterexamples that seem to be the same as in the recent preprint of Arlin and Christensen that you mention.
...

2

This is a translation of Dmitri Pavlov's answer into a more intrinsic, more geometric, and more elementary language. In particular, I will show that the étale topos of a positive-dimensional variety is never the topos of a topological space.
If $X$ is a topological space, then the associated topos $E = \mathbf{Sh}(X)$ is generated by subobjects of the final ...

answered May 24 at 2:03

R. van Dobben de Bruyn

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2

Consider the maps
\begin{align*}
i \colon K_0(\mathscr A) &\to K_0\big(K^{\text{b}}(\mathscr A)\big) & & & \chi \colon K_0\big(K^{\text{b}}(\mathscr A)\big) &\to K_0(\mathscr A)\\
[A] &\mapsto\big [A[0]\big] & & & \big[K^*\big] &\mapsto \sum_i (-1)^i \big[K^i\big].
\end{align*}
It is clear that $i$ is well-defined, and ...

answered May 18 at 16:49

R. van Dobben de Bruyn

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1

See Problem 4.23 and Problem 4.24 (with proofs) of Ulrich Bunke's Differential cohomology.
The underlying abstract machinery for computing homotopy (co)limits
via homotopy (co)ends is presented by
Sergey Arkhipov and Sebastian Ørsted
in Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories.

1

I am sure even your (6) is correct, but am a bit lazy today to check things carefully, so let me answer your question for polynomial rings.
If $I\subset R$, a graded ideal, it is immediate that one can pick a minimal set of generators for $I$ which are homogeneous. With your hypothesis, these become a regular sequence after localizing at the `irrelevant' ...

1

Let $A=k[x]/(x^2)$ and $S$ the simple $A$-module for a field $k$. If $\eta\colon 0\to \tau_{A^e}(A) \to X \to A\to 0$ is the almost split sequence ending in $A$ over $A^e$, then $S\otimes_A X$ is a semisimple module. It follows that $S\otimes_A \eta$ is a split exact sequence. Hence Question 2 is not always true.
There is an analogue of this for group ...

1

Hochschild homology is a derived invariant, and in particular a quasi-isomorphism invariant, since quasi-isomorphic algebras are derived invariant. Your algebra is non-unital, I assume, since $1$ is usually a non-trivial cycle in general.
In any case, however, you can consider the Hochschild cyclic complex of $A$, call it $C_*(A)$, which has an internal ...

1

One way to construct injectives on a presheaf category $[\mathscr C^{\operatorname{op}},\mathbf{Ab}]$ is to consider the forgetful functor
$$i^* \colon \big[\mathscr C^{\operatorname{op}},\mathbf{Ab}\big] \to \big[\mathscr C^{\operatorname{disc,op}},\mathbf{Ab}\big]$$
induced by the inclusion $i \colon \mathscr C^{\operatorname{disc}} \to \mathscr C$ (where $...

answered May 3 at 23:41

R. van Dobben de Bruyn

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