On the Grothendieck groups of toric stacks

On the Grothendiec k groups of toric stac ks Zheng Hua Marc h 2, 2022 1 In tro duction In this note, w e prov e that the G rothendiec k group of a smo o t h complete toric Deligne-Mumford stac k is torsion free. This statemen t holds when the generic p oin t is stac ky . W e a lso construct an example of op en t oric stac k with torsion in K-theory . This is a part of the author’s P h.D t hesis. A similar result has been prov ed b y Goldin, Harada , Holm, Kimura a nd Kn utson in [GHHKK] using sy mplectic methods. 2 Grothen d iec k groups of re d uced stac ks Let N b e a free a b elian group o f rank d and N R = N ⊗ R . Giv en a complete simplicial fan Σ in N R , one c ho oses an in tegral elemen t v i in eac h of the one-dimensional cones o f Σ. This defines a stac ky fan Σ in the sense of [BCS]. W e denote the corresp onding toric D eligne-Mumford stac k b y X Σ . Recall the G rothendiec k gro up K 0 ( X Σ ) is defined to b e the free ab elian group generated by all formal com binations of coheren t shea ves on X Σ mo dding out b y the short exact s equences. Each elemen t v i corresp onds to a toric in v arian t divisor E i . This divisor E i determines an in v ertible sheaf O ( E i ). W e denote its equiv alent class in K 0 ( X Σ ) b y R i . The ring structure of K 0 ( X Σ ) is giv en b y tensor pro duct of coheren t shea v es. K-theory o f smo oth toric stac ks has b een studied in [BH ]. In part icular they computed K 0 ( X Σ ) explicitly by writing out its generator s and relations. Theorem 2.1. [BH] L et B b e the quotient of the L aur ent p olynomial ring Z [ x 1 , x − 1 1 , . . . , x n , x − 1 n ] by the i d e al gener ate d by the r elations 1 • Q n i =1 x h m,v i i i = 1 for any dual ve ctor m ∈ M = H om ( N , Z ) , • Q i ∈ S (1 − x i ) = 0 for any set S ⊆ [1 , . . . , n ] such that { v i | i ∈ S } ar e not c ontaine d in any c one of Σ . Then the map f r om B to K 0 ( X Σ ) which sends x i to R i is an isomorphism o f rings. Our main result is the following. Theorem 2.2. Th e Gr othendie ck gr oup K 0 ( X Σ ) o f a c omplete smo oth toric Deligne-Mumfor d stack X Σ is a fr e e Z mo dule. Pr o o f. W e denote the La uren t po lynomial ring Z [ x 1 , x − 1 1 , . . . , x n , x − 1 n ] by R . Let A = R/I , where I is generated by Q i ∈ S (1 − x i ) = 0 f or an y set S ⊆ [1 , . . . , n ] suc h that { v i | i ∈ S } are not con tained in any cone of Σ. And B = A/J , where J is generated by n Lauren t p olynomials h j = Q n i =1 x h m j ,v i i i − 1 where m j is an in tegral basis of M . First w e w an t to replace h j b y g j = Q > 0 x i − Q < 0 x − i . They generate the same ideal J but this collection a v oids n egat ive p ow- ers. T o prov e B is a free Z module w e need to show that the m ultipli- cation map B → pB is an injection for an y prime p . Let K ( g 1 , . . . , g d ) and K ( g 1 , . . . , g d , p ) b e the Koszul complexes for sequenc es g 1 , . . . , g d and g 1 , . . . , g d , p of elemen ts of the ring A . Thes e t w o Koszul complexes are re- lated b y the follo wing lemma. Lemma 2.3. [E] L et φ : K ( g 1 , . . . , g d ) → K ( g 1 , . . . , g d ) b e the map of multi- plic ation by p . Then K ( g 1 , . . . , g d , p ) e quals C one ( φ )[ − 1] . Her e C one me ans mapping c one of c omplexes. Pr o o f. See corollary 17.11. of [E]. According to this lemma, w e get a long exact sequenc e of cohomology groups: . . . − − − → H i ( K ( g 1 , . . . , g d , p )) − − − → H i ( K ( g 1 , . . . , g d )) φ − − − → H i ( K ( g 1 , . . . , g d )) − − − → H i +1 ( K ( g 1 , . . . , g d , p )) − − − → . . . (2.1) 2 W e will sho w that all the cohomology gro ups of K ( g 1 , . . . , g d ) and K ( g 1 , . . . , g d , p ) v anish except at one p osition. More precisely , the only non v anis hing piece of (2.1) is: 0 − − − → H n ( K ( g 1 , . . . , g d , p )) ∼ = B p − − − → H n ( K ( g 1 , . . . , g d )) ∼ = B − − − → H n +1 ( K ( g 1 , . . . , g d , p )) ∼ = B /pB − − − → 0 T o pro v e this w e need a result ab out Cohen-Macaula y prop erties of Stanley- Reisner rings. Theorem 2.4. L et A ′ = Z [ x 1 , . . . , x n ] /I . Ring A ′ is Cohen- Mac aulay. Pr o o f. If we mak e a c hange of v ariables x i to 1 − x i , then w e see that A ′ is nothing but t he Stanley-Reisner ring asso ciat ed to supp orting po lytop e of Σ. It is a general fact that the Stanley-Reisner ring of p olytop es are CM ov er an y field(See Chapter 5 of [BrHe]). F urt hermore one can sho w it is actually CM o ver Z (See Exercise 5.1.25 of [Br He ]). W e will sk etc h the solution of this exercise in the follo wing remark. Remark 2.5. Consider the flat morphism Z → A ′ . F or an y maximal ideal q ⊂ A ′ , we ha v e q ∩ Z = ( p ). In order to sho w A ′ is CM it suffices t o c hec k it for eac h fib er, i.e. A ′ q /pA ′ q is CM for all the maximal ideal q . If ( p ) is not (0) then A ′ q /pA ′ q = ( A ′ ⊗ Z /p Z ) q . It is CM b ecause Stanley-Reisner ring o v er the field is CM. So w e just need to show that for any maximal ideal q , the restriction q ∩ Z is not (0). Supp ose this is the case, w e will hav e an inclusion Z → A ′ / q . Ho w ev er, since w e a ssume q ∩ Z = (0), the field A ′ / q m ust ha v e c haracteristic zero. But this con tra dicts the fact that A ′ is finitely generated o v er Z b ecause Q is not finitely generated ov er Z . Corollary 2.6. The rin g A is Cohen-Mac aulay. Pr o o f. Because A is a lo calization of A ′ and b eing CM ring is a lo cal prop erty , A is CM by Theorem 2.4. Remark 2.7. It f ollo ws from the general theory of Stanley Reisner ring (Theorem 5 . 1 . 16 of [BrHe]) t ha t A ′ has Krull dimension d + 1. Lemma 2.8. [E] Supp ose M is a finitely gen er ate d mo dule over ring A and I = ( x 1 , . . . , x n ) ⊂ A is a pr op er ide al. If depth ( I ) = r then H i ( M N K ( x 1 , . . . , x n )) = 0 for i < r , while H r ( M N K ( x 1 , . . . , x n )) = M /I M . Lemma 2.9. The quotient A/J is a finitely gener ate d ab elian gr oup. 3 Pr o o f. Let k b e a n y field and f b e an arbitrary map from A/J t o k . Maximal ideals of A/J are in one to one corresp ondence with suc h map f . W e w an t to solve for ( x 1 , . . . , x n ) tha t satisfy equations in ideal I and J in the field k . Recall elemen ts of ideal I a r e in form of Q i ∈ S (1 − x i ) for any subset S ⊆ [1 , . . . , n ] suc h that one dimensional rays v i , i ∈ S are not contained in an y cone of Σ. So x i equals 1 outside some cone σ . Then equations in J reduce t o Q v i ∈ σ x h m,v i i i =1. W e can c ho ose the dual v ector m suc h that h m, v i i = 0 for all but o ne i . Say h m, v i i = d i . The n um b er d i only dep ends on the fan but not on the field k . This implies that 1 − x d i i maps to 0 for an y map f from A/J to k , i.e. 1 − x d i i is con tained in an y maximal ideal of A/J . Because A/J is a finitely generated Z alg ebra the Jacobson radical coincides with nilra dical. So (1 − x d i i ) N =0 fo r any i . W e can pic k a large enough in teger N uniformly for any x i suc h that there exists a Z basis consisting of monomials with pow ers o f eac h x i b et w een 0 and N d i . This pro ves the statemen t of the lemma. By theorem 2.4, remark 2.7 and lemma 2.9 w e can pro v e: Corollary 2.10. The id e al J = ( g 1 , . . . , g d ) has depth d . Pr o o f. Because A is CM, by the definition of CM rings depth ( J ) = codim ( J ). The quotien t A/J is finitely generated ov er Z , therefore, of Krull dimension one. By remark 2.7 codim ( J ) = d and depth ( J ) = d . This corollary ab ov e together with lemma 2.8 imply the Koszul complex K ( g 1 , . . . , g d ) ha s only one nonzero cohomology H d ( K ( g 1 , . . . , g d )) = B = A/J . On the other hand, the lemma 2 .9 imples B /pB is a finite dimensional v ector space ov er Z /p . By similar argumen t with the corollary ab o v e w e get depth ( J, p ) = d + 1. Then H i ( K ( g 1 , . . . , g d , p )) = B /p B when i = d + 1 and zero o t herwise. Now b y applying the long exact sequence (2.1) we pro ve the m ultiplication map b y p is an injection. This finish the pro of of theorem 2.2. Remark 2.11. The pro of of theorem 2.2 can b e generalized to the non complete toric stac ks satsifying a condition called “shellabilit y”. This is a com binatorial condition on the underlying simplicial complex of the toric stac k(See [BrHe] fo r details of this definition). It is prov ed in [BrHe] that Stanley-Reisner rings of shellable simplicial complexes are Cohen-Macaula y . Ho w ev er, w e will see in Chapter 4 that Grothendiec k groups of op en toric stac ks are not neces sarily free. 4 3 Grothen d iec k groups of non-reduce d s tac ks No w w e remo v e the a ssumption that N is a free ab elian group. Then the corresp onding toric stac k will hav e non trivial stabilizer at t he generic p oint. W e will generalize theorem 2.2 to this setting. Recall the deriv ed Gale dual of the homomorphism β : Z n → N is the homomorphism β ∨ : ( Z n ) ∨ → D G ( β ). When N is torsion free, D G ( β ) is the Picard group. The g eneral definition of DG ( β ) in volv es a pro jectiv e resolution of N . W e refer to [BCS] for details. Theorem 2.1 can b e generalized to the case when N has torsion. Notice the ring Z [ x 1 , x − 1 1 , . . . , x n , x − 1 n ] /J is the represen tation ring of t he algebraic group H om ( D G ( β ) , C ∗ ) when N is torsion free. If N has torsion then H om ( D G ( β ) , C ∗ ) maps to ( C ∗ ) n with finite k ernel. After replacing Z [ x 1 , x − 1 1 , . . . , x n , x − 1 n ] /J b y the represen tatio n ring of H om ( D G ( β ) , C ∗ ) w e can generalize Theorem 2.1 to no n reduced case(See section 6 of [BH] for more details). Theorem 3.1. L et N b e a finitely gener ate d ab elian g r oup a n d Σ is a stacky fan in N . T h e Gr othendie ck Gr oup K 0 ( X Σ ) is a fr e e Z mo dule. Pr o o f. Let’s denote t he N f r e e for the quotien t N/ tor sion ( N ) and X r ed for the reduced stac k asso ciated to N f r e e . Recall the Grothendiec k g r oup K 0 ( X Σ ) is the quotien t of represen tation ring of H om ( DG ( β ) , C ∗ ) b y the ideal I generated b y Stanley-Reisner relations. Let’s denote the Gale dual group of the reduced stac k X r ed b y DG ( β r ed ). The quotient map N → N f r e e induces an inclusion on Gale dual gro ups D G ( β r ed ) → D G ( β ), whose cokerne l is isomorphic to tor sion ( N ). Now we see the G rothendiec k groups K 0 ( X Σ ) a nd K 0 ( X r ed ) a r e isomorphic to the g roup rings Z [ D G ( β )] and Z [ D G ( β r ed )]. If w e fix a lifting from tor sion ( N ) to D G ( β ), then w e get a cose t decomposition D G ( β ) = ⊔ y ∈ tor sion ( N ) ( y D G ( β r ed )). This induce a coset decomp osition of t he group ring Z [ D G ( β )] suc h t ha t each coset is isomorphic with Z [ D G ( β r ed )]. Since Z [ DG ( β r ed )] is tor sion free b y theorem 2.2, w e prov e the theorem. 4 Grothen d iec k groups of non co mplete stac ks Theorem 2.1 holds for no n complete toric stac ks to o. But our pro o f for freeness of K-theory relies on the shellabilit y of the underlying simplicial complex of the toric stac k. There are man y no n complete t o ric stac ks whose underlying simplicial complexe s a r e not shellable. F or example , w e can take 5 P 1 × P 1 . D enote its fo ur toric in v arian t divis ors b y E 1 , E 2 , E 3 and E 4 . Let p oin t P (resp. Q ) b e the in tersection of E 1 and E 2 (resp. E 3 and E 4 ). Sim- plicial complex of P 1 × P 1 \{ P , Q } is not shellable. Actually , there are examples of non complete toric stac ks suc h that their Grothendiec k groups hav e torsions. The following example is due to Lev Boriso v. Example 4.1. Let’s take a dimension fiv e w eighted pro jectiv e stac k P ( 1 , 1 , 1 , 1 , 2 , 2). Denote its toric in v arian t divisors by E 1 , E 2 , . . . , E 6 , where E 1 , . . . , E 4 ha v e w eigh ts one and E 5 , E 6 ha v e w eigh ts t w o. L et X be the substac k P (1 , 1 , 1 , 1 , 2 , 2) \{ ( E 1 ∩ E 2 ∩ E 3 ∩ E 4 ) ∪ ( E 5 ∩ E 6 ) } . By theorem 2.1 K 0 ( X ) = Z [ t, t − 1 ] h (1 − t ) 4 , (1 − t 2 ) 2 i It is easy to c hec k that t (1 − t ) 2 is a torsion elemen t. References [BCS] L.A. Borisov, L. Chen, G.G. Smith, The o rb ifold Chow ring of toric Deligne-Mumfor d stacks. J. Amer. Math. So c. 18 (20 05), no. 1, 193– 215. [BH] L.A. Boriso v, R.P . Horja, O n the K -the ory of smo oth toric DM stacks. Sno wbird lectures on string geometry , 21–42 , Contemp. Math., 401, Amer. Math. So c., Pro vidence, RI, 2006. [BrHe] W. Br uns, J. Herzog, Cohen - Mac aulay rings . Cam bridge St udies in adv anced mathematics 39. Cam bridge Univ. Press, Cam bridge, 1993. [E] D. Eisen bud, Commutative Algebr a with a View T owar d Algebr aic Ge ometry , G TM 1 50. [GHHKK] R. Goldin, M. Harada, T. Holm, T. Kim ura, A. K n utson. MSRI talk on workshop in Combinatorial, Enumer ative and T oric Ge o m - etry by T a r a Holm . Departmen t of Mathematics, Univ ersit y of Wiscons in- Madison, Madison, WI, 53706, U.S. h ua@math.wisc.edu 6