/ Roman Power Operations and Central Maps in Representation Theory by Gus Lonergan Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2018 Massachusetts Institute of Technology 2018. All rights reserved. A uthor..... Department of Mathematics April 27, 2018 Certified by Accepted by. Signature redacted fv Signature redacted Bezrukavnikov Professor of Mathematics Thesis Supervisor *-J William P. Minicozzi II Chairman, Department Committee on Graduate Theses MASSACHUSETS INSTITUTE OF TECHNOLOGY MAY 3 0 2018 LIBRARIES ARIUE
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Power Operations and Central Maps in Representation Theory by Gus Lonergan Submitted to the Department of Mathematics on April 27, 2018, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract The theme of this thesis is the novel application of techniques of algebraic topology (specifically, Steenrod's operations and Smith's localization theory) to representation theory (especially in the context of the geometric Satake equivalence). In Chapter 2, we use Steenrod's construction to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We also demonstrate the corresponding result for the K-theoretic version of the quantum Coulomb branch. In Chapter 3, we develop the theory of parity sheaves with coefficients in the Tate spectrum, and use it to give a geometric construction of the Frobenius-contraction functor. In Chapter 4, we discuss some related results, including a geometric construction of the Frobenius twist functor, and also discuss future directions of research. The content of Chapter 3 is joint work with S. Leslie. Thesis Supervisor: Roman Bezrukavnikov Title: Professor of Mathematics 3
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Acknowledgments I would like to thank my advisor, Roman Bezrukavnikov, for teaching me so much and supporting me so patiently for the last five years. I would also like to thank the many other mathematicians I have had the pleasure to communicate with during this period, including P. Achar, D. Ben-Zvi, T. Braden, A. Braverman, V. Drinfeld, W. Dwyer, P. Etingof, V. Ginzburg, S. Gunningham, N. Harman, D. Juteau, S. Leslie, I. Loseu, G. Lusztig, S. Makisumi, C. Mautner, H. R. Miller, D. Nadler, V. Nandakumar, V. Ostrik, S. Raskin, K. Vilonen, M. Viscardi, D. Vogan, G. Williamson, Z. Yun. Interacting with you has been fantastically stimulating and encouraging, and I am very grateful for your time and interest. Before coming to MIT, I was inspired to do mathematics by the following people (in reverse chronological order): I. Grojnowski, I. Smith, L. Scott, J. Evans, J. Saxl, N. Turner, A. Kingston, C. Lonergan. Thank you for believing in me. Finally, thank you to C. Lonergan (again!), P. Lonergan, H. Lonergan and J. Louveau for your love: it means everything to me. 5
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. Contents 1 Introduction 11.... 2 Steenrod operators, the Coulomb branch and the Frobenius twist 2.1 Introduction............................... 2.1.1 2.1.2..d..onstructin......................... 2.1.3....... t..... 2.1.4..unctors...... 2.2 Steenro d's construction............... 2.2.1 O verview.................. 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 Steenrod's construction........... Six functors................ Tate's construction............. Borel-Moore homology........... Steenrod operations............. Artin-Schreier................ 2.3 Coulon b branch.................. 2.3.1 Prelude: Frobenius-constant quantizations 2.3.2 Formal neighborhoods.......... 15 15 2.3.3 Global groups; pro-smoothness. 2.3.4 2.3.5 2.3.6 Beilinson-Drinfeld Grassmannians; reasonableness Jet bundles; placidity................ Equivariance.................... 7
............ 2.3.7 Dimension theories... 2.3.8 Notational remark... 2.3.9 Borel-Moore homology. 2.3.10 The branch....... 2.3.11 The central map.... 2.3.12 Linearity......... 2.3.13 Centrality........ 2.3.14 Completion of proof.. 2.3.15 Closing remarks..... 2.4 K-theoretic version....... 2.4.1 K-theory and K-homology 2.4.2 Adams operations.... 2.4.3 Proof of Theorem 2.4.1. 51 55 56 65 67 70 71 78 80 82 82 87 89.......... 3 Parity sheaves and Sn 3.1 Introduction... 3.1.1...... 3.1.2...... 3.1.3...... 3.1.4...... 3.1.5...... 3.1.6...... 3.1.7...... 3.1.8...... 3.2 The Tate category 3.2.1...... ith theory 93........... 93........... 93........... 94........... 95........... 96........... 96........... 96........... 98........... 99........... 100........... 100. 3.2.2 Periodicity 3.2.3 Derived invar~iants... 3.2.4 Tate cohomo logy.............. 10 1........... 10 1........... 103 3.2.5 Tate complex........... 103 8
3.2.6 P arity............................... 104 3.3 Sheaves.............................. 106 3.3.1 Disclaimer............................. 106 3.3.2................................... 106 3.3.3 Tate cohomology sheaves..................... 107 3.3.4 The six functors.......................... 109 3.3.5 Horns as colimits......................... 109 3.3.6 Modular reduction........................ 111 3.3.7 Addendum............................. 112 3.3.8 The Tate hypercohomology spectral sequence......... 113 3.3.9 Support.............................. 115 3.3.10 The Smith functor........................ 117 3.4 Tate-parity sheaves............................ 119 3.4.1 Reminder on parity sheaves................... 119 3.4.2 Parity conditions......................... 119 3.4.3 Tate-parity complexes...................... 120 3.4.4 Tate-parity sheaves........................ 121 3.4.5 Single stratum case........................ 122 3.4.6 JMW redux............................ 123 3.4.7 Modular reduction revisited................... 126 3.4.8 A technical remark........................ 129 3.4.9 Smith functor revisited...................... 129 3.4.10 Lifting............................... 130 3.5 An application to geometric representation theory........... 131 3.5.1 Geometric Satake equivalence.................. 131 3.5.2 Parity sheaves and tilting modules............... 133 3.5.3 Relation to Frobenius contraction................ 134 4 Some related results 137 4.1 Introduction................................ 137 9
..... 4.2 Equivariant cohomology................ 4.2.1 Background................... 4.2.2 Description.................... 4.2.3 Relation with the center of distribution algebra 4.3 Frobenius contraction.................. 4.4 Frobenius twist..................... 137 137 139 144 145 147 10
Chapter 1 Introduction During my PhD, I have been trying to answer three main questions, as summarized in the table below. Question Asker Solved? Describe the center of the distribution algebra of a reduc- tive algebraic group in positive characteristic R.B. No Interpret Steenrod operations for geometric representa- tion theory G.W. Yes! 1 Understand the geometric meaning (via geometric Satake, see [31]) of the Frobenius twist functor Y.T. Sort of The first question is still very much work in progress; see [23] for almost everything that is known. However, this question is important because, following the suggestion of R.B. that the center should be knowable 'via geometric Satake', it got me thinking about the affine Grassmannian Gr, and in particular about what sort of interesting centrality phenomena might occur there. The study of commutativity phenomena associated with the affine Grassmannian is ancient. After all, the affine Grassmannian is nothing more than an algebro-geometric model for the based loop group Q 1 K of a compact Lie group K. This is an example of a 'double loop space', so it has a homotopy-commutative group structure, and in 'Maybe my interpretation is not complete but I seem to have made a good start. 11
particular its homology has the structure of commutative ring. Even better: Q 1 K is K-equivariantly homotopy-commutative, and accordingly its K-equivariant homology is also a commutative ring. More recently [1], this homotopy-commutativity has been cast in algebro-geometric terms, and this is a key ingredient in the geometric Satake equivalence. Evidently centrality phenomena are redundant in a world where everything is commutative. Luckily, the homotopy-commutativity of double loops breaks when we work equivariantly with respect to the loop rotation action. Indeed, the K x S1- equivariant homology of Q 1 K is a strictly non-commutative ring, as long as K is non-trivial. For instance, if K = S' then this ring is the following algebra: Hfxs' (Q 1 K) = Z[h]Kx, 0)/([0, x] = h). Even though this ring is non-commutative, it is a classic observation that in characteristic p it contains a large center, generated by xp and OP. In fact, this is the primordial example of the p-center phenomenon [6, 41, lifting the Frobenius, which is important in representation theory and algebraic geometry. It is natural to ask whether there is a topological reason for this, and indeed there is. It is essentially provided by the Kudo-Araki-Dyer-Lashof operations [271,[14]; however as far as I know, the fact that these are shadows of a centrality phenomenon is a new observation. On equivariant parameters, the Kudo-Araki-Dyer-Lashof operations are nothing more than the Steenrod operations, which brings us neatly to the second question. The main observation to make here is that on equivariant parameters the total Steenrod operator, appropriately 2 normalized, is the same thing as the Artin-Schreier map. This is exactly what we hope to see for the p-center phenomenon. In Chapter 2, we explain these facts in more detail, and apply the Kudo-Araki-Dyer-Lashof (Steenrod) technique to find large centers in a wide variety of interesting rings in representation theory 3. Unfortunately, the distribution algebra is not one of them! It should not be very surprising, given the above, that the answer to the third 2 i.e., naturally. 3 This is supposed to justify my claim to have answered the second question. 12
question involves the same sorts of considerations, and indeed it does, as explained in Chapter 4. However, this answer is not completely satisfactory, since it is not obviously compatible with the alternative answer provided by the Finkelberg-Mirkovic conjecture [161, nor has it been applied yet in any representation-theoretically meaningful way 4 yet, hence the status: 'sort of'. Finally, a spin-off result. A classic (naive) approach to understanding the Frobenius twist geometrically is via the embedding of Gr in itself by raising the loop parameter to its pth.power. Of course, topologically this is just the pth-power map from the group Q 1 K to itself. Pushing forward along this map is related to, but fundamentally different from, the Kudo-Araki-Dyer-Lashof (Steenrod) process which gave us the Frobenius twist. In fact it seems, representation theoretically, to be more closely related to the operation of applying the Frobenius twist and then tensoring with the square of the Steinberg representation, rather than the Frobenius twist itself. This claim is justified in Chapter 3, where (joint with S. Leslie) we demonstrate that pulling back along this pth-power embedding corresponds, modulo some Smiththeoretic considerations, to the Frobenius contraction functor of [191, which is adjoint to the functor claimed. 4 e.g. understanding stalks of Frobenius twists. 13
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Chapter 2 Steenrod operators, the Coulomb branch and the Frobenius twist 2.1 Introduction 2.1.1 This chapter is about power operations. Homological algebra Steenrod's construction Power operations Coulomb branch Frobenius-constant quantizations A power operation is an enhanced version of a p th -power map. One of the most famous examples is Steenrod's operations [341, a cornerstone of algebraic topology. In Section 2.2, we will give an account of Steenrod's construction in the language of derived categories. In these terms, the construction itself is very simple, and it yields not only Steenrod's cohomology operations but also operations in Borel-Moore homology, which are presumably related to the Kudo-Araki-Dyer-Lashof operations 15
[271,[14]. A reader who knows about equivariant constructible derived categories on complex algebraic varieties will be able to understand these constructions even if they do not know any homotopy theory. Perhaps this is an advantage. 2.1.2 In Section 2.3, we will introduce a different type of power operation, introduced by Bezrukavnikov and Kaledin [4], which is an important tool in non-commutative algebraic geometry. Such a power operation is known as a Frobenius-constant quantization. Essentially, a Frobenius-constant quantization of a commutative algebra A over F, is a 1-parameter flat deformation Ah of A in associative algebras which has a large center; see Subsection 2.3.1 for a precise definition which also justifies regarding such a thing as a power operation. The main example is the Weyl algebra Fp [h](x, )/([0, x] = h) which contains xp, OP in its center. 2.1.3 We will then illustrate a general method to apply Steenrod's construction to produce Frobenius constant quantizations. It is not completely clear just how general this method may be, but heuristically it ought to work whenever the multiplication in Ah is somehow related to, if not directly inherited from, the homotopy-commutative multiplication of a based loop group. The example which we use to illustrate the method is the quantum Coulomb branch of Braverman-Finkelberg-Nakajima [10] - or rather, its natural characteristic p version. That is, we prove: Theorem 2.1.1. For any complex reductive algebraic group G, and finite-dimensional representation N of G, and any odd prime p, the quantum Coulomb branch is a Frobenius-constant quantization. The Coulomb branch is the G-equivariant Borel-Moore homology of a certain alge- 16
braic space R; the quantum Coulomb branch is obtained by switching on loop-rotation equivariance. The key geometric insight behind the Theorem is that, following ideas of Beilinson-Drinfeld [1], one may deform the space R with its pp-action by loop rotation, to lzp with its pp-action by permuting the factors cyclically. 2.1.4 This is already quite a broad class of examples. For instance, it includes partially spherical rational Cherednik algebras, see [9], [38]. It is expected that the same underlying geometry will lead to the discovery of large centers of related algebras. In fact, in Section 2.4 we indicate how the same underlying geometry shows that the K- theoretic version of integral quantum Coulomb branch, which is itself a q-deformation of the K-theoretic version of the Coulomb branch, admits a large center when q is evaluated at any complex root of unity (not necessarily of prime order). Essentially the only difference with the homological case is to replace Steenrod's construction with a so-called 'Adams construction' which is to Adams operations as Steenrod's construction is to Steenrod's operations. Remark 2.1.2. It seems likely that the analogous statement holds for the elliptic version of the quantum Coulomb branch. That is, following [18], [17], there is a theory E11G of G-equivariant elliptic homology which takes values in coherent sheaves over some non-affine scheme (identified with Sll(*)) of equivariant parameters. The scheme of equivariant parameters of C* is a chosen elliptic curve. In our situation, we expect Ell (O)C*( 7 Z) to be a ring ind-coherent sheaf on EllGxC*(*) = gllg(*) X Ellc*(*), whose restriction to 0 e Ellc*(*) is the commutative algebra Ell G(R). G(O Then, the analogous statement will be an embedding of Ell* (7Z) in the center of the restriction of Ell*(O)xC*(R) to any torsion point of 8llf*(*). The proof would be essentially identical, but one first needs to develop theories of power operations and specialization in equivariant elliptic homology. Warning 2.1.3. The proof of Theorem 2.1.1 relies quite heavily on the theory of placid ind-schemes, dimension theories etc., see [32]. The first half of Section 2.3 17
simultaneously reviews this theory and introduces the examples which are relevant for us. As such it is written to be reasonably convincing, with the key facts explained in full detail, but with some details missing. All of the details are available in [32j, which the reader is strongly recommended to read. 2.2 Steenrod's construction 2.2.1 Overview Let p be an odd prime number, and let pp be the group of complex pth roots of unity. Let R be a commutative ring. Let k be a field of characteristic p, and let F : k -+ k be the Frobenius map. Let X be a topological space and let Db(X, R) denote the bounded derived category of sheaves of k-modules on X. We write XPP for Map(pp, X). Following Steenrod [34], we construct a functor St : Db(X, R) -+ D'(X"4, R) where D (Xt'P, R) denotes the bounded pp-equivariant derived category of sheaves of R-modules on XPP. This functor is not linear or triangulated, but nonetheless if we take R = k, compose with restriction to the diagonal and apply to morphisms between shifted constant sheaves we obtain linear maps F* Hn(X, k) -+ H2(X, k) ~ HZ(X, k) 0 Hi(Bpp7, k) (2.2.1) i+j=pn for each n > 0. Recall that H*(Byp, k) = k[a, h] is the super-polynomial algebra in one variable a of degree 1 and one variable h of degree 2. Here h is the first Chern class of the tautological complex line bundle on Byp arising from the embedding, c C*. The direct sum of the maps of equation 2.2.1 is not in the most naive sense an algebra homomorphism. This fact led Steenrod to introduce certain correction factors which make it so; his famous cohomology operations are then defined to be the coefficients of the resulting algebra homomorphism in the monomial basis of 18
k[a, h]. However, the sum of maps of equation 2.2.1 does give a homomorphism of super-graded algebras H* (X, k)u III,* (X, k) ~- H* (X, k) [a, h] (2.2.2) where H*(X, k)g) denotes the Frobenius twist of H*(X, k). Naively one might think that this is just the p-dilation of F*H*(X, k). This is wrong: rather, the natural and correct definition of the Frobenius twist of an algebra A in any symmetric monoidal category over k is as the Tate cohomology: A () := Ht0 (A@& P) where the symmetric monoidal structure endows A "P with the structure of Mpequivariant algebra. In the case of the super-graded k-algebra H*(X, k), the underlying super-graded k-module of this construction is the same as the p-dilation of F*H*(X, k), but the multiplication differs by a signi, removal of which is part of the purpose of Steenrod's correction factors. We prefer therefore to use Steenrod's operations in their raw form - that is, without the correction factors and packaged as in equation 2.2.2. This has the advantage of revealing the fundamental connected between Steenrod's operations and the Artin- Schreier map, which is obscured by the correction factors: Fact 2.2.1. Let X = BT for some complex torus T. Then the Picard group of X is canonically isomorphic to the character lattice X'(T) of T, and the cohomology ring is the polynomial algebra H*(X, Z) = Symz X'(T) with X'(T) in degree 2. This is equal to the ring O(tz) of polynomial functions on the canonical Z-form of the scheme t = Lie(T). Likewise we have 'When p = 3 mod 4. H*(X, k) = 0(tk) 19
where tk denotes the canonical k-form of t. Under this identification, the map of equation 2.2.2 factors as 0 (tk)) 0s" (t,) h]c 0 (tk)[a, h] where ASh corresponds, on the level of k-points, to the k -equivariant family, parameterized by h e k, of additive maps of free k-modules k tk - I( 1 )O tk i Xi 0 vi -+ i(xi - hp- 1 x) 0 vi for a basis {vj} of tf. This family interpolates between the usual Artin-Schreier map for h = 1 and the Frobenius map for h = 0. Remark 2.2.2. The appearance of ASh in the topological setting was the first indication that Steenrod's construction might be related to the theory of Frobenius-constant quantizations, where ASh plays a central role, see Fact 2.3.2. 2.2.2 Steenrod's construction Recall that p is an odd prime, pp is the group of complex pth roots of unity, R is a commutative ring, k is a field of characteristic p and X is a topological space. We denote by Cb(X, R), Db(X, R) the (bounded) cochain, derived categories of sheaves of R-modules on X. If Y is a topological -space with an action of p, we denote by Cb (Y, R), D' (Y, R) the corresponding pp-equivariant categories. Since pp is a finite group, these are the same as the (bounded) cochain, derived categories of pp-equivariant sheaves of R-modules on Y. Consider the functor of pth external tensor power Cb(X, R) O C(XtP, R). It sends quasi-isomorphisms to quasi-isomorphisms, and so descends to a (not trian- 20
gulated) functor Db(X, R) O Db(XLP, R) by the universal property of derived categories. Notice that the cochain-level functor factors as jp: Cb(X, R) C,)(X C,bR) Cb(XP, R). To make this explicit, we first choose an isomorphism the result will be independent of this choice. Write o for the generator of, corresponding to 1 under the isomorphism. Then, for a complex AO, we give the complex (A-' = A" [Z... [ AP) the pp-equivariant structure by letting the generator o act by the direct sum of the canonical isomorphisms of sheaves A"' Z...[Rx Alp ~: a-* (Ah 12...G AZ XP A"') each twisted by the sign (-1)"l. The sign twist is the natural (Koszul) choice which makes the action of p, commute with the differential. Moreover, given a chain map f : A*- Be, fop is automatically a y,-equivariant chain map. Since the functor Cb (XA", R) -+ Cb(Xip, R) reflects quasi-isomorphisms, it follows immediately that Stc descends to a functor StD as below: [Zp : D b(x, R) St D b (XILP, R) -+-) D b(x"p, R). Writing E for the suspension functor, we have StDE EpStD. Also, StD is not triangulated, nor additive or even linear. The following two propositions control the failure of linearity. 21
Proposition 2.2.3. Suppose given two parallel morphisms f, g :A - B' in Db(X, R). Then the morphism StD(f + 9) - StD(f) - StD(9) StD(A) -- StD(B') is an induced map. That is, there exists some non-equivariant map h: (A')/P --+ (B')ZAP such that the equivariant map Av(h) =, xhx' : StD(A) -- StD(B') xc-pp is equal to StD(f + g) - StD(f) - StD)(9) Proof. Let us right away replace A*, B' by isomorphic objects so that f, g become genuine maps of complexes. Let f, g denote the constant functions yp -+ {f, g} with respective values f, g. Then pp acts freely on {f, g} IP - {f, g}; choose a set {hi,..., h,} of orbit representatives (n = (2P - 2)/p). Then each hi determines a non-equivariant map (A) -+ (B')2fP, hence so does their sum h. Then, we have (f + g) 'AP - fpia - g2jp = t Y hx- 1 where, by definition xhx- 1 is the composition: xhx- 1 : (A')EAp ~ x*(a*)p >() x*(b')e MP ~ (BO)ZP where the two isomorphisms are given by the equivariant structures. I Proposition 2.2.4. StD is Frobenius-multiplicative with respect to the action of the multiplicative monoid R on hom-sets. That is, StD determines a functor StD : IndR Db(X, R)- Db(XIP, R) 22
which respects multiplication by R. Here the category on the left is obtained from Db(X, R) by regarding each hom-set as a set with an action of the multiplicative monoid R and inducing along the pth-power map of monoids R -+ R. Note that IndR Db(X, R) is not an additive category in general. However, suppose that R = k and k is perfect. In that case, the Frobenius map of monoids is actually a map of rings F, and is moreover bijective. Write M : k -mod -+ k -set for the forgetful functor, where k -set denotes the category 2 of sets with action of the multiplicative monoid k. We have the following: Lemma 2.2.5. Suppose that k is a perfect field of characteristic p. Then we have M o F* Ind om. Proof. Indeed, in that case both F* and Indk are equivalent to functors which do not change the underlying abelian group/set, and only change the way that k acts. El It follows that if k is a perfect field of characteristic p, we have produced a k- multiplicative functor StD : F*Db(X, k) -+ D'(XPP, k). Since F*Db(X, k) is triangulated, we find this statement somewhat nicer that the version for general R. 2.2.3 Six functors We are mainly concerned with the case where X is the (Borel) quotient EG Y of a complex algebraic variety Y by the action of some affine algebraic group G, and R is a Noetherian ring of finite homological dimension. In this case we will replace the category Db with its constructible analogue D'. That is, any G-equivariant constructible sheaf on Y descends to a sheaf on X, and Db(X, R) is the thick subcategory 2 The reader may prefer to replace this by its full subcategory of all k -sets with a unique stable point. 23
of Db(X, R) generated by all such sheaves. We will usually write DGG(Y, R) instead of Db(X, R). The constructions of the previous section preserve constructibility, so we have a Steenrod construction S :D,G (Y, R) -+ DebC',,(YAP,R) Recall that we have the six functor formalism for constructible derived categories. We assume that the reader is familiar with this material, but remind him/her of the standard notation: for a G-equivariant algebraic map f : Y - Y', we have the adjoint pairs of exact functors f*: Dc,(Y', R)a De',G * R and also a pair of bi-exact bifunctors f! e,(y, R)--I D',G(Y D,G(Y, R) x De,G(Y, R)-* D,(YG R) '=om(-, -) : Di,G(Y, R)" 2 x D,,G(Y, R) -+ DG(Y, R) related by a tensor-hom adjunction. There is also a Verdier duality functor D, and an exceptional tensor product 0!, which can be written in terms of the other functors, as can the external tensor product x. We call the collection of all of these functors the six plus functors. Notice that GAP x pp is also an affine algebraic group, so the six functor formalism exists for the target category of StD. Also if f : Y -+ Y' is G-equivariant then f A : YA -* (Y')a is G"A x pp-equivariant. The following fact is essentially a consequence of the same fact for O: Proposition 2.2.6. Steenrod's construction is compatible with the six functor for- 24
malism. That is, we have canonical isomorphisms (fp)*st StDf Sf" )*StD (f Ap )*StD S StDf* (f )! StD StD (fn)!std StDf StD() StD ( -- StD(-- -9 Wom(StD(-), StD(-) StD om(-,-) commuting with any and all adjunction morphisms of the six functor formalism. We have the same compatibilities with functors St'D. Remark 2.2.7. Many of the six plus functors are defined in much more general contexts than the constructible derived category. For instance f*, f. are defined in complete generality, and their compatibilities with Steenrod's construction hold in that generality. One expects that, in some sense, any time any of these functors is defined it is compatible with Steenrod's construction. But we wish to avoid making any precise statement along these lines. 2.2.4 Tate's construction Note that the object StD(E"R) is canonically isomorphic to EPR with its trivial p- equivariant structure. Here R is the constant sheaf. Since a degree n cohomology class is just a morphism R - ER in Db(X, R), we thus obtain a Frobenius-multiplicative map of multiplicative R-sets: StH : H"(X, R) Hp"(X-*,R). This map is not Frobenius-linear, but as in Proposition 2.2.3, its deviation from additivity is by a class induced from HP (XMP, R). To see how these maps interact with multiplication, we make the following definition. Definition 2.2.8. Let (C, *, 1, e, a, s) be a symmetric monoidal abelian category 25
enriched over some commutative ring R, i.e. C is an abelian category, * is a biexact R-linear functor C x C -- C, 1 is an object of C, e is a pair of equivalences 1 *Id ~ Id ~ Id * -1, a is an associativity constraint for * and s is a commutativity constraint for *, satisfying natural compatibilities. Let A be an object of C. Then s determines an action of pp on A* P, and we define A(') := 0 1 u (AP) kera*a, (1-0)/ ima*a,(n). This is the so-called Tate construction. For a morphism f : A -+ B the morphism f *A : A* a - B* 4 is pp-equivariant and so induces a morphism AM -B B1) so that (-)(1) becomes a functor. Lemma 2.2.9. (-)(1) is additive over Z. Proof. First we show that (-)(1) is linear over Z. Suppose f, g : A - B are two parallel morphisms in C. Let f, g denote the constant functions pp - {f, g} with respective values f, g. Then pp acts freely on {f, g}iip - {f, g}; choose a set {h,..., h,} of orbit representatives (n = (2P - 2)/p). Then each hi determines a non-equivariant map A* Pp B*"P, and we have n (f + g)*pp - f*" -*p = Z Zxhix- 1. XEpp i=1 Restricting to kera* M (1-0-), this becomes ((f + g) * " - f*"p - g* "P)kerA*AP(1-a-) = YE hi =N hi xe/pp i=1 i=1 n n which factors through imb* P(N) as required. Next we show that (-) (1) preserves direct sums. Let A, B denote the constant functions pp -- {A, B} with respective values A, B. Then pp acts freely on {A, B}'"p - {A, _}; choose a set { C1,..., Cn } of orbit representatives. Then each Ci determines 26
an object of C, and as a pp-module in C we have (A 0 B) *" ~1 A*F /1 B *F/- CiIPP]. n The result then follows from the fact that HO, (k[pp]) = 0 in R-mod. E spaces. Let SVectk denote the symmetric monoidal category of Z /2-super graded k-vector Lemma 2.2.10. Suppose that R = k is a field of characteristic p and that C admits a super-fiber functor C -+ SVectk. Then (-)(') is exact, monoidal and Frobenius-linear over k. In the case C = SVectk, (-)(1) is equivalent to the functor kof(-) which tensors the k-linear structure along the Frobenius map F : k -+ k. Proof. Since Tate's construction commutes with the fiber functor, it is enough to take C = SVectk, where it is a simple calculation using bases. E Now suppose that A, B are objects of C. We have a pp-equivariant isomorphism (A * B)*PP ~ (A*"A * B*!/P). We have also the natural inclusions kera*pp(1 - o) * kerb*pp(1 - -) - kera*mppb*ip(1- a) ima* PP (N) * kerb* PP(1 - ) -- ima* PP*B* PP (N) kera* PP(1 -- a) * ima*(i (N) - ima* P *B*,uP (N) which induce a map A 1 ) * Bl) -) (A * B)( 1 ). Suppose that (A, 1 A, ma) is a unital ring in C. Then AM 1 still has a multiplication (1) ma(1) : AM 1 * AM 1 -+ (A * A)( 1 ) A^ A(). Also, there is a canonical isomorphism ker 1 *- (1-o) 2 1, hence a canonical surjection 1-1 1) which determines a map LA() I 11 ~(l) 'A (') 27
One may check that this makes AM) into a ring, and moreover that AM 1 is associative or commutative if A is. The following lemma explains how this looks in the main example. Lemma 2.2.11. Suppose A = RC-Z/2 Aj is a unital ring in C = SVectk. Then the ring structure on A) corresponds under the identification AM) = k OF A to the ring structure with unit 1 OF IA and multiplication ma (r 0 a, r' 0 a') = (-1 (P)rr' 0 ma(a, a) for a e Aj, a' e Aj and r, r' e k. Proof. The isomorphism of k OF Ai with (A))i sends the element r 0 a to the class of r. a (... 0 a. The natural map p times (A(')); i (A ()1% H- uol ((Ai) *P) 0 Hlttj ((Aj) * tp) -> up ((Ai (9 Aj)**P) sends the class of to the class of a D... 0 a 9a'0... D a' p times p times (1)j (P) (a (9 a') (9... (9 (a 0 a'), p times since it entails permuting the xth copy of Ai with the yth copy of A 3 for every p X > y > 1. Arguing the same way, we have the following: Proposition 2.2.12. Let A be a Hopf algebra in SVectk. Then AM) is naturally a Hopf algebra in SVectk. It has multiplication and unit given as in 2.2.11, comultiplication given by AA( (r 0 a) = r 0 (-1)(P) d*a da(a), 28
counit given by ea(1(r0 a) = r(ea(a))p and antipode given by SAM)(r 0 a) = r & SA(a). Moreover the functor (-)(1) on SVectk upgrades to functor (-)(1) : A-comod - A(') -comod. For an A-comodule M, the A( 1 )-comodule structure on M( 1 ) is given by Am() (r 0 m) = r 0 (--1)() de AmdeM(m) Example 2.2.13. Suppose 0 is a commutative Hopf algebra in SVectk. Then the monoidal category C of 0-comodules is symmetric. Taking pth powers gives a (Frobenius) map of Hopf algebras FO : (9(1) - 0. Then, Tate's construction on C factors as 0 -comod 41 )0(l) -comod 0 -comod. For instance, we could take 0 to be the ring of functions O(Gm) on the multiplicative group G, over k, concentrated in degree 0 e Z /2. Then O(Gm) -comod contains as a full subcategory over SVectk the category of Z-super graded vector spaces, and Tate's construction there is isomorphic to the functor which applies k OF (-) and multiplies degrees by p. Recall we have the Frobenius-multiplicative maps of multiplicative k-sets StH : H (X, k) Hg(XtP, k). We view cohomology rings as commutative ring objects of Z-super graded vector spaces; in particular we can apply functor (-)(1) to them. By Lemma 2.2.5, if k is perfect then it gives a map of Z-super graded k-sets Stex : H*(X, k)) -- H* (XP, k). The following fact is immediate from the constructions. 29
Proposition 2.2.14. Stex respects the multiplicative k-monoidal structures. Remark 2.2.15. If k is not perfect, then the map H*(X, k) -) H1*,(XAP, k) of Z /2- super graded sets respects the multiplicative monoidal structures up to the sign change of Lemma 2.2.11. There is presumably an appropriate non-linear version of Tate's construction which would allow us to say that we really have a certain Z-super graded k-monoid H*(X, k)() and a map of monoids H*(X, k)() -- H* (XtP, k), but we prefer for simplicity not to do it. 2.2.5 Borel-Moore homology We return to the setting of Subsection 2.2.3. Let w denote the G-equivariant dualizing complex on Y with coefficients in R. We have a canonical isomorphism 3 StD(W) w. By definition, the G-equivariant Borel-Moore homology of Y is HBM,G(Y R) := HomDh(YR) En.). See Subsection 2.3.9 for more about this. Altogether H BM,G(Y, R) form a Z-super graded HG(Y, R)-module; in particular it is a module for HG,(*, R). By functoriality we have the non-linear maps StBM : H BMG(Y R) - H BM,GPP x PP(YPe, R). This is a map of StH-monoids. Its discrepancy from additivity it averaged from HBM,GP (YtLp, R). If R is a perfect field k of characteristic p, we can say that we have a non-linear graded map of Stex-monoids: St ~ ~ ~ ~ 3 B AP,(,k)N +HSE''" (YPP, k ). 3 here of course the second w denotes the GJPP x lpp-equivariant dualizing complex on YP with coefficients in R 30
2.2.6 Steenrod operations For simplicity let's assume k to be perfect from now on. Let us compose Ste, with the restriction map to the diagonal: Sti, : H* (X, k) (1) s'ex H,*,(XW"", k) HI H*(X, k) ~ *X )a,] This is again a map of multiplicative k-monoids. Tautologically we have Stin,(x) = XP mod (a, h). Also Stin is compatible with pull-back maps in cohomology in the natural way. Since induction commutes with restriction, the difference between Sti,(x + y) and Sti,(x) + Sti,(y) is induced from a cohomology class z e He(X; k). Since P, acts trivially on X, that means that it is equal to pz = 0, so Sti,, is linear. That is, we have a map of super-commutative k-algebras Sti, : H*(X, k) () -+ H*(X, k)[a, h]. Remark 2.2.16. The coefficients of h', ah' in Sti, are not the Steenrod operations. More precisely, they are the Steenrod operations only up to some non-zero scalars. Even more precisely, let x e H'(X, k) and let p = 2q + 1. Consider (-1)q"("-l)2(q!) -"Stin (X). where x is viewed as a degree pn element of H* (X, k) ). The coefficient of h' in this expression vanishes unless m = 2(p - 1)(n - 2s) for some s such that 2s < n, in which case that coefficient is equal to (-1)sPs(x) where PS is the sth Steenrod operation. Similarly, the coefficient of ah' in that expression vanishes unless m = (p - 1)(n - 2s) - 1 for some s such that 2s < n, in which case that coefficient is equal to (-1)s+13Ps(x) where 3 is the Bockstein operation. 31
2.2.7 Artin-Schreier We indicate how the Artin-Schreier map comes naturally out of the above considerations. First note that if n is even then the number (- 1)qn(n-l)/2 (q!)~" boils down to (-1)n/2. It is a standard fact that on a degree 2 class x we have P 0 (x) = x, P 1 (x) = xp, and higher powers vanish. Therefore Sti,(x) = xp - h x +h-2(x). Let X = BT for some compact torus T. Since its cohomology is supported in even degrees, the Bockstein operator acts as zero and Stin, on the level of k-cohomology, is exactly the h-artin-schreier map 0 (t') (1) As"> )0(t,) [h]c c- (tk)[a, h] as defined in Fact 2.2.1. Recall that if G is a compact Lie group with maximal torus T, and p is large enough with respect to the Weyl group of G, then the projection BT - BG induces an inclusion H*(BG, k) -- H*(BT, k) which is identified with O(tk //W) -- 0(t4). The h-artin-schreier map induces a map on subspaces 0(te //W)(1 -"h) 0(tk //W)[h] which is also important in the theory of Frobenius-constant quantization. The point is that this map is also induced by Stin, since it is compatible with pullbacks. 32
. It is entertaining to show more directly how ASh arises, without relying on any outside facts about Steenrod operations. We can reduce to the rank one case T = S 1 Let b denote the degree 2 generator (first Chern class of tautological line bundle) of BS 1 ; we need to show that Sti,(b) = bp - hp- 1 b. Let C, c S' denote the cyclic group of order p, considered as distinct from pp. Consider the projection BC, -+ BS'. It induces an injective map k [b] -+ k[s, b] in cohomology, where s is a degree 1 generator. By functoriality it is enough to prove the equality when b is regarded as a cohomology class of BC,. Note that amongst degree 2p elements of k[b, h], the desired element bp - hp-b is the unique one which gives 0 when we set b to any multiple in Fp of h, and gives bp when we set h = 0. The latter statement is automatic, so we have to check the former. So fix some t e F,. Having chosen an isomorphism p, ~ C,, t determines a group homomorphism pp -~ CP. The constant sheaf EZk of BC, is contained in the full subcategory Db(k[C]-mod) = Dc,(*) c D(BC, ). Our coefficients are k, which we drop from the notation. It is easier for our purpose to work in Db(k[C, ] -mod). Compatible with the functor StD out of D(BC, ) we have the functor StD: Db(k[Cp] -mod) -. Db(k[pp x (Cp)'P] -mod) This is then composed with the diagonal restriction Db(k[pp x (Cp)xP]-mod) A Db(k[pp x CP] -mod). By definition Sti,(b) is given by applying that composition to the morphism k b+ k[2], 33
where k is the trivial Cp-module. We want further to set b = th; this corresponds to restricting along the map pp idxt pp xcp. Write (id x t)* : Db(k[pp x C] -mod) -- Db(k[pp]-mod) for the corresponding restriction map. We need to show that (id x t) * o A* o St(b) = 0. But actually there is an isomorphism of functors (id x t)* 0 A* 0 StD : StD 0 where i* is the forgetful functor Db(k[Cp]-mod) = Dc,(*) -+ D(*). Indeed for an object A* of Db(k[Cp]-mod), the underlying complex of both functors is (A*)P, and the automorphism which sends each summand At' (D... 0 A P to itself by 1 0D r 0 2t 0... 0 o(p-)t intertwines the two actions of pp. Here o is some generator of pp. But the functor St o i* kills b, since i* does. 2.3 Coulomb branch 2.3.1 Prelude: Frobenius-constant quantizations Let k be a field of characteristic p and let C be a symmetric monoidal category over k. The reader may assume that C is the category of comodules of some commutative Hopf algebra in SVectk. Let A be a commutative (and associative) algebra in C. Let F : A(' -+> A be the Frobenius map. Let Q be an augmented commutative algebra in C with augmentation c : Q -> k. Following [41 we make the following definition: 34
Definition 2.3.1. C such that AQ 0Q k = A. 1. A Q-quantization of A is a flat associative Q-algebra AQ in 2. A Frobenius-constant Q-quantization of A is a Q-quantization AQ of A together with a map FQ A'M -' Z(AQ) of algebras which lifts the Frobenius map, i.e. such that e o FQ = F. Here Z(AQ) denotes the center of AQ. The main example for us is the following. We take K to be some Gm-equivariant algebraic group in Vectk, and view 0 O(K x Gm) as a Hopf algebra in SVectk concentrated in degree 0. We take C = 0 -comod. Let h be a basis vector of the 1-dimensional representation of K x Gm in which K acts trivially and Gm acts with weight 2. Let Q = k[h]. In this case, we will call a Q-quantization simply an h- quantization, or just a quantization if the meaning is clear. Fact 2.3.2. 1. Let X be a smooth affine algebraic variety over k. Then the ring of asymptotic crystalline differential operators, Dh(X), is a canonical h - quantization of 0(T*X). Here G, acts trivially on ((X) and on vector field with weight 2. Let a be a vector field on X. Then OP acts as a derivation on 0(X), so that OP - ONPI annihilates ((X) for a unique vector field O[PI. Then Dh(X) has a canonical Frobenius-constant structure determined by FA : x - xp X e O(X) a -- O - hp- 1 0LA a e Vect(X). 2. Let J be a smooth algebraic group over k. Then F as above is K = J x J- equivariant (induced by left and right regular actions). In particular if we take invariants for the left factor, we obtain a Frobenius-constant structure for the quantization l'lh(j) of O(Lie( J)* ). 3. Let T be a complex torus and let TV be the Langlands dual split torus over k, 35
that is: T' = Spec(k[X.(T)]) where X.(T) is the cocharacter lattice of T and k[x.(t)] is its group algebra. We have canonical identifications O(*) = ON(t) Uh(tv) = 9 (tk X Ga). If we take Spec of the Frobenius-constant structure we recover the h-artin- Schreier map Fh = ASh : tk x Ga -+ t of Fact 2.2.1. Remark 2.3.3. If a commutative algebra and its quantization contain in a natural way H (*, k) for some complex reductive group G with maximal torus T, then when searching for a Frobenius-constant structure it is natural to look for one which is compatible with the h-artin-schreier map. 2.3.2 Formal neighborhoods Let X be a smooth complex curve and let S be a finite set. Given a commutative ring R and an R-point x of Xs, we denote the coordinates of x by x, (s e S), write lf(x,) for the graph of x, in XR and write I(x,) for its ideal. We write As(x) for the formal neighborhood of the union of the graphs of x, (s e S). That is, As(x) is the direct limit in affine schemes over X: As(x) = coli As,i(x) where As'j(X) = Spec (OxR 36 ses I(xY).
Given a subset S' c S and an R-point x of XS we will write for the S'-punctured formal neighborhood, i.e. the complement of the union of the graphs of x, (s e S') in As(x). As a sheaf of algebras on As(x), O(As'(x)) has an exhaustive increasing filtration: F1 O(As'(x)) = o(as(x)). H I(xs)-. ses' Suppose we have S" c S' ( S and x e XS(R). The inclusion S' c S defines a projection f : Xs XS', and we will occasionally write for Az''(f(x)). We have a closed embedding AS"(x) - As"(x). Note however that this is in a sense non-uniform in x: for instance if for every s e S there exists an s' e S' such that x, = xe, then the embedding is an isomorphism; and conversely. This is essentially the fact underlying Beilinson-Drinfeld's 'fusion' Grassmannian [1]. We will make more of this when we discuss co-placid morphisms, see Example 2.3.14. For notational simplicity, we frequently remove commas and braces from S, S', and also drop the part (x), when it is clear which point we refer to. So for example the expression: becomes: A12' 37
2.3.3 Global groups; pro-smoothness Now fix an affine algebraic group G over C. Consider the following functor from commutative rings to groups over XS: Gs(R) := {(x, f)jx e XS(R), f : As(x) -+ G}. Then Gs is represented by the limit of an inverse system of smooth affine group schemes over XS: Gs = lim Gs,i such that each transition morphism is a smooth homomorphism. Here Gs,j may be taken to represent the functor Gs,(R) = {(x, f)ix e Xs(R), f : As,j(x) -+ G}. Later, the notation Gs,j may represent a piece of some other cofiltered system presenting Gs; we will refer to the specific group above by Map(As,j, G). The fact that each transition morphism is smooth is directly verified using the valuative criterion. Indeed let Spec(R) be a square-zero thickening of Spec(R). A commutative diagram Spec(R) Gs,i41 Spec(R) -+ Gs,i is the same thing as a point 35 e XS(R), with residue x e XS(R), and a commutative diagram Asi(x) -+ As,i+ 1 (x) As,i(X) ->G. This determines a morphism P -+ G where P is the appropriate pushout in affine schemes. Since As,i(x) is equal to the intersection of As,i+ 1 (x) with As,i(z) in- 38
side Asj+ 1 (3), and As,j+ 1 (z) is a square-zero thickening of As,i+ 1 (x), it follows that As,i+ 1 (3) is a square-zero thickening of P. Therefore since G is smooth we can extend P - G to As,i+1 (X) -- G, as required. Note that Gs,o = XS so in particular each Gs,j is smooth over Xs. Now fix x e XS(C). It partitions S into subsets S 1,..., Sn, according to coincidence amongst the coordinates. Write ym for the coordinate x, for any s e S,, and zm for the C-point of Xsm with coordinates ym. We have As,j(x) = Spec (OX /~X" 1 I(ym)iSmI) = U$X" Spec (Ox /I(ym)iSm I). Therefore we have nl n G s,j xxs {x} = Gsm,i xxsm {zm} = 17 G{m),iismi xximi {ym}. m=1 m=1 The smooth transition map G{rm,(i+1)IsmI xx(m} {ym} G{m},jismI xximi {ym} is surjective for all i > 0 and has a unipotent kernel for all i > 1. It follows that Gs,i+1 -- Gs,j has the same property. Thus Gs is a prosaic affine group scheme over XS in the following sense: Definition 2.3.4. 1. A scheme T over B is said to be pro-smooth over B if it can be written as the limit of a inverse system of schemes T smooth over B and with smooth transition morphisms. If T is pro-smooth then it is formally smooth (in particular flat) over T. 2. An affine groupoid scheme g over B is pro-smooth over B if it can be written as the limit of an inverse system of affine groupoid schemes gi over B whose structure maps to B are both smooth, and which has smooth transition homo- 4 morphisms 3. In (2) and (3) we can upgrade to the property of being a pro-smooth cover by 4 1s this the same thing as an affine groupoid scheme g over B such that both structure morphisms 9 -- B are pro-smooth in the sense of (2)? 39
demanding that each transition map and structure map is a smooth cover. 4. Let the affine groupoid scheme 9 = limiezp 9i over B be a pro-smooth cover. Then each 9i is the fpqc quotient over B of 9 by some pro-smooth affine subgroup Ki. We say that 9 is prosaic if the Ki can be chosen to be also prounipotent. 5. Let g be an affine group scheme over the same base B. Then 9 is said to be pro-smooth, a pro-smooth cover, prosaic over B if it is so when regarded as a groupoid. From now on, 'groupoid' will mean 'affine pro-smooth covering groupoid', unless it is clear from the context that this is not the case. All examples of groupoids will actually be prosaic. Remark 2.3.5. Recall the construction of Gs. If the affine algebraic group G is replaced by an arbitrary smooth affine variety T over C, we get a pro-smooth affine variety Ts over XS in exactly the same way. 2.3.4 Beilinson-Drinfeld Grassmannians; reasonableness We also consider the functor Gi'(R) := {(x, f)ix e Xs(R), f : Ass' - G}. Then GS' is represented by an ind-affine ind-scheme, formally smooth over XS. It is a group in ind-schemes over Xs, but not an inductive limit of group schemes. It is a reasonable ind-scheme in the following sense (taken from [131): Definition 2.3.6. 1. An ind-scheme T is reasonable if it admits a reasonable presentation, that is an expression T = colim T 40 jej
where j is some (countable) filtered indexing category, and the transition morphisms in the filtered system of schemes (Tj)jEJ are all finitely presented (f.p.) closed embeddings 5. Note that any two reasonable presentations admit a common refinement, so that the category of reasonable presentations of T is filtered. 2. A closed subscheme of a reasonable ind-scheme T is reasonable if it is a term in some reasonable presentation of T. 3. A morphism U -+ T of reasonable ind-schemes is co-reasonable if for some, equivalently any, reasonable presentation T = colim T of T, the presentation U = colim U XT T of U as an ind-scheme is reasonable. Warning: this is not a relative version of reasonableness for ind-schemes. Example 2.3.7. 1. Let T be a reasonable ind-scheme and let U - T be either ind-f.p. or an ind-flat cover. Then U -+ T is co-reasonable. 2. In the case of G', one reasonable presentation is given as follows. Fix a finite set {a,..., an} of generators of O(G). Then set j = Zao and set GS'' to be the closed subscheme of GS' which on the level of R-points is given by GS''-(R) {(x, f)ix e XS(R), f : A -+ G, ak o f e HOF 0(As')}. Here we have taken Gs'" = Gs. The left- and right-regular actions of the subgroup Gs preserve the inductive structure, meaning that each GS"' has a free action on both sides by Gs over Xs, even though it is not itself a group. Moreover the fpqc quotient Gs"/Gs is of finite-type over XS, and flat, although generally quite singular. The result is that the fpqc quotient GS'/Gs has the structure of ind-finite-type ind-flat ind-scheme over XS. In particular, it is reasonable, and G' -G s is an ind-flat cover and thus co-reasonable. 5 That is, they have finitely generated ideal sheaves. 41
On R-points, we may identify Gs'/Gs(R) x c Xs(R) =1(x, E, f) S a principal G-bundle over As(x) f a trivialization of C over As'() Here the symbol '/ -' means 'taken up to isomorphism', i.e. we identify two R-points (x, 8, f), (x', E', f') if x = x' and there exists an isomorphism of S with S' which intertwines f, f'. Such an isomorphism is unique if it exists. The following fact is due to [1]: Lemma 2.3.8. 1. G '/Gs is ind-projective over Xs if and only if G is reductive. 2. Gs'/Gs is ind-reduced if and only if G has no non-trivial characters. Remark 2.3.9. Ultimately we are concerned only with the analytifications of these ind-schemes, so point (2) appears merely for interest's sake. But point (1) is crucial for the definition of convolution in Borel-Moore homology. We may re-identify the R-points of Gi' in a way more compatible with the above identification of Gs'/Gs(R): x e XS(R) GSI(R) =-(x, S, f, g) S a principal G-bundle over As(x) f a trivialization of over As'S W. g a trivialization of S over A'(x)j Notice that the inclusion S' c S induces a closed embedding AS'' - As" for any S" c S'. This in turn induces restriction homomorphisms GS" -+ GI". These maps are co-reasonable. One readily checks by looking at points that the induced maps Gs"/Gs -XS x s Gs''/Gs' 42