PDF Satoshi Mochizuki - 広島大学

A generalization of elementary divisor theory

Satoshi Mochizuki

Contents

1 Main theorems and motivation 4

2 Spirit of elemetary divisor theory 9

3 What makes a multi-complex exact? 13

4 Weak absolute geometric presentation theorem 23

5 A¹-homotopy invariances, generic isomorphisms 28

2

Conventions.

Throughout the lecture, we utilize the letterA,B,C,S andX to de- note a commutative noetherian ring with unit, an essentially small abelian category, a category, a finite set and the affine sheme associated with A respectively.

For any non-negative integer m, let us denote the totally ordered set of integers k such that 0≤ k ≤ m with the usual order by [m].

1 Main theorems and motivation

4

Main theorems.

1 Abstract Buchsbaum-Eisenbud theorem for multi- complexes.

2 Weak geometric presentation theorem

For any strictly regular closed immersion Y ,→X, we have a de- rived equivalence

X_w^Y_M ^Tot→ X_Top^Y .

3 (In progression work) Caborn-Fossum, Dutta chow groups problem

If A is regular local, thenCh_d(X) = 0 for any d < dimX.

The statement 1 is an assertion about objects and morphisms in B involves a natural number n.

• n= 1

• → •^a → •^b

If ba is a monomorphism, then a is a monomorphism.

• n= 2

• ^//

^I

• ^//

^II

•

• ^//

^III

• ^//

^IV

•

• ^// • ^// •,

If the big square I+II+III+IV is a Cartesian square and if all mor- phisms in the diagram above are monomorphisms, then the square I is Cartesian.

6

•The CDF problem is a variant of the following classical theorem.

Theorem.

If A is a unique factorization domain, then PicX = 0.

• The CDF problem is known for variousA by utilizing structure theorems over a base

or

weight argument of Adams operations.

Ideally, the CDF problem should be proven from the following con- jectural statement which I call

an absolute geometric presentation theorem.

If A is regular local, then the canonical map

X_M^p → X_Top^p

is a derived universally homeomorphism for any0 ≤ p≤ dimX.

Compare with the weak geometric presentation theorem with an absolute geometric presentation theorem.

8

2 Spirit of elemetary divisor theory

Sorting out modules by Systems of matrices with equivalence relations.

10

Syzygy.

Systems of matrices = complexes of finitely generated free mod- ules.

Equivalence relation = quasi-isomorphism.

Bourbaki-Iwasawa-Serre theory.

Equivalence relation = pseudo-isomorphism.

Here a homomorphism of A-modules f : M → N is a pseudo- isomorphism if Codim kerf ≥ 2and Codim Cokerf ≥ 2.

Today’s lecture.

Modules= TT-pure weight modules.

Systems of matrices = Koszul cubes.

Equivalence relation = totalized quasi-isomorphism. (A¹-homotopy equivalence, generic isomorphism).

What makes a comlex exact?

Buchsbaum-Eisenbud theorem.

For a complex of free A-modules of finite rank.

F_• : 0 → F_s →^ϕs F_s−1 ^ϕ→ → · · · →^s−1 F₁ →^ϕ1 F₀ → 0, setr_i =

∑s j=i

(−1)^j−irankF_j. Then the following are equivalent:

(1) F_• is a resolution ofH0(F_•).

(2) gradeI_ri(ϕ_i) ≥ i for any 1 ≤ i ≤ s where I_ri(ϕ_i) is the r_i-th Fitting ideal of ϕi.

12

3 What makes a multi-complex exact?

Definitions.

An S-cube inC is a contravariant functor P(S)^op(→^∼ [1]^Sop) → C.

For any U ∈ P(S) and k ∈ U,

• x_U(:= x(U)) vertex of x at U.

• d^k,x_U (= d^k_U) := x(U ∖{k},→U) k-boundary map at U.

S = {1} S = {1,2} S = {1,2,3}

x_S

d¹_S

x_∅

x_S ^d

2 S //

d¹_S

x_{1}

d¹_{1}

x_{2}

d²_{2}

// x_∅

xS

d3

S //

d2 S

{{

vvvvvvvvvv ^x{1,2}

d1 {1,2}

d2 {1,2}

{{

wwwwwwww

x{1,3} //

d1 {1,3}

x{1}

x{2,3}

d2 {2,3}

{{

vvvvvvvvv

d3 {2,3}

// x{2}

d2

{{ {2}

wwwwwwwww

x{3}

d3 {3}

// x∅

14

Example 1

For a family of morphisms x := {xs ds

→x}^s∈S inB, we put

Fibx_U :=







x if U = ∅

x_s if U = {s}

xt₁ ×^xxt₂ ×^x· · · ×^xxt_r if U = {t1,· · · , tr}

Definition.

An S-cube x in B isfibered if the canonical map x →Fib{x_{s} ^d

s{s}

→ x_∅}^s∈S

is an isomorphism.

Example 2

For a family of elements f_S = {f_s}s∈S inA, we put Typ(f_S)U := A and d^s_U = fs

for any U ∈ P(S) and s ∈ U.

(Notice that Tot Typ(f_S) is the Koszul complex associated with f_S.)

Faces and homology of cubes Definitions.

k ∈ S, x: S-cube

B^k, F^k : P(S ∖{k}) → P(S) F^k : U 7→U

B^k : U 7→ U ⊔ {k}

• xB^k: backsidek-face of x.

• xF^k: frontside k-face of x.

• H^k₀(x) := Coker(xB^k →xF^k): k-direction 0-th homology of x.

S = {1,2}

x_S ^//

x_{1}

⇒

H²₀(x)_{1}

x_{2} ^//

⇓

x_∅ H²₀(x)_∅

H¹₀(x)_{2} ^// H²₀(x)_∅

16

Admissibility

Definition.

We say that an S-cube x in B is admissible if

(1) its boundary morphism(s) is (are) monomorphism(s) and (2) if for every k inS, H^k₀(x) is admissible.

• We can prove that any admissible cube is fibered.

• We can prove that an S-cube x inB is admissible iff (1) all faces of the S-cube x are admissible and

(2) Hk(Totx) = 0 for any k > 0.

Example 1.

For a commutative diagram of monomorphisms in B

• ^//

•

• ^// •,

the diagram above is admissible iff it is Cartesian.

Example 2.

For any family of elementsf_S = {f_s}s∈S inA,Typ(f_S)is admissible iff f_S is a regular sequence in any order.

Admissibility

= a higher analogue of the notion about Cartesian squares

= a categorical variant of the notion about regular sequences.

18

Double cubes Definitions.

A double S-cube in C is a contravariant functor x : [2]^Sop → C.

For any i ∈ [1]^S, we put

e_i : [1]^S → [2]^S, j 7→ i+j and Out : [1]^S →[2]^S, j 7→2j.

Example.

x_(2,2) ^//

^I

x_(2,1) ^//

^II

x_(2,0)

x_(1,2) ^//

^III

x_(1,1) ^//

^IV

x_(1,0)

x_(0,2) ^// x_(0,1) ^// x_(0,0)

I= xe_(1,1), II= xe_(1,0), III= xe_(0,1), IV= xe_(0,0) and I+II+III+IV = xOut.

ABE theorem Theorem.

Let x be a double S-cube in B. We assume that the following conditions hold.

• The S-cube xOut is admissible.

• All boundary morphisms of the double S-cube x are monomor- phisms.

• If #S ≥ 3, all faces of the S-cube xe_T are admissible for any proper subsetT of S.

Then the S-cube xe_S is also an admissible S-cube.

20

Adjugates of cubes

From now on, let B be the category ofA-modules.

Definitions.

An adjugate of an S-cube x in B is a pair (a,d^∗) consisting of a family of elements a = {as}^s∈S in A and a family of morphisms d^∗ = {d^t_T^∗ : x_T∖{t} →x_T}T∈P(S),t∈T inB which satisfies the following two conditions.

• We have the equalities d^t_Td^t_T^∗ = (at)_x

T∖{t} and d^t_T^∗d^t_T = (at)_x

T for any T ∈ P(S) and t ∈ T.

• For any T ∈ P(S) and any distinct elements a and b ∈ T, we have the equality d^b_Td^a_T^∗ = d^a_T^∗_∖{b}d^b_T∖{a}. Namely, the following dia- gram is commutative.

x_T∖{a} ^d

a∗ T //

d^b_T∖{a}

x_T

d^b_T

x_T∖{a,b}

d^a_T^∗_∖{b}

// x_T∖{b} .

(2) An adjugate of an S-cube (a,d^∗) is regular if a forms xT- regular sequence in any order for any T ∈ P(S).

Example.

BE theorem for cubes Corollary.

If an S-cube x in B admits a regular adjugate, then x is admissi- ble.

Proof.

For S = {1,2}, we apply the ABE theorem to the double cube below.

xS d¹_S

//

d²_S

x_{2} ^d

1∗ S //

d²_{2}

xS d²_S

x_{1}

d¹_{1}

//

d²_S^∗

x_∅

d¹_{^∗_1}

//

d²_{^∗_2}

x_{1}

d²_S^∗

x_S

d¹_S

// x_{2}

d¹_S^∗

// x_S.

22

4 Weak absolute geometric presentation theorem

Koszul cubes

In this section, letf_S = {fs}^s∈S be a family of elements in Awhich forms a regular sequence in any order and x an S-cube inB.

Definition.

x is Koszul (associated with f_S) if for any T ∈ P(S) and any k ∈ T,

(1) xT is a finitely generated projective A-module and (2) d^k_T is injective and

(3) There exists a non-negative integer m_k such that f_k^mkCokerd^k_T = 0.

We denote the category of Koszul cubes associated with f_S by Kos^f_A^S.

•We can prove that any Koszul cube is admissible by BE theorem for cubes.

A morphism between koszul cubes (associated with f_S) a : x →y is atotalized quasi-isomorphismifH₀Totais an isomorphism.

Example.

Typ(f_S) is a Koszul cube.

24

Solid devices Definition.

A solid deviceE = (E, w) is a pair of a category E and a class of morphisms w in E which satisfies the following axioms.

• Eis an exact category.

•(E, w)and(Eop, wop )are categories with cofibrations and weak equivalences.

•(Extensional axiom).For any commutative diagrams admissible exact sequences inE,

x // //

a

y

b //// ^z

c

x′ // // ^y′ //// ^{z′ ,}

ifaandcare inw, thenbis also inw. Let us writeEwfor the full subcategory ofEconsisting of those objectxsuch that the canonical morphism0→xis inw. ThenEwnaturally becomes an exact category.

•(Solid axiom).For any morphismf:x→yinw, the complexConef= [xf

→y]is connected to a bounded complexes onEwby a zig-zag of quasi-isomorphisms.

•(Fibrational axiom).The canonical inclusion functorsEw→ Eand(E, i)→(E, w)induce the sequence having a homotopy type of fibration sequence iS• Ew→iS• E →wS• E

whereimeans the class of all isomorphisms andSmeans the Segal-WaldhausenS-construction.

• We will functorially associate a solid device E with (1) the non-connective K-spectrum K(E) and

(2) thebounded derived categories D^b(E) which is a triangulated category.

Example 1.

Y ,→X:closed subset. We put

X_Top^Y := (Perf^Y_X,qis), X_Top^p := (Perf^p_X,qis)

wherePerf^Y_X is the category of perfect complexes whose cohomo- logical support are in Y and Perf^p_X := ∪

CodimY≥p

Perf^Y_X. X_Top^Y and X_Top^p are solid devices.

Example 2.

Y ,→X:regular closed immersion of codimension r.

A perfect O^X-module F on X is a TT-weight r (supported on Y) if SuppF ⊂ Y and Tor-dimF ≦r.

Let us denote the exact category of TT-weight r modules sup- ported on Y by Wt^Y_X.

We put

X_TT^Y := (Wt^Y_X,isom).

X_TT^Y is a solid device.

Example 3.

Y = V(f_S). We put

X_w^Y_M := (Kos^f_A^S,tq)

where tq is the class of totalized quasi-isomorphisms in Kos^f_A^S. X_w^Y_M is a solid device.

26

WGP theorem Theorem.

ForY = V(f_S), the canonical map

X_w^Y_M ^Tot→ X_Top^Y is derived equivalence.

Proof.

X_w^Y_M ^{Tot //}

H0Tot^EEEEE""

EE

EE X_Top^Y

X_TT^Y

I

<<

yy yy yy yy

In the derived commutative diagram above, the mapI is a derived equivalence by Hiranouchi-M.

To prove that H₀Tot is a derived equivalence, one of the key in- gredient is giving an algorithm about inductive resolution process of pure weight modules by Koszul cubes.

5 A¹-homotopy invariances, generic isomorphisms

28

Regularity and A ¹ -homotopy invariance

In this section, let A be a regular local ring.

Lemma.

Let f_S = {fs}^s∈S be a regular sequence in A and we put Y = V(f_S). Then the canonical mapA → A[t]induces an isomorphism of K-groups

K_n(X_w^Y_M)→^∼ K_n(X[t]^Y_wM).

• In particular, for any z(t) ∈ X[t]^Y_wM, [z(0)] = [z(1)]in K₀(X_w^Y_M).

Generic isomorphisms

Let0 ≤ p≤ dimAbe an integer. We propose the following asser- tions.

(αp)For any regular sequence f1,· · · , fp inA, the canonical inclu- sion functor

X_Top^{V(f1,···,fp)},→X_Top^{V(f1,···,fp−1)}

induces the zero map on their Grothendieck groups.

(βp) For any regular sequence f_S such that #S = p, the Grothendieck group K₀(X_w^fS_M) is generated by Koszul cubes of rank one.

Lemma.

(1) The assertion (βp) implies (αp).

(2) The assertions (αp) (0≤ p≤ dimA) imply the CDF problem.

(3) Let g be an element in A and f_S := {f_s}s∈S a regular se- quence of A such that f_S is still a regular sequence in Ag and we put Xg := SpecAg and Y = V(f_S). Assume (α#S+1), then the canonical localization map A → Ag induces an isomorphism of Grotheindieck groups

K₀(X_w^Y_M)→^∼ K₀(X_g^Y_wM).

• We will prove (β_p) by descending induction of p.

30

How to prove (β _p )?

Notations.

• For an element λ in A, and for 1 ≤ i ̸= j ≤ n, eⁿ_ij(λ) or simply e_ij(λ) will denote the n×n matrix such that the diagonal entries are one, the(i, j) entry is λ and the other entries are zero.

• For elements a₁,· · · , a_n in A, we write diag(a₁,· · · , a_n) for the diagonal matrix whose (i, i) entry is a_i.

Let x ∈ X_w^Y_M. Let us put n := rankx. By fixing the bases of all vertexes of x, we write A^s = (a^s_ij) for the matrix description of d^x_s : x_{s} → x_∅ for each s ∈ S. We put d^s := gcd{a^s_1j; 1 ≤ j ≤ n} and b^s_1j = a^s_1j/d^s. Then for any s in S, there exists a subset T_s ⊂ {2,· · · , n− 1, n} such that c^s₁₁ = b^s₁₁+ ∑

j∈Ts

b^s_1j is prime to f_s. Now we put g := ∏

s∈S

c^s₁₁, c^s_j1 := a^s_j1 + ∑

l∈Ts

a^s_jl (2≤ j ≤ n) and

B(t)^s = diag(d^s,1,1,· · · ,1)

∏n j=2

e_1j(−c^s_j1 c^s₁₁t)







b^s₁₁ b^s₁₂ · · · b^s_1n a^s₂₁ a^s₂₂ · · · a^s_2n ... ... · · · ...

a^s_n1 a^s_n2 · · · a^s_nn







∏

j∈Ts

e_j1(t).

Notice that B(0)^s = A^s and the first column ofB(1)^s is of the form





 d^sc^s₁₁

0





 { ^{⊕n B}→^{(t)s ⊕n}}