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Sheaves MAT4215 — Vår 2015 Sheaves Notes 1—Sheaves MAT4215 — Vår 2015 Sheaves Warning: Version prone to errors. Version 0.07—lastupdate:3/28/1510:46:24AM The concept of a sheaf was conceived in the German camp for prisoners of war called Oflag XVII where French officers taken captive during the fighting in France in the spring 1940 were imprisoned. Among them was the mathematician and lieutenant Jean Leray. In the camp he gave a course in algebraic topology (!!) during which he introduced some version of the theory of sheaves. He was aimed at calculating the cohomology of a total space of a fibration in terms of invariants of the base and the fibres (and naturally the fibration). To achieve this in addition to the concept of sheaves, he invented spectral sequences. After the war Henri Cartan and Jean Pierre Serre developed the theory further, and finally the theory was brought to the state as we know it today by Alexandre Grothendieck. Definition of presheaves Let X be a topological space. A presheaf (preknippe) of abelian groups on X consists of two sets of data: Sections over open sets: For each open set U X an abelian group F (U),also ⇤ ✓ written Γ(U, F ). The elements of F (U) are called sections (seksjoner) of F over U. Restriction maps: For every inclusion V U of open sets in X agrouphomo- ⇤ ✓ morphism ⇢U : F (U) F (V ) subjected to the conditions V ! ⇢U = ⇢V ⇢U W W ◦ V for all sequences W V U of inclusions of open sets in X. The maps ⇢U are ✓ ✓ V called restriction maps (restriksjonsavbildninger) and if s is a section over U,the U restriction ⇢ (s) is often written as s V . V | The notion of a presheaf is not confined to presheaves of abelian groups. One may speak about presheaves of sets, rings, vector spaces or whatever you want: Indeed, for any category C one may define presheaves with values in C. The definition goes just like for abelian groups, the only difference being that one requires the “spaces” of sections F (U) over open sets U be objects in the category C and not abelian groups, and of course all the restriction maps are required to be morphisms in C.Onemayphrasethis definition purely in categorical terms by introducing the small category openX of open sets in X whose objects are the open sets, and the morphisms are the inclusion maps between open sets. With that definition up our sleeve, a presheaf with values in the category C is just a contravariant functor F : open C X ! —1— Notes 1—Sheaves MAT4215 — Vår 2015 We are certainly going to meet sheaves with a lot more structure than mere the structure of abelian groups—e.g., like sheaves of rings—but they will all have an un- derlying abelian group, so we start with those. That said, sheaves of sets play a great role in mathematics, and in algebraic geometry, so we should not completely wipe them under the rug. Most results we establish for sheaves of abelian groups can be proved mutatis mutandis for sheaves of sets as well as long as it can be formulated in terms of sets. Hartshorn includes a third axiom when defining a presheaf. He requires that F ( )= ; 0 , but most other texts do not include that axiom. It follows from the sheaf axioms { } that F ( )=0whenever F is a sheaf, see the comment further down. ; Definition of sheaves A sheaf (knippe) of abelian groups F on X is a presheaf of abelian groups on X satisfying the two following requirements: ⇤ Locality axiom: Let Ui i I be an open cover of the open set U and let s be a { } 2 section of F over U.Iftherestrictionsofs to Ui all vanish, i.e., one has s U =0 | i for all i,thens =0. ⇤ Gluing axiom: Let Ui i I be an open cover of the open set U.Givensectionssi { } 2 over Ui matching on the intersections Uij = Ui Uj, i.e., si U U = sj U U ,then \ | i\ j | i\ j there is a section s of F over U satisfying s U = si. | i The locality condition says that sections are determined locally; that is, if two sections coincide on all open sets of a covering, they are equal. The gluing axiom says that locally given sections (i.e., si over Ui) matching where they can match (i.e., over the intersections Ui Uj)canbepatchedtogethertoaglobalsection(i.e., asection \ over U = i Ui). This of course applies to functions of any type on X,andindeed, sheaves of functions are simple and familiar examples of sheaves. S An alternative formulation of the sheaf axioms There is a nice alternative way of formulating the two sheaf axioms. They are equivalent to the following sequences all being exact: ↵ ⇢ 0 / F (U) / F (Ui) / F (Ui Uj) (1) i i,j \ Q Q where as usual Ui i I is an open cover of the open set U,andthemaps↵ and ⇢ { } 2 are defined by ↵(s)=(s U )i I and ⇢((si)i I )=(si U U sj U U )i, j, and where the | i 2 2 | i\ j − | i\ j indices of the second product run over I I. The locality axiom for the cover Ui is ⇥ { } equivalent to ↵ being injective and the gluing axiom to Im ↵ =Ker⇢. Of course in the definition of the map ⇢ where we take difference between the rest- rictions of the sections si working with sheaves of abelian groups is essential. However, when working with sheaves not being sheaves of abelian groups, e.g., sheaves of sets, —2— Notes 1—Sheaves MAT4215 — Vår 2015 the map ⇢ must be replaced with two maps, ⇢01 and ⇢2 where ⇢1((si)i I )=(si U U )i, j 2 | i\ j ⇢2((si)i I )=(sj U U )i, j. 2 | i\ j We underline that the indices i, j run through the product I I.Inthiscasethe ⇥ sequence ( 1)takestheform ⇢1 ↵ / / F (U) i F (Ui) / i,j F (Ui Uj), (2) ⇢2 \ Q Q and to say that it is exact amounts to saying that ↵ is injective and that if ⇢1(σ)=⇢2(σ) if and only if σ lies in the image of ↵.Onesaysthat↵ is the equalizer (ekvilisatoren) of the the to maps ⇢1 and ⇢2. First and familiar examples Take X = R and let C be the sheaf whose sections over an open set U is the ring of continuous real valued functions on U,andthere- striction maps are just the good old restriction of functions. Then C is a sheaf of rings (functions can be added and multiplied), both the sheaves axioms are obviously satisfied. You should convince yourself that this is true. For a second familiar example let X C be any open set. On X one has the sheaf ✓ OX of holomorphic functions. That is, for any open U X the sections Γ(U, OX ) is ✓ the ring of holomorphic (i.e., complex analytic) functions on U.Onecanrelaxthe condition of holomorphy to get the larger sheaf KX of meromorphic function in X.It contains OX ,andthesectionsoveranopenU are the meromorphic functions on U. In a similar way, one can get smaller sheafs contained in OX by imposing vanishing conditions on the functions. For example if a X is any point, one has the sheaf 2 denoted OX ( a) of holomorphic functions vanishing at a.Asthenameindicatesthe − sections of OX ( a) over U are holomorphic functions in U,andifa U,onerequires − 2 additionally that they should vanish at a. Convince yourself that this indeed is a sheaf. A third example, which at least is familiar to students having followed the course Algebraic Geometry I, is highly relevant for us. This time X is a variety over the algebraically closed field k,andOX is the sheaf whose sections over the Zariski-open set U are the regular functions in U. Convince yourself that this indeed is a sheaf of rings. The empty set again The empty set is always around, and to develop a theory properly one has to deal with it. This is not always clear how to do, and from time to time loud arguments among mathematicians erupt. For critics of academia this is gefundenes Fressen: Loud arguments about nothing! It follows from the sheaf axioms that F ( )=0when F is an abelian sheaf. May be the best thing is just to believe ; it, but anyhow here here follows the argument: Use the sequence above for the empty covering of the empty set (You don’t need any open sets to cover nothing), and then use that the empty product is 0! Problem .. If F is a sheaf of sets, what is your guess at F ( ) being? X ; —3— Notes 1—Sheaves MAT4215 — Vår 2015 Problem .. Let X C be an open set, and assume a1,...,an are distinct points in ✓ X and n1,...,nr be natural numbers. Define Γ(U, F ) to be the set of those function f Γ(U, OX ) that vanishes to an order at least ni at ai when ai U.ShowthatF is 2 2 asheafofrings. X Problem .. Let X C be an open set, and assume a1,...,an are distinct points in X ✓ and n1,...,nr be natural numbers. Define Γ(U, F ) to be the set of those meromorphic functions f Γ(U, KX ) holomorphic away from the ai’s and having a pole order 2 bounded by ni at ai.ShowthatF is a sheaf. Is it a sheaf of rings. X Problem .. Let X = R.IfonedefinesΓ(U, B) to be the ring of continuous and bounded functions on the open set U R.ShowthatB is a presheaf but not a sheaf.
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