# Construction of M B, M Dol, M DR

Size: px
Start display at page:

Transcription

1 Construction of M B, M Dol, M DR Hendrik Orem Talbot Workshop, Spring 2011 Contents 1 Some Moduli Space Theory Moduli of Sheaves: Semistability and Boundedness Geometric Invariant Theory Moduli of Semistable Sheaves Λ-Modules The Moduli of Higgs Bundles & Vector Bundles with Flat Connection Moduli of Λ-Modules Moduli of Higgs Bundles Moduli of Vector Bundles with Flat Connection Betti Moduli Space The Betti Moduli Space over Spec C Interlude: Local Systems of Schemes The Relative Betti Moduli Space Bibliography 11 1 Some Moduli Space Theory I will begin this section, following [4], by sketching the general theory needed to construct moduli spaces of sheaves. This will be followed by some comments on Λ-modules, of which vector bundles with flat connection and Higgs bundles are examples. This will set the stage for constructing some of the moduli spaces of interest to us in these notes. 1.1 Moduli of Sheaves: Semistability and Boundedness Throughout, X will be a projective scheme over Spec C, and we fix a very ample invertible sheaf O X (1). The Hilbert polynomial of a coherent sheaf E is the unique polynomial p(e, x) Q[x] such that p(e, n) = dim H 0 (X, E(n)) for all n sufficiently large. We say that a coherent sheaf E on X is of pure dimension d if, for all coherent subsheaves F E, we have that the dimension of the support of E and the dimension of the support 1

2 of F are both equal to d. Support of E here means the set of points where the stalk of E is nonzero. The degree of the Hilbert polynomial of E is the dimension of the support of E. In [4], Simpson defines two different notions of semistability for coherent sheaves. First, if we write p(e, n) = r nd d! + a nd 1 (d 1)! + then we call r = r(e) the rank of E and a(e) the degree of E. If E is locally free (i.e., a vector bundle), then this coincides with the usual notion of degree. We now say that E is µ-semistable if, for all coherent F E, we have a(f)/r(f) a(e)/r(e). We call E p-semistable if instead for sufficiently large n we have p(f, n) r(f) p(e, n) r(e) for all coherent F E. A family of coherent sheaves on X is bounded if there exists some finite type C-scheme T and a coherent sheaf F on T X such that the family of sheaves is contained in {F Spec k(t) X : t T a closed point}. The idea is that constructing moduli spaces for a family of sheaves requires that the set isn t too large ; requiring that the family arise as restrictions of a fixed coherent sheaf imposes this [2]. If X is projective over S, where S is a finite type C-scheme, we say that a p-semistable sheaf E on X/S with Hilbert polynomial P is a coherent sheaf E on X which is flat over S and which, on each fiber X s over a closed point s S, is p-semistable with Hilbert polynomial P and of pure dimension d. The following basic result will be needed below. Theorem 1.1 ([4], Corollary 1.6). The set of p-semistable sheaves on X over S is bounded. 1.2 Geometric Invariant Theory This subsection summarizes the basic ideas of Mumford s geometric invariant theory, as described in [4]. Let Y be a functor from schemes to sets, Y a scheme, and ϕ : Y Y a natural transformation (viewing Y as its functor of points); we say that Y is corepresented by Y if, given any scheme W and natural transformation ψ : Y W, there is a unique map of schemes f : Y W such that ψ = f ϕ. A more general notion, and the one we will use to characterize the moduli spaces below, is the notion of Y universally corepresenting the functor Y : if V Y is a map of schemes, we define the fiber product of functors [V Y Y ](S ) = V (S ) Y (S ) Y (S ). Now say that Y universally corepresents Y if V corepresents V Y Y for all schemes V Y. We say that a map of functors is a local isomorphism if it induces an isomorphism of sheafifications in the étale topology. The importance of this notion for us is that, if there exists a local isomorphism between two functors, then a scheme universally corepresents one functor if and only if it universally corepresents the other; in constructing the moduli space of semistable sheaves, we will first show that the desired moduli functor is locally isomorphic to a functor with a moduli space whose construction is easier. 2

3 Let G be a reductive algebraic group (i.e., trivial unipotent radical) which acts on a scheme Z (for now, take the base scheme to be Spec C). Now define the quotient functor Y to be Y (S ) = Z(S )/G(S ). Given a G-invariant map of schemes ϕ : Z Y, Y is a categorical quotient if it corepresents Y, and a universal categorical quotient if it universally corepresents Y. With G and Z as above, suppose that we additionally have an equivariant line bundle L on G. Given z Z, we call z semistable if, for some n, there exists a G-invariant section f H 0 (Z, L n ) G such that f(z) 0 and {x : f(x) 0} is affine. The following important result will be used in the construction in 1.3. Theorem 1.2 ([4], Proposition 1.11). With the above notation, there exists a universal categorical quotient ϕ : Z ss Y of the semistable points of Z under the action of G. 1.3 Moduli of Semistable Sheaves In these notes, we ll be interested in a moduli space which universally corepresents the following functor: take X to be projective over a finite type C-scheme S. Then M (O X, P ) is the functor which associates to an S-scheme S the set of semistable sheaves on X S of pure dimension d with Hilbert polynomial P, where X is obtained from X by base change to S. Sketching the construction of the moduli space associated to this functor requires some preliminaries. First, the Hilbert scheme Hilb(W, P ) is a scheme which represents the following functor. Fix a coherent sheaf W on X and polynomial P. Then for any σ : S S, the S -valued points of Hilb(W, P ) are the isomorphism classes of quotients on X, σ W F 0 where F is flat over S with Hilbert polynomial P. Moreover, the fiber of Hilb(W, P ) over any closed point s S is Hilb(W s, P ). It is a theorem of Grothendieck that this scheme is projective over S. Let V be a finite dimensional vector space. Then we can see that the group Sl(V ) acts naturally on Hilb(V W, P ). This group action preserves the morphism Hilb(V W, P ) S. Now we return to the functor of interest to us, M (O X, P ). Let X be the scheme obtained from X S via base-change to S. Define W = O X ( N) for a fixed N, and let V = C P (N). The S -valued points of Hilb(V W, P ) are the set of pairs (E, α) where E is a coherent sheaf on X which is flat over S with Hilbert polynomial P and α is a morphism α : V O S H 0 (X /S, E(N)) such that the image generates E(N). Define Q 1 Hilb(V W, P ) to be the subset with the property that E has pure dimension d and is p-semistable; it is a nontrivial result which I will not prove that Q 1 is open. Because the family of p-semistable sheaves on the fibers with Hilbert polynomial P is bounded by Theorem 1.1, we may choose the above N sufficiently large to guarantee the following: 1. Every p-semistable sheaf with Hilbert polynomial P appears as a point of Q 1 ; 3

4 2. Every p-semistable sheaf E with Hilbert Polynomial P has H 0 (X /S, E(N)) locally free over S with rank P (N), where the H 0 (X /S, F) denotes the direct image of the sheaf F on the base; 3. Formation of the H 0 commutes with base change. Given the above choice of N, we now set Q 2 to be the open subset of Q 1 where the map α is an isomorphism. Summarizing, we now have that Q 2 represents the functor which associates to an S-scheme S the set of all (E, α) with E p-semistable on X with Hilbert polynomial P and α : V O S = H 0 (X /S, E(N)). Notice that there is an action of the corresponding functors, i.e., Sl(V ) acts on Q 2. Since Q 2 Q 1, there is a natural map of functors Q 2 M (O X, P ) (given an (E, α) in Q 2 (S ), simply forget α); this map is invariant under the action of Sl (V ). Thus the map descends to a map of quotient functors Q 2/Sl(V ) ϕ M (O X, P ) where the left hand side takes S to the set-theoretic quotient of Q 2 (S ) by Sl (V ). Theorem 1.3 ([4], Theorem 1.21 (1)). The above map ϕ is a local isomorphism. Proof. (Sketch) By definition, we argue étale-locally. Because we may choose local frames for H 0 (X /S, E(N)), the natural transformation Q 2/Gl(V ) M (O X, P ) is a local isomorphism (i.e., the group acts by locally identifying the possible choices of frame, which is certainly étale local). However, in this action the center G m acts trivially on Q 2, so we in fact have an action of PGl(V ) on Q 2. Now note that the maps Gl(V ) PGL(V ) and Sl(V ) PGl(V ) are étale-locally surjective, hence the corresponding map on the quotient Q 2/Sl(V ) Q 2/PGl(V ) is also a local ismorphism. Thus we obtain the following diagram M (O X, P ) Q 2/Gl(V ) Q 2/Sl(V ) Q 2/PGl(V ) where we have that all but the top arrow are local isomorphisms, hence the top arrow is also. From our generalities on geometric invariant theory, we know that in order to construct a scheme M(O X, P ) universally corepresenting M (O X, P ) it thus suffices to construct a scheme which universally corepresents the quotient Q 2/Sl(V ). In order to apply Mumford s results, which will give us a scheme universally corepresenting the quotient functor, we will exhibit Q 2 as the set of semistable of a scheme on which Sl(V ) acts. In particular, let Hilb(V W, P, d) be the closure of the points of Hilb(V W, P ) with the property that the sheaf E is of pure dimension d. By definition, we have Q 2 as a subset of this scheme. The desired fact now follows from a technical and lengthy calculation [4]: 4

5 Theorem 1.4. The scheme Q 2 is the set of semistable points of Hilb(V W, P, d) under the action of Sl(V ). The above results now give us the existence of M(O X, P ) as a scheme via Theorem 1.2. I will close this section by stating some further results, whose proof requires considerable additional work and will not be discussed (see [4], Theorem 1.21). Theorem 1.5. The scheme M(O X, P ) is projective and its points represent equivalence classes of semistable sheaves, where E 1 E 2 when the associated graded sheaves are equal, gr(e 1 ) = gr(e 2 ) (i.e., M(O X, P ) parametrizes Jordan equivalence classes of semistable sheaves). 1.4 Λ-Modules In this section, I will define Λ-modules, which are sheaves of modules over a sheaf of rings of differential operators. Higgs bundles are examples of Λ-modules, and the following section will discuss the construction of the moduli space of Λ-modules in some generality. This follows 2 of [4]. A sheaf of rings of differential operators on X over some base scheme f : X S (taken to be Noetherian over C) is a sheaf of O X -algebras Λ over X together with a filtration Λ 0 Λ 1 such that: 1. Λ = i Λ i ; 2. Λ i Λ j Λ i+j ; 3. The image of the given map O X Λ is precisely Λ 0 ; 4. f 1 (O S ) O X is contained in Z(Λ); 5. The left and right O X -module structures on the quotients Gr i (Λ) coincide; 6. All of the sheaves Gr i (Λ) are coherent; 7. The sheaf Gr(Λ) = i=0gr i (Λ) is generated by Gr 1 (Λ) in the sense that Gr 1 (Λ) OX OX Gr 1 (Λ) Gr i (Λ). Now we can define a Λ-module as a sheaf E of left Λ-modules on X which is coherent as a sheaf of O X -modules. There are two examples of central importance to these notes, described below. We may view vector bundles with flat connection as Λ-modules under appropriate circumstances: assume that f : X S is smooth, and take Λ = D X/S, the usual sheaf of relative differentials. The order filtration on differentials is known to satisfy the properties required of a sheaf of rings of differential operators (indeed, the axioms are modeled on D). It is a basic result of the theory of D-modules that any sheaf of O X -coherent modules E over Λ will be locally free of finite rank, i.e., the sheaf of sections of a vector bundle. The action 5

6 of Λ defines a map from differential operators to endomorphisms of E such that 2 = 0, and thus a flat connection [1]. A related choice of Λ yields Higgs bundles as an example of Λ-modules: let Λ = Λ Higgs denote Sym (T (X/S)). This is related to the previous choice in that, for smooth varieties over C, Λ Higgs is the associated graded algebra of D X/S above. Now we define Λ-modules to be Hitchin pairs (E, ϕ) where E is an O X -coherent sheaf and ϕ : E Ω X/S E has the additional property that, when the matrices defining ϕ are written out in a local trivialization, they pairwise commute (i.e., if ϕ = ϕ i dz i in local coordinates z i, then the ϕ i commute [3]). This latter condition is typically written ϕ ϕ = 0. ϕ is sometimes called a Higgs field. If we impose the additional condition that E is locally free, the above Hitchen pair is called a Higgs bundle. The realization of these as Λ-modules will be described when we address their moduli. 2 The Moduli of Higgs Bundles & Vector Bundles with Flat Connection I will begin by discussing the moduli space of Λ-modules, as in [4]. As vector bundles with flat connection and Higgs bundles are both examples of Λ-modules, this will give us existence of both of the moduli spaces. 2.1 Moduli of Λ-Modules Consider X projective over S, where S is finite type over C, and denote by X s = X S s the fiber over a geometric point s S. Let Λ be some sheaf of differential operators on X and Λ s the corresponding sheaf of differential operators on X s obtained via base change. We call a Λ-module E p-semistable (or respectively µ-semistable) if 1. E is flat over S; 2. the restrictions to geometric fibers E s are of pure dimension d and p-semistable (resp. µ-semistable) Λ s -modules; 3. each restriction E s has the same Hilbert polynomial. A crucial fact in the construction of the moduli spaces of interest, as in the construction of the moduli of general semistable sheaves, is the following boundedness result. Theorem 2.1 ([4], Proposition 3.5). The set of µ-semistable Λ s -modules on geometric fibers X s with a fixed Hilbert polynomial P is bounded. The construction here is very similar to that given in 1.3. Again we begin with a Hilbert scheme with an action of Sl(V ), and construct the desired scheme as the set of semi-stable points of a subscheme. Theorem 2.2. Fix a polynomial P and let N be sufficiently large. There exists a scheme Q, quasi-projective over S, which represents the functor that associates to an S-scheme S 6

7 the set of isomorphism classes of pairs (E, α) where E is p-semistable Λ-module with Hilbert polynomial P on X and α is a map α : (O S ) P (N) H 0 (X /S, E(N)). The idea of the proof is to begin with the Hilbert scheme of quotients Λ k OX O X ( N) P (N) E 0. The procedure of finding appropriate subschemes of this, eventually obtaining the desired Q, is similar to (but much more involved than) what happened in 1.3, so I will omit it. Let M (Λ, P ) denote the functor which associates to an S-scheme S the set of isomorphism classes of p-semistable Λ -modules on X over S with Hilbert polynomial P. Just as in 1.3, one can show that the points of Q are semistable for the action of Sl(V ), so we obtain the following. Theorem 2.3 ([4], Theorem 4.7). The universal categorical quotient M(Λ, P ) = Q/Sl(V ) exists as a scheme and universally corepresents M (Λ, P ). The scheme is a quasi-projective variety and the geometric points represent the Jordan equivalence classes of p-semistable Λ-modules with Hilbert polynomial P on fibers X s. 2.2 Moduli of Higgs Bundles Let Λ = Λ Higgs. Before proceeding with the following description of Higgs bundles. Theorem 2.4 ([5], Lemma 6.5). A Hitchin pair on X over S is the same thing as an O X - coherent Λ-module on X. In particular, a Higgs bundle on X over S is the same thing as an O X -coherent, locally free Λ-module on X. Then Theorem 2.3 immediately gives us that the functor M Higgs (X/S, P ), associating to an S-scheme S the set of isomorphism classes of p-semistable Hitchin pairs on X over S with Hilbert polynomial P is universally corepresented by a scheme M Higgs (X/S, P ) = M(Λ Higgs, P ). Its points parametrize Jordan equivalence classes of p-semistable Hitchin pairs with Hilbert polynomial P on fibers X s. Now let s consider the Dolbeault moduli space: let M Dol (X/S, n) be the functor which associates to an S-scheme S the set of isomorphism classes of p-semistable Higgs bundles on X over S with vanishing Chern classes and Hilbert polynomial np 0, where P 0 is the Hilbert polynomial of O X. One can prove that the assumption that the Higgs bundles were locally free was unnecessary (i.e., if we had defined M Dol (X/S, n) to simply give us Hitchin pairs rather than Higgs bundles, we would have the same functor). From this description we can see that there exists a scheme M Dol (X/S, n) M Higgs (X/S, np 0 ) universally corepresenting M Dol (X/S, n). Its points correspond to direct sums of µ-stable Higgs bundles with vanishing rational Chern classes on the fibers (µ-stable means that we take the inequality defining µ- semistability to be strict); see [5], Proposition 6.6 and Corollary 6.7 7

8 2.3 Moduli of Vector Bundles with Flat Connection The description of vector bundles with flat connection as Λ-modules given in 1.4 allows us to immediately apply our results about moduli of Λ-modules for Λ = D X/S. We thus have the desired theorem about the functor M DR (X/S, n), which assigns to an S-scheme S the set of isomorphism classes of vector bundles with flat connection on X /S of rank n: Theorem 2.5 ([5], Theorem 6.13). If X is smooth and projective over S, then there is a scheme M DR (X/S, n), quasi-projective over S, which universally corepresents M DR (X/S, n). 3 Betti Moduli Space In this section, I will sketch the construction of the Betti moduli space associated to representations of π 1 (X an, x). I will follow the treatment of [5]. 3.1 The Betti Moduli Space over Spec C Given a finitely generated group Γ, we will consider the set of complex representations of degree n: R(Γ, n) = {Hom (Γ, Gl(n, C))}. I claim that this has the structure of a scheme, as follows. If we write Γ as generated by γ 1,..., γ k modulo the relations W, then clearly R(Γ, n) is the subset of the k-fold product of Gl(n, C) consisting of those (m 1,..., m k ) such that r(m 1,..., m k ) = 1 for each r W. Thus we have a Zariski-closed subset of the affine scheme Gl(n, C) Gl(n, C), hence is also an affine scheme. Recall that the Jordan-Hölder Theorem states that the quotients in a composition series of a representation do not depend on the choice of composition series. Thus it makes sense to define two representations to be Jordan equivalent if their associated graded representations are isomorphic, i.e., if they have the same sets of composition factors. Theorem 3.1. There exists a universal categorical quotient R(Γ, n) M(Γ, n) under the action of Gl(n, C). M(Γ, n) is an affine scheme of finite type over C, and its closed points correspond to the Jordan equivalence classes of representations. The construction, though I will not prove this statement, is given by letting M(Γ, n) = Spec B, where B is the ring of invariants inside the coordinate ring of R(Γ, n). Now if we consider X a connected, smooth, projective variety over C and choose x X, then we can form the Betti representation space R B (X, x, n) = R(Γ, n) where Γ = π 1 (X an, x). The Betti moduli space, M B (X, n) is defined analogously. It does not depend on the choice of basepoint, as change of basepoint corresponds to conjugation, compatible with the action of Gl(n, C). 8

9 3.2 Interlude: Local Systems of Schemes A local system of schemes Z over a topological space T is a functor from the category of schemes over C to the category of sheaves of sets over T satisfying the following condition: there is a covering α U α = T so that on each U α, the action of Z on open subsets V of U α is represented by a scheme Z(V ) (that is, for fixed V U α, Z(V ) is a functor from schemes to sets, and we require this to be representable), and if W V U α, the restriction map Z(V ) Z(W ) is an isomorphism. We can thus be assured that the stalks Z t = lim t V Z(V ) will be schemes. With basic point-set and covering space topology results, one may prove the following. I will sketch the direction which will be used later. Theorem 3.2 ([5], Lemma 6.2). π 1 (T, t) acts on Z t by C-scheme automorphisms. If T is connected and locally simply connected, then Z Z t defines an equivalence of categories between the category of local systems of schemes over T and the category of schemes with an action of π 1 (T, t). Proof. (Sketch) If we have π 1 (T, t) acting on a scheme Z t and T has a universal cover T, then form the constant local system Z on T given by Z(U)(S) = Z t (S) for U open and connected. For U open now in the base T, we set Z(U)(S) to be the invariants under the action of π 1 (T, t) inside Z(Ũ)(S); here Ũ denotes the inverse image of U under the covering map. All restriction maps are defined to be the identity, so we obtain a local system of schemes Z on T, as desired. We may define the total analytic space Z (an) of a local system of schemes as follows. Given a local system of schemes Z on T, choose an open covering of T = T i such that Z Ti is a constant local system of stalk Z i. For each overlap T i T j, we have an isomorphism Z i = Zj (obtained via comparison with the stalk of Z at a point in the intersection); these isomorphisms obey the cocycle compatibility condition with respect to triple overlaps, so we may form Z (an) by gluing together Zi an complex analytic space corresponding to Z i ). 3.3 The Relative Betti Moduli Space T i along these relations (here Z an i denotes the We now generalize the construction in 3.1 to the situation where X is smooth and projective over S, a finite type C-scheme. Let S and the fibers X s be connected, and choose basepoints t S and x X t. Denote by Γ the group π 1 (Xt an, x). I will describe an action of π 1 (S an, t) on the scheme M B (Γ, n) which, by Theorem 3.2, gives us a local system of schemes over S an ; this local system will give us the desired relative moduli space. Let Aut(Γ) denote the group of automorphisms, Inn(Γ) its inner automorphisms, and Out(Γ) = Aut(Γ)/Inn(Γ). Define a map π 1 (S an, t) Out(Γ) as follows. First note that the map f an corresponding to f : X S is a fibration. Given a loop σ : [0, 1] S an based at 9

10 t S, we obtain another fibration σ (X an ) over [0, 1]: [0, 1] X an = σ (X an ) X an X t [0, 1] f an σ S an t It is a fibration over [0, 1], hence trivial. As σ is a loop based at t, the fibers over 0 and 1 give two copies of X t {point}, and hence a self-homeomorphism of X t. We thus get an action of π 1 (S an, t) on π 1 (X t, x) which is well-defined up to inner automorphism (the self-homeomorphism of X t might move the basepoint x, but change of basepoint corresponds to an inner automorphism). This gives the desired map π 1 (S an, t) Out(Γ). Notice that Aut(Γ) acts on R B (X t, x, n) for each n, and that this descends to an action of Inn(Γ) on M B (X t, n), as we obtain M B from R B via modding out by conjugation. Combining this with the above observation, we have an action of π 1 (S an, t) on M B (Γ, n), which, by Theorem 3.2, gives a local system of schemes M B (X/S, n) over S an. We call this local system of schemes the relative version of the Betti moduli space. It has the property (justifying its status as a moduli space) that M B (X/S, n) s = M B (X s, n), i.e., the stalk of this local system of schemes over s S is the (usual, non-relative) Betti moduli space of the fiber X s. 10

11 References [1] R. Hotta, K. Takeuchi, and T. Tanisaki. D-modules, perverse sheaves, and representation theory. Springer, [2] D. Huybrechts and M. Lehn. The geometry of moduli spaces of sheaves. Cambridge Univ Pr, [3] C.T. Simpson. Higgs bundles and local systems. Publications mathématiques, 75:5 95, [4] C.T. Simpson. Moduli of representations of the fundamental group of a smooth projective variety I. Publications mathématiques de l IHES, 79(1):47 129, [5] C.T. Simpson. Moduli of representations of the fundamental group of a smooth projective variety. II. Publications Mathématiques de l IHÉS, 80(1):5 79,

### We can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr )

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a pre-specified embedding in a projective space. However, an ample line bundle

### 1. Algebraic vector bundles. Affine Varieties

0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

### 1 Notations and Statement of the Main Results

An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main

### Math 248B. Applications of base change for coherent cohomology

Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the so-called cohomology and base change theorem: Theorem 1.1 (Grothendieck).

### h M (T ). The natural isomorphism η : M h M determines an element U = η 1

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 7 2.3. Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition 2.15. Let M : Sch Set be a moduli

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves

### Algebraic varieties and schemes over any scheme. Non singular varieties

Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two

### HARTSHORNE EXERCISES

HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing

### where Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset

Classification of quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring

### Chern classes à la Grothendieck

Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces

### Smooth morphisms. Peter Bruin 21 February 2007

Smooth morphisms Peter Bruin 21 February 2007 Introduction The goal of this talk is to define smooth morphisms of schemes, which are one of the main ingredients in Néron s fundamental theorem [BLR, 1.3,

### Synopsis of material from EGA Chapter II, 5

Synopsis of material from EGA Chapter II, 5 5. Quasi-affine, quasi-projective, proper and projective morphisms 5.1. Quasi-affine morphisms. Definition (5.1.1). A scheme is quasi-affine if it is isomorphic

### Quotient Stacks. Jacob Gross. Lincoln College. 8 March 2018

Quotient Stacks Jacob Gross Lincoln College 8 March 2018 Abstract These are notes from a talk on quotient stacks presented at the Reading Group on Algebraic Stacks; meeting weekly in the Quillen Room of

### NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY

NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY RUNE HAUGSENG The aim of these notes is to define flat and faithfully flat morphisms and review some of their important properties, and to define the fpqc

### Lectures on Galois Theory. Some steps of generalizations

= Introduction Lectures on Galois Theory. Some steps of generalizations Journée Galois UNICAMP 2011bis, ter Ubatuba?=== Content: Introduction I want to present you Galois theory in the more general frame

### Representations and Linear Actions

Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category

### Math 249B. Nilpotence of connected solvable groups

Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C

### THE KEEL MORI THEOREM VIA STACKS

THE KEEL MORI THEOREM VIA STACKS BRIAN CONRAD 1. Introduction Let X be an Artin stack (always assumed to have quasi-compact and separated diagonal over Spec Z; cf. [2, 1.3]). A coarse moduli space for

### LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS

LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 16. Symplectic resolutions of µ 1 (0)//G and their deformations 16.1. GIT quotients. We will need to produce a resolution of singularities for C 2n

### Synopsis of material from EGA Chapter II, 4. Proposition (4.1.6). The canonical homomorphism ( ) is surjective [(3.2.4)].

Synopsis of material from EGA Chapter II, 4 4.1. Definition of projective bundles. 4. Projective bundles. Ample sheaves Definition (4.1.1). Let S(E) be the symmetric algebra of a quasi-coherent O Y -module.

### MATH 233B, FLATNESS AND SMOOTHNESS.

MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)

### THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3

THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3 KAZUNORI NAKAMOTO AND TAKESHI TORII Abstract. There exist 26 equivalence classes of k-subalgebras of M 3 (k) for any algebraically closed field

### AN INTRODUCTION TO AFFINE SCHEMES

AN INTRODUCTION TO AFFINE SCHEMES BROOKE ULLERY Abstract. This paper gives a basic introduction to modern algebraic geometry. The goal of this paper is to present the basic concepts of algebraic geometry,

### The Hilbert-Mumford Criterion

The Hilbert-Mumford Criterion Klaus Pommerening Johannes-Gutenberg-Universität Mainz, Germany January 1987 Last change: April 4, 2017 The notions of stability and related notions apply for actions of algebraic

### Mathematics 7800 Quantum Kitchen Sink Spring 2002

Mathematics 7800 Quantum Kitchen Sink Spring 2002 4. Quotients via GIT. Most interesting moduli spaces arise as quotients of schemes by group actions. We will first analyze such quotients with geometric

### ABSTRACT NONSINGULAR CURVES

ABSTRACT NONSINGULAR CURVES Affine Varieties Notation. Let k be a field, such as the rational numbers Q or the complex numbers C. We call affine n-space the collection A n k of points P = a 1, a,..., a

### QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS

QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS SAM RASKIN 1. Differential operators on stacks 1.1. We will define a D-module of differential operators on a smooth stack and construct a symbol map when

### The moduli stack of vector bundles on a curve

The moduli stack of vector bundles on a curve Norbert Hoffmann norbert.hoffmann@fu-berlin.de Abstract This expository text tries to explain briefly and not too technically the notions of stack and algebraic

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### The derived category of a GIT quotient

September 28, 2012 Table of contents 1 Geometric invariant theory 2 3 What is geometric invariant theory (GIT)? Let a reductive group G act on a smooth quasiprojective (preferably projective-over-affine)

### Motivic integration on Artin n-stacks

Motivic integration on Artin n-stacks Chetan Balwe Nov 13,2009 1 / 48 Prestacks (This treatment of stacks is due to B. Toën and G. Vezzosi.) Let S be a fixed base scheme. Let (Aff /S) be the category of

### APPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP

APPENDIX 2: AN INTRODUCTION TO ÉTALE COHOMOLOGY AND THE BRAUER GROUP In this appendix we review some basic facts about étale cohomology, give the definition of the (cohomological) Brauer group, and discuss

### Cohomology jump loci of local systems

Cohomology jump loci of local systems Botong Wang Joint work with Nero Budur University of Notre Dame June 28 2013 Introduction Given a topological space X, we can associate some homotopy invariants to

### SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN

SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN Abstract. Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free O X -bimodule of rank

### SUMMER COURSE IN MOTIVIC HOMOTOPY THEORY

SUMMER COURSE IN MOTIVIC HOMOTOPY THEORY MARC LEVINE Contents 0. Introduction 1 1. The category of schemes 2 1.1. The spectrum of a commutative ring 2 1.2. Ringed spaces 5 1.3. Schemes 10 1.4. Schemes

### ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset

### An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations

An overview of D-modules: holonomic D-modules, b-functions, and V -filtrations Mircea Mustaţă University of Michigan Mainz July 9, 2018 Mircea Mustaţă () An overview of D-modules Mainz July 9, 2018 1 The

### ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES

ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasi-coherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open

### 0.1 Spec of a monoid

These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.

### arxiv:math/ v3 [math.ag] 10 Nov 2006

Construction of G-Hilbert schemes Mark Blume arxiv:math/0607577v3 [math.ag] 10 Nov 2006 Abstract One objective of this paper is to provide a reference for certain fundamental constructions in the theory

### PERVERSE SHEAVES. Contents

PERVERSE SHEAVES SIDDHARTH VENKATESH Abstract. These are notes for a talk given in the MIT Graduate Seminar on D-modules and Perverse Sheaves in Fall 2015. In this talk, I define perverse sheaves on a

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES

CHEAT SHEET: PROPERTIES OF MORPHISMS OF SCHEMES BRIAN OSSERMAN The purpose of this cheat sheet is to provide an easy reference for definitions of various properties of morphisms of schemes, and basic results

### Néron models of abelian varieties

Néron models of abelian varieties Matthieu Romagny Summer School on SGA3, September 3, 2011 Abstract : We present a survey of the construction of Néron models of abelian varieties, as an application of

### Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti

Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY E-mail address: alex.massarenti@sissa.it These notes collect a series of

### Coherent sheaves on elliptic curves.

Coherent sheaves on elliptic curves. Aleksei Pakharev April 5, 2017 Abstract We describe the abelian category of coherent sheaves on an elliptic curve, and construct an action of a central extension of

### MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f

### Vector Bundles vs. Jesko Hüttenhain. Spring Abstract

Vector Bundles vs. Locally Free Sheaves Jesko Hüttenhain Spring 2013 Abstract Algebraic geometers usually switch effortlessly between the notion of a vector bundle and a locally free sheaf. I will define

### The Picard Scheme and the Dual Abelian Variety

The Picard Scheme and the Dual Abelian Variety Gabriel Dorfsman-Hopkins May 3, 2015 Contents 1 Introduction 2 1.1 Representable Functors and their Applications to Moduli Problems............... 2 1.2 Conditions

### Lecture 9 - Faithfully Flat Descent

Lecture 9 - Faithfully Flat Descent October 15, 2014 1 Descent of morphisms In this lecture we study the concept of faithfully flat descent, which is the notion that to obtain an object on a scheme X,

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 25 RAVI VAKIL CONTENTS 1. Quasicoherent sheaves 1 2. Quasicoherent sheaves form an abelian category 5 We began by recalling the distinguished affine base. Definition.

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24

FOUNDATIONS OF ALGEBRAIC GEOMETR CLASS 24 RAVI VAKIL CONTENTS 1. Normalization, continued 1 2. Sheaf Spec 3 3. Sheaf Proj 4 Last day: Fibers of morphisms. Properties preserved by base change: open immersions,

### COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY

COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski

### Notes on p-divisible Groups

Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete

### MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY

MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY VICTORIA HOSKINS Abstract In this course, we study moduli problems in algebraic geometry and the construction of moduli spaces using geometric invariant theory.

### COMPLEX ALGEBRAIC SURFACES CLASS 9

COMPLEX ALGEBRAIC SURFACES CLASS 9 RAVI VAKIL CONTENTS 1. Construction of Castelnuovo s contraction map 1 2. Ruled surfaces 3 (At the end of last lecture I discussed the Weak Factorization Theorem, Resolution

### Two constructions of the moduli space of vector bundles on an algebraic curve

Two constructions of the moduli space of vector bundles on an algebraic curve Georg Hein November 17, 2003 Abstract We present two constructions to obtain the coarse moduli space of semistable vector bundles

### GALOIS DESCENT AND SEVERI-BRAUER VARIETIES. 1. Introduction

GALOIS DESCENT AND SEVERI-BRAUER VARIETIES ERIC BRUSSEL CAL POLY MATHEMATICS 1. Introduction We say an algebraic object or property over a field k is arithmetic if it becomes trivial or vanishes after

### MIXED HODGE MODULES PAVEL SAFRONOV

MIED HODGE MODULES PAVEL SAFRONOV 1. Mixed Hodge theory 1.1. Pure Hodge structures. Let be a smooth projective complex variety and Ω the complex of sheaves of holomorphic differential forms with the de

### MODULI TOPOLOGY. 1. Grothendieck Topology

MODULI TOPOLOG Abstract. Notes from a seminar based on the section 3 of the paper: Picard groups of moduli problems (by Mumford). 1. Grothendieck Topology We can define a topology on any set S provided

### AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES

AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES YEHAO ZHOU Conventions In this lecture note, a variety means a separated algebraic variety over complex numbers, and sheaves are C-linear. 1.

### MA 206 notes: introduction to resolution of singularities

MA 206 notes: introduction to resolution of singularities Dan Abramovich Brown University March 4, 2018 Abramovich Introduction to resolution of singularities 1 / 31 Resolution of singularities Let k be

### Introduction to Chiral Algebras

Introduction to Chiral Algebras Nick Rozenblyum Our goal will be to prove the fact that the algebra End(V ac) is commutative. The proof itself will be very easy - a version of the Eckmann Hilton argument

### Systems of linear equations. We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K,

Systems of linear equations We start with some linear algebra. Let K be a field. We consider a system of linear homogeneous equations over K, f 11 t 1 +... + f 1n t n = 0, f 21 t 1 +... + f 2n t n = 0,.

### ALGEBRAIC GROUPS JEROEN SIJSLING

ALGEBRAIC GROUPS JEROEN SIJSLING The goal of these notes is to introduce and motivate some notions from the theory of group schemes. For the sake of simplicity, we restrict to algebraic groups (as defined

### PICARD GROUPS OF MODULI PROBLEMS II

PICARD GROUPS OF MODULI PROBLEMS II DANIEL LI 1. Recap Let s briefly recall what we did last time. I discussed the stack BG m, as classifying line bundles by analyzing the sense in which line bundles may

### DERIVED CATEGORIES OF COHERENT SHEAVES

DERIVED CATEGORIES OF COHERENT SHEAVES OLIVER E. ANDERSON Abstract. We give an overview of derived categories of coherent sheaves. [Huy06]. Our main reference is 1. For the participants without bacground

### Tunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society

unisian Journal of Mathematics an international publication organized by the unisian Mathematical Society Ramification groups of coverings and valuations akeshi Saito 2019 vol. 1 no. 3 msp msp UNISIAN

### Lecture 6: Etale Fundamental Group

Lecture 6: Etale Fundamental Group October 5, 2014 1 Review of the topological fundamental group and covering spaces 1.1 Topological fundamental group Suppose X is a path-connected topological space, and

### Infinite root stacks of logarithmic schemes

Infinite root stacks of logarithmic schemes Angelo Vistoli Scuola Normale Superiore, Pisa Joint work with Mattia Talpo, Max Planck Institute Brown University, May 2, 2014 1 Let X be a smooth projective

### Section Higher Direct Images of Sheaves

Section 3.8 - Higher Direct Images of Sheaves Daniel Murfet October 5, 2006 In this note we study the higher direct image functors R i f ( ) and the higher coinverse image functors R i f! ( ) which will

### NOTES ON PROCESI BUNDLES AND THE SYMPLECTIC MCKAY EQUIVALENCE

NOTES ON PROCESI BUNDLES AND THE SYMPLECTIC MCKAY EQUIVALENCE GUFANG ZHAO Contents 1. Introduction 1 2. What is a Procesi bundle 2 3. Derived equivalences from exceptional objects 4 4. Splitting of the

### Logarithmic geometry and moduli

Logarithmic geometry and moduli Lectures at the Sophus Lie Center Dan Abramovich Brown University June 16-17, 2014 Abramovich (Brown) Logarithmic geometry and moduli June 16-17, 2014 1 / 1 Heros: Olsson

### GEOMETRIC INVARIANT THEORY AND SYMPLECTIC QUOTIENTS

GEOMETRIC INVARIANT THEORY AND SYMPLECTIC QUOTIENTS VICTORIA HOSKINS 1. Introduction In this course we study methods for constructing quotients of group actions in algebraic and symplectic geometry and

### Introduction and preliminaries Wouter Zomervrucht, Februari 26, 2014

Introduction and preliminaries Wouter Zomervrucht, Februari 26, 204. Introduction Theorem. Serre duality). Let k be a field, X a smooth projective scheme over k of relative dimension n, and F a locally

### Algebraic Varieties. Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra

Algebraic Varieties Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra Algebraic varieties represent solutions of a system of polynomial

### PART II.1. IND-COHERENT SHEAVES ON SCHEMES

PART II.1. IND-COHERENT SHEAVES ON SCHEMES Contents Introduction 1 1. Ind-coherent sheaves on a scheme 2 1.1. Definition of the category 2 1.2. t-structure 3 2. The direct image functor 4 2.1. Direct image

### DESCENT THEORY (JOE RABINOFF S EXPOSITION)

DESCENT THEORY (JOE RABINOFF S EXPOSITION) RAVI VAKIL 1. FEBRUARY 21 Background: EGA IV.2. Descent theory = notions that are local in the fpqc topology. (Remark: we aren t assuming finite presentation,

### HYPERSURFACES IN PROJECTIVE SCHEMES AND A MOVING LEMMA

HYPERSURFACES IN PROJECTIVE SCHEMES AND A MOVING LEMMA OFER GABBER, QING LIU, AND DINO LORENZINI Abstract. Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique

### CHAPTER 1. AFFINE ALGEBRAIC VARIETIES

CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the

### BEZOUT S THEOREM CHRISTIAN KLEVDAL

BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this

### Deformation Theory. Atticus Christensen. May 13, Undergraduate Honor s Thesis

Deformation Theory Undergraduate Honor s Thesis Atticus Christensen May 13, 2015 1 Contents 1 Introduction 3 2 Formal schemes 3 3 Infinitesimal deformations 7 4 Scheme with no deformation to characteristic

### ALGEBRAIC GROUPS J. WARNER

ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic

### Logarithmic geometry and rational curves

Logarithmic geometry and rational curves Summer School 2015 of the IRTG Moduli and Automorphic Forms Siena, Italy Dan Abramovich Brown University August 24-28, 2015 Abramovich (Brown) Logarithmic geometry

### Geometric motivic integration

Université Lille 1 Modnet Workshop 2008 Introduction Motivation: p-adic integration Kontsevich invented motivic integration to strengthen the following result by Batyrev. Theorem (Batyrev) If two complex

### BASIC MODULI THEORY YURI J. F. SULYMA

BASIC MODULI THEORY YURI J. F. SULYMA Slogan 0.1. Groupoids + Sites = Stacks 1. Groupoids Definition 1.1. Let G be a discrete group acting on a set. Let /G be the category with objects the elements of

### On the Hitchin morphism in positive characteristic

On the Hitchin morphism in positive characteristic Yves Laszlo Christian Pauly July 10, 2003 Abstract Let X be a smooth projective curve over a field of characteristic p > 0. We show that the Hitchin morphism,

### 1 Existence of the Néron model

Néron models Setting: S a Dedekind domain, K its field of fractions, A/K an abelian variety. A model of A/S is a flat, separable S-scheme of finite type X with X K = A. The nicest possible model over S

### Lecture 2 Sheaves and Functors

Lecture 2 Sheaves and Functors In this lecture we will introduce the basic concept of sheaf and we also will recall some of category theory. 1 Sheaves and locally ringed spaces The definition of sheaf

### arxiv: v1 [math.ra] 5 Feb 2015

Noncommutative ampleness from finite endomorphisms D. S. Keeler Dept. of Mathematics, Miami University, Oxford, OH 45056 arxiv:1502.01668v1 [math.ra] 5 Feb 2015 Abstract K. Retert Dept. of Mathematics,

### NOTES ON THE CONSTRUCTION OF THE MODULI SPACE OF CURVES

NOTES ON THE CONSTRUCTION OF THE MODULI SPACE OF CURVES DAN EDIDIN The purpose of these notes is to discuss the problem of moduli for curves of genus g 3 1 and outline the construction of the (coarse)

### Theta divisors and the Frobenius morphism

Theta divisors and the Frobenius morphism David A. Madore Abstract We introduce theta divisors for vector bundles and relate them to the ordinariness of curves in characteristic p > 0. We prove, following

### ON DEGENERATIONS OF MODULI OF HITCHIN PAIRS

ELECRONIC RESEARCH ANNOUNCEMENS IN MAHEMAICAL SCIENCES Volume 20, Pages 105 110 S 1935-9179 AIMS (2013) doi:10.3934/era.2013.20.105 ON DEGENERAIONS OF MODULI OF HICHIN PAIRS V. BALAJI, P. BARIK, AND D.S.

### INTRODUCTION TO GEOMETRIC INVARIANT THEORY

INTRODUCTION TO GEOMETRIC INVARIANT THEORY JOSÉ SIMENTAL Abstract. These are the expanded notes for a talk at the MIT/NEU Graduate Student Seminar on Moduli of sheaves on K3 surfaces. We give a brief introduction

### BRIAN OSSERMAN. , let t be a coordinate for the line, and take θ = d. A differential form ω may be written as g(t)dt,

CONNECTIONS, CURVATURE, AND p-curvature BRIAN OSSERMAN 1. Classical theory We begin by describing the classical point of view on connections, their curvature, and p-curvature, in terms of maps of sheaves

### VERY STABLE BUNDLES AND PROPERNESS OF THE HITCHIN MAP

VERY STABLE BUNDLES AND PROPERNESS OF THE HITCHIN MAP CHRISTIAN PAULY AND ANA PEÓN-NIETO Abstract. Let X be a smooth complex projective curve of genus g 2 and let K be its canonical bundle. In this note

### Azumaya Algebras. Dennis Presotto. November 4, Introduction: Central Simple Algebras

Azumaya Algebras Dennis Presotto November 4, 2015 1 Introduction: Central Simple Algebras Azumaya algebras are introduced as generalized or global versions of central simple algebras. So the first part

### 1 Flat, Smooth, Unramified, and Étale Morphisms

1 Flat, Smooth, Unramified, and Étale Morphisms 1.1 Flat morphisms Definition 1.1. An A-module M is flat if the (right-exact) functor A M is exact. It is faithfully flat if a complex of A-modules P N Q