aa r X i v : . [ m a t h . AG ] D ec ACYCLIC COMPLEXES AND 1-AFFINENESS
DENNIS GAITSGORY AND SAM RASKIN
Abstract.
This short note is an erratum to [Gai], correcting the proof of one of its main results.It includes some counterexamples regarding infinite-dimensional unipotent groups and affine spacesthat may be of independent interest. A : “ colim n A n is not where wework relative to a field k of characteristic 0. The proof in loc. cit . is not correct, and the purposeof this note is to correct it.Along the way, we also give some (possibly new) counterexamples on representations of A considered as additive group, and the formal completion of a pro-infinite dimensional affine space atthe origin. These counterexamples will be a categorical level down, i.e., they will reveal pathologicalbehavior for QCoh rather than
ShvCat .1.2.
What is the problem in [Gai] ? In § loc. cit ., there is a claim that something is “easyto see,” with no further explanation. In fact, the relevant claim is not true. This failure invalidatesthe argument about 1-affineness given in loc. cit .1.3. Structure of the argument.
There has only ever been one successful strategy to provingthat a prestack is not 1-affine: showing that 1-affineness would imply some functor is co/monadic(usually by computing some tensor product of DG categories and applying the Beck-Chevalleyformalism), and then showing that the relevant functor is not conservative.We will exactly follow this strategy. Namely, in § § A is not 1-affine.The reader who is most invested in the non-1-affineness of A might prefer to read the statementof Theorem 1 and then skip ahead to 1.8, returning to read the proofs of the other results afterseeing their application.1.4. Invariants and ind-unipotent groups.
Our first main result is the following.
Theorem 1.
Γ :
QCoh p BA q Ñ Vect is not conservative.
In other words, the functor of invariants for ind-unipotent groups is not conservative, even thoughit is for unipotent groups.Moreover, the construction will produce a non-zero representation in the heart of the t -structurewhose invariants vanish, and whose higher group cohomologies vanish as well. That is to say, thisis an essential issue, not the sort resolved by some kind of renormalization procedure. Date : October 9, 2018. Recall, however, that there are some remarkably similar spaces that are A dR is 1-affine. This result is generalized (in a more sophisticated setup) in [Ras]. We refer to loc. cit . for the definition of this notion. We also follow the notational conventions of loc. cit ., soeverything should be understood in the derived sense. Note that BA is ind-smooth, in particular ind-flat, and therefore its QCoh indeed has an obvious t -structure. A “ k r t , t , . . . s , so Spec p A q is a pro-infinite dimensional affine space. Theorem 2.
There exists ‰ V P A – mod ♥ with the properties that: ‚ Each t i acts on V nilpotently (in fact, with square zero). ‚ Tor Aj p V, k q “ for all j , where k is equipped with the A -module structure where each t i actsby zero. The proof can be found in § Proof that Theorem 2 implies Theorem 1.
Since each t i acts on V nilpotently, the induced actionon V _ ‰ t i acting on V _ commute. Therefore, V _ has a canonical structure as an object of QCoh p BA q ♥ , since this is the abelian category ofa non-derived vector spaces equipped with Z ą -many locally nilpotent and pairwise commutingoperators.Finally, observe that: H j p Γ p BA , V _ qq “ Ext j QCoh p BA q p k, V _ q “ Tor Aj p k, V q _ “ (cid:3) Construction of V , and a heuristic. We now construct V from Theorem 2, and explainwhy the relevant Tor vanishes.Namely, V has a basis v S indexed by finite subsets S Ď Z ą , and we define t i ¨ v S “ v S zt i u if i P S and t i ¨ v S “ i R S . Clearly t i acts by zero on V .To see that Tor A p V, k q “
0, it suffices to see that there are no morphisms f : V Ñ k in A – mod ♥ .For any such f , we should show that f p v S q “ S as above. Since S is finite and Z ą isinfinite, we can find i R S . Then t i v S Yt i u “ v S , so f p v S q “ t i f p v S Y i q “ t i acts on k by zero. (A similar argument also works for Tor .)1.7. We now show that the higher Tors vanish as well. Proof of Theorem 2.Step . First, we rewrite the representation V in a more conceptual way.For each n ą
0, let I n Ď A be the ideal generated by t , . . . , t n and all t i for i ą n .Then we claim V “ colim n A { I n , where the structure morphisms: A { I n Ñ A { I n ` are given by multiplication by t n ` . Indeed, the relevant structure morphism sends A { I n to V bymapping the “vacuum vector” 1 P A { I n to v t ,...,n u in the right hand side, which is an injectionwhose image identifies with the subspace of V spanned by v S for S Ď t , . . . n u Ď Z ą . Step . Next, let m “ p t , t , . . . q Ď A , and observe that the morphism: We avoid geometric notation here so that A has a unique meaning in this text. And not merely locally nilpotently. Here and always we use b A for the derived tensor product, i.e., for L b A . CYCLIC COMPLEXES AND 1-AFFINENESS 3 A { I n b A A { m t n ` ¨ ÝÝÝÑ A { I n ` b A A { m is zero, i.e., is canonically nullhomotopic; indeed, this follows from the fact that multiplication by t n ` is zero on A { m .Passing to the colimit over n and using our earlier expression for V , we see that V b A A { m “ (cid:3) Application to non-1-affineness of A . We now show that A is not 1-affine. Proof of Theorem 2.4.5 from [Gai] . We follow the beginning of the proof of the theorem from [Gai]9.3.2.Namely, it suffices to show that the canonical functor:
Vect b QCoh p A q Vect Ñ QCoh p Spec p k q ˆ A Spec p k qq is not an equivalence.Let G m act on A by scaling. The above functor is a morphism of QCoh p G m q -module categories.By 1-affineness of BG m , it is equivalent to show that the above functor is not an equivalenceafter passing to G m -equivariant categories. Then using the “shift of grading trick” (c.f. [AG] A.2)and 1-affiness of BG m again, we see that it is equivalent to show that the functor: Vect b lim n ` Sym p k ‘ n r´ sq – mod ˘ Vect Ñ lim n p k b Sym p k ‘ n r´ sq k q – mod “ lim n Sym p k ‘ n r´ sq – mod “ QCoh p BA q is not an equivalence, where all structure maps in the limits are induced by the projections k ‘ n ` Ñ k ‘ n (so are given by tensor product functors everywhere).By the Beck-Chevalley formalism, the (discontinuous) right adjoint to the canonical functor: Vect “ Vect b Vect
Vect Ñ Vect b lim n Sym p k ‘ n r´ sq – mod Vect (which is induced by the canonical functors
Vect Ñ Sym p k ‘ n r´ sq – mod sending k to to the freemodule Sym p k ‘ n r´ sq , i.e., the trivial representation in QCoh p BA n q ) is monadic (c.f. [Gai] Lemma9.3.3).Therefore, it suffices to show that the right adjoint to the corresponding functor: Vect Ñ lim n p k b Sym p k ‘ n r´ sq k q – mod “ QCoh p BA q is not conservative (in particular, not monadic). But this functor corresponds to group cohomology,and Theorem 1 says that it is not conservative. (cid:3) Here
QCoh p G m q is equipped with the convolution monoidal structure. DENNIS GAITSGORY AND SAM RASKIN
References [AG] Dima Arinkin and Dennis Gaitsgory. Singular support of coherent sheaves and the geometric Langlands con-jecture.
Selecta Mathematica , 21(1):1–199, 2014.[Gai] Dennis Gaitsgory. Sheaves of categories and the notion of 1-affineness. In T. Pantev, C. Simpson, B. To¨en,M. Vaqui´e, and G. Vezzosi, editors,
Stacks and Categories in Geometry, Topology, and Algebra: , ContemporaryMathematics. American Mathematical Society, 2015.[Ras] Sam Raskin. On the notion of spectral decomposition in local geometric Langlands. arXiv preprintarXiv:1511.01378.
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