I noticed only now that the entry *bimodule* is in bad shape and needs some attention. For the moment I have added here a mentioning of the 2-category of algebras, bimodules and intertwiners and a pointer to the Eilenberg-Watts theorem.

Added to noetherian ring a homological chacaterization: a ring is Noetherian iff arbitrary direct sums of injective modules are injective.

]]>added reference to dendroidal version of Dold-Kan correspondence

]]>the entry *Dold-Kan correspondence* and several related entries had long been asking for *connective chain complex*. I have now started something there. Of course this can be much further expanded on…

at *additive functor* there was a typo in the diagram that shows the preservation of biproducts. I have fixed it.

Also formatted a bit more.

]]>After scanning a bunch of literature, my favorite survey of the Adams spectral sequence is now this gem here:

- Dylan Wilson
*Spectral Sequences from Sequences of Spectra: Towards the Spectrum of the Category of Spectra*, lecture at 2013 Pre-Talbot Seminar (pdf)

created some minimum at *lim^1 and Milnor sequences*

(sorry for the title, I am undecided as to which single singular term best to use here)

]]>added a little bit more to *split exact sequence*.

I am beginning to split off from *fiber sequence* an entry *long exact sequence in homology* (also splitting off all the related redirects, such as long exact sequence in cohomology etc).

started universal coefficient theorem

]]>At *projective resolution* I have

spelled out the

*Definition*in lots of detail;spelled out statement and proof of the

*existence of resolutions*in full detail.

at *Künneth theorem* I have spelled out statemennt and detailed proof of the Künneth theorem in ordinary homology.

following public demand, I added to *tensor product of chain complexes* a detailed elementary discussion of the tensor product $I_\bullet \otimes I_\bullet$ of the (normalized) chain interval with itself, and how it gives chains on the cellular square: in *Square as tensor product of interval with itself*.

added to *perverse sheaf* a paragraph on the issue with and the origin of the terminology, here.

slightly edited *AT category* to make the definition/lemma/proposition-numbering and cross-referencing to them come out.

Probably Todd should have a look over it to see if he agrees.

]]>created *Amitsur complex*

I have corrected and expanded my note (at *4-sphere*: here) of the result of Roig-Saralegi 00, p. 2 on minimal rational dg-models of the following maps over $S^3$

induced from the “suspended Hopf action” of $S^1$ on $S^4$.

My aim in extracting this is to rename the generators given in Roig-Saralegi 00, p. 2 such as to make their degrees and their pattern more manifest. I hope I got it right now:

$\array{ \text{fibration} & \array{\text{vector space underlying} \\ \text{minimal dg-model}} & \array{ \text{differential on} \\ \text{minimal dg-model} } \\ \array{ S^4 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3\rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{deg = 2p + 4}{ \underbrace{ f_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_4 & \mapsto 0 \\ f_{2p+6} & \mapsto h_3 \wedge f_{2p + 4} \end{aligned} \right. \\ \array{ S^0 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3\rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{ deg = 2p }{ \underbrace{ f_{2p} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_0 & \mapsto 0 \\ f_{2p+2} &\mapsto h_3 \wedge f_{2p} \end{aligned} \right. \\ \array{ S^4//S^1 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3 , f_2 \rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{ deg =2p + 4 }{ \underbrace{ f_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_2 & \mapsto 0 \\ f_{2p+4} & \mapsto h_3 \wedge f_{2p + 2} \end{aligned} \right. }$ ]]>For $(V^\bullet,d)$ a cochain complex in characteristic zero, the cohomology of its graded symmetric dg-algebra should be the graded symmetric algebra on its cohomology

$H^\bullet(Sym(V^\bullet,d)) \;\simeq\; Sym( H^\bullet(V,d) ) \,.$(now recorded at *symmetric algebra*: here)

I just want to cite this from the literature. This must be a textbook fact (in char = 0 at least); but the only reference I find is this MO discussion here.

Anyone has a more canonical citation (for char = 0) at your fingertips?

]]>I have expanded a bit at *Serre-Swan theorem*: gave it an actual Idea-section, mentioned more variants (over general ringed spaces, in higher geometry) and added more references.

Needed to be able to point to *contractible chain complex* and discovered that we didn’t have an entry for that, so I quickly created one.

for the purposes of having direct links to it, I gave a side-remark at *stable Dold-Kan correspondence* its own page: rational stable homotopy theory, recording the equivalence

I also added the claim that under this identification and that of classical rational homotopy theory then the derived version of the free-forgetful adjunction

$(dgcAlg^{\geq 2}_{\mathbb{Q}})_{/\mathbb{Q}[0]} \underoverset {\underset{U \circ ker(\epsilon_{(-)})}{\longrightarrow}} {\overset{Sym \circ cn}{\longleftarrow}} {\bot} Ch^{\bullet}(\mathbb{Q})$models the stabilization adjunction $(\Sigma^\infty \dashv \Omega^\infty)$. But I haven’t type the proof into the entry yet.

]]>stub for model structure on dg-Lie algebras

]]>am starting [[model structure on dg-coalgebras]].

In the process I

created a stub for [[dg-coalgebra]]

and linked to it from [[L-infinity algebra]]

started *projective module*

(will need to move some material around with *projective object*. Also, I am splitting off now *projective resolution* from *resolution* )

I have expanded a good bit the discussion of the classical Adams spectral sequence. In order to simplify the cross-referencing, for the moment I have added the material *not yet* at *classical Adams spectral sequence* nor at *May spectral sequence* but at

But mostly I did expand on details of the May spectral sequence applied to the classical case, for instance I have spelled out some of the computations that go into the identification of the differential on the second page (e.g. here).

]]>