Wise choice of notation and terminology can help clarify thinking. I would like to make wise choices, constrained somewhat by convention (although I may try to selectively push the envelope), and by lack of ability to hold everything in my head. Here are some conventions I may try to follow (although not religiously).

- A, B, C: rings. I believe using A for a ring comes from french: A is for anneaux. Elements of A might be called some flavor of a, and similarly for B and C.
- i, j, k, l, m, n, p, q: integers. p is often prime.
- p and q may be points too.
- k, K: Different flavors of K are for fields. k is often the ground field. K(A) is the fraction field of an integral domain. K(X) is the function field of an integral scheme X. kappa(p) is the residue field of a point. (Discussion on mathoverflow tentatively convinced me to go with this.)
- L: field
- M is a matrix
- U is an open subset or subscheme
- W, X, Y, Z: schemes. w is a point of W. X = Spec A, Y = Spec B.
- Caligraphic font (\mathcal) is for quasicoherent sheaves.
- Fraktur font is for prime ideals (especially m, n, p, q). m and n are maximal ideals.
- Currently, categories whose names are english words are in bold, but general categories are just in \mathcal. That’s not very consistent. (Perhaps all categories should be in bold?)

**Changes I’m considering**

Here are some changes I am leaning toward. I likely won’t implement the changes until summer 2011 when more of the notes are posted, so please comment (pro and con, and either here or by email, anonymously or not).

I’m now strongly leaning toward using “open embedding” instead of “open immersion”, and similarly for “closed embedding” and “locally closed embedding”. Enough others have used it that there is sufficient cover. (I’ve most recently seen it in Rob Lazarsfeld’s wonderful “Positivity in Algebraic Geometry I”.) Brian Conrad argues otherwise, and has written to me: “Is there any algebraic geometry reference which uses “open embedding” for “open immersion” in settings where schemes seriously intervene (i.e., not just reduced varieties over a field, which is what I assume comes up for Lazarsfeld’s book) … Please be very careful about changing widely-used terminology unless you’ve checked that the proposed switch is consistent with many references which treat algebraic geometry in a degree of generality comparable to what you are doing in the book; otherwise it can create confusion much as Hartshorne’s textbook has done with the notion of “coherent”.” On the other hand, Dan Abramovich and Georges Elencwajg are I think supportive. (Update some time before March 2012: the change to “embedding” is now made.)

For points of projective space: [0 : 1 : 2] instead of [0 ; 1 ; 2] (and [0,1,2] and (0,1,2) etc.). Ezra Miller made a good argument that the colons suggest ratios. I would have to be cautious of spacing. (Update some time before March 12: I now use [0,1,2].)

August 20, 2010 at 2:00 am

The stacks project uses f.f.(A) for the fraction field of a domain A and Q(R) for the total quotient ring of a ring R. In addition it uses R(X) to denote the ring of rational functions on a scheme X and, if X is integral sometimes k(X) or K(X). I think it makes sense to distinguish the case where there may be zero divisors from the case where there are none.

The stacks project uses the “mathcal” font to indicate sheaves. Functors are usually denoted F, G, H. Since a sheaf is a presheaf (i.e., a functor) there is some overlap…

Please do not denote by \underline{Hilb} the sheaf defined by Hilb! In older texts you can find the following notation: use Hilb(X/S) to denote the set of closed subschemes flat and of finite presentation over S, use \squigglyunderline{Hilb}_{X/S} for the functor T |—> Hilb(X_T/T), use \underline{Hilb}_{X/S} for a scheme which represents this sheaf (if it exists). Of course we can’t \squigglyunderline, and I think using \underline is not so great either. Maybe we can use \mathcal{Hilb}_{X/S} to denote the functor? Anyway, I suggest to avoid using underline if at all possible (it just looks bad). If X is a scheme, then just use h_X to denote the functor represented by X and not \underline{X}. This is less important than the Hilb thing above since underlining a single letter with no part sticking out below the base line (what’s that called?) doesn’t look as bad…

August 23, 2010 at 6:19 pm

About f.f. vs. K vs. Q: I agree that it is wise to separate the case when there might be zero-divisors. I earlier used “FF” (which I liked because it could stand for both “function field” and “fraction field”), but some people (Brian Conrad maybe?) didn’t like it, so the mathoverflow discussion led me to favor flavors of K (so “K(X)” and “K(A)” and “\kappa(p)” and “\operatorname{char} k =0”). How about K’s in general for fields, and Q when one wants to allow possible zero-divisors?

I’ve been following the \mathcal convention for sheaves, and have clarified that above.

On Hilb: I’m happy with what you say. I already (I think) use h_X. Because one uses h_X and not \mathcal{h}_X, why not \operatorname{Hilb}_{X/S} rather than \mathcal{Hilb}_{X/S}?

About underline: how do you denote locally constant sheaves (taking values in some set S)? This isn’t so relevant for the notes, but they come up as examples.

August 23, 2010 at 11:24 pm

Well, I do not agree with everything that Brian says! Of course the really important thing is to make sure everybody knows what you are talking about. I think using Q(A) for rings, f.f.(A) when A is a domain, R(X) when X is a scheme (as in EGA I section 7), and K(X) when X is integral are all suffiently clear. By the way working with total quotient rings and with R(X) for general X is rather tricky. Should this be avoided in a first year coarse?

The stacks project tries to keep the latex commands down to a view very simple ones; there are no \operatorname commands in there yet, and no macros at all. Part of the idea behind this is that it should be clear what output the latex will produce from looking at it.

You are right with your comment about locally constant sheaves. Hmm… I may have to make an exception for this one!

August 26, 2010 at 7:24 pm

Response to Johan: I use macros precisely to make quick systemic changes, so FF can be easily changed. I’ll still tentatively change it to K, since I think there is no confusion. I can’t remember if Q(A) or Q(X) comes up at all, but if it does, it is as an aside (so it shouldn’t distract people learning for the first time). I’ll try to be careful when we reach those sections.

February 10, 2011 at 9:13 am

[…] proposed notational changes (“[1:2:3]” and “open embedding”) to the post on Notation, so please comment […]

February 10, 2011 at 9:28 pm

I think I raised this objection before, but if you go through with this proposed “open embedding” change (which I urge you to reconsider — it will confuse students who go on to read scheme-related sources later), how will you distinguish a scheme morphism that is an isomorphism onto an open subscheme from one that is merely a homeomorphism onto an open subset (perhaps with different scheme structure)? Or more importantly the “closed” variant (which comes up a lot)? This is why it seems good to have two versiosn: “open embedding” for the topological aspect, and “open immersion” for the scheme aspect.

I genuinely don’t understand the rationale to change terminology that is widely used and not really a problem in its current form (especially for situations where schemes really intervene in an essential way). By way of analogy, differential geometers use “smooth map” with an entirely different meaning than algebraic geometers, so do you advocate that we also change “smooth morphism” to “submersion” to avoid that confusion too? I hope not.

February 12, 2011 at 8:32 am

I’m totally out of my depth commenting here, but I’m going to do so anyway.

I don’t think Brian Conrad’s analogy holds strongly, because the term “embedding” is much more universally applicable than the term “smooth.” In almost any branch of mathematics (except algebraic geometry, as the terminology currently stands), the notions “embedding,” “inclusion,” and “sub___” are practically interchangeable. One wants to be able to say “if you embed X in Y, then X is a subobject [or substructure] of Y.” Divorcing these terms in this one subdiscipline seems bizarre.

February 14, 2011 at 6:36 am

Dear Charles: In differential geometry submanifolds need not be embedded. More specifically, within the theories of Lie groups and foliations “submanifold” is just (injectively) immersed and not embedded (the Lie group/Lie alg. correspondence wouldn’t hold otherwise, and similarly for Frobenius integrability). So “submanifold” and “embedding” aren’t synonymous there. I would have thought that “inclusion” is too ambiguous to have universal meaning in differential geometry (e.g., could be reasonably applied to injective immersions as well as to embeddings). It’s true that embeddings in differential geometry are “substructures”, but the same is true for the weaker notion of (injective) immersion in some important contexts, and likewise for closed/open immersions in algebraic geometry.

February 14, 2011 at 7:27 am

Commenting out of your depth can be embarrassing sometimes, but it is a great way to improve one’s knowledge and understanding.

February 22, 2011 at 2:03 pm

I found your comment helpful to the discussion, and I’m sure Brian did too!

February 13, 2011 at 7:59 am

Dear Ravi,

I completely agree with Brian here. There is a standard terminology, every source uses it, and it is not a source of confusion. Why not just stick with it?

Best wishes,

Matt

March 14, 2011 at 10:31 am

Dear Matt,

I disagree that every source uses it, but I suspect every arithmetic geometry source uses it, given that you and Brian are on the same page! And that may be convincing enough…

February 22, 2011 at 1:37 pm

A further comment from Tony is here and is copied below for readers’ convenience.

March 16, 2011 at 9:01 am

Perhaps I am mistaken, but I think Spec Q -> Spec Z would be considered an immersion by your definition, but not by the usual one. (More generally, this works for many examples of the form Spec k(x) -> X, where k(x) is the residue field of the non-closed point x in X.)

March 16, 2011 at 10:58 am

Charles, that’s a cool point; I didn’t realize it. I’m not sure where Tony’s definition was from (or if it was just a proposed definition).

March 11, 2011 at 10:07 am

Here is some data, which I will update as I get more (and as people tell me more). I won’t bother listing the use of “immersion”, as this is certainly fairly standard. Fulton’s Intersection Theory uses “imbedding” (see p. 426). Lazarsfeld’s “Positivity” uses “embedding”. (I prefer “locally closed immersion” to “immersion” to avoid confusion, but I admit that I sometimes say “finite type sheaf” when I should say “finite type quasicoherent sheaf”.)

People often like a verb to get across the notion like “we can embed P^2 blown up at 6 points in P^3”. The (perhaps archaic) phrases “Veronese embedding” and “Segre embedding” show that the use of “embedding” was at least common in the past.

I asked Rob Lazarsfeld when he came to speak at our joint seminar with Berkeley and Davis, and he agreed with me that “embedding” is in common use on the geometric side of the subject. I suspect that the split is between those on the more geometric side (where we constantly map things into other things) and the arithmetic side.

March 31, 2011 at 7:32 am

I asked Rob Lazarsfeld when he came to speak at our joint seminar with Berkeley and Davis, and he agreed with me that “embedding” is in common use on the geometric side of the subject. I suspect that the split is between those on the more geometric side (where we constantly map things into other things) and the arithmetic side.

Based on what he said, I’m continuing to lean toward using “closed embedding”, “open embedding”, and “locally closed embedding”, and that this is not new language (although it may be unfamiliar to some, and perhaps in the minority). Furthermore, this will not cause confusion, whereas “immersion” will and does cause confusion with people in neighboring fields on the geometric side of the subject. (But I’m still far from implementing it.)

April 17, 2011 at 11:00 am

Does anyone have strong feelings on using “schematic” (the “adjectival form of scheme”) rather than “scheme-theoretic”? (Example: “schematic pre-image”.) I’ve heard this used, and I like it better than “scheme-theoretic, and no one should be confused.

Update Jan 11, 2012: “scheme-theoretic” still stands.

May 24, 2011 at 7:13 am

I think “schematic” suffers from the same defects as “categorical”: It has a completely different meaning in standard English which in some cases might be applicable (particularly when the adjective shows up in titles, abstracts, and the like). I don’t think it is a terrible defect, but I personally prefer “scheme-theoretic.”

May 31, 2011 at 10:55 am

Another (very minor) thought I had here: I personally like the parallelism in statements such as, “X is a complete intersection set-theoretically but not scheme-theoretically.” However, this is clearly a matter of taste, and a very minor one at that.

June 9, 2011 at 10:48 am

Charles, those are both good points, and enough to make me hold off making any changes. I’m still torn.

April 17, 2011 at 11:42 am

A tiny question: If (X,O_X) is a ringed space, then we know what an O_X-module is. (I believe it is right to say O_X-module, not sheaf of O_X-modules.) It can be helpful to have a presheaf version of the same notion. Precisely: If (X,O_X) is a space with a presheaf of rings (which may in fact be a sheaf), then we can define a “presheaf O_X-module” in the obvious way: it is a presheaf of abelian groups F, with an O_X-action. (This can be useful even when O_X is a sheaf.) Is there better terminology than “presheaf O_X-module”? (This isn’t too important, but it isn’t just idle curiosity.)

Example: the right definition of extension by 0 (to be given in 3.6.G in the post of Apr. ~23 — thanks to Jason Ferguson for catching the error in the previous version) involves defining a presheaf, and sheafifying. We want i_! of an O-module to be an O-module. We’d want to say that the “presheaf” (let me call it i_!^{pre}) is a “presheaf O-module”, and thus (by an argument people should think through) i_! is an O-module. The awkward terminology makes the exposition (and hence the thinking) awkward.

April 20, 2011 at 8:00 am

I much prefer the phrase “line bundle” to “invertible sheaf”. I propose using the phrase “line bundle” without apology (after an initial apology) for invertible sheaf. Translation: when I mean “sheaf of sections of a line bundle”, I will elide the “sheaf of sections of a” when it should be clear from the context. (I will of course mention the phrase “invertible sheaf”.) If there are enough shouts of dismay, I won’t do this. My arguments in favor: (i) “line bundle” is geometrically suggestive, (ii) the insight embedded in the phrase “invertible sheaf” is much less useful (I’ve never really cared that in the monoid of O-modules, these are the invertible ones, (iii) this is absolutely common usage, and (iv) there seems to me to be no possibility of confusion, and finally (v) if people are no longer ashamed of using this phrase for this notion, the world will be a better place.

April 20, 2011 at 9:18 am

Of course you are free to declare “line bundle” to be synonymous with “invertible sheaf”. I prefer “invertible sheaf on X” or “invertible O_X-module”, because “line bundle” (sometimes) has another meaning. For example, in the stacks project we go between quasi-coherent sheaves and vector bundles by the Spec(Sym^*( – )) construction which is contravariant. In this terminology a “line bundle on X” is the vector bundle associated to an invertible O_X-module. I.e., a line bundle is a certain type of scheme over X and not a sheaf. Moreover, with this convention, the line bundle does not correspond to its “sheaf of sections”, but rather to the dual of that.

April 20, 2011 at 2:16 pm

Hi Johan,

Is there some reason you went with Spec(Sym^*(-)) rather than Spec(Sym^*(-^{\vee}))? I agree that the Spec(Sym^*(-)) construction is important and more fundamental — that’s not my quibble. But when you say “the vector bundle corresponding to a locally free sheaf” F, might you not say that it is Spec(Sym^* F^{\vee})?

April 21, 2011 at 7:01 am

Here is follow-up after a day’s thought. To set the terms of the discussion: (a) the shortest way to define “the vector bundle associated to a [finite rank] locally free sheaf” is Spec Sym^*(-). (b) The shortest way to define “the locally free sheaf associated to a vector bundle” is to take the sheaf of sections. These constructions do not commute, so we can either (i) just accept this, or (ii) take the “dual” of (a), or (iii) take the “dual” of (b). I’m proposing (ii) (which I *think* might be conventional), and I think you are proposing (iii), although you may be proposing (i).

Now suppose X is a smooth subvariety of A^n_k. We have a standard geometric sense (which can be made precise easily) of what the tangent bundle means. The tangent plane to p (a k-valued point) in X is naturally interpreted as an affine space in A^n_k. There is also an accepted definition of tangent (locally free) sheaf. According to your definition, the vector bundle associated to the tangent sheaf is the *cotangent* bundle, and vice versa. This seems non-ideal.

The same problem arises with (co)normal bundles and other things. When deciding what to call these sheaves, the ancients seemed to be voting (with their terminology) for (ii).

As always, I’m open to being convinced, and like figuring out the right path.

April 21, 2011 at 6:05 pm

Ha! I was proposing (a) and its inverse. I want to try and avoid ever taking the “sheaf of sections”. The reason for using (a) is that it gives an anti-equivalence of categories between the category of quasi-coherent O_X-modules and the category of vector bundles over X (see Lemma Tag 062N). In EGA and the stacks project the Spec(Sym^*(-)) of any quasi-coherent sheaf is called a vector bundle. I’m willing to change that (and I *certainly* do not suggest you follow this convention) if enough people complain to me.

I agree that it is a little funny that the tangent bundle of a smooth variety is the vector bundle associated to the sheaf of differentials, but I’ll live with that, especially because for me the phrase “is associated to” is synonymous with “is the Spec(Sym^*()) of”.

By the way, isn’t it better to use “sheaf of derivations”, or even “sheaf of vector fields” instead of “tangent sheaf”?

April 23, 2011 at 10:47 am

I see! What you propose is consistent, and fine by me. I’m still happy with what I propose too, and perhaps someone else will convince one of us to change. People will have to be cautious, but on this issue they have to be cautious anyway.

If I understand you correctly, you have a more expansive definition of vector bundle than I do — to me, they are associated (in one of two ways!) to a locally free sheaf. It makes me a little nervous, because to me “bundle” invokes some image of “fibration”. But I’m not too concerned. (I vaguely remember Jason Starr using “cone” for the notion you are using.) Okay, must run, so I’ll cut this short

May 24, 2011 at 7:27 am

Something recently occurred to me about this. If I’m not mistaken, for a locally free sheaf E on X, the scheme B = Spec Sym^* E^{\vee} represents a natural functor on schemes over X. Specifically, for a scheme p: Y -> X over X, an X-morphism to B is equivalent to a global section of p^* E. (The same functor can be defined even if E is not locally free, but may not be representable.) This, in my opinion, weighs in favor of (ii).

[Note: If U is an open subscheme of X, then X-morphisms U -> B are precisely sections of the map from an open subset of B to U. In this case, the description above states that such sections correspond naturally to sections of the sheaf E over U.]

June 9, 2011 at 10:50 am

Thanks Charles — that helps keep me convinced.

(Update January 11, 2012: This issue remains unresolved.)April 21, 2011 at 4:34 am

I define the Jacobian matrix (Exercise 13.1.E in the March 31 2011 version) in the transpose of the way it is usually defined in the mathematical literature (although perhaps not the algebro-geometric literature). See for example

wikipedia’s definition. (In general, I have a lot of confidence in wikipedia in describing the conventional definitions of things.) I know many people will argue correctly that it is good practice to follow convention, and that needlessly going against convention needlessly causes confusion. In this case I am leaning toward using this nontraditional definition, because it makes things notationally cleaner later on, and because (at least from how I read wikipedia) there is a reasonable minority which goes with this transpose definition. Please complain or comment if you want to. The fact that no one has written (here or in email) to query my definition suggests that no one was confused or upset.April 27, 2011 at 8:57 am

Something I mentioned in comments on some other post that I want to also make sure is here, so I don’t forget: projective coordinates are tentatively going to be [a,b,c] (commas not colons or semicolons).

September 29, 2011 at 10:32 am

I’ve now implemented this (which was of course no fun), which means people are free to complain if I’ve left a semicolon in. But if someone now complains that they prefer colons or semicolons instead of commas, I will hunt them down.