added pointer to:

- Wolfgang Lück,
*Equivariant Cohomological Chern Characters*, International Journal of Algebra and ComputationVol. 15, No. 05n06, pp. 1025-1052 (2005) (arXiv:math/0401047, doi:10.1142/S0218196705002773)

added pointer to:

- Wolfgang Lück,
*Chern characters for proper equivariant homology theories and applications to K- and L-theory*, Journal für die reine und angewandte Mathematik, Volume 2002: Issue 543 (doi:10.1515/crll.2002.015, pdf)

added pointer to:

- Victor Buchstaber, A. P. Veselov,
*Chern-Dold character in complex cobordisms and abelian varieties*(arXiv:2007.05782)

added the original references on the Chern character in K-theory:

Friedrich Hirzebruch, Section 12.1 of:

*Neue topologische Methoden in der Algebraischen Geometrie*, Ergebnisse der Mathematik und Ihrer Grenzgebiete. 1. Folge, Springer 1956 (doi:10.1007/978-3-662-41083-7)Armand Borel, Friedrich Hirzebruch, Section 9.1 in:

*Characteristic Classes and Homogeneous Spaces, I*, American Journal of Mathematics Vol. 80, No. 2 (Apr., 1958), pp. 458-538 (jstor:2372795)

added pointer to

- Peter Hilton,
*General cohomology theory and K-theory*, London Mathematical Society Lecture Note Series 1, Cambridge University Press (1971) (doi:10.1017/CBO9780511662577)

both for the Chern character on K-theory, and for the Chern-Dold character, and for the proof that the latter reduces to the former

]]>added pointer to

- Helge Maakestad,
*Notes on the Chern-character*(arXiv:math/0612060)

(has this been published?)

]]>have added pointer to Atiyah-Hirzebruch, and organized the references in two subsections, one for Chern char on K-theory, one for Chern-Dold char on generalized cohomology

]]>added pointer to

- V. M. Buchstaber,
*The Chern–Dold character in cobordisms. I*, Mat. Sb. (N.S.), 1970 Volume 83(125), Number 4(12), Pages 575–595 (mathnet:3530)

In the section *For spectra and generalized cohomology* (here) I have tried to make more explicit what is going on, and which authors find which words for this situation:

For $E$ a spectrum and $E^\bullet$ the generalized cohomology theory it represents

$E^\bullet(X) \;\simeq\; \pi_{-\bullet} Maps(X,E)$the *$E$-Chern-character* or *Chern-Dold character for $E$* is simply the map induced by rationalization over the real numbers

i.e. is

$\label{ChernDoldCharacter} chd \;\colon\; E^\bullet(X) \;\simeq\; \pi_{-\bullet}Maps(X,E) \overset{ \pi_{-\bullet}Maps(X,L_{\mathbb{R}}) }{\longrightarrow} \pi_{-\bullet}Maps(X,E_{\mathbb{R}}) \;\simeq\; E^\bullet_{\mathbb{E}}(X) \;\simeq\; H^\bullet(X, \pi_{\bullet}(E)\otimes_{\mathbb{Z}}\mathbb{R}) \,.$The very last equivalence in (eq:ChernDoldCharacter) is due to Dold 56 (reviewed in detail in Rudyak 98, II.3.17, see also Gross 19, Def. 2.5).

One place where this neat state of affairs (eq:ChernDoldCharacter) is made fully explicit is Lind-Sati-Westerland 16, Def. 2.1. Many other references leave this statement somewhat in between the lines (e.g. Upmeier 14) and, in addition, often without reference to Dold (e.g. Hopkins-Singer 02, Sec. 4.8, Bunke 12, Def. 4.45, Bunke-Gepner 13, Def. 2.1, Bunke-Nikolaus 14, p. 17)

Beware that some authors say *Chern-Dold character* for the full map in (eq:ChernDoldCharacter) (e.g. Upmeier 14, Lind-Sati-Westerland 16, Def. 2.1), while other authors mean by this only that last equivalence in (eq:ChernDoldCharacter) (eq. Rudyak 98, II.3.17, Gross 19, Def. 2.5).

]]>

added pointer to

- John Lind, Hisham Sati, Craig Westerland, Section 2.1:
*Twisted iterated algebraic K-theory and topological T-duality for sphere bundles*, Ann. K-Th. 5 (2020) 1-42 (arXiv:1601.06285)

where the nature of Chern-Dold is made fully explicit.

]]>I had removed it, but the redirects-bug keeps it around: It’s not in the source :-)

]]>What’s with the random ’spring’ in here ?

]]>made explicit that the $E$-Chern character in Hopkins-Singer is based on the “Dold-Chern character”, made that term redirect here, and added some references on it:

Albrecht Dold,

*Relations between ordinary and extraordinary homology*, Matematika, 9:2 (1965), 8–14; Colloq. algebr. Topology, Aarhus Universitet, 1962, 2–9 (mathnet:mat350), reprinted in: J. Adams & G. Shepherd (Authors),*Algebraic Topology: A Student’s Guide*(London Mathematical Society Lecture Note Series, pp. 166-177). Cambridge: Cambridge University Press (doi:10.1017/CBO9780511662584.015)Yuli Rudyak, II.7.13 in:

*On Thom Spectra, Orientability, and Cobordism*, Springer 1998 (doi:10.1007/978-3-540-77751-9){#Upmeier14} Markus Upmeier,

*Refinements of the Chern-Dold Character: Cocycle Additions in Differential Cohomology*, J. Homotopy Relat. Struct. 11, 291–307 (2016). (arXiv:1404.2027, doi:10.1007/s40062-015-0106-y){#Gross19} Jacob Gross,

*The homology of moduli stacks of complexes*(arXiv:1907.03269)

I have expanded the Idea-section at *Chern character*; rewording (making more succinct, basically) the statement about the Chern-character on generalized (Eilenberg-Steenrod) cohomology and adding its generalization to the Chern character in cohesion, given by the shape of the Maurer-Cartan form:

with pointers to Bunke et al, where this is proven.

Maybe I find time to further exand later.

]]>No, on the contrary, I believe that you have more practical experience with RH than I do. I am on foundational side in algebraic geometry so I just talk more and think more on the foudnational aspect. But I do very little hands on algebraic geometry.

The wikipedia article is quite good. I would be careful not to merge GRH and HRH for the simple reason that these are two different statements. GRH implies HRH in the situations in which GRH applies.

]]>I had thought that GHRR was just another name for GRR. My apologies if this is incorrect; I came to this impression after reading the wikipedia entry for "GHRR". ]]>

My past experience is that writing a neutral information like references and so on, is harmless, but writing complicated tricky section once done wrong way is very hard to improve (read takes more time to imrpove than to write from the beginning), especially when you care not to hurt the structure from your predecessor (in fact a number of pages where I could contribute I never did just because there are too difficult to tackle, being created in difficult style to start with). So I absolutely did not ask not to write but asked for the care; this is one of the most important entreis. One of the voluems of SGA is dedicated solely to Riemann-Roch. I have created most of the entries on algebraic geometry in nlab and of course would like help from new people, but with similar care and dilligence – so what is wrong with the warning – spent several months last Winter and Spring preparing some grounds on the related circle of questions in nlab e.g. the very related topic of index theory (e.g. entry Fredholm operator), then bibliography stubs like slope filtration, just to have some grounds. I could of course just go to the middle of the topic and write uncarefully rambling statements but this I would not dare.

I hoped Kevin will write but with care as the topic is tricky. For example, at the most trivial level talking Grothendieck-Hirzebruch-Riemann-Roch is already strange. There is Hirzebruch-Riemann-Roch and there is Grothendieck-Riemann-Roch, the first is corollary of the second in those setups where both apply. In some geometric situations one has HRR but not GHR, as the stronger version does not hold or is difficult to prove or even formulate. There is no Grothendieck-Hirzebruch-Riemann-Roch and I hope it will not appear as an artifact of nlab.

]]>Zoran, I’m not sure I understand why you are being so fussy here. Rhetorical question: is every page you write perfectly polished and written with great care, with proofs in conformance with nPOV as you understand the term?

If Kevin is motivated to write something on Grothendieck-Hirzebruch-Riemann-Roch, then by all means he should proceed full steam ahead, without worrying about whether he (or Grothendieck) has really gotten to the bottom of it according to Zoran. As we all know, nLab pages generally improve over time anyway (with the help of people like Zoran).

]]>I took nPOV in the widest possible sense of a natural proof. So far such a proof of Grothendieck-Riemann-Roch has not being found. I know a couple of people who have some insight in the right direction but did not get to the bottom yet.

]]>Is the goal [in general, or of nCafe/nLab/nForum] to make everything fit into the nPOV?

No. We want all the good content that we can get. But we take the liberty of allowing us to afterwards go through it and reformulate some of it from the nPOV.

If you feel energetic, don’t let the nPOV issue stop you from adding stuff. Just create a a subsection with a descrptive title and go.

]]>Is the goal [in general, or of nCafe/nLab/nForum] to make everything fit into the nPOV?

Maybe some answers: nPOV

]]>I haven't read any of it beyond the abstract, but this paper of Toen might be in the right direction? ]]>

It is difficult to go to the bottom when Riemann-Roch theorems are considered. In other words it is difficult to take NPOV as the subject is still not well understood. I mean all the proofs are still a hack (in agreement with Grothendieck’s unsatisfaction with his own proof, published by Jean-Pierre Serre). So please be careful when writing, it is easy to make a mess in such a difficult topic.

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