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A Scheme? Tom Gannon Communicated by Cesar E THE GRADUATE STUDENT SECTION WHAT IS… a Scheme? Tom Gannon Communicated by Cesar E. Silva Schemes pop up everywhere in mathematics. But getting But what exactly is an affine scheme? Affine schemes a grasp on this ubiquitous concept is not so easy. And the are the full manifestation of a simple idea: If 퐴 is a bare definition might leave you wondering why a scheme commutative ring, we should view the elements of that is defined the way it is. The fol- ring as functions instead of points. If you’re thinking lowing interpretation introduces about rings like ℚ(푖) or ℤ/6ℤ, this can be a little weird, The idea of important elements in the defini- but if you do the construction by first imagining rings tion of a scheme and helped me like ℝ[푥, 푦] it might be more clear. But the beautiful part a scheme understand the basic idea. I hope it is, this idea works for any ring, meaning that any ring can borrows can help you too. be viewed as functions on some space. Schemes have played a funda- For the remainder of this article, fix a commutative from the mental role in algebraic geometry ring 퐴. You might ask, “What space could this ring 퐴 ever since they were introduced by be a function on?” To motivate this, let’s do an example idea of a Alexander Grothendieck [Gr]. The from basic high school algebra. If you want to see where 3 2 idea of a scheme borrows from the the polynomial 푝(푥) ∶= 푥 + 푥 + 푥 vanishes, your natural 2 manifold. idea of a manifold in differential instinct would be to factor: 푝(푥) = 푥(푥 + 푥 + 1). Then if 2 topology. Recall that a manifold is 푝(푥) = 0, you know that either 푥 = 0 or 푥 +푥+1 = 0. Put in a topological space locally diffeomorphic to an open slightly more abstract terms, if the function 푝(푥) vanishes 푛 at some point, then either the function 푥 vanishes at that subset of ℝ , subject to some additional topological 2 considerations to avoid pathologies. Now, if you have a point or the function (푥 + 푥 + 1) vanishes at that point. This is an important property we’d like our topological function defined on an open neighborhood of a point in space to satisfy: ℝ푛, you can take the derivative of that function at that The “Vanishing” Property: If 푓, 푔 ∈ 퐴 and the product point. So a manifold is a topological space that locally 푓푔 vanishes at a point 푝 in the topological space, then looks like “the place to do differential calculus.” In a simi- either 푓 vanishes at 푝 or 푔 vanishes at 푝. lar way, a scheme is a topological space that locally looks This definition looks suspiciously close to the definition like “the place to do algebra/find zeros of functions.” But of a prime ideal of a ring. Recall that a prime ideal 픭 of that’s a pretty long name, so let’s call “the place to do a ring 퐴 is an ideal where if 푓, 푔 ∈ 퐴 and 푓푔 ∈ 픭, then algebra/find zeros of functions” an affine scheme instead. either 푓 ∈ 픭 or 푔 ∈ 픭. So if we assume that the “points” (Note for advanced readers: There’s actually more to it of our special topological space are prime ideals of the than just a topological space. There is a sheaf of functions ring 퐴, then we have an obvious choice for what it might associated to the space as well, but we won’t go into that mean for a function 푓 ∈ 퐴 to vanish at the point 픭. We’ll here.) say 푓 vanishes at 픭 if 푓 ∈ 픭 (or equivalently, it is zero in the ring 퐴/픭). Tom Gannon is a graduate student in algebraic geometry at the Given a ring 퐴, define the spectrum of 퐴, written University of Texas at Austin. His email address is gannonth@math. 푆푝푒푐(퐴) to be the set of prime ideals of 퐴. (Again, for utexas.edu. advanced readers: We’ve just defined a scheme in the For permission to reprint this article, please contact: category of sets, but you can put additional structure [email protected]. on your spectrum so that it becomes an object in the DOI: http://dx.doi.org/10.1090/noti1603 category of locally ringed spaces.) Now I’ve promised a 1300 Notices of the AMS Volume 64, Number 11 THE GRADUATE STUDENT SECTION topology on this space, and any good topology in a This is about half of the definition of an affine scheme. subject relating to “vanishing” should have the topology Continuing with the “rings are functions on some natural related to vanishing. And luckily for us, it does. We space” interpretation, we can develop a sheaf on the define a set of points (which are prime ideals, but we’re space 푆푝푒푐(퐴) so that we can, for example, talk about the thinking of them as “points”) to be closed if it is of the 푥2+푦2 function 푥−3 on the places where 푥 − 3 doesn’t vanish. form 푉(푆) = {픮 ∈ 푆푝푒푐(퐴) ∶ 푆 ⊂ 픮}, for some set of Sheaves are beyond the scope of this short piece, which “functions” 푆 ⊂ 퐴 (meaning, “every element of 푆 vanishes is intended only to give you a good picture of what the at 픮.”) Checking this forms a topology on 푆푝푒푐(퐴) is not topological space of an affine scheme looks like. And I too hard and is a good exercise. hope it has also given you a new interpretation of rings! This is a good definition for affine schemes (which, recall, Any ring can Further Reading was meant to be related to [Gr] A. Grothendieck (with J. Dieudonne), Eléments de determining where functions be viewed as Géométrie Algébrique, Publ. Math. IHES, Part I:4 (1960); Part vanish) because in some sense II:8 (1961); Part III:11 (1961), 17 (1963); Part IV:20 (1964), 24 the definition is “maximal.” functions on (1965), 28 (1966), 32 (1967). MR 0217083 That is, any set of elements [Ha] R. Hartshorne, Algebraic Geometry, Grad. Texts in in the ring 퐴 that can be the some space. Math. 52, Springer-Verlag, New York. Heidelberg, 1977. set of functions vanishing at a MR 0463157 point is a prime ideal (since it has the “Vanishing” Property). Moreover, functions dis- EDITOR’S NOTE. This article is a revision of Gannon’s tinguish the points in the topological space. That is, if post “The ‘Idea’ of a Scheme” on the AMS Graduate Stu- two points 픭, 픮 in the space are distinct, then there is a dent Blog blogs.ams.org/mathgradblog. A related function that vanishes on one of the points, but not on article “WHAT IS...an Anabelian Scheme?” by Kirsten the other. Seeing why this is true is a good check on your Wickelgren appeared in the March 2016 issue of understanding of the concept. However, given two points the Notices, www.ams.org/notices/201603/rnoti- 픭, 픮 ∈ 푆푝푒푐(퐴), it isn’t necessarily the case that you can p285z.pdf. find a function 푓 ∈ 퐴 vanishing at 픭 but not at 픮. This is a good exercise to check your understanding of the material, and an example is also provided below. Example: Consider the ring 퐴 ∶= ℂ[푥, 푦]. What are ABOUT THE AUTHOR some points in 푆푝푒푐(퐴)? Certainly if you choose some 푎, 푏 ∈ ℂ, then 픭 ∶= (푥 − 푎, 푦 − 푏) is a maximal ideal, When not doing mathematics, Tom since the quotient field 퐴/픭 is ℂ, a field, and thus it Gannon can often be found in is a prime ideal. A good way to think about points of the kitchen cooking new recipes, the form (푥 − 푎, 푦 − 푏) is to think of the points of the or can be found in the kitchen form (푎, 푏) ∈ ℂ2 (equivalently, the places on the 푥푦 cooking old recipes. plane where 푥 − 푎 = 0 and 푦 − 푏 = 0). The function 푝(푥, 푦) ∶= 푥4 + 푥3푦 − 푥푦 − 푦2 ∈ 퐴 vanishes at 픭 since Tom Gannon 푝(푥) = 푟(푥, 푦)(푥 − 2) + (−푥 − 푦)(푦 − 8) ∈ 픭 Photo Credit where Photo of Tom Gannon courtesy of Tom Gannon. 푟(푥, 푦) = 푥3 + 2푥2 + 4푥 + 푥2푦 + 2푥푦 + 4푦. Also, 푝(푥, 푦) vanishes in our normal sense because 푝(2, 8) = 24 + 26 − 24 − 26 = 0. So we can conclude that vanishing in our new sense generalizes vanishing in the old sense. Now the question becomes, what else can we get from this scheme construction? One quick payoff from the definition of affine schemes is that we can talk about functions vanishing at places that aren’t points in the traditional sense. For example, consider the polynomial 푞(푥, 푦) ∶= 푥3 − 푦. It is an irreducible polynomial by Eisenstein’s Criterion, so the ideal 픮 ∶= (푥3 −푦) is another point in 푆푝푒푐(퐴). This has an obvious interpetation in ℂ2. It’s the curve 푦 = 푥3. And more amazingly, notice that 푝(푥, 푦) ∶= 푥4 + 푥3푦 − 푥푦 − 푦2 = (푥3 − 푦)(푥 + 푦), so 푝(푥, 푦) ∈ 픮. We can interpret it to mean the function 푝(푥, 푦) vanishes on the entire curve. That is very pretty. December 2017 Notices of the AMS 1301.
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