June Massachusetts Institute of Technology All rights reserved. William P. Minicozzi II Chairman, Department Committee on Graduate Theses


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1 / 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 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 LIBRARIES ARIUE
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3 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 Frobeniusconstant quantization. We also demonstrate the corresponding result for the Ktheoretic 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 Frobeniuscontraction 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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5 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. BenZvi, 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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7 . Contents 1 Introduction Steenrod operators, the Coulomb branch and the Frobenius twist 2.1 Introduction d..onstructin t unctors Steenro d's construction O verview Steenrod's construction Six functors Tate's construction BorelMoore homology Steenrod operations ArtinSchreier Coulon b branch Prelude: Frobeniusconstant quantizations Formal neighborhoods Global groups; prosmoothness BeilinsonDrinfeld Grassmannians; reasonableness Jet bundles; placidity Equivariance
8 Dimension theories Notational remark BorelMoore homology The branch The central map Linearity Centrality Completion of proof Closing remarks Ktheoretic version Ktheory and Khomology Adams operations Proof of Theorem Parity sheaves and Sn 3.1 Introduction The Tate category ith theory Periodicity Derived invar~iants Tate cohomo logy Tate complex
9 3.2.6 P arity Sheaves Disclaimer Tate cohomology sheaves The six functors Horns as colimits Modular reduction Addendum The Tate hypercohomology spectral sequence Support The Smith functor Tateparity sheaves Reminder on parity sheaves Parity conditions Tateparity complexes Tateparity sheaves Single stratum case JMW redux Modular reduction revisited A technical remark Smith functor revisited Lifting An application to geometric representation theory Geometric Satake equivalence Parity sheaves and tilting modules Relation to Frobenius contraction Some related results Introduction
10 Equivariant cohomology Background Description Relation with the center of distribution algebra 4.3 Frobenius contraction Frobenius twist
11 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 algebrogeometric 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 homotopycommutative group structure, and in 'Maybe my interpretation is not complete but I seem to have made a good start. 11
12 particular its homology has the structure of commutative ring. Even better: Q 1 K is Kequivariantly homotopycommutative, and accordingly its Kequivariant homology is also a commutative ring. More recently [1], this homotopycommutativity has been cast in algebrogeometric 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 homotopycommutativity 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 noncommutative ring, as long as K is nontrivial. 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 noncommutative, 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 pcenter 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 KudoArakiDyerLashof 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 KudoArakiDyerLashof 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 ArtinSchreier map. This is exactly what we hope to see for the pcenter phenomenon. In Chapter 2, we explain these facts in more detail, and apply the KudoArakiDyerLashof (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
13 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 FinkelbergMirkovic conjecture [161, nor has it been applied yet in any representationtheoretically meaningful way 4 yet, hence the status: 'sort of'. Finally, a spinoff 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 pthpower map from the group Q 1 K to itself. Pushing forward along this map is related to, but fundamentally different from, the KudoArakiDyerLashof (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 pthpower 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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15 Chapter 2 Steenrod operators, the Coulomb branch and the Frobenius twist 2.1 Introduction This chapter is about power operations. Homological algebra Steenrod's construction Power operations Coulomb branch Frobeniusconstant 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 BorelMoore homology, which are presumably related to the KudoArakiDyerLashof operations 15
16 [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 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 noncommutative algebraic geometry. Such a power operation is known as a Frobeniusconstant quantization. Essentially, a Frobeniusconstant quantization of a commutative algebra A over F, is a 1parameter flat deformation Ah of A in associative algebras which has a large center; see Subsection 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 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 homotopycommutative multiplication of a based loop group. The example which we use to illustrate the method is the quantum Coulomb branch of BravermanFinkelbergNakajima [10]  or rather, its natural characteristic p version. That is, we prove: Theorem For any complex reductive algebraic group G, and finitedimensional representation N of G, and any odd prime p, the quantum Coulomb branch is a Frobeniusconstant quantization. The Coulomb branch is the Gequivariant BorelMoore homology of a certain alge 16
17 braic space R; the quantum Coulomb branch is obtained by switching on looprotation equivariance. The key geometric insight behind the Theorem is that, following ideas of BeilinsonDrinfeld [1], one may deform the space R with its ppaction by loop rotation, to lzp with its ppaction by permuting the factors cyclically 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 qdeformation of the Ktheoretic 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 socalled 'Adams construction' which is to Adams operations as Steenrod's construction is to Steenrod's operations. Remark 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 Gequivariant elliptic homology which takes values in coherent sheaves over some nonaffine 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 indcoherent 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 The proof of Theorem relies quite heavily on the theory of placid indschemes, dimension theories etc., see [32]. The first half of Section
18 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 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 kmodules 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 ppequivariant derived category of sheaves of Rmodules 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 superpolynomial 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 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
19 k[a, h]. However, the sum of maps of equation does give a homomorphism of supergraded 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 pdilation 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 supergraded kalgebra H*(X, k), the underlying supergraded kmodule of this construction is the same as the pdilation 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 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 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 Zform of the scheme t = Lie(T). Likewise we have 'When p = 3 mod 4. H*(X, k) = 0(tk) 19
20 where tk denotes the canonical kform of t. Under this identification, the map of equation factors as 0 (tk)) 0s" (t,) h]c 0 (tk)[a, h] where ASh corresponds, on the level of kpoints, to the k equivariant family, parameterized by h e k, of additive maps of free kmodules 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 ArtinSchreier map for h = 1 and the Frobenius map for h = 0. Remark The appearance of ASh in the topological setting was the first indication that Steenrod's construction might be related to the theory of Frobeniusconstant quantizations, where ASh plays a central role, see Fact 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 Rmodules on X. If Y is a topological space with an action of p, we denote by Cb (Y, R), D' (Y, R) the corresponding ppequivariant categories. Since pp is a finite group, these are the same as the (bounded) cochain, derived categories of ppequivariant sheaves of Rmodules on Y. Consider the functor of pth external tensor power Cb(X, R) O C(XtP, R). It sends quasiisomorphisms to quasiisomorphisms, and so descends to a (not trian 20
21 gulated) functor Db(X, R) O Db(XLP, R) by the universal property of derived categories. Notice that the cochainlevel 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 ppequivariant 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 quasiisomorphisms, 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
22 Proposition 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 nonequivariant map h: (A')/P + (B')ZAP such that the equivariant map Av(h) =, xhx' : StD(A)  StD(B') xcpp 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 nonequivariant 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 StD is Frobeniusmultiplicative with respect to the action of the multiplicative monoid R on homsets. That is, StD determines a functor StD : IndR Db(X, R) Db(XIP, R) 22
23 which respects multiplication by R. Here the category on the left is obtained from Db(X, R) by regarding each homset as a set with an action of the multiplicative monoid R and inducing along the pthpower 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 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 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 Gequivariant 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
24 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 Gequivariant 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 biexact 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 tensorhom 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 Gequivariant then f A : YA * (Y')a is G"A x ppequivariant. The following fact is essentially a consequence of the same fact for O: Proposition Steenrod's construction is compatible with the six functor for 24
25 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 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 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 Frobeniusmultiplicative map of multiplicative Rsets: StH : H"(X, R) Hp"(X*,R). This map is not Frobeniuslinear, 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 Let (C, *, 1, e, a, s) be a symmetric monoidal abelian category 25
26 enriched over some commutative ring R, i.e. C is an abelian category, * is a biexact Rlinear 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, (10)/ ima*a,(n). This is the socalled Tate construction. For a morphism f : A + B the morphism f *A : A* a  B* 4 is ppequivariant and so induces a morphism AM B B1) so that ()(1) becomes a functor. Lemma ()(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 nonequivariant map A* Pp B*"P, and we have n (f + g)*pp  f*" *p = Z Zxhix 1. XEpp i=1 Restricting to kera* M (10), this becomes ((f + g) * "  f*"p  g* "P)kerA*AP(1a) = 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
27 an object of C, and as a ppmodule 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 Rmod. E spaces. Let SVectk denote the symmetric monoidal category of Z /2super graded kvector Lemma Suppose that R = k is a field of characteristic p and that C admits a superfiber functor C + SVectk. Then ()(') is exact, monoidal and Frobeniuslinear over k. In the case C = SVectk, ()(1) is equivalent to the functor kof() which tensors the klinear 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 ppequivariant 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 * (1o) 2 1, hence a canonical surjection 11 1) which determines a map LA() I 11 ~(l) 'A (') 27
28 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 Suppose A = RCZ/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 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 , comultiplication given by AA( (r 0 a) = r 0 (1)(P) d*a da(a), 28
29 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) : Acomod  A(') comod. For an Acomodule M, the A( 1 )comodule structure on M( 1 ) is given by Am() (r 0 m) = r 0 (1)() de AmdeM(m) Example Suppose 0 is a commutative Hopf algebra in SVectk. Then the monoidal category C of 0comodules 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 Zsuper 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 Frobeniusmultiplicative maps of multiplicative ksets StH : H (X, k) Hg(XtP, k). We view cohomology rings as commutative ring objects of Zsuper 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 Zsuper graded ksets Stex : H*(X, k))  H* (XP, k). The following fact is immediate from the constructions. 29
30 Proposition Stex respects the multiplicative kmonoidal structures. Remark 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 There is presumably an appropriate nonlinear version of Tate's construction which would allow us to say that we really have a certain Zsuper graded kmonoid H*(X, k)() and a map of monoids H*(X, k)()  H* (XtP, k), but we prefer for simplicity not to do it BorelMoore homology We return to the setting of Subsection Let w denote the Gequivariant dualizing complex on Y with coefficients in R. We have a canonical isomorphism 3 StD(W) w. By definition, the Gequivariant BorelMoore homology of Y is HBM,G(Y R) := HomDh(YR) En.). See Subsection for more about this. Altogether H BM,G(Y, R) form a Zsuper graded HG(Y, R)module; in particular it is a module for HG,(*, R). By functoriality we have the nonlinear maps StBM : H BMG(Y R)  H BM,GPP x PP(YPe, R). This is a map of StHmonoids. 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 nonlinear graded map of Stexmonoids: St ~ ~ ~ ~ 3 B AP,(,k)N +HSE''" (YPP, k ). 3 here of course the second w denotes the GJPP x lppequivariant dualizing complex on YP with coefficients in R 30
31 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 kmonoids. Tautologically we have Stin,(x) = XP mod (a, h). Also Stin is compatible with pullback 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 supercommutative kalgebras Sti, : H*(X, k) () + H*(X, k)[a, h]. Remark The coefficients of h', ah' in Sti, are not the Steenrod operations. More precisely, they are the Steenrod operations only up to some nonzero 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
32 2.2.7 ArtinSchreier We indicate how the ArtinSchreier map comes naturally out of the above considerations. First note that if n is even then the number ( 1)qn(nl)/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 +h2(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 kcohomology, is exactly the hartinschreier map 0 (t') (1) As"> )0(t,) [h]c c (tk)[a, h] as defined in Fact 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 hartinschreier map induces a map on subspaces 0(te //W)(1 "h) 0(tk //W)[h] which is also important in the theory of Frobeniusconstant quantization. The point is that this map is also induced by Stin, since it is compatible with pullbacks. 32
33 . 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  hpb 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
34 where k is the trivial Cpmodule. 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 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 Prelude: Frobeniusconstant 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
35 Definition C such that AQ 0Q k = A. 1. A Qquantization of A is a flat associative Qalgebra AQ in 2. A Frobeniusconstant Qquantization of A is a Qquantization 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 Gmequivariant 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 1dimensional 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 Qquantization simply an h quantization, or just a quantization if the meaning is clear. Fact 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 Frobeniusconstant 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 Frobeniusconstant 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
36 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 Frobeniusconstant structure we recover the hartin Schreier map Fh = ASh : tk x Ga + t of Fact Remark 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 Frobeniusconstant structure it is natural to look for one which is compatible with the hartinschreier map Formal neighborhoods Let X be a smooth complex curve and let S be a finite set. Given a commutative ring R and an Rpoint 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).
37 Given a subset S' c S and an Rpoint 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 nonuniform 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 BeilinsonDrinfeld's 'fusion' Grassmannian [1]. We will make more of this when we discuss coplacid morphisms, see Example 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
38 2.3.3 Global groups; prosmoothness 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 squarezero 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
39 side Asj+ 1 (3), and As,j+ 1 (z) is a squarezero thickening of As,i+ 1 (x), it follows that As,i+ 1 (3) is a squarezero 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 Cpoint 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 A scheme T over B is said to be prosmooth 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 prosmooth then it is formally smooth (in particular flat) over T. 2. An affine groupoid scheme g over B is prosmooth 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 prosmooth cover by 4 1s this the same thing as an affine groupoid scheme g over B such that both structure morphisms 9  B are prosmooth in the sense of (2)? 39
40 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 prosmooth cover. Then each 9i is the fpqc quotient over B of 9 by some prosmooth 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 prosmooth, a prosmooth cover, prosaic over B if it is so when regarded as a groupoid. From now on, 'groupoid' will mean 'affine prosmooth covering groupoid', unless it is clear from the context that this is not the case. All examples of groupoids will actually be prosaic. Remark 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 prosmooth affine variety Ts over XS in exactly the same way BeilinsonDrinfeld Grassmannians; reasonableness We also consider the functor Gi'(R) := {(x, f)ix e Xs(R), f : Ass'  G}. Then GS' is represented by an indaffine indscheme, formally smooth over XS. It is a group in indschemes over Xs, but not an inductive limit of group schemes. It is a reasonable indscheme in the following sense (taken from [131): Definition An indscheme T is reasonable if it admits a reasonable presentation, that is an expression T = colim T 40 jej
41 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 indscheme T is reasonable if it is a term in some reasonable presentation of T. 3. A morphism U + T of reasonable indschemes is coreasonable if for some, equivalently any, reasonable presentation T = colim T of T, the presentation U = colim U XT T of U as an indscheme is reasonable. Warning: this is not a relative version of reasonableness for indschemes. Example Let T be a reasonable indscheme and let U  T be either indf.p. or an indflat cover. Then U + T is coreasonable. 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 Rpoints 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 rightregular 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 finitetype over XS, and flat, although generally quite singular. The result is that the fpqc quotient GS'/Gs has the structure of indfinitetype indflat indscheme over XS. In particular, it is reasonable, and G' G s is an indflat cover and thus coreasonable. 5 That is, they have finitely generated ideal sheaves. 41
42 On Rpoints, we may identify Gs'/Gs(R) x c Xs(R) =1(x, E, f) S a principal Gbundle over As(x) f a trivialization of C over As'() Here the symbol '/ ' means 'taken up to isomorphism', i.e. we identify two Rpoints (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 G '/Gs is indprojective over Xs if and only if G is reductive. 2. Gs'/Gs is indreduced if and only if G has no nontrivial characters. Remark Ultimately we are concerned only with the analytifications of these indschemes, so point (2) appears merely for interest's sake. But point (1) is crucial for the definition of convolution in BorelMoore homology. We may reidentify the Rpoints 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 Gbundle 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 coreasonable. One readily checks by looking at points that the induced maps Gs"/Gs XS x s Gs''/Gs' 42
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