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

started universal coefficient theorem

]]>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.

]]>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 *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.

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)

]]>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).

]]>Might anyone have a pdf copy of Peter May’s thesis for me, the document that, I gather, expands over the published version May 66 by a discussion specific to the Steenrod algebra?

]]>added the statement that every abelian group admits a free resolution of length 2, here.

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

]]>I have spelled out the derivation of the Gysin sequence from the Serre spectral sequence at *Gysin sequence* .

expanded/polished a little the Definition at *exact couple*

gave *Atiyah-Hirzebruch spectral sequence* a minimum of an Idea-section and added a minimum paragraph with pointers to applications to D-brane charges in string theory here, also on the D-brane charge page itself here

created an entry *braid lemma* with the statement and the application to the long exact sequence of a triple in (generalized) homology.

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

]]>What would be a source for an explicit derivation – for $E$ an $E_\infty$-ring and $X$ a spectrum – that the classical $E$-Adams tower corresponding to the canonical $E$-Adams resolution $N(X \wedge E^{\wedge^\bullet})$ is equivalently the Tot-tower of the cosimplicial spectrum $X \wedge E^{\wedge^\bullet}$? That’s not immediate, is it?

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