My main interest has been the study of the local cohomology with support in the Schubert variant in G/B. In 1981, it was proved by Brylinski-Kashiwara in [12] that in the category of holonomic B-equivariant leftDX modules, which are quasi-coherent OX modules (denoted DX−mod), that the simple modules are parametrized by Schubert varieties, and if we let L(w) ∈ DX −mod denote the simple module with support in the Schubert variant X(w) and codim(X(w)) =cw that there is an inclusion. The main question we investigate in Sections 4.6 and 4.7 is to find [HXj (w)(OX) :L(vi)], and we will give a full description of the above in one of these three cases.
One of the other two cases will not be considered, but the last one will be fully considered in the case Zi = {1,2,. We generalize his result to the example of X irreducible smooth varieties and Y ⊂ X closed and irreducible, and give a condition on Y which guarantees that Hcodim(YY )(OX) is simple in DX-mod. This number is interesting because it locally gives a lower bound on the minimum number of generators of the ideal bundle X(a1, . . . , ar).
Notation
Acknowledgments
So, combining this definition with (2.2), we get that cd(I) gives a lower bound for the minimal set of generators forI.
Local cohomology of sheaves
We will need some exact sequences of local cohomologies and these are the lemmas below. We will also need some conditions for the annihilation of local cohomology bundles. In this section, we want to define the module of differential operators and consider some of its properties.
A ring A1 is a left fraction ring of A with respect to S if there exists a ring homomorphism Θ :A→A1 such that. We define a right fraction ring of A with respect to S in a similar way only replace Θ(s)−1Θ(a) by Θ(a)Θ(s)−1 and with that and denote it asAS−1. End(RU)(e)(RU) is a corresponding right and left fraction ring of D(R) with respect to U.
The sheaf of differential operators
M ∈ DX−modifM has a structure as a leftDX and Mis quasi-coherent module with respect to the induced structure OX =Dif0(OX). If F,G are quasi-coherent in OX -mod, then Difm(F,G) and Dif(F,G) are also quasi-coherent with respect to the left and right OX-module structure, and ifU ⊂X is open and affine. Given x ∈ X and U ⊂ X open and affine such that x∈U then (DX)x is both a left and right fraction ring of D(OX(U)) with respect to x andD(OX,x )≃( DX)x as rings.
Since OX =Dif0(OX) ⊂ DX is a bundle of subcycles OX,x ⊂ (DX)x is a subring and since Θ((x)c) ⊂ OX,x follows from the first property, we have proved that (DX )x is the left ring of the fraction D(A) with respect to x. By the same reference, we obtain that this subcategory of DX-mods is closed under inclusion, quotients, and expansions and if M ∈ DX-mod is holonomic, it has finite length and OX is holonomic. Given M ∈ DX −mod, we will denote DX ⊗DX(X) M(X) as the sheaf associated with the presheaf.
Kashiwaras equivalence
To prove the first part that follows, for given U ⊂X open∀V ⊂Xopen and U∩Y ⊂V ⊂U the restriction map, all that remains to be proved. By tensoring with ⊗OYωY, it follows from Proposition 3.2.5 that there is an isomorphism inmod− DY. Using the same spectral sequence argument used at the end of the proof of Lemma 3.3.4, we get that the two spectral sequences.
After tensorizing both sides with ⊗OYωY, it follows from Proposition 3.2.5 that we only need to prove that there is an isomorphism in mod− DY ∀j ≥ codim(Y). Let us begin by proving that there is an isomorphism in mod − R HomOX(i∗(OY),HjZ(OX)O. Let K(X) be equal to the quotient of the free abelian group generated by all sheaves in the DX−modby subgroup that are generated by all expressions of the form F − F1− F2 whenever there is an exact sequence in DX −mod.
G/B in general
According to Kazhdan-Lusztig conjecture we proved in [1], [12] that the simple modules in DX−mod are parametrized by the Schubert varieties and further, if we let L(w) be the simple module parametrized by X(w) (meaning that Supp(L(w)) =X(w)), that in the Grothendieck group. So if M ∈ DX −mod, it follows since M has a finite decomposition series, that∃aw∈Nso. Given M ∈ DX−mod, we call (4.5) the character formula for M and to find it, we need to know allaw.
The first rule below follows as a consequence of Theorem 4.1 in [4], where one must be aware, that he gives an outline of the proof at the end of his proof and the second is a consequence of Theorem 8.5 in [12 ].
G/B versus G/P
Sl n /B
So, if only the first part of the statement is true, the second follows due to Lemma 4.7.3. Then we define F∗eM ∈R−modas as an R-module, which as an abelian group is just R, but r.m := rpem, where we have used the usual structure R on M on the right. If R is F-finite, say that R is strongly F-regular if for every c∈R not in any minimal prime of R there exists e≥0 such that it maps R-modules.
The absolute Frobenius morphism on X is the morphismF :X→X of schemes, which is the identity on the set of points, but is the associated map of sheaves. For an effective Cartier divisor D, we may denote the associated portion of the line bundle OX(D). We say X is stably Frobenius distributed along D, if there is a positive integer e, such that the map is inOX−mod.
If D′ ≤D and X is stable Frobenius split along D, then X is stable Frobenius split along pD, and X is also stable Frobenius split along D'. X is stably Frobenius partitioned along some abundant effective divisor such that the open setX\D is locally strongly F-regular. Since regular local rings are strongly F-regular, which is proved in [26], we get that if X is smooth, X is globally F-regular if and only if X is stable Frobenius divided along an abundant effective divisor.
Since OX(D) is a locally free bundle in OX−mod, we get that the map inOX −mod. By Proposition 5.1.3, we must show for all D effective Cartier divisors over Y, that Y is Frobenius stably divided along D. It then follows from Proposition 5.1.3, that there is an integer positive, such that the map is in OX -mod.
Schubert varieties and F-regularity
But by combining Lemma 5.1.4 and Proposition 5.1.3, we obtain that Z(w) is a stable Frobenius split along any divisor of the formPr. The module structure on the left side is the one derived from the left module structure Re, and in the tensor product we use the right module structure Re, so that r⊗Rr1m=r1per⊗Rm∀r, r1 ∈R, m∈M. By Theorem 5.2.2, X(w) is globally F-regular, so OX(w),x is strongly F-regular ∀x∈X(w) and then the proposition follows.
If char(k) = 0, then the coefficients in the proposition above would be equal to the inverse Kazhdan-Lusztig polynomials evaluated in 1. The two main ingredients of the proof are the Grothendieck-Fätter complex on Gr(r, n) and result given in [ 10]. In the rest of this section, we will write cd(Y) instead of cdGr(r,n)(Y), and we will also use Lemma 2.2.10 often without giving a reference.
According to the discussion at the beginning of section 4.3, wI is the longest element in the subgroup of W generated by sα1. To prove Theorem 6.4.1, the following Theorem together with the Lemma above and induction will prove to be sufficient. Its proof is rather technical and can be found in the next section.
If only one could get a similar result in a general setting, we might be able to answer the question above, but we have not been able to achieve this. We would like to generalize our arguments to find cdGr(r,n)(X(a1, . . . , ar)), but, as the example at the end of Section 6.2 shows, our methods must. So in the summation in proposal A.1.4 is among these, we have to do a search.
We want to know the kernel of this ring homomorphism and its image, and in case R is a finitely generated k-algebra, this can be achieved. That ⇒ is true follows from the definition of Dm(R). and the lemma follows by induction in s = Pn. Since R is a finitely generated k-algebraR=k[x1,. The proof of this implication is an induction proof of the order of the differential operator. r >0, and it is proven for all differential operators of order less than
This is an induction proof in m = Pn. and we have proved the first rule of the Lemma. We will only prove the first equality in the second rule, since the proof of the second is the same.
As a consequence of Lemma 3.2.8, its support is a union of B-orbits and thus a union of Schubert varieties and then ⇒ is a consequence of (6.3) in section 6.1. In [10] Bruns and Schw¨anzl found the cohomological dimension of the ideal generated by all r ×r minors in an n×m polynomial ring defined over a field of characteristic zero. The Proposition below simply uses this result to find cd(Xs), which is therefore known.
If the rate were a codimension, this could never be the case thanks to Theorem 4.1 in [4]. Let's finish this chapter by showing why our above methods sometimes fail when we are in a different situation. Let us take X(2, a, n) with a >3 and thereby show that the methods used above need to be improved.
To do this, we first find the parabolic Kazhdan-Lusztig polynomials, which have been described in Appendix A.1. We obtain the following corollary, whose proof is exactly identical to the proof of Lemma 6.3.3. So now we know all Kazhdan-Lusztig polynomials and using this information we get the following Lemma which is crucial to find the sign formula for HcXwr,k,nwI(w . r,k,nwI)(OG/B).
The proof is an induction proof inr, and we must therefore first prove it forr = 2. We have not dealt with this last possibility, but we still have a conjecture. But since we were unable to generalize these arguments further, they are not included.
It is an A-algebra, which is free as an A-module and is generated by {Tw}w∈W and the multiplication is defined as.
