Isomorphisms Between Local Cohomology Modules As Truncations of Taylor Series
aa r X i v : . [ m a t h . A C ] M a y ISOMORPHISMS BETWEEN LOCAL COHOMOLOGY MODULES ASTRUNCATIONS OF TAYLOR SERIES
JENNIFER KENKELA
BSTRACT . Let R be a standard graded polynomial ring that is finitely generated over afield, and let I be a homogenous prime ideal of R . Bhatt, Blickle, Lyubeznik, Singh, andZhang examined the local cohomology of R / I t , as t grows arbitrarily large. Such rings areknown as thickenings of R / I . We consider R = F [ X ] where F is a field of characteristic0, X is a 2 × m matrix, and I is the ideal generated by size two minors. We give concreteconstructions for the local cohomology modules of thickenings of R / I . Bizarrely, theselocal cohomology modules can be described using the Taylor series of natural log.
1. I
NTRODUCTION
Let R be a graded Noetherian commutative ring and I a homogeneous ideal of R . Foreach integer t >
1, the rings R / I t are referred to as thickenings of R / I . The canonical surjec-tion from R / I t + to R / I t induces a degree-preserving map on local cohomology modules: H k m ( R / I t + ) j −→ H k m ( R / I t ) j Our focus is on local cohomology modules supported in the maximal ideal, m .In general, much of the work that has been done on local cohomology modules examineswhether the module is or is not zero. While they are useful, local cohomology modulesare defined homologically, and thus are difficult to work with concretely. They tend to belarge and, even when derived from simple rings, are rarely explicitly described. In a paperof Bhatt, Blickle, Lyubeznik, Singh, and Zhang, the authors examined when the inducedmaps on local cohomology modules in a fixed degree are isomorphisms for large valuesof t [BBL + +
19, DM17, DM18, DS18].In this paper, we will build on the work of [BBL + I . In the ring R / I t , such an infinite series behaves like a finitesum, as all but finitely many terms are zero. We use this idea to understand isomorphismsbetween local cohomology modules as truncations of a power series. In particular, we showthat in characteristic 0, H m ( R / I t ) is generated as a vector space over the base field by anelement represented by the Taylor series of natural log. This gives an explicit descriptionof the isomorphisms guaranteed in [BBL + p >
0, and show that the rank of H m ( R / I t ) grows arbitrarily large along an infinite subsequence of natural numbers t .This paper is structured as follows:(1) In Section 2, we set the scene by introducing a determinantal variety over a field ofcharacteristic 0. We recall the results of [BBL +
19] and verify this variety satisfiesthe hypotheses there given.
The author was supported by NSF grant DMS 1246989. (2) In Section 3, we introduce our protagonist, the local cohomology module of thick-enings of a determinantal variety. We show that, in degree 0, the vector space rankof this module is 1.(3) In Section 4, we explicitly construct an element of the local cohomology describedin Section 3. Based on our results from Section 3, we are ensured that we have abasis for this module as a vector space.(4) In Section 5, we consider the same variety over a field of positive characteristic.We explicitly construct elements of the module H m ( R / I t )) for all t >
1. In thatcontext, isomorphisms do not exist.2. I
SOMORPHISMS A RE G UARANTEED TO E XIST
For the entirety of this paper, we examine the ring R = F [ X ] where X is a 2 × F is a field: R = F (cid:20) u v wx y z (cid:21) . Let I be the ideal generated by the size two minors of the matrix X , that is, the elements; ∆ = vz − wy , ∆ = wx − uz , ∆ = uy − vx . The field F will be of characteristic 0, except in Section 5.In this section, we will repeat the theorem of [BBL +
19] that guarantees an isomorphismbetween local cohomology modules of thickenings. We will then verify that the ring R andthe ideal I satisfy the hypotheses when the cohomological index, k , is equal to 3. Theorem 2.1. [BBL + Let X be a closed lci subvariety of P n over a field of characteris-tic 0, defined by a sheaf of ideals I . Let X t ⊂ P n be the t-th thickening of X, i.e., the closedsubscheme defined by the sheaf of ideals I t . Let F be a coherent sheaf on P n that is flatalong X. Then, for each k < codim ( Sing X ) , the natural mapH k ( X t + , O X t + ⊗ O P n F ) −→ H k ( X t , O X t ⊗ O P n F ) is an isomorphism for all t ≫ . In particular, if X is smooth or has at most isolatedsingular points, then the map above is an isomorphism for k < dim X and t ≫ . Rather than considering this theorem in the setting of sheaf cohomology, as it originallyappears, we consider the local cohomology setting:
Theorem 2.2. [BBL + Let R be a standard graded polynomial ring that is finitely gener-ated over a field R of characteristic 0. Let m be the homogeneous maximal ideal of R. LetI be a homogeneous prime ideal such that R / I is a locally complete intersection on the set
Spec R \{ m } , and let k be a natural number such that k < dim ( R ) − height ( Sing R / I ) .Then, for a fixed natural number j, the maps between the modules H k m ( R / I t + ) j andH k m ( R / I t ) j are isomorphisms for sufficiently large t. We first determine in which indices the hypotheses of Theorem 2.2 apply. For a generaltreatment of determinantal rings, see [BV88]. In particular, Proposition 1.1 of [BV88]gives that the ring R / I has dimension 4, and Theorem 2.6 gives that the localization of R / I at a prime ideal p is regular if and only if p is not equal to the maximal ideal m .Since R / I is regular when localized away from the maximal ideal, the ring R / I is alocally complete intersection on the punctured spectrum. On the other hand, ( R / I ) m is notregular, so we have that Sing ( R / I ) = m and thus height ( Sing ( R / I )) is the dimension of R / I , which is 4. Therefore Theorem 2.2 applies for cohomological indices k ≤ SOMORPHISMS BETWEEN LOCAL COHOMOLOGY MODULES AS TRUNCATIONS OF TAYLOR SERIES 3
3. T
HE RANK OF H m ( R / I t ) IS ONE
Hochster and Eagon showed that determinantal rings are Cohen-Macaulay in [HE71],specifically, in our case, the local cohomology modules H k m ( R / I ) are zero at every cohomo-logical index k =
4. However, the successive thickenings, R / I t , are not Cohen-Macaulayfor all t greater than 1; see [DEP80].As is stated in Proposition 7.24 of [BV88], the depth of the ring R / I t is at least threefor all t . Since the rings R / I t are not Cohen-Macaulay for all t ≥
2, this implies that thedepth of R / I t must be exactly three. Therefore the module H m ( R / I t ) is nonzero for allthickenings with t ≥
2. As we shall see, H m ( R / I t ) is a rank 1 vector space over F for each t ≥
2. Towards proving this, we first recall the following theorem.
Theorem 3.1. [LSW16]
Let R = Z [ X ] be a polynomial ring, where X is an m × n matrixof indeterminates. Let I d be the ideal generated by the size d minors of X. If ≤ d ≤ min ( m , n ) and d differs from at least one of m and n, then there exists a degree-preservingisomorphism H mn − d + I d ( Z [ X ]) ∼ = H mn m ( Q [ X ]) . Applying Theorem 3.1 when X is an 2 × I is the idealgenerated by size 2 minors of X gives:(3.1.1) H I ( Z [ X ]) ∼ = H m ( Q [ X ]) We are considering the ring F [ X ] where F is some field of characteristic 0, not the ring Z [ X ] . However, we claim the above isomorphism implies that the modules H I ( F [ X ]) and H m ( F [ X ]) are isomorphic. To see this, first tensor both sides of Equation 3.1.1 with themodule F [ X ] to get(3.1.2) F [ X ] ⊗ Z [ X ] H I ( Z [ X ]) ∼ = F [ X ] ⊗ Z [ X ] H m ( Q [ X ]) . Note that, since F is a field of characteristic 0, the module F [ X ] is flat over Z [ X ] . Lemma 3.2. [ILL + Let I be an ideal of a ring R, and let M be an R-module. Then ifR −→ S is flat, there is a natural isomorphism of S-modulesS ⊗ R H jI ( M ) ∼ = H jIS ( S ⊗ R M ) . Applying Lemma 3.2 to Equation 3.1.2 gives that F [ X ] ⊗ Z [ X ] H I ( Z [ X ]) ∼ = H I F [ X ] ( F [ X ] ⊗ Z [ X ] Z [ X ]) ∼ = H I ( F [ X ]) . Similarly F [ X ] ⊗ Z [ X ] H m ( Q [ X ]) ∼ = H m ( F [ X ] ⊗ Z [ X ] Q [ X ]) ∼ = H m ( F [ X ]) . Proposition 3.3.
The module H m ( R / I t ) is an F -vector space of rank 1 for all t.Proof. Let ω R denote the canonical module of R and let E R ( R / m ) denote the injective hullof the residue field, R / m . As R is a Gorenstein ring of dimension 6, the canonical module ω R is isomorphic to R ( − ) . The injective hull E R ( R / m ) is isomorphic to H m ( R )( − ) .Local duality gives that H m ( R / I t ) ∨ ∼ = Ext R ( R / I t , ω R ) , JENNIFER KENKEL where ( − ) ∨ indicates Hom R ( − , E R ( R / m )) .An R -homomorphism in Hom R ( H m ( R / I t ) , E R ( R / m )) j is determined by the preimage ofelements in E R ( R / m ) of degree zero. Therefore the rank of Hom R ( H m ( R / I t ) , E R ( R / m )) j is equal to the rank of H m ( R / I t ) − j . We thus haverank H m ( R / I t ) j = rank (cid:0) H m ( R / I t ) ∨ (cid:1) − j = rank ( (cid:0) Ext R ( R / I t , ω R ) (cid:1) − j )= rank ( (cid:0) Ext R ( R / I t , R ( − )) (cid:1) − j = rank ( (cid:0) Ext R ( R / I t , R ) (cid:1) − j − , i.e., the rank of H m ( R / I t ) j is the same as the rank of Ext R ( R / I t , R ) − ( j + ) . Note that onecan define a different local cohomology module as the direct limit of these Ext-modules : H I ( R ) = lim t −→ ∞ Ext R ( R / I t , R ) . Since Theorem 2.2 guarantees an eventual isomorphism, the rank of the module H m ( R / I t ) j must equal the rank of H I ( R ) j + for sufficiently large t . Therefore, we can compute therank of H m ( R / I t ) j as an F -vector space by instead calculating the rank of H I ( R ) with adegree shift:(3.3.1) rank H m ( R / I t ) j = rank H I ( R ) − j − for sufficiently large t .Recall that Equation 3.1.2 gave H I ( F [ X ]) ∼ = H m ( F [ X ]) as graded modules. Therefore,the rank of H I ( F [ X ]) in any degree is equal to the rank of H m ( F [ X ]) in that degree. Sincethe rank of H m ( R / I t ) j equals the rank of H I ( R ) − j − from Equation 3.3.1, we haverank H m ( R / I t ) j = rank H m ( R ) − j − In particular, when j = H m ( R / I t ) = rank H m ( R ) − . As R is a regular ring in six variables, the top local cohomology module is well-understood: H m ( R ) = uvwxyz F [ u − , v − , w − , x − , y − , z − ] . It follows from the above description that the rank of H m ( R ) − equals 1 as an F -vectorspace. (cid:3)
4. A N E LEMENT OF L OCAL C OHOMOLOGY I S D ESCRIBED U SING N ATURAL L OG One can determine the module H m ( R / I t ) by considering elements in the ˇCech complexon the generators x = u , v , w , x , y , z of the maximal ideal, m . As R / I is a four dimensional ring, the maximal ideal m could begenerated up to radical by only four elements. However, we chose to use six variables, forthe sake of symmetry.Thus, we expect to find an element of ˇ C ( x , R / I t ) , that is, an element of ( R / I t ) uvw ⊕ ( R / I t ) uvx ⊕ ( R / I t ) uvy ⊕ . . . ⊕ ( R / I t ) xyz SOMORPHISMS BETWEEN LOCAL COHOMOLOGY MODULES AS TRUNCATIONS OF TAYLOR SERIES 5 that maps to 0 in ˇ C ( x , R / I t ) but is not in the image of ˇ C ( x , R / I t ) . Note that the presenceof only two variables in the denominator of a particular component of ˇ C ( x , R / I t ) does notguarantee that the element is in the image of ˇ C ( x , R / I t ) , as we shall see.Surprisingly, the isomorphism H m ( R / I t + ) −→ H m ( R / I t ) can be elegantly under-stood in terms of truncations of the formal power series of natural log. Theorem 4.1.
Let R = F (cid:20) u v wx y z (cid:21) , i.e., R is the ring of polynomials in 6 indetermi-nates over a field of characteristic 0. Let I be the ideal generated by size two minors of (cid:20) u v wx y z (cid:21) and let ∆ = vz − wy , ∆ = wx − uz , ∆ = uy − vx . Then the identity ln (cid:18) wyvz uzwx vxuy (cid:19) = ln ( ) = gives the following identity in the fraction field of the I-adic completion of R: ∞ ∑ m = m (cid:18) ∆ vz (cid:19) m + ∞ ∑ m = m (cid:18) ∆ wx (cid:19) m + ∞ ∑ m = m (cid:18) ∆ uy (cid:19) m = . The t th truncation of this Taylor series yields the generator for H m ( R / I t ) . While the above identity in the fraction field of the I -adic completion is the heart ofour local cohomology module element, it must be finessed slightly into the form of anelement in ˇ C ( x , R / I t ) . We will show that Table 1 describes such an element, which wewill henceforth refer to as η . We will prove the above theorem in two steps. First, inSubsection 4.1, we will show that η maps to 0 in ˇ C ( x , R / I t ) . Second, in Subsection 4.2,we will show that η is not in the image of ˇ C ( x , R / I t ) .4.1. The Element η Vanishes.
First we show that the element given by Table 1 is a co-cycle, that is, the image of η is 0 in ˇ C ( x , R / I t ) . Proof.
Let ˆ R denote the I -adic completion of R . Then, in the fraction field of ˆ R , one has0 = ln ( )= ln (cid:18) wyvz uzwx vxuy (cid:19) = ln (cid:18)(cid:18) − (cid:18) − wyvz (cid:19)(cid:19) (cid:16) − (cid:16) − uzwx (cid:17)(cid:17) (cid:18) − (cid:18) − vxuy (cid:19)(cid:19)(cid:19) = ln (cid:18)(cid:18) − (cid:18) vz − wyvz (cid:19)(cid:19) (cid:18) − (cid:18) wx − uzwx (cid:19)(cid:19) (cid:18) − (cid:18) uy − vxuy (cid:19)(cid:19)(cid:19) = ln (cid:18)(cid:18) − (cid:18) ∆ vz (cid:19)(cid:19) (cid:18) − (cid:18) ∆ wx (cid:19)(cid:19) (cid:18) − (cid:18) ∆ uy (cid:19)(cid:19)(cid:19) = ln (cid:18)(cid:18) − (cid:18) ∆ vz (cid:19)(cid:19) + ln (cid:18) − (cid:18) ∆ wx (cid:19)(cid:19) + ln (cid:18) − (cid:18) ∆ uy (cid:19)(cid:19)(cid:19) = ∞ ∑ m = m (cid:18) ∆ vz (cid:19) m + ∞ ∑ m = m (cid:18) ∆ wx (cid:19) m + ∞ ∑ m = m (cid:18) ∆ uy (cid:19) m . JENNIFER KENKEL T ABLE
1. Element in H m ( R / I t ) Component of ˇ C ( x , R / I t ) Component in the I -adic completion R uvw R uvx R uwx R vwx R uvy R uwy R vwy R uxy − t − ∑ m = m ( ∆ uy ) m ln (cid:0) uyvx (cid:1) R vxy t − ∑ m = m ( ∆ vx ) m ln (cid:0) uyvx (cid:1) R wxy t − ∑ m = m ( ∆ wx ) m − ∑ k = t − k ( ∆ wy ) m ln (cid:0) uzwx (cid:1) + ln (cid:0) wyvz (cid:1) R uvz R uwz R vwz R uxz − t − ∑ m = m ( ∆ uz ) m ln ( uzwx ) R vxz − t − ∑ m = m ( ∆ vz ) m + ∑ k = t − k ( ∆ vx ) m ln (cid:16) vzwy (cid:17) + ln (cid:0) uyvx (cid:1) R wxz t − ∑ m = m ( ∆ wx ) m ln (cid:0) uzwx (cid:1) R uyz t − ∑ m = m ( ∆ uy ) m − ∑ k = t − k ( ∆ uz ) m ln (cid:16) vxuy (cid:17) + ln (cid:0) uzwx (cid:1) R vyz − t − ∑ m = m ( ∆ vz ) m ln (cid:16) vzwy (cid:17) R wyz t − ∑ m = m ( ∆ wy ) m ln (cid:16) vzwy (cid:17) R xyz ∆ i , are in the ideal I , in ˇ C ( x , R / I t ) , the elements of the form t − ∑ m = m (cid:18) ∆ vz (cid:19) m are exactly equal to the truncations of the sum, ∞ ∑ m = m (cid:18) ∆ vz (cid:19) m . Therefore, an element of ˇ C ( x , R / I t ) has the same image in ˇ C ( x , R / I t ) as would an ele-ment in ˇ C ( x , lim t −→ ∞ R / I t ) , that is, an element in the ˇCech complex of the I -adic comple-tion of R . We also record the completion of the element in Table 1.The element η of Table 1 is nonzero only in components of ˇ C ( x , R / I t ) with one ofthe variables, u , v , and w (i.e., variables in the first row) and two of the variables x , y , or z (i.e., variables in the second row) inverted. Thus, the image of η in all components of SOMORPHISMS BETWEEN LOCAL COHOMOLOGY MODULES AS TRUNCATIONS OF TAYLOR SERIES 7 ˇ C ( x , R / I t ) with only one of the variables x , y , or z inverted will certainly be 0; that is, theimage of η is 0 in the components ( R / I t ) uvwx , ( R / I t ) uvwy and ( R / I t ) uvwz .By symmetry, it suffices to check that the image of η is 0 in the three components, ( R / I t ) uvxy , ( R / I t ) uxyz , and ( R / I t ) uwxy .1. In ( R / I t ) uvxy , η maps to: t − ∑ m = m (cid:18) ∆ uy (cid:19) + t − ∑ m = m (cid:18) ∆ vx (cid:19) , which agrees with the infinite sum ∞ ∑ m = m (cid:18) ∆ uy (cid:19) + ∞ ∑ m = m (cid:18) ∆ vx (cid:19) in the I -adic completion of the module ( R / I t ) uvxy . We have that the sum equalsln (cid:18) vxuy (cid:19) + ln (cid:16) uyvx (cid:17) = ln (cid:18) vxuy (cid:19) − ln (cid:18) vxuy (cid:19) = .
2. The image of η in ( R / I t ) uxyz is t − ∑ m = m (cid:18) ∆ uy (cid:19) m − t − ∑ m = m (cid:18) ∆ uz (cid:19) m − t − ∑ m = m (cid:18) ∆ uy (cid:19) m + t − ∑ m = m (cid:18) ∆ uz (cid:19) m =
03. Finally, the image of η in ( R / I t ) uwxy is t − ∑ m = m (cid:18) ∆ uy (cid:19) m + t − ∑ m = m (cid:18) ∆ wx (cid:19) m − t − ∑ k = k (cid:18) ∆ wy (cid:19) m which, in R / I t is the same as ∞ ∑ m = m (cid:18) ∆ uy (cid:19) m + ∞ ∑ m = m (cid:18) ∆ wx (cid:19) m − ∞ ∑ k = k (cid:18) ∆ wy (cid:19) m = ln (cid:18) vxuy (cid:19) + ln (cid:16) uzwx (cid:17) − ln (cid:18) vzwy (cid:19) = ln (cid:18) vxuy (cid:19) + ln (cid:16) uzwx (cid:17) + ln (cid:18) wyvz (cid:19) = ln (cid:18) vxuzwyuywxvz (cid:19) = ln ( )= . (cid:3) The Element η Is Not a Coboundary.
The element given by Table 1 is not a cobound-ary, that is, the element η is not the image of an element from ˇ C ( x , R / I t ) . Proof.
Give R the following multi-grading:deg ( u ) = ( , , , ) deg ( x ) = ( , , , ) deg ( v ) = ( , , , ) deg ( y ) = ( , , , ) deg ( w ) = ( , , , ) deg ( z ) = ( , , , ) The generators of the ideal I are homogeneous with respect to this multi-grading, and henceone obtains a grading on R / I t . The element, η ∈ ˇ C ( x , R / I t ) is homogeneous of degree ( , , , ) . JENNIFER KENKEL
Suppose for the sake of contradiction that the element, η were a coboundary, that is, inthe image of ˇ C ( x , R / I t ) . The map from ˇ C ( x , R / I t ) to ˇ C ( x , R / I t ) is degree-preserving.Therefore, an element of ˇ C ( x , R / I t ) mapping to the given element would necessarily bedegree ( , , , ) in each component.Consider the R uv component of ˇ C ( x , R / I t ) . Since deg ( u n v m ) = ( − n , − m , , ) , in or-der for an arbitrary element deg ( au n v m ) of ( R / I t ) uv with a ∈ R / I t to be of multi-degree ( , , , ) , it must be that deg ( a ) = ( n , m , , ) , or a =
0. But since a ∈ R , this means a = λ u n v m . Thus, any degree ( , , , ) element in R uv is of the form λ u n v m u n v m for λ ∈ F . Thus any multi-degree ( , , , ) element in R uv is, in fact, in the field F .The same argument shows that all elements in R uw , R vw , R xy , R xz , R yz , R ux , R vy , and R wz in multi-degree ( , , , ) are scalars from the field F .Note that η , is 0 in the R uvx component. Since the elements in R ux and R uv are scalarsin F , the preimage of η in the R xv component must also be a scalar. Using the fact that η is 0 in the R uvy , R uwx , R uwz , R vwz , and R vwy components, a similar argument shows that thepreimage of η would be forced to be a scalar in every component of ˇ C ( x , R ) .Thus, if the element η were in the image of an element of ˇ C ( x , R ) , it would be theimage of an element that consisted of scalars in each component. However, ∆ uy is not in F . (cid:3) Note that the arguments in 4.2 were independent of the characteristic of the ground field.5. E
XPLORATIONS IN C HARACTERISTIC p > p >
0, the situation is remarkably different.We shall consider the same setting but over a field of positive characteristic. Let X be a2 × R be the ring F [ X ] for a field of prime characteristic p > R = F (cid:20) u v wx y z (cid:21) . As before, let I be the ideal generated by size two minors of the matrix X , and let ∆ = vz − wy , ∆ = wx − uz , ∆ = uy − vx . In the characteristic 0 case, Proposition 3.3 guaranteed that the local cohomology module H m ( R / I t ) is an F -vector space of rank 1 for all t ≥
2. In the characteristic p > H m ( R / I t ) grow arbitrarily large on a subsequence of t in the natural numbers.In Subsection 5.1, we construct elements of H m ( R / I t ) . The construction proceeds byshowing in Subsection 5.2 that the given elements are not boundaries, then showing inSubsection 5.3 that the given elements are cycles.5.1. Elements Of Local Cohomology Modules In Positive Characteristic.
We seek toconstruct elements of H m ( R / I t ) when the ground field is characteristic p >
0. The element η from Table 1 is no longer defined when the characteristic of the field is positive, as η isdefined using the fraction m for arbitrary m ≤ t − SOMORPHISMS BETWEEN LOCAL COHOMOLOGY MODULES AS TRUNCATIONS OF TAYLOR SERIES 9 T ABLE
2. Elements in ˇ C ( x , ˆ R ) in characteristic p > C ( x , R / I t ) R uvw R uvx R uwx R vwx R uvy R uwy R vwy R uxy xu q − m ( α m − ) R vxy yv q − m ( α m − ) R wxy zw q − m ( − γ m − β m + ) R uvz R uwz R vwz R uxz xu q − m ( − γ m ) R vxz yv q − m ( α m − + β m − )) R wxz zw q − m ( − γ m ) R uyz xu q − m ( − α m − γ m + ) R vyz yv q − m ( β m − ) R wyz zw q − m ( β m − ) R xyz Theorem 5.1.
Let q be the largest power of p such that q ≤ t − , and let q be the smallestpower of p such that q + q ≥ t. Further, suppose that m is a positive integer with < m ≤ qsuch that q | m. Then rank H m ( R / I t ) ≥ (cid:22) qq (cid:23) − . In particular, whenever t = p e + for some e, we have that the rank of H m ( R / I t ) is atleast t − .Proof. The proof of Theorem 5.1 proceeds by concrete construction; we will demonstrate2 j qq k − H m ( R / I t ) .First, let α = vxuy , β = wyvz , γ = uzwx . Note that 1 − α = ∆ uy , − β = ∆ vz and 1 − γ = ∆ wx . We claim the elements of Table 2 andTable 3 are nonzero elements of H m ( R / I t ) whenever q , q , and m satisfy the hypothesesof Theorem 5.1. Furthermore, the elements of Table 2 and Table 3 are distinct when m = q .We will refer to the elements in Table 2 as η , m and the elements in Table 3 as η , m , with m ranging over all integers less than q that are divisible by q as in Theorem 5.1.We consider the elements η , m . The argument for the elements η , m is symmetric.5.2. The Elements η , m Are Not Coboundaries.
We first consider the case that m = q and so q − m ≥
1. We claim these elements are not in the image of ˇ C ( x , R / I t ) . T ABLE
3. Elements in ˇ C ( x , ˆ R ) in characteristic p > C ( x , R / I t ) R uvw R uvx ux q − m ( − α m ) R uwx ux q − m ( − γ m ) R vwx ux q − m ( − α m − γ m + ) R uvy vy q − m ( α m − ) R uwy vy q − m ( − α m − β m + )) R vwy vy q − m ( β m − ) R uxy R vxy R wxy R uvz wz q − m ( − γ m − β m + ) R uwz wz q − m ( γ + ) R vwz wz q − m ( β m − ) R uxz R vxz R wxz R uyz R vyz R wyz R xyz η , m is in the image of ˇ C ( x , R / I t ) for some arbitrary m = q thatsatisfies the hypotheses of Theorem 5.1. Using the multi-grading introduced in Section 4.2,every component of the elements η , m is of degree ( , , , q − m ) , where 1 ≤ q − m < q .Since the maps between ˇCech complexes are degree preserving, they would need to be theimage of elements also of degree ( , , , q − m ) . Consider R xy , R xz , and R yz . There are noelements of degree ( , , , q − m ) in each of these components whenever q − m ≥ η , m were in the image of an element in ˇ C ( x , R / I t ) , the R uvx component would be asum of the elements in the R uv , R ux , and R vx components. In the R uv , R ux , and R vx compo-nents, the multi-degree elements of degree ( , , , q − m ) where q − m ≥ R uv : (cid:16) yv (cid:17) q − m F ⊕ (cid:16) xu (cid:17) q − m F R ux : (cid:16) xu (cid:17) q − m F R vx : (cid:16) yv (cid:17) q − m F h uyvx i . The entry in the R uvx component of η , m is 0. As (cid:0) yv (cid:1) q − m F ⊕ (cid:0) xu (cid:1) q − m F do not containpolynomials in F [ uyvx ] , in order for the sum of components from R uv , R ux , and R vx to be zero,the element in the R vx component could not be a polynomial in F and must be of the form SOMORPHISMS BETWEEN LOCAL COHOMOLOGY MODULES AS TRUNCATIONS OF TAYLOR SERIES 11 λ (cid:0) yv (cid:1) q − m where λ is some scalar from F . Thus, if η , m is in the image of an element inˇ C ( x , R / I t ) , that element is a constant from F in the R vx component.Similarly, the element in R wvz comes from a sum of the elements in the R wv , R wz , and R vz components. Since the entry in the R uvy component is 0, there must not be a nonzeropower of wyvz in the R vz component. The entry in R vz then must be λ ′ (cid:0) yv (cid:1) , where λ ′ is somescalar from F .But then consider η , m in the R vxz component (cid:16) yv (cid:17) q − m (cid:18) α m − − β m + (cid:19) . If η , m were a coboundary, then the above would have to be a sum of elements from R vx , R vz , and R xz components. We have already established there are no elements of de-gree ( , , , q − m ) in R xz , so it must only be from R vx and R vz . However, by the aboveargument, the only possible elements in those components are in yv F . As η , m in the R vxz component is not in yv F , it cannot be a coboundary.5.3. The Elements η , m Are Cocycles.
We show that the elements given by Table 2 areindeed cocycles, that is, the images of η , m and η , m are 0 in ˇ C ( x , R / I t ) . The argumentfor the elements given by Table 3 works similarly.By symmetry, it suffices to check that the image of η , m is 0 in the three components, ( R / I t ) uvxy , ( R / I t ) uxyz , and ( R / I t ) uwxy .First consider the image of η , m in the ( R / I t ) uxyz component: − (cid:16) xu (cid:17) q − m ( α m − ) + (cid:16) xu (cid:17) q − m (cid:18) − γ m (cid:19) − (cid:16) xu (cid:17) q − m (cid:18) − α m − γ m + (cid:19) = (cid:16) xu (cid:17) q − m (cid:18) − α m + + − γ m − + α m + γ m − (cid:19) = . Second, consider the image of η , m in the ( R / I t ) uwxy component: − (cid:16) xu (cid:17) q − m ( α m − ) + (cid:16) zw (cid:17) q − m ( − γ m − β m + )= − (cid:16) xu (cid:17) q − m α m + (cid:16) xu (cid:17) q − m + (cid:16) zw (cid:17) q − m − (cid:16) zw (cid:17) q − m γ m − (cid:16) zw (cid:17) q − m β m + (cid:16) zw (cid:17) q − m = − (cid:16) xu (cid:17) q − m (cid:18) vxuy (cid:19) m + (cid:16) xu (cid:17) q − m + (cid:16) zw (cid:17) q − m − (cid:16) zw (cid:17) q − m (cid:16) uzwx (cid:17) m − (cid:16) zw (cid:17) q − m (cid:18) vzwy (cid:19) m + (cid:16) zw (cid:17) q − m = − x q v m u q y m + (cid:16) xu (cid:17) q − m + (cid:16) zw (cid:17) q − m − z q u m w q x m − v m z q w q y m + (cid:16) zw (cid:17) q − m . Since m is divisible by q by hypothesis, write m = ζ q for some natural number ζ . Thenthe above equals ( wx ) q ( − u ζ q y ζ q + v ζ q x ζ q ) + ( uz ) q ( u ζ q y ζ q − v ζ q x ζ q ) u q w q x m y m = ( wx ) q ( − u ζ y ζ + v ζ x ζ ) q + ( uz ) q ( u ζ y ζ − v ζ x ζ ) q u q w q x m y m = − ( wx ) q ( v ζ x ζ − u ζ y ζ ) q + ( uz ) q ( u ζ y ζ − v ζ x ζ ) q u q w q x m y m = ( − ( wx ) q + ( uz ) q ) ( v ζ x ζ − u ζ y ζ ) q u q w q x m y m = ( − wz + uz ) q ( v ζ x ζ − u ζ y ζ ) q u q w q x m y m . The polynomial ( x − y ) divides ( x ζ − y ζ ) for any natural number ζ . Let ϕ ζ ( x , y ) be thepolynomial such that ϕ ζ ( x , y )( x − y ) = ( x ζ − y ζ ) . Then the above is ( − wz + uz ) q (( uy − vx ) ϕ ζ ( uy , vx )) q u q w q x m y m = ( ∆ ) q ( ∆ ) q ϕ ζ ( uy , vx ) q u q w q x m y m . By hypothesis, q + q ≥ t . So this is indeed 0 in the R uwxy component of ˇ C ( x , R / I t ) .Third and finally, consider the image in R uvxy . We will use the fact that m is divisible by q , so again, let ζ be the natural number such that m = ζ q . (cid:16) xu (cid:17) q − m (cid:18) − (cid:18) vxuy (cid:19) m (cid:19) − (cid:16) yv (cid:17) q − m (cid:16)(cid:16) uyvx (cid:17) m − (cid:17) = (cid:16) xu (cid:17) q − m − (cid:18) vxuy (cid:19) ζ q ! − (cid:16) yv (cid:17) q − m (cid:18)(cid:16) uyvx (cid:17) ζ q − (cid:19) = (cid:16) xu (cid:17) q − m − (cid:18) vxuy (cid:19) ζ ! q − (cid:16) yv (cid:17) q − m (cid:18)(cid:16) uyvx (cid:17) ζ − (cid:19) q = (cid:16) xu (cid:17) q − m (cid:18) uy (cid:19) m (cid:16) ( uy ) ζ − ( vx ) ζ (cid:17) q − (cid:16) yv (cid:17) q − m (cid:18) vx (cid:19) m (cid:16) ( uy ) ζ − ( vx ) ζ (cid:17) q . SOMORPHISMS BETWEEN LOCAL COHOMOLOGY MODULES AS TRUNCATIONS OF TAYLOR SERIES 13
Then the above is (cid:16) xu (cid:17) q − m (cid:18) uy (cid:19) m ( uy − vx ) q ϕ q ζ ( uy , vx ) − (cid:16) yv (cid:17) q − m (cid:18) vx (cid:19) m ( uy − vx ) q ϕ q ζ ( uy , vx )= ∆ q ϕ q ζ ( uy , vx ) (cid:18)(cid:16) xu (cid:17) q − m (cid:18) uy (cid:19) m − (cid:16) yv (cid:17) q − m (cid:18) vx (cid:19) m (cid:19) = ∆ q ϕ q ζ ( uy , vx ) (cid:18) uv (cid:19) q − m (cid:18) ( xv ) q − m (cid:18) uy (cid:19) m − ( uy ) q − m (cid:18) vx (cid:19) m (cid:19) = ∆ q ϕ q ζ ( uy , vx ) (cid:18) uv (cid:19) q − m (cid:18) uyvx m (cid:19) (( vx ) q − ( uy ) q )= ∆ q ϕ q ζ ( uy , vx ) (cid:18) uv (cid:19) q − m (cid:18) uyvx m (cid:19) ( vx − uy ) q = ∆ q + q ϕ q ζ ( uy , vx ) (cid:18) uv (cid:19) q − m (cid:18) uyvx m (cid:19) . Recall that q + q ≥ t , so this is indeed 0 in the R uvxy component of ˇ C ( x , R / I t ) .Now consider the case that m = q and so q − m =
0, which is recorded in Table 4. In thiscase, the element is in multi-degree ( , , , ) . Recall that the argument from Section 4.2is characteristic free, so to show this element is not a coboundary.Thus we have demonstrated 2 ⌊ qq − ⌋ linearly independent elements of H m ( R / I t ) when the base field is characteristic p > (cid:3) T ABLE
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