The perverse t-struture Milan Lopuhaä Marh 15, 2017 1 The perverse t-struture The goal of today is to define the perverse t-struture and perverse sheaves, and to show some properties of both. In his talk Ben already defined D b (X, Q l ), where X is separated of finite type over a field k in whih l is invertible. We also want to onsider a similar ategory over the omplex numbers. Definition 1.1. Let X be a omplex algebrai variety. The ategory D b (X an, Q) is the full subategory of K D(X an, Q) that are bounded and are loally onstant with respet to some algebrai stratifiation of X, i.e. there is some finite deomposition X = i X i of X into loally losed subshemes of X suh that for every i and every integer n the sheaf j i Hn K is loally onstant, where j i : X i X is the inlusion morphism. Throughout this talk, X is either of the type of Bens's talk, or a omplex algebrai variety. Definition 1.2. Let X be as before. Then a omplex K D b is in p D b, 0 if for every point x X with inlusion i x : Spe(κ(x)) X and every j > dim(x) we have H j (i XK) = 0. Similarly, a omplex K D b is in p D b, 0 if for every point x X with inlusion i x : Spe(κ(x)) X and every j < dim(x) we have H j (i! XK) = 0. Remark 1.3. Another way to formulate the perverse t-struture is: where D is the Verdier duality. Remark 1.4. Let K D b, and let U B p D b, 0 B p D b, 0 K p D b, 0 K p D b, 0 dim supph i B i i, dim supph i DB i i, j X be an open subset of X. Let F j K p D b, 0 j! K p D b, 0 and i X p D b, 0 ; and i! X p D b, 0. i X be its omplement. Then If U is dense and universally smooth (i.e., (U k) red is smooth over k), then this simplifies to K p D b, 0 K p D b, 0 j K D b, dim(u) j! K D b, dim(u) and i X p D b, 0 ; and i! X p D b, 0. 1
Theorem 1.5. Let X be as above. Then ( p D b, 0, p D b, 0 ) is a t-struture on D. b Remark 1.6. This definition depends on the so-alled middle perversity, i.e. the funtion X Z given by x dim(x). Another suh a funtion satisfying some onditions is alled a perversity, and different perversities give rise to different t-strutures. In this seminar we will only disuss the middle perversity. Proof of Theorem 1.5. We hek the three properties of t-strutures: 1. If K p D b, 0 and L p D b,>0, then Hom(K, L) = 0: We prove this by indution on dim(x). It is lear for X zerodimensional. Suppose X is of dimension n and that the statement has been proven for all dimensions smaller than n. From the distinguished triangle j! j! K K i i K we get an exat sequene Hom(i i K, L) Hom(K, L) Hom(j! j! K, L). By the adjuntions we know, the first term is isomorphi to Hom(i K, i! L), whih is zero beause of the indution hypothesis. Also, the last term in the sequene is isomorphi to Hom(j B, j! C), whih again is zero, beause j B D b, dim(u) and j! C D b,> dim(u). 2. p D b, 0 p D b, 1, p D b, 1 p D b, 0 : this is lear. 3. There exists a distinguished triangle A K B with A p D b, 0 and B p D b,>0 : again we proeed by indution on dim(x). If dim(x) = 0, the proposition is lear. Now assume X has dimension d and that the statement has been proven for all shemes of dimension < d. Let K be a omplex in D b, and let U be an essentially smooth (i.e. (U k) red is smooth over k) dense open subsheme of X on whih the H n K are loally onstant. Then τ dim(u) K U p D b, 0 τ > dim(u) K U p D b,>0 (U), so τ dim(u) K U K U τ > dim(u) K U (U) and is a distinguished triangle on U that defines the required distinquished triangle on U. Let F be the omplement of U; then dim(f ) < dim(x), so by the indution hypothesis the perverse t- struture is indeed a t-struture on F, and there exists a distinguished triangle p τ 0 K F K F p τ >0 K F that is our required distinguished triangle on F. Now, by what we know of reollements, we an glue the standard t-struture on U and the perverse t-struture on F to find a t-struture on D b, and in this t-struture a triangle A K B that lifts the two triangles mentioned before. Although the t-struture we get in this way is not the perverse t-struture, the triangle is still distinguished, and in fat A p D b, 0 and B p D b,>0 by Remark 1.4 Definition 1.7. The abelian ategory of perverse sheaves Perv(X) is the heart of the perverse t-struture. 2
2 t-exat funtors Definition 2.1. Let D 1 and D 2 be two t-strutured ategories with hearts C 1 and C 2, and let f : C D be a morphism of ategories; then we denote p f = H 0 f ɛ. Here H 0 = τ 0 τ 0. Definition 2.2. Let D 1 and D 2 be two t-strutured ategories with hearts C 1 and C 2, and let f : C D be amorphism of triangulated abelian ategories. We say that f is left t-exat if f(d 0 1 ) D 0 2, right t-exat if f(d 0 1 ) D 0 2, and t-exat if both are true. Lemma 2.3. Let D 1 and D 2 be two t-strutured ategories with hearts C 1 and C 2, and let f : C D be a morphism of ategories. Proof. 1. If f is (left, right) t-exat, then p f is (left, right) exat. 2. If f is left (right) t-exat and K is an element of D 0 1, then p fh 0 K H 0 fk. 3. Suppose f has a left adjoint g : D 2 D 1. Then g is right t-exat if and only if f is left t-exat, and in this ase ( p g, p f) form an adjoint pair. 4. If both f and some h: D 2 D 3 are (left, right) t-exat, then h f is as well, and p (h f) = p h f. 1. Let 0 X Y Z be a short exat sequene in C 1 ; then fx, fy, fz D 2 0, so the long exat ohomology sequene gives 0 H 0 fx H 0 fy H 0 fz. The statement on right exatness is dual to this one. 2. If K D 0 1, then H0 K K τ >0 K is a distinguished triangle, hene so is fh 0 K fk fτ >0 K. Sine fτ >0 K is an element of D 2 >0, the long exat sequene gives an isomorphism H 0 fh 0 K H 0 fk. 3. Suppose f is left t-exat, and let U D 1 >0 and V D 0 2. Then Hom(gV, U) = Hom(V, fu) = 0. Sine this is true for all U, one has τ >0 gv = 0, hene gv D 0 1, so g is right t-exat. For A C 1 and B C 2, we find H 0 gb = τ 0 gb and H 0 fa = τ 0 fa; this gives a funtorial isomorphism Hom(H 0 gb, A) = Hom(gB, A) = Hom(B, fa) = Hom(B, H 0 fa). 4. The first point is trivial; furthermore for every A C 1 one has by point 2. p (h f)a = H 0 hfa = H 0 hh 0 fa 3
3 t-exatness in the geometri setting Proposition 3.1. Let X be as before, let U X be a Zariski open on X, and let F Consider the perverse t-struture on all shemes. j 1. j! and i are right t-exat, j and i! are left t-exat, and j (= j! ) and i (= i! ) are t-exat. 2. There are adjuntions ( p i, p i ), ( p i!, p i! ), ( p j!, p j! ), ( p j, p j ). 3. The ompositions p j p i, p i p j!, p i!p j are zero. 4. For A Perv(F ) and B Perv(U) one has 5. For A Perv(X) the sequenes Hom( p j! B, p i A) = Hom( p i A, p j B) = 0. p j! p j A A p i p i A 0 i X be its losed omplement. and are exat. 0 p i p i! A A p j p j A Proof. 6. p i, p j! and p j are fully faithful, i.e. the natural transformations p i p i id p i!p i and p j p j id p j p j! are isomorphisms. 1. By definition of the perverse t-struture, j = j! is t-exat, i is right t-exat, and i! is left t-exat. By Lemma 2.3.3 i = i! is t-exat, j! is right t-exat, and i! is left t-exat. The result now follows from Lemma 2.3.1. 2. This now follows diretly from Lemma 2.3.3. 3. This follows from j i = 0 (et). 4. This is a diret onsequene of the former statement. 5. This follows from the fat that and are distinguished. j! j A A i i A i i! A A j j A 6. This follows from the fat that these are isomorphisms without the p. Proposition 3.2. Let f : X Y be a quasifinite morphism. Then f! and f are right t-exat, and f! are f are left t-exat. 4
Proof. Let K D b (X). One has that K D b, 0 (X) if and only if dimsupph i K i. f! is exat on sheaves, so H i Rf! K = f! H i K, so SuppH i f! K = f(supp H i K) and dim Supp H i f! K = dim Supp H i K, whih proves that f! is right t-exat. The proof for f is similar, and the result for f! and f follow by adjuntion. Theorem 3.3. Let X and Y be separated shemes of finite type over k, and let l be a prime invertible in k. If f : X Y is an affine morphism, the funtor f : D b (X, Q l ) D b (Y, Q l ) is right t-exat. Proof. Let F be a onstrutible sheaf (on X or Y ), and let d(f ) be the smallest integer suh that F p D d(f ). If K is an objet in the derived ategory, then we define d(k) = sup(i + d(h i K)); then again d(k) is the smallest d suh that K p D d(k) ; hene K p D 0 if and only if all the H i K[ i] are. Hene we need to show that if F is a sheaf suh that d(f ) d, then d(r i f F ) d i; but this is proven in (SGA4, XIV 3.1). Corollary 3.4. With the notation as above, f! is left t-exat. Proof. This follows from Verdier duality. Corollary 3.5. If f : X Y is quasi-finite and affine, the funtors f! and f are t-exat. 5