On relative log de Rham-Witt complex
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KonditionszahlHomomorphismusGeradeArithmetisches MittelTabelleVerträglichkeit <Mathematik>Residuumt-TestGerichteter GraphMathematikMaß <Mathematik>NichtunterscheidbarkeitMomentenproblemRechter WinkelPhysikalisches SystemNatürliche ZahlKomplex <Algebra>Projektive EbeneMorphismusZahlensystemEinfacher RingModelltheorieResultanteArithmetischer AusdruckFlächeninhaltMereologieRelativitätstheorieGradientVertauschungsrelationDifferentialJensen-MaßDerivation <Algebra>Algebraisches ModellTheoremPaarvergleichMonoidLogarithmusMinimalgradKohomologieVorzeichen <Mathematik>BetafunktionVorlesung/Konferenz
08:36
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15:17
Gleitendes MittelTrigonometrische FunktionLeistung <Physik>Algebraische StrukturLogarithmusBetafunktionTeilbarkeitNatürliche ZahlModulformEinfacher RingQuotientAussage <Mathematik>Projektive EbeneKlassische PhysikKomplex <Algebra>DiagrammMorphismusPaarvergleichGefrierenDivisionEntscheidungstheorieMaßerweiterungAbgeschlossene MengeMultiplikationsoperatorVorzeichen <Mathematik>OrtsoperatorArithmetisches MittelEreignishorizontSortierte LogikKonvexe MengeElement <Gruppentheorie>Rechter WinkelVorlesung/Konferenz
21:58
AchtArithmetisches MittelWurzel <Mathematik>Einfacher RingMonoidGlattheit <Mathematik>Güte der AnpassungIsomorphieklasseMorphismusKomplex <Algebra>TheoremKörper <Algebra>Charakteristisches PolynomGeometriePunktIdeal <Mathematik>Projektive EbeneAlgebraische StrukturPerfekte GruppeLeistung <Physik>QuotientFaserbündelBeweistheorieDiagrammLogarithmusResultantePaarvergleichStellenringMathematikModulformGerichteter GraphPhysikalische TheorieProzess <Physik>WasserdampftafelMomentenproblemMereologieZentralisatorFlächeninhaltMaßerweiterungUnrundheitDivergente ReiheModelltheorieSummierbarkeitFunktionalEnergiedichteVarianzResiduumSpezifisches VolumenMaß <Mathematik>Vorlesung/Konferenz
30:27
ResultantePaarvergleichTheoremAlgebraische StrukturRangstatistikFresnel-IntegralAnalysisVorlesung/Konferenz
31:28
Algebraische StrukturVarietät <Mathematik>RelativitätstheoriePaarvergleichLogarithmusDivisorEreignisdatenanalyseAssoziativgesetzGeradeSchwebungRechter WinkelVorlesung/Konferenz
32:22
IsomorphieklasseÜbergangswahrscheinlichkeitAnalogieschlussMorphismusLogarithmusKomplex <Algebra>p-GruppeKurvenintegralAlgebraische StrukturGruppenoperationTorsionEinfacher RingTermModulformBeweistheorieKonditionszahlResultantePaarvergleichTheoremPhysikalische TheorieGibbs-VerteilungStellenringRechenbuchDifferenteWurzel <Mathematik>WürfelMultiplikationsoperatorGerichteter GraphDimensionsanalyseVorlesung/Konferenz
37:23
TensorAlgebraische StrukturEinfacher RingMultiplikationsoperatorElement <Gruppentheorie>Folge <Mathematik>GruppenoperationNatürliche Zahlp-GruppeRelativitätstheorieResultanteLogarithmusPoisson-KlammerEinsSortierte LogikMathematische LogikKörper <Algebra>Lokales MinimumKartesische KoordinatenEuler-WinkelMengenlehreLie-GruppeDifferenteKlasse <Mathematik>OrdinalzahlOrdnungsreduktionVorlesung/Konferenz
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FunktionalSummierbarkeitElement <Gruppentheorie>MereologieModulformMultiplikationsoperatorp-GruppeTeilmengePolynomringRechenbuchAlgebraische StrukturEinfacher RingDifferentialKomplex <Algebra>Algebraisches ModellIndexberechnungFunktorRechter WinkelRelativitätstheorieObjekt <Kategorie>SterbezifferLoopTopologieNumerische MathematikMomentenproblemSummengleichungGruppenoperationVorlesung/Konferenz
47:47
Vollständiger VerbandPartitionsfunktionElement <Gruppentheorie>ModulformModelltheorieRelativitätstheorieMereologieGewicht <Ausgleichsrechnung>SummierbarkeitDifferentialp-GruppeMinimumRichtungMultiplikationsoperatorRechenbuchKonditionszahlKartesische KoordinatenDreiStandardabweichungMaßerweiterungPhysikalischer EffektSpezielle unitäre GruppePunktspektrumGruppenoperationIsomorphieklasseVorlesung/Konferenz
50:43
GeradeFolge <Mathematik>Partielle DifferentiationEntscheidungstheorieVorzeichen <Mathematik>Inhalt <Mathematik>DifferenteInjektivitätGrundraumMomentenproblemDiagrammPartitionsfunktionMereologieSummierbarkeitGruppenoperationGewicht <Ausgleichsrechnung>DifferentialTermPaarvergleichExakte SequenzKomplex <Algebra>Vorlesung/Konferenz
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TeilbarkeitQuotientRechenbuchSummandFreie GruppeBeweistheorieQuotientenraumSummierbarkeitIdeal <Mathematik>MereologieIndexberechnungElement <Gruppentheorie>Radikal <Mathematik>GegenbeispielInverser LimesTorsionÜbergangswahrscheinlichkeitPaarvergleichLokales MinimumDruckspannungFunktionalSpezielle unitäre GruppeMultiplikationsoperatorResultanteFamilie <Mathematik>ExistenzsatzRechter WinkelMaß <Mathematik>UnordnungGeradeParametersystemWärmestrahlungSortierte LogikRichtungVektorraumWarteschlangeVorlesung/Konferenz
Transkript: Englisch(automatisch erzeugt)
00:00
Thank you very much. First, I'd like to thank the organizers for the invitation.
00:20
And it's a great honor for me to give a talk celebrating Professor Ufagaba's 60th birthday. And, yeah. Okay, so the title of my talk is Relative Log-Dramatic Complex.
00:49
In the talk, I will first explain the definition of Relative Log-Dramatic Complex, which is a log-relative version of the dramatic complex of El-Z,
01:03
and which is also a log-version of the relative dramatic complex of language zinc, and which is introduced by a former student of mine, Matsue. And I will explain the comparison theorem
01:22
between relative log-dramatic cohomology and relative log-crystalline cohomology, which generalizes most of the previously known results by El-Z, RangaZinc, Yoto Kato, and Matsue. And this is a joint work
01:49
with Kazuki Hirayama, who is also a former student of mine.
02:01
And throughout this talk, we fixed a small p, a prime number. And first, I give the definition and our main result.
02:35
First, I will fix some terminologies, maybe which already appeared, maybe.
02:43
A log-link is a triple consisting of R and large p and alpha, where R is a commutative ring with a unit,
03:02
and large p is a fine monoid. Monoid is commutative, and fine means a finite regenerative integral monoid. And alpha is a monoid homomorphism from p to R.
03:28
Yes, where R is realized as a monoid by multiplication. And usually, I will simply write the log-link like Rp, and we omit to write alpha.
04:15
And next... And for a log-link, R, large p, alpha,
04:31
its bit log-link, which I denote by WmR large p,
04:43
this is a triple. This is the log-link, consisting of the bit m over R and p and beta,
05:01
where beta is a composition of from p by alpha to R, and then we compose it with a tight mu to WmR. And then, assume that we have a morphism of log-link
05:23
from Rp to Sq, a map of morphism of log-links. Then, this is a map between its bit log-links,
05:48
which is compatible with respect to m. And then, we define the notion of log, then log f be pro-complex.
06:05
This is a projective system, a projective system, E, m dot,
06:20
d, m dot, and d log m, where m is a natural number greater than or equal to 1, a projective system of log differential graded commutative,
06:42
commutative means hyper-commutative, graded commutative algebra,
07:09
algebras over WmSq,
07:23
on WmSq over WmRp, maybe it would be better to write this also as a projective system, satisfying the following three conditions.
07:41
So this means it starts with, everything is WmRp linear? Yes, yes, yes, yes. And WmSq maps to the zeros? Yes, yes, yes.
08:00
This is the derivation. And is it only in non-negative degrees? Non-negative degrees. Non-negative degrees. And is it strictly anti-commutative? Strictly anti, in the sense of previous talk, yes. Thank you. And satisfying the, yes. First, yeah, as Professor Garbo said, we have a morphism from,
08:29
ah, I'm sorry, I forgot, I'm sorry, I'm sorry. We forgot to write the important maps equipped with, yes, yes, maps
08:45
F from the E m plus one dot to E m dot, and V from E m dot to E m plus one dot. Maybe we should write that also as a system,
09:02
more of a projective system of gradient Wm groups, satisfying the following three conditions.
09:22
And the first condition says that we have a map from WmS to E m zero, and we assume that this is compatible with F and V. And secondly, F is a map of gradient algebra,
09:56
over F on between.
10:05
And thirdly, we have the following set of equations, F V is P, F D V is D, where I omit the subscript m and superscript dot, but you can guess which index should be here.
10:24
And F D V is D, and yes, I forgot. Yeah, but the third one is, I forgot. F D tai-hi-mu. F D tai-hi-mu is tai-hi-mu to the P minus one to the D tai-hi-mu.
10:41
And projection formula, V of omega F eta, omega F eta is equal to the V omega times eta. And finally, the F D log Q, Q is an element in large Q, is equal to D log Q.
11:01
So you assume that there is a map D log, which is what you mean by differential, you assume that there is a D log map. Yes, D log map is a map Q to the E m one.
11:20
And D D log is equal to zero, and D log D is, D log is a log derivation. Pair D log m and D zero m is a derivation, and D m one composed with D log m is equal to zero.
11:45
D log m, D log, this pair is a log derivation in the sense of log geometry.
12:03
Okay, so yes, this is a definition, and then it is a problem, of course, based on the work of Yuzi and Landau zinc.
12:21
And this form is due to Matsue, that there exists the initial object. Ah, I'm sorry, that F G, probably I should say that log F V pro complex on, something over something on,
12:41
S Q over R P. When R P to S Q is fixed, then we have the category of log F V pro complexes on S Q over R P, and then there exists the initial object
13:03
in the category of... And R is above C something times S Q. Pardon? R. Just for the existence, I don't think we need it. No, but you need the divided power.
13:21
Ah, yes, I'm sorry. Yes, yes, yes. Sorry, R is Z bracket P algebra. I'm sorry, thank you very much. And there exists the initial object in the category of log F V pro complexes
13:41
on S Q over R P. And this is denoted by W M omega dot S Q over R P.
14:03
And D and D log. Yeah, we omit to write M M dot here. And N is called... N is called the relative logarithmic complex
14:25
on S Q over R P. And then next, as a next step,
14:42
the people there have shown that it signifies U Z, lambda Z, and a much way,
15:00
has shown that if we are given F, the map from X M to I N, this is a morphism of fine log schemes on which P is nilpotent.
15:31
And this is canonically, in some canonically in natural sense that C form complex is on
15:43
eta side of X, which is equal to the eta side of the bit schemes. Such that, such that,
16:04
eta locally, I will write, eta locally, it is written
16:21
as the previous one. If X M to X M to I N is written as a spec induced by the morphism of log rings like this. So this is the
16:40
quasi-coherent? I'm sorry, quasi-coherent? Yes. On bit schemes. And yes, this is the definition. And next I will explain briefly the comparison map.
17:04
So let F as before, as in the proposition definition, and yes, O M, let O M is the structure crystal on the log-crystalline side of X M over bit M over I N.
17:29
And U M, then we have the map, I'm sorry, quiz, quiz in topos, and then U M, we denote by U M the canonical projection
17:42
from the crystalline side to eta side, because I'm in France, I'd like to read this, eta O,
18:01
eta, and then, yes, log, log, yeah, probably I stress it. No, this is a classical eta, classical eta. And yes, and then we have
18:21
canonical, we can define canonical map from the R U M star O M to the, yes, W M, our reactive log-dramptile complex, which I call the comparison map
18:42
in this talk. I will not recall the definition in general case, but if F is logarithm and there exists a following diagram,
19:08
a diagram which I write here, X M is O bar Y N, and this is map F, and this is in the
19:23
V to log scheme, and I ask, P is nilpotent here? P is nilpotent, yes. As before, I mean that includes the assumption that P is nilpotent. All right. I'm sorry that I, I should write,
19:43
I'm sorry, yes. The crystal inside is a kind of log divided power thickening compatible with P, something like this. Divided power thickening should be exactly close to the margin, and yes, with respect to, with respect to the, yes,
20:01
the idea of defining this. This is, this has a canonical log structure. And PD structure, and with respect to this, this. On the base. On the base. And yes, with respect to classical eta topology.
20:22
And if F is assumed to be logarithm, so we can take locally, eta locally, the logarithm's lift, and if we, locally enough, we have this diagram,
20:41
this is over here. And then, C is the following map. C is from R U M star O M, but this is, this is written as a logarithm complex of X M O M M
21:01
over the beta M O Y N. And this restricts to the beta logarithm complex of beta M over beta X M, beta M X M over beta M Y N.
21:21
And I forgot to say this thing, but our reactive logarithm beta complex is a quotient of this. Actually, the quotient of the PD version of this, but yeah, anyway. And so this is a map.
21:45
Y M is equal to, yeah, I forgot, I forgot to say this thing also, but if M is equal to one, then this is, this is all the, I think, equal. So it tells you, you are alluding to exactly, you, you impose a D of the divided powers
22:01
of the four-usual formula, or? Yes, yes, yes. D over X divided N is equal to X divided N minus one D X, something like that, yeah. So it is a quotient of the usual? Yes, yes, yes, yes, yes. So it is okay to write also like this, but okay.
22:34
And this comparison map, is it right there, is it long, or? This is the main result,
22:41
But under some assumptions, so. Does it prove it's well-defined globally, or? Yes, yes, yes, yes. Yes, yes, yes, yes. simple shaft technique, I'll use it here.
23:00
Then, yes, okay, okay, then, I give our main result. I'm going to work with the diagram. I assume that F
23:23
from X, M to Y, N is the morphism of FS-log schemes, I'm sorry, are logarithms
23:44
saturated morphism of FS-log, or morphism of FS-log schemes
24:00
on which P is nilpotent. And this would be nice if we
24:20
proved the theorem, and only this assumption, but unfortunately, I assume more of the following assumption. So just to remind myself, saturated includes integral, which includes exact, yes? Yes, the following.
24:46
We call this as this is condition on Y, N. It's locally around any
25:01
geometric point small y of large y, there exists Q, an FS monoid, and there exists some ideal J.
25:21
This is a radical ideal. A radical means that the root of J is equal to J. And there exists some R, contact ring, and such that
25:40
there exists some morphism from Y, N to the spec of the R, Q monoid ring divided by Z, they are generated by J, and with monoid Q, there exists a strict smooth morphism.
26:04
Smooth morphism like this. Such that because this is a strict smooth morphism, this induces a chart from Q, Y, to N, and we assume this is good at Y.
26:22
Good at Y means that this induces an isomorphism with Q and N divided by O, Y cross at small y. So, we have to assume the local structure of Y, N.
26:45
yes. Then then says that was isomorphism.
27:02
So, there is a kind of base change question here that you are asking locally and bad in something good? You mean the general case and the defect or something? this yeah, I don't know the base change question or the
27:21
reactive lockdown with the complex. So, this is one of the problems. Even in the absence, I don't still know. Oh yes, the following things are not sufficiently flat. Yeah, I yeah, that this kind of problem. Okay, so you formulated
27:41
this efficiently derived sense. Oh, maybe, yeah, maybe, but I have not considered in derived derived versioning yet. I don't know. yeah, maybe. Don't you think it's probably true without that assumption? Do you
28:01
probably think it's true without the star assumption or not? Not completely clear. But yeah, I hope so, but I have no idea. Okay, then C is a projection and I will
28:29
explain previous works on that. Of course, this kind of theorem is the first of proven by Yuzi. In this case, there was no log structure yet, so N,
28:40
N are trivial and Y was a perfect scheme of characteristic P. yes, this is generalized by a lambda-zinc.
29:02
In lambda-zinc case, lambda-zinc is a trivial case, M and N are trivial and Y is in general. And in trivial log case, of course, this assumption is automatically satisfied, so it's nothing. And for log-drambit
29:20
complex, there is a work of Yodo Kato. They treat the case where Y is a spec K, a perfect field of characteristic P. P is
29:40
always fixed, so P. And but any log structure, I think. And in this case, the log-smooth morphism of Kato type. And yes,
30:00
yes, which I'd like to explain now. So with another definition of drambit denoted by W M
30:20
or small omega-dot. And but actually we can prove that they coincide under this assumption. And the comparison is compatible with this isomorphism.
30:42
our result so when M and M and F S log structures, our result contains
31:00
a Yodo Kato. But in Yodo Kato, they proved the theorem when M and N are just fine log structures, not necessarily F S log structures. strictly speaking, our result does not free contain the result of Yodo Kato.
31:21
And I also mentioned the result of Matsue in the paper he defined the relative logarithm Matsue treated the case that two cases M is
31:41
trivial and X over Y is smooth and M is associated to a normal crossing divisor a relative normal crossing divisor on X over Y. This is one case he proved the comparison. And the second case is that N is
32:00
the log structure associated to the zero map from N to the semi-stable variety.
32:22
And in these two cases Matsue proved the comparison theorem. And actually in these cases that this condition is satisfied. our result generalizes that of Matsue.
32:47
And we can prove this isomorphism by taking that this log-f b pro-complex and then we have a map from here to here and both
33:01
admit the suggestion from the log-dram complex and we know the kernel from the log-dram complex. So in the case
33:21
of the characterization of the lambda-8 complex is primary method you can also do it without the x you can have an problem that is the b or n. I think he used this definition I think he used b only b so
33:40
this f-b definition is invented by Langasinghe and the universal problem is only the written analog of the definition in this setting also. Okay. Yeah. What about all the all the structure
34:00
theory like canonical filtration with various exact sequences? No, not yet. So you have a lot of detailed things. Yeah, but I don't use this
34:21
filtration such kind of theory. Anyways, it uses the local calculation in terms of complex integral forms so there is some precise structure of the you need to okay. Anyway, so the okay.
34:41
yeah yeah we of course we need to calculate the structure but yeah. So so I always sketch the proof.
35:07
So we may work eta locally and first we reduced the simple case so we may
35:20
assume that f is the morphism like this spec of r p divided by j r p with log structure p to the spec of
35:40
r of q divided by j r of q with log structure q associated to the map from q to p this is an injective
36:01
saturated morphism map of f is monoid and such
36:20
that q group and p group and the coca and p group of q group these are torsion free torsion free z module and this is the map from q
36:40
to p and as before j is a radical ideal
37:02
yes saturated is integral saturated is integral you can assume this thing localized because it's mod p to r
37:20
yes r is a ring actually after reducing to this case we can work over z bracket p ring but yeah you can assume that p is nilpotent or you should assume that p is nilpotent or otherwise yes yes yes exactly
37:40
yes exactly p should be nilpotent or in this result p should be nilpotent or yeah and then if we reduce to this case we can work over the definition it is
38:00
w m omega dot s q over r p and yes first we teach the case j is empty and this is in some sense essential case and
38:20
in this case the omega dot w m r q tensor over z q with
38:40
z p and that was a log structure p I'm sorry I write like this over w m or yes r q r q q bit log ring over r q q and the target is our
39:02
relative I'm sorry w w m omega dot over r p p I'm sorry I write like this over r q q and I will denote
39:20
simply this over omega dot p over q for simplicity I will write here as w m omega dot p over q
40:02
and then yes I need to prepare some more so p we have a taking bar
40:21
to the bit time bar p and for an element x here we denote it by the large x to small x but scheme large x will not appear in the sequence so this is harmless and yes and also we have a map
40:41
from p to by d log w m omega 1 p over q and this induces a map from q group and q group is torsion free so I fix this
41:10
and again for yes for small x
41:20
in p p 1 over p this I think has appear in some talk this is an element consisting of elements in y in p group tensor over z I with z 1 over p such
41:40
that z i with z 1 over p that z i z 1 p such
42:06
that z i with natural number such that p to the u times x in p and
42:27
we define the x part of the log-drambit complex to be the sub-module
42:41
actually sub-complex generated by the elements of the following form v over u x over the y times x to the p to the u x times x dot i
43:01
in some large i over d log x i and its derivative d over v over u p to u x times x times i in large
43:20
i d log x i and where x is fixed and y is an element of m minus u x r and i is a subset of the possibly empty r from set from 1 to r r is here there p is isomorphic to r and
43:41
the x part is a sub sub write the v is up this follows the way of writing of language it's awful sounds awful maybe ok
44:00
what is it d times d d yeah so so yeah we can write like this yes yes yeah just i follow the way of writing but if you don't like it so we can write like this
44:21
yeah but then it would be disaster to change how more people so ok and
44:41
then by with some calculation we can prove that w m dot p over q is equal to some of its x parts x runs through p one over
45:00
p over x parts but we would like to prove that this is a direct sum yes to do so one sees the following
45:21
so we can consider the same drumbeat complex with p and q replaced by p-group and q-group and if we write
45:42
precisely this is a log drum bit complex like this but it is canonically equal to the
46:03
drum bit complex without log structure and this is treated by langer and zinc and this this ring is relatively a lowland polynomial ring over this ring
46:20
and then langer and zinc introduced a notion of basic bit differentials and proved that any element is written
46:49
in a algebra and I think it is also written in paper but more short and yeah and yeah and so for this we know better
47:00
and then so we compare them then one we can see the following so we have w m omega dot p over q and this maps by functionality to
47:20
this one and this was a sum of x and its x parts and this also has a sum of x parts but x is now ran through the
47:41
p-group 1 over p but here we can check that the relation between the x parts and the basic bit differentials and we can prove that this map respects
48:01
the x parts so we have this map for each x in p 1 over p
48:29
and this is first we consider the case q is
48:40
trivial and in this case we use basic bit differentials I said that the element is written in a unique way as a sum of basic bit differentials and we construct the map in the opposite direction by defining
49:00
the image of each basic bit differentials and check that this gives the converse if I have time maybe I don't have time but I'm sorry I don't have time if I had time I would explain some calculation but I
49:21
omit it and then this shows that this is injective in particular and then
49:41
q is trivial and p is concrete so the sum of the bottom is equal to 8 or is it some partition yeah I mean
50:01
this map is all elements in p group one you don't need some partition yeah yeah no no I don't think so
50:21
it is written on the image of this form and yeah this vanishes you need some model you need some partition oh the partition yes so this is so the weight is fixed and
50:42
the so this x part is some of basically the differential of fixed weight so the partition moves and sum is with respect to weights yes and then next we we
51:03
we concept map comparison map and yeah again we compare with the case before you didn't explain so you claimed yes
51:20
you do it the comparison of the to show the direct sum you have to compare the terms of the composition and you said that somehow you do it when q is trivial yeah first we construct a map when q is trivial and we prove that this is
51:40
a map for general q in this setting with j empty yeah this is so we have something like how to say exact sequence which is valid for log if drum we have an exact sequence of drum complex if we divide over something
52:01
and we have a similar formula for relative log and yeah this is proven by much way and yes and we have this diagram and this is injective this is that map and yes and this is injective also this is essentially and we can prove the
52:20
injectivity here here we we should use crucially and that this is saturated and yes and then then we see that image c is injective I'm sorry c is injective and image c is
52:40
actually a direct we can check this map completely and we see that it is a map so image is a direct where p prime is q 1 over p plus p
53:02
and then image is a direct so finally we should check that the other part the remaining parts are cyclic but in this case because we have shown already that this p our x part is
53:20
isomorphic to the x part in lambda-zinc sense and so this is a cyclic for x in p 1 over p which is not contained in p prime by the we know the effect of applying d to basically
53:40
to differentials very well so this is if we use lambda-zinc calculation we finish the proof and this is a case where j is empty and maybe
54:06
in remaining 4 minutes I will explain very briefly the case j general
54:20
and in the case general actually our map c is the quotient of some quotient map between some quotient of the map in the previous case
54:48
radical yes and we need to so we need to
55:04
know what i dot and i prime dot for this we introduce the following terminology
55:20
x an element in p is called j minimal if one cannot write x as a sum of y and z when
55:41
y is j and z is in p and this is the definition of an element in large p but we need to define the j minimality for an element in p 1 over p and the element x in p 1 over p
56:00
is called j minimal if for any n in j such that p to n x is in p this element is j minimal
56:23
and actually we can prove using the radicalness j is radical that for any is equivalent for some n and so we see that
56:40
these two definitions are compatible for an element x in p and then we prove that i prime dot is
57:00
a direct sum of x parts such that x is not j minimal and i dot
57:20
is mapped by c to x in p prime so q 1 over p such that x is not j minimal j is an ideal in q
57:43
j is an ideal in q q q i'm sorry j is a radical in q q is in p
58:00
and so by dividing out we see that this is a direct sum of x parts in which an index is j minimal and also for this and so this is the image of a direct factor and the other direct factors is acyclic by the same reasons before so this finishes the proof and if we drop the assumption that j is
58:21
a radical or if we drop the assumption yeah a constant star it seems that by constructing this quotient the non-trivial this summand can be have to say divided in non-trivial way so we need more
58:40
we would need more calculation to prove beyond this case thank you very much so we have an example that this is not this quotient
59:01
is not direct sum so this is this is divided non-trivially but if both hand sides are divided non-trivially this is possible that it remains to be quasi-sythomorphic so I don't know the situation so I don't have
59:20
a counterexample for the comparison so under your assumptions suppose you take a projective limit of the wn yes and get the wn omega i is it p torsion 3
59:40
probably yes but probably yes probably yes maybe is it known and so is it religious the language so you mean the language case or or or or or oh oh
01:00:02
In my case, R may not be reduced, but in the... Well, if R is not reduced, then I guess it can be shorter. Ah, R is not reduced. OK, so maybe... So, OK means what? So, if R is not reduced, then P-torsion can happen. But if R is reduced, it is...
01:00:21
Reduced to R... In P, is it that... Yeah, yeah, yeah, I know. I know. So, you have a reference for this? Sorry, in the R is... Is it that this limit is P-torsion free? No, I don't think so. Because I don't think you know the kernel of D under the new R.
01:00:47
You don't have the... You are still part of the local structure, but not much. So you want to know that the kernel needs major F on the... Yeah, yeah, I have not yet explored it. First, you want to know that it's P-torsion free.
01:01:02
So you know this canonical situation, and you want to see it as injective, essentially. Yeah, yeah, I have not yet explored it. This, I think, is not known in P-torsion. By long as ink? Yeah, no, I don't think so, yeah. Sorry, yeah, thank you.
01:01:20
Yeah, it would be... I should explore, yeah.