Algebraic Stacks TCC course, Autumn 2014 Michael Groechenig with contributions by: Alexander Betts Andrea Petracci Claudio Onorati Craig Smith December 15, 2014 Contents 1 Introduction 3 1.1 What is a stack? . .3 1.2 What is algebraic geometry? A functorial approach. .7 2 Faithfully flat descent 13 2.1 Tensor products and flatness . 14 2.1.1 A reminder on tensor products and base change . 14 2.1.2 Base change invariant properties . 17 2.1.3 Flatness . 18 2.2 Faithfully flat descent . 20 2.2.1 Basic properties . 20 2.2.2 Descending modules . 22 2.2.3 Descent for ring homomorphisms . 24 3 Sheaves and stacks 24 3.1 Sheaves . 25 3.1.1 Topological spaces . 25 3.1.2 Grothendieck topologies and sheaves . 26 3.1.3 Three examples . 28 3.2 Stacks . 29 3.2.1 Groupoids as generalised sets . 29 3.2.2 Stacks as groupoid-valued sheaves . 30 3.2.3 Examples . 33 3.2.4 Fibre products in 2-categories . 35 1 4 Algebraicity 36 4.1 Schemes . 37 4.1.1 Affine schemes . 37 4.1.2 Open subfunctors of affine schemes . 39 4.1.3 Schemes . 40 4.2 Algebraic spaces and stacks . 45 4.2.1 Finite presentation . 45 4.2.2 Smooth and ´etalemorphisms . 47 4.2.3 Various Grothendieck topologies . 48 4.2.4 Algebraic spaces . 49 4.2.5 Algebraic stacks . 49 4.2.6 Presentations . 50 4.2.7 Base change . 51 5 Torsors and quotient stacks 52 5.1 Group objects . 52 5.2 Torsors . 54 5.2.1 Definition and basic properties . 54 5.2.2 Faithfully flat descent revisited . 57 5.2.3 The tautological trivialisation . 58 5.3 Algebraicity of quotient stacks . 59 5.3.1 Algebraicity of the stack of torsors . 59 5.3.2 Quotient stacks . 59 5.4 Philosophy: quotient stacks and equivariant objects . 60 6 The stack of G-bundles on a curve 61 6.1 Preliminaries . 61 6.1.1 Curves and vector bundles on curves . 61 6.1.2 Recollection of cohomology of coherent sheaves . 61 6.1.3 Stacks which are locally of finite presentation . 62 6.1.4 A criterion for surjectivity . 62 6.1.5 Quot schemes . 62 6.2 The stack of vector bundles on a curve . 63 6.2.1 Families of vector bundles . 63 6.2.2 The atlas . 63 6.3 The stack of G-bundles on a curve . 65 6.3.1 Stacks of G-bundles as mapping stacks . 65 6.3.2 Presentability statements . 66 7 Smooth algebraic stacks 69 7.1 Smooth morphisms . 69 7.2 The cotangent complex . 73 7.2.1 Dual numbers and the tangent space . 73 7.2.2 K¨ahlerdifferentials and basic deformation theory . 76 7.2.3 The cotangent sheaf for smooth Deligne-Mumford stacks . 77 7.2.4 Towards the cotangent complex for smooth algebraic stacks . 78 2 8 Quasi-coherent sheaves 79 8.1 Quasi-coherent sheaves on prestacks . 79 8.1.1 A stacky definition of quasi-coherent sheaves . 80 8.1.2 The category of quasi-coherent sheaves as a limit . 82 8.1.3 Stackification and quasi-coherent sheaves . 83 8.2 BG and G-representations . 84 8.2.1 The Barr-Beck Theorem . 84 8.2.2 Quasi-coherent sheaves on BG: the proof . 87 1 Introduction This introduction (just like the whole course) aims to explain two things: the meaning of the word stack, and its connection with algebraic geometry. We begin by discussing three typical examples of stacks, two of which take place in the category of topological spaces, while the third is of arithmetic nature. 1.1 What is a stack? Glueing of vector bundles The archetypical example of a stack is well-known to every mathematician. In order to describe the data of a rank n vector bundle E on a topological space X one often resorts to a so-called cocycle description. This amounts to choosing an open covering fUigi2I of X, and for each pair (i; j) of indices, a vector bundle automorphism φij of the trivial vector bundle n (Ui \ Uj) × R on the open subset Ui \ Uj. One then hopes to find a vector bundle E on X, together with vector bundle isomorphisms (called local trivializations) n ∼= φi : Ui × R −! EjUi ; such that the identities −1 φij = φi ◦ φj (1) are satisfied on Ui \ Uj. A necessary condition for the existence of (E; (φi)i2I ) is that the cocycle condition φij ◦ φjk = φik (2) is satisfied for every triple (i; j; k). Note that this identity implies for i = j = k that φii = id, and −1 hence for i = k that φji = φij . It is essential to the theory of topological vector bundles that the cocycle condition is also sufficient. Indeed, we can define E by glueing the trivial vector bundles n Ui × R with respect to the isomorphisms φij: let Ee be the disjoint union of the topological spaces n n n Ui × R ; we define an equivalence relation, by declaring (x; v) 2 Ui × R and (y; w) 2 Uj × R as equivalent if x = y and v = φij(w). The topological quotient space will be denoted by E and has a canonical map E / X, and canonical trivializations (φi)i2I , which satisfy the identity (1) above. In modern language we would say that the theory of vector bundles on topological spaces is a stack. Let's be more precise and actually nail down what kind of mathematical object the stack 3 is in this example. It is the following assignment, which associates to an open subset U ⊂ X the groupoid of vector bundles on U. Groupoid is a fancy word for a simple thing. In fact, it simply denotes a category where every morphism is invertible. Hence, the groupoid of vector bundles on U denotes the category of vector bundles on U, where we discard non-invertible morphisms. Remark 1.1. It is possible to formulate a theory of stacks (or rather sheaves of categories), which takes values in honest categories and not just groupoids. However, it will not be necessary for us to pursue that level of generality. As we have seen in the example of vector bundles on topological spaces, we only need vector bundle isomorphisms to describe the glueing data. Hence it is sensible to discard all the other morphisms of vector bundles. The cocycle condition (2) can be elegantly reformulated. This requires the usage of fibre products. Definition 1.2. For continuous maps of topological spaces f : X / Z and g : Y / Z, we denote by X ×Z Y the subspace of X × Y , consisting of tuples (x; y) with f(x) = g(y). We denote by a Y = Ui; i2I and by π : Y / X the canonical map, induced by the inclusions Ui ,! X. Let Ee be the trivial rank n vector bundle on Y . The fibre product Y ×X Y is equivalent to the disjoint union a Ui \ Uj: (i;j)2I2 The cocycle (φij)(i;j)2I2 is captured by an isomorphism of vector bundles on Y ×X Y , namely ∗ ∼= ∗ φ: p2Ee −! p1E:e (3) Here, we use the suggestive notation which denotes the projection to the first component of a fibre product X ×Z Y / X by p1, and similarly for the projection to the second component. Let's pause a second to see that this is indeed equivalent to a collection of automorphisms (φij) of the trivial vector bundles on Ui \ Uj. The trivial vector bundle always pulls back to the trivial vector bundle, no matter which map we considered. Hence, φ is really an automorphism of the trivial vector bundle on Y ×X Y , which is, as remarked above, equivalent to the disjoint union of all the spaces Ui \ Uj. The cocycle condition amounts to the identity of vector bundle isomorphisms on Y ×X Y ×X Y : ∗ ∗ ∗ ∗ ∗ p12φ ◦ p23φ = p13φ: p3Ee / p1E:e (4) The family of local trivializations (φi)i2I amounts to ∗ n π E =∼ Ee = Y × R : (5) We will re-encounter a similar formulation below. 4 Descending vector bundles to quotients One of the first non-trivial vector bundles one encounters in topology is a M¨obiusstrip of infinite width over the circle S1. A rigorous description of this bundle can be given as follows. At first we observe the existence of a homeomorphism 1 ∼ 1 S /µ2 = S ; 1 where µ2 = f1; −1g denotes the group of second order roots of unity, acting on S ⊂ C multi- 1 1 2 plicatively. The homeomorphism is induced by the µ2-invariant map S / S , z 7! z . Although both spaces are homeomorphic, we will distinguish between both sides, and denote the canonical projection by 1 ∼ 1 ∼ 1 π : Y = S / X = S /µ2 = S : 1 1 The M¨obiusbundle E on S can be defined by extending the action of µ2 on S to the total space Ee 1 1 of the trivial rank 1 bundle S × R / R. Namely, we define the action of ζ 2 µ2 on (z; λ) 2 S × R to be ζ · (z; λ) = (ζx; ζλ): 1 ∼ 1 The quotient E/µe 2 will be denoted by E. It defines a rank 1 vector bundle on S /µ2 = X = S . One observes that π∗E =∼ S1 ×R, which should be compared to equation (5). But there are even more similarities with the glueing procedure described in the proceeding subsection. Let's take a look at the fibre products Y ×X Y and Y ×X Y ×X Y .