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Smoothness In Zariski Categories A Proposed Definition and a Few Easy Results. John Iskra jiskra@ehc.edu Department of Mathematics & Computer Science Emory and Henry College Emory, Virginia 24327 USA Smoothness In Zariski Categories p.

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SLIDE 1

Smoothness In Zariski Categories

A Proposed Definition and a Few Easy Results.

John Iskra

jiskra@ehc.edu

Department of Mathematics & Computer Science Emory and Henry College Emory, Virginia 24327 USA

Smoothness In Zariski Categories – p. 1/29

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SLIDE 2

Outline

  • Context

Smoothness In Zariski Categories – p. 2/29

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SLIDE 3

Outline

  • Context
  • Philosophy

Smoothness In Zariski Categories – p. 2/29

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SLIDE 4

Outline

  • Context
  • Philosophy
  • Definition

Smoothness In Zariski Categories – p. 2/29

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SLIDE 5

Outline

  • Context
  • Philosophy
  • Definition
  • A Few Easy Results

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SLIDE 6

Context

In 1992, Yves Diers published his inspirational book “Categories of Commutative Algebras”. As he himself says in the Introduction to “Categories of Commutative Algebras” his work stands in a line of advances on the problem of classifying categories. He specifically wants to understand categories which can are very similar to the category of commutative rings with identity.

Smoothness In Zariski Categories – p. 3/29

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SLIDE 7
  • Z. Luo

Was one of those who were inspired. In his work, available at www.geometry.net/cg, Z. Luo built upon Diers work in the dual situation. He argued that this side was the geometric, , and Diers was the algebraic.

Smoothness In Zariski Categories – p. 4/29

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SLIDE 8

Smoothness

Smoothness, I think, is traditionally understood as a geometric property

Smoothness In Zariski Categories – p. 5/29

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SLIDE 9

So, In Spite of the Title of My Talk

I’d like to talk about Smoothness as a geometric property and, so, for the most part, use Luo’s language.

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SLIDE 10

Equivalence

Dier’s Zariski category is roughly dual to Luo’s left coherent ’analytic geometry’. Among other properties, it’s possessed of a strict initial object a category in which limits commute with finite sums locally disjunctable, reducible, and perfect.

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SLIDE 11

Grothendieck

Firmly established the importance of nilpotents when he defined three related types of morphisms:

  • Net - or unramified

Smoothness In Zariski Categories – p. 8/29

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SLIDE 12

Grothendieck

Firmly established the importance of nilpotents when he defined three related types of morphisms:

  • Net - or unramified
  • Lisse - or smooth

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SLIDE 13

Grothendieck

Firmly established the importance of nilpotents when he defined three related types of morphisms:

  • Net - or unramified
  • Lisse - or smooth
  • Etale - or étale (slack....)

Smoothness In Zariski Categories – p. 8/29

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SLIDE 14

Grothendieck’s Definition

Consider the commutative diagram X

f

W

w

S W ′

w′ j g

with j a strong unipotent mono.

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SLIDE 15

If there exists

X

f

W

w h

S W ′

w′ j g

  • at most one such h, then f is net

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SLIDE 16

If there exists

X

f

W

w h

S W ′

w′ j g

  • at most one such h, then f is net
  • at least one such h, then f is smooth

Smoothness In Zariski Categories – p. 10/29

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SLIDE 17

If there exists

X

f

W

w h

S W ′

w′ j g

  • at most one such h, then f is net
  • at least one such h, then f is smooth
  • exactly one such h, then f is étale

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SLIDE 18

Definition

A strong unipotent (Luo’s definition) mono is approximately the equivalent of a closed morphism in algebraic geometry induced by a morphism with unipotent kernel. Luo defines a unipotent morphism to be one which has no non-zero pullback isomorphic to zero. Geometrically, its image is not disjoint with anything.

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Clearly

Sticking with Grothendieck’s definition, if a category were to have no proper strong monic unipotents, every arrow would be étale.

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Dier’s Problem:

These three types of morphisms play an exceptionally important role in algebraic geometry. However, so-called reduced categories - that is categories without nilpotents satisfy the axioms for a Zariski category.

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SLIDE 21

Subtext

Can we develop a more widely meaningful definition

  • f these concepts, so that perhaps this axiomatic and

categorical approach provide us with fresh insight into these concepts?

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SLIDE 22

Dier’s Solution - for net and étale

(In Luo’s language)

  • A morphism f : X → S is net if the

corresponding diagonal ∆ : X → X ×S X is a local isomorphism.

Smoothness In Zariski Categories – p. 15/29

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SLIDE 23

Dier’s Solution - for net and étale

(In Luo’s language)

  • A morphism f : X → S is net if the

corresponding diagonal ∆ : X → X ×S X is a local isomorphism.

  • f : X → S is étale if it is net and coflat.

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SLIDE 24

This is a very nice approach. However,

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The definition of Smooth is miss- ing

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Problems with finding a defini- tion

  • One approach would be to find some sort of map

which approximates the sort of super-denseness which unipotent maps possess.

Smoothness In Zariski Categories – p. 18/29

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SLIDE 27

Problems with finding a defini- tion

  • One approach would be to find some sort of map

which approximates the sort of super-denseness which unipotent maps possess.

  • We could try to adapt the work of Anders Kock

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SLIDE 28

Problems with finding a defini- tion

  • One approach would be to find some sort of map

which approximates the sort of super-denseness which unipotent maps possess.

  • We could try to adapt the work of Anders Kock
  • We could try to understand what ’locally linear’

means given Diers and Luo’s frame work.

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SLIDE 29

Philosophy

In Calculus we teach students that to say that a function is differentiable is to say that in a sufficiently small neighborhood, the function is linear.

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SLIDE 30

This has several difficulties

  • In most cases of interest (i.e ’integral objects’),

no neighborhoods are really small

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This has several difficulties

  • In most cases of interest (i.e ’integral objects’),

no neighborhoods are really small

  • It wasn’t clear to me how to define ’linear’ in

categorical terms

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SLIDE 32

However we define ’smooth’ it ought to be

  • Local

Smoothness In Zariski Categories – p. 21/29

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SLIDE 33

However we define ’smooth’ it ought to be

  • Local
  • Linked to Dier’s definitions of net and étale

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SLIDE 34

However we define ’smooth’ it ought to be

  • Local
  • Linked to Dier’s definitions of net and étale
  • Geometrically consistent with how we

understand differentiability in more familiar settings, like Calculus

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SLIDE 35

However we define ’smooth’ it ought to be

  • Local
  • Linked to Dier’s definitions of net and étale
  • Geometrically consistent with how we

understand differentiability in more familiar settings, like Calculus

  • As in algebraic geometry we want S[T] → S to

be smooth

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My Proposal

An arrow f : X → S will be called smooth if it is coflat and if there exists a unipotent analytic cover {Ui}i∈I of X so that for all i ∈ I there exists r > 0 so that Ui

ui

X

f

  • r S′

φ

S commutes where

r S′ is the product of r

cogenerators and the arrow Ui →

r S′ is étale.

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SLIDE 37

Unfortunately,

It turns out it was also Grothendieck’s idea first. In fact, in Grothendieck’s Universe, the two are equivalent as long as f is finitely presented

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SLIDE 38

Some Preliminary Results:

  • Compositions of smooth are smooth

Smoothness In Zariski Categories – p. 24/29

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SLIDE 39

Some Preliminary Results:

  • Compositions of smooth are smooth
  • The pullback of smooth by a monic is again

smooth

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SLIDE 40

Some Preliminary Results:

  • Compositions of smooth are smooth
  • The pullback of smooth by a monic is again

smooth

  • Net plus smooth is étale

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SLIDE 41

Some Preliminary Results:

  • Lisse is a local property

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Some Preliminary Results:

  • Lisse is a local property
  • Lisse is not reflected by pullbacks

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Some Preliminary Results:

  • Lisse is a local property
  • Lisse is not reflected by pullbacks
  • Étale implies lisse

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SLIDE 44

Some Preliminary Results:

  • Lisse is a local property
  • Lisse is not reflected by pullbacks
  • Étale implies lisse
  • Lisse doesn’t imply étale

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SLIDE 45

Some Preliminary Results:

  • Lisse is a local property
  • Lisse is not reflected by pullbacks
  • Étale implies lisse
  • Lisse doesn’t imply étale
  • S → S is lisse where S is a cogenerator

Smoothness In Zariski Categories – p. 25/29

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SLIDE 46

Questions

  • Is Grothendieck’s E.G.A definition is equivalent

to mine in a Zariski category?

Smoothness In Zariski Categories – p. 26/29

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SLIDE 47

Questions

  • Is Grothendieck’s E.G.A definition is equivalent

to mine in a Zariski category?

  • Develop a nice, useful definition of perfect field.

Regularity vs. Smoothness...

Smoothness In Zariski Categories – p. 26/29

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SLIDE 48

Questions

  • Is Grothendieck’s E.G.A definition is equivalent

to mine in a Zariski category?

  • Develop a nice, useful definition of perfect field.

Regularity vs. Smoothness...

  • Develop classification of maps which are not

smooth and catalog the effect that blow-ups have

  • n such.

Smoothness In Zariski Categories – p. 26/29

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A Hopeful Sign

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LWSR

Suppose V → W is a map of varieties. Then there exists blow-ups ˜ V → V and ˜ W → W so that the diagram ˜ V V ˜ W W commutes and the canonical map ˜ V → ˜ W is flat.

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SLIDE 51

A proof of LSWR would complete the proof of desingularization of 3 dimensional varieties and give a big leg up on the desingularization of those in higher dimension.

Smoothness In Zariski Categories – p. 29/29