 Okay. Yeah. So thank you very much for the introduction and invitation. It's really a tremendous honor for me to speak here at this conference. I first had the opportunity to meet Luke maybe, I think maybe it was about three or four years ago. And since then I've had a lot of exchanges with him and I really appreciate how kind and generous and encouraging. He's always been one of my last trips before the pandemic. I was in Paris and Luke helped hosted me for a bunch of the trip. And Luke's mathematics has certainly had a great impact on me both from exchanges with him and from from studying his work directly. So yeah, so thank you, Luke and happy birthday. So what I want to talk about, I want to talk about some results, I guess, in the area of logarithmic algebraic geometry. So this is a subject of which Luke is of course, one of the principal architects. And in particular, I want to talk about some versions of log-syntomic chronology and some comparisons with paedic nearby cycles. And start by talking about logarithmic homology theories in general. So what I'm going to talk about is joint work in progress with Bargov-Baud and Dustin Klosson. So I want to talk about log-syntomic homology and sort of an integral version of log-syntomic homology. But to get there, I want to start by first just talking about logarithmic paedic homology theories in general. So I think somehow the starting point for this is the following example. So if x is a smooth variety over a field k and d sitting inside x is a normal crossings divisor and j is the open inclusion from u meaning x minus d sitting inside x, then one constructs the logarithmic Diron complex, sorry k is a field, constructs the logarithmic Diron complex and make up our star of x log d. And so this is, I guess this is a quasi-coherent chief of differential-created algebra on x and basically it's generated, well, so this is, sorry, so I guess I should say that it's a sub-sheaf of the Diron complex of u. So it's a sub-sheaf of the Diron complex of u and it's a sub-sheaf which is generated by, well, differential forms on x as well as sort of dxi over xi where these come from local equations of the divisor d. Right, and so I guess it's a, so this is a definition of the logarithmic Diron complex in this context and in characteristic zero it gives a, it computes the Diron cohomology of the open complement which is one reason this is a good definition. Right, so this definition I guess it generalizes to the theory of log structures. So, right, so maybe I should say this computes cohomology u in characteristic zero. Okay, so Cato and Fontaine Lucie define more generally the notion of a log structure on a scheme, so on a scheme x. So it's given by a sheaf of monoids which I'll write as m sub-x on the etal side of x and a map of sheaves of monoids. So I'll call this alpha which goes from m sub-x to to the monoid which is just the structure sheaf of x but with the monoid structure given by multiplication instead of addition with the property that alpha inverse of the units is isomorphic to the units via alpha and so then this structure is called a log scheme. So for example, if we're in the setting as above so if d sitting inside x is the normal crossings divisor then we have a log structure given by m sub-x is going to be those functions which are invertible on on u. Okay, so I want to fix some notation here so if we have a log structure like this some notation for further on so if we have a log structure this is also going to be we also obtain a sort of a well so we also define or we notate m sub-x bar to be the quotient of m sub-x by o-x o-x cross so this is sort of measuring the the non-triviality of the log structure and I guess usually we're going to be in a setting where o-x cross is acting freely on m-x so this is again some sort of sheaf of monoids and now m sub-x bar is naturally going to come with a map so so so from the structure map of the log structure m sub-x comes with a map from m sub-x to the following following sheaf of monoids which is the sheaf of monoids of sorry line bundles with a section so this is I guess the this is represented by a1 mod gm and effective effective cartier divisor right so this will be useful later on okay so so in general I guess it's a sort of general phenomenon that a lot of homology theories that one can define in the world of world of schemes naturally can be extended to the to the category of log schemes and so the example is right so so maybe the first example is that if you're given a log scheme over a field then one can define its log diram complex so I guess we had called this omega upper star of x comma m sub-x and this is well this is defined sort of similarly as the diram complex is defined similarly via generators and relations so and you start with the diram complex well I guess I'll say this sort of informally it's and then you add you sort of formally add objects d log of m for every dm divided by m for every element m of this this sheaf of monoids so this is supposed to be d log of m and then you sort of enforce the you sort of impose the the appropriate relations for for the diram complex and for what d log of m is supposed to satisfy so this generalizes the example of the logarithmic diram complex so this is the first example but there are many other piatic homology theories for which there are also also log log versions so for example there's log crystalline homology developed by kato so let's say we have a log scheme over fp then can define the log crystalline homology and this is defined right so sorry so maybe what I should say is that this is going to be some sort of lift of well in in nice situations of the log diram co-comology to zp from fp now and this is defined and this log crystalline homology so I guess this is absolute log crystalline homology is defined using well I mean crystalline homology is defined using the crystalline site and one sort of defines a log version of the crystalline site to to define log crystalline homology so adapt definition of a crystalline site um so I guess there's the example there's also the examples two other examples that I want to keep in mind but maybe I'm going to come back to sort of later our symptomic homology uh so this is going to be some sort of filtered for danious eigenspace of uh of log diram homology but in in mixed characteristic uh and piatic nearby cycle homology so this is for schemes over zp and this is going to be the homology the etel homology of the rigid generic fiber so for formal schemes over over zp so these are also two examples you want to keep in mind and we'll come back to more later but I guess the most powerful of these piatic theories is given by prismatic homology and so a logarithmic version of prismatic homology has been developed by by koshikawa so so given a prism a comma i well then prismatic homology is some homology theory for formal schemes over amod i that lives in modules over a and one can one can generalize this to log schemes so can take the prismatic homology of log formal schemes um and actually so koshikawa considers a say more general situation where where a comma i is is also allowed to have some sort of log structure um and uh and this this homology theory is is uh i mean there are a lot of you know structure theorems in prismatic homology like the hud shape comparison and so forth and these these have adaptations to the log setting where sort of differential forms you have log differential forms um so there's a there's a log prismatic homology and this was also preceded by um work of uh sasnovicius and koshikawa in the case of semi-stable schemes uh over oc and where the prism is is a prism a comma oc um right so um so this is an important special case where i was done earlier um right so so in general why you know why does one want what is the motivation for having these log um log theories well at least part of the motivation is that uh i mean these various types of phoetic homology theories um are are related to each other in various ways uh which it's governed by i guess part of the subject of of uh phoetic hodge theory um and having so a lot of these sort of comparisons can be extended to the log setting um and so so so there are relations that you can you can obtain between between phoetic homology theories and in the log setting uh as a result so so let me just explain a corollary and I guess this is a corollary of the theory of log prismatic homology um so let's say um right so so let's see the uh complete algebraically closed non-archimedean field ring of integers oc and residue field k and let x be a smooth proper scheme over oc uh and let d be a normal crossings divisor and then the conclusion is that if you take the the dimension of the etalchromology groups of x minus d excuse me is it a relative normal crossing divisor sorry is it a relative normal crossing divisor yes thank you thank you yes relative normal crossings divisor thanks uh so then the the dimension of the etalchromology uh the dimension of the etalchromology of the of the complement is going to be bounded by the dimension of the diromcomology uh of of the special fiber with log poles uh along the reduction of the special fiber of d so dimension over k so um right so this is the type of comparison theorem that that follows from the theory of prismatic homology actually the the a of homology so this is is proved by batmorrow and schultz uh when d is empty and and sort of a log consequence of the log theory is sort of being able to make some more examples like this and i guess also we talked about torsion sorry torsion also uh yeah yes i think yes you have bounds for the yeah the amount of the amount of torsion in terms of the homology so yeah so the first thing i want to do in this talk is i want to explain how one can build these homology theories so as i mentioned um so for example um log i guess log crystalline homology and log prismatic homology are defined uh using logarithmic uh sort of logarithmic versions of the original site theoretic constructions um but what i want to what i want to explain is that there's also a way that one can sort of reconstruct these logarithmic theories uh from the non logarithmic versions um and so this this will become useful when i uh when i want to talk about this log symptomic homology later on um and that's and the strategy is to use this uh use this construction uh use a use a construction in in logarithmic geometry called the called the infinite root stack so the first goal is to construct or to give an alternate construction of these theories um from the non logarithmic versions using uh using a general construction called the infinite root stack um okay so this is a construction which is due to talpa and vestoli uh and it takes those inputs so so let x comma m sub x be a fine and saturated log scheme um and so then what they do is they construct a stack that they call uh so then there is a stack uh which uh so let's we'll denote it um the infinite root stack of this log structure um let's denote it like this uh and there's a stack this the stack comes with a map to x um which is the isomorphism where the log structure is trivial and somehow the stack is encoding uh is sort of encoding the geometry of of the log structure um um so let me give the basic example the basic example is that if you have a one uh and the log structure is given by the origin so it's given by the divisor at zero um so then the infinite root stack of a one comma zero it's going to be the inverse limit uh it's going to be the inverse limit over n of the stack a one mod mu n uh inverse limit of n under divisibility uh and right so I guess what's going on is that the transition maps are they're raising to the appropriate power on a one and they're raising to the appropriate power on mu n so raising to powers um so I guess right so I guess we can think of this as it's it's like spec of k brackets x to the one over n modulo mu n at at each stage how does this compare to the koumari tall site uh yeah thanks for the for that question I will I will come back to that in in just a in just a moment actually so um yeah it's closely related um okay so but yeah maybe let me just let me just do two more examples um so right so this example can sort of be generalized so uh so suppose the log structure uh comes from a from an effective Cartier um sorry comes from a divisor uh from so from a single divisor so let's say a smooth divisor um so uh right so log structure on x so then that effective Cartier divisor is classified by a map to a one mod gm uh and then uh so this infinite root stack of x comma m sub x or we could also call it x comma d uh is given by uh so what you do is you form the fiber product of x with so you have a one mod gm mapping by multiplication by n to a one mod gm and you you take that fiber product so you're taking an nth root of the of the divisor of the blind bundle with section given um well given by this uh the structure and you form the inverse limit of n along to visibility um right so so these are two basic examples um and in general the phenomenon that happens that you sort of see here is that when you take this infinite root stack of x comma m sub x uh mapping to x then uh the fibers at a point well x and x uh is given by the classifying stack of uh so if you have a if you have a point x in in in the base scheme x then uh uh then there's a rank r which corresponds to the so then if you let r to be the rank of the monoid m sub x bar the sort of residual monoid um then the fiber of this this this map from the infinite root stack to x is given by the classifying stack of the tape module of uh the group scheme of the group scheme of all roots of unity so take the group scheme of all roots of unity form its tape module and uh take its classifying stack to the to the power of r so so that's what the fiber looks like at at each point um right so I think this is uh this is uh analogous to the construction and partially motivated by the construction of the kato nakayama space for for a complex log structure where this would be replaced by a circle um okay so so let me just give the definition let me just explain that the definition in general so uh so if you're given a map from t to x where x is the log scheme if you want to lift it to the infinite root stack of x comma m sub x then what you need to do is well so the map from t to x is is giving you so f yields uh a map from f upper star of mx bar uh to the to the groupoid of of line bundles with section so I guess this is like a symmetric monoidal functor and what you need to do is you need to you need to make this map infinitely divisible so in order to lift it to give a lift need to make this map so let's call this map phi need to make phi divisible i.e uh so let me move this down need to extend it over the rationalization of the monoid f upper star of mx bar um and so that's exactly what we saw sort of going on earlier which is that you you know you had like a log structure given by a single divisor and so that's that's like a effective cartier divisor that's like a line bundle with a section and you want it you want to take the roots of that structure okay um but so so when one formulates it in this way that's on top of in the Stoley's paper um it's it's a definition that that's not sort of it's not really in terms of the the given divisors really in terms of the log structure okay so um right so the general sort of idea or sort of principle is that uh a homology theory uh with uh for a log scheme so uh is given by its value on the stack with and the stack with no log structure and i guess this is this should say with profinite coefficients um so i guess this is sort of a general sort of idea in this in this business um so yeah so you want to go back to um so so right so this this idea has at least a couple of antecedents in the literature um so one of them so to go back to to Arthur's question uh is that this is related to um this is related to kato's comor flat site not quite the comor atoll site because we're going to be taking my p power roots sorry so the antecedents are given by kato's comor flat site so i believe the the comor flat site is closely related to like the maybe the fppf or finite flat site of this infinite root stack i think this is discussed in the top of the Stoley paper so i believe it should be possible to phrase this construction in terms in terms of the comor flat site instead of the stack as well um and another antecedent is uh is Olson's work on the stack of log structures and in Olson's setup as uh so instead of instead of uh instead of forming this infinite root stack there's there's a map from x to the stack of log structures and one considers some sort of relative theory um and so i believe that's closely related to the setup because this the sort of infinite root stack and i think it's it's i think you can you can build it by um by sort of well sort of by making by starting with the the map to the stack of log structures and sort of making that infinitely divisible uh or sort of perfect and that will you know so so that will that will sort of make some comparison so haven't worked out you know details here but i i believe that this should be closely related to the setup of Olson uh where you where you sort of use the fact that making something perfect is going to get rid of sort of differential forms um okay but so um but basically the example that um wanted to um sort of an illustration of this principle is the following calculation um so let's let's consider the case where x is smooth and d sitting inside x as a divisor uh is a normal crossings divisor right so i guess it's a it's a relative normal crossings divisor if we're over some base some given base um and then there's the following right so there's a following calculation so if you consider the infinite root stack uh of x comma d um so this is right so i should say this is not quite it's not quite it's not quite an algebraic stack because you get these i mean you get these like big uh stabilizers but it's not so far from one um and in particular so we can do something like take the differential forms on it and so a calculation is that if we take um global sections of uh differential forms on this this object then this is naturally the same as differential forms uh on x with log pulse along d and this is with profite net coefficients so excuse me so when you say stack of course you have to specify the topology so for example you can use since you're with all the finances you can say f p q c topology but then you want to do a differential form so this not so i'm not sure what what the foundation so for for a usual i think stack of course we use that it's given by a scheme model who uh smooth something i mean smooth group way but here of course you have to work in some with bigger things maybe formally smooth so i mean so so i think one way of yeah sorry sorry i think one way of saying this oh so is is a stack in the flat topology for example sorry you mean f p q c uh sure or you could say some sort of in the centomic topology yeah f p q c topology or some sort of in centomic topology and right so yeah so thanks for that question so i i should yeah i should probably clarify what i mean here um so one way of saying what i mean by differential forms is that uh so um i guess it's a theorem of uh bar gov that uh if you if you form the wedge powers of the co-attention complex they are sheaves in the flat topology so it's a theorem of thought that um wedge powers of the co-attention complex say on rings are sheaves uh in the flat uh in the yeah f p q c topology and so it's going to make sense to evaluate these on any um on any stack like um like this um but in fact it's a little so i said i did say differential forms here and the reason is that uh i guess you can sort of regarded as a stack i think in the i think i want to say in the in smooth topology so so you can really just say differential forms instead of the co-attention complex um um sorry i didn't particularly catch what is that there's a quiet topology so it's a joke a quiet uh okay so i uh thanks yeah i don't i don't um not completely sure i think that what i understand is that that that the kumar flat topology is closely related to this construction but i didn't know about fontans um quite topology a lot thanks i'll have to yeah i'll show look that up um yeah but this is kind of a fun sort of yeah sorry so just to go back to also to open this question i mean this is kind of a fun sort of calculation to do like if you have i mean if you have k brackets x and you have the log structure given by x and so so in fact for what i'm saying is you don't actually have to go to the infinite root stack you can do the p typical version where you take extract p power roots and then what you're considering is you're considering something like spec of k brackets x to the one over p to the infinity and you're quotienting by uh by the group scheme which is uh uh the tape module of mu p to the infinity um and so when you when you go up to mu p to the infinity or when you go up to k brackets x to the one over p to the infinity differential forms go away there but the this group scheme is going to have a cotangent complex and so when you form the and when you form the limit um when you form the stacky cotangent complex sort of from the limit of these things you end up recovering this log um uh cotangent complex um yeah so um so um yeah so i just want to go back to this um this principle so in particular it's going to follow that one can uh can define sort of log prismatic homology or log crystalline homology or the others uh of say so if we're in the setup of a prism a comma i of an a mod i formal log scheme uh in terms of the infinite root stack as um but there's a bit of an asterisk here which is that in general the construction of the infinite root stack um if you if you really sort of follow it it it it's going to involve some sort of tensor products and those tensor products might not be derived um and so i think that doesn't isn't going to cause a problem for some sort of etal theory but for prismatic homology that's going to be an issue so this might not this might not work in general um so if you want to define log prismatic homology i think this is going to this kind of thing is going to work very well this is going to work well in say a very nice situation if you have a formally smooth a mod i scheme with log structure given by a relative normal crossings divisor um but it might not quite give you the right thing in in general if you do this because of these um these higher tours um but what you can you can what you can do then is um is is you can this is sort of the fundamental case uh and and then you can sort of animate in general so so you can define log prismatic homology or log crystalline homology in in this special case using the infinite root stack and then you can animate in general so here animate means that um um um you sort of simplistically resolve a log ring over a mod i um and so i yeah i want to i want to emphasize this this this idea of taking a homology theory um so so simplistically resolve a log ring over a mod i over sort of their by sort of very special cases and the special cases could be something like you know a mod i brackets x1 through xn and y1 through ym where these are giving these variables are giving you the log structure um and then and then you sort of animate in general to to define it for for a general um log um log ring over a mod i or log scheme um and this yeah so you want to emphasize that this this construction um of of uh of animation in this way uh for for piatic homology theories um goes back to uh goes back to luke in in in the context of derived dirom homology so this is as in eluce's construction of derived dirom homology so right so as a result one can recover this construction of of log for example this construction of log prismatic homology and the sort of various types of comparison results uh using the infinite root stack um in this manner um so i but and it is yeah also for like crystalline or dirom homology and yeah i just want to emphasize that the the essential feature that to do this is that you have a homology theory on the one hand you need some sort of flat descent so as as a no first question um so that you can um you can you can evaluate it on a stack in a good way and the second is that it behaves well with respect to animation so anything that's sort of built out of the cotangent complex um is going to behave well um with respect to animation so yeah so when you animate the the you are seeing about some geometric object but within in the like the cotangent complex where we don't have the results by uh some free polynomial thing but geometrically it changes the spec here you work with something maybe you want to to complete it along the original locus or do something so or do you do you take this and so do you have some simplification resolution and do you do you complete it along the kernel to the original thing or do something to localize the original thing or it doesn't matter no i don't think i don't think you need to i mean so i think for example in this in this case i think so it's a i guess an insight so i mean this is sort of i guess comparable to we're using like the fact that if you um so this is an insight of uh of bot that if you take derived dirameclomology in the piatic setting uh then your it gives you the right theory say for more generally for smooth objects by the by the cartier you know by the cartier at least and more awesome and so yeah so yeah so maybe i should also add uh i should also say that um so here yeah so you don't need to do any sort of further completion i mean you need to do the iatic p comma iatic completion as usual but um so here we use i mean we essentially use the hodge tape they're into cut here for log the dog's moving to the cutty type right yes yes um yeah so i mean in general it's going to be i mean so you're going to get a theory that is you know controlled so i mean this like this this type of theory you're going to get if you have if you have a theory that's controlled by the cotangent complex then this this sort of procedure is going to produce an analogous theory that's controlled by uh well gabber's uh log cotangent complex so use the hodge tape filtration and it's i guess it's a bot that log piatic derived dirameclomology works so it gives you the right answer for smooth schemes not just for polynomial for smooth algebra so another natural thing i just to recall from some discussions i don't remember the explanation but when you do derived dirameclomology as far as i remember you have some big complex and either you can take the total with direct sum or direct product it gives you different things or the same thing so what is the answer so just about derived dirameclomology it's not so you have your your so either you do it in characteristic p or when it's important to maybe over said i'm not sure so you have some nice implicit resolution to take the complex of each and then you put it out of course it is not in the first quadrat it is another one so the total complex is there are two ways at least to define the total complex so they give you different theories yes so sorry so i think i think what you're saying is the question of like the hodge filtration but i think it's right so i think it was the insight of bar gov that if you that in the in this patic setting it's it's not going to matter at least at least if i understand your question correctly i mean if you form derived so i think i think that the so i think the distinction is equivalently between derived dirameclomology where you i think you you don't where you don't complete with respect to the hodge filtration and when you do and the answer is that first move things that doesn't make a difference which is not true i mean it's not true in characteristic zero but it is true in characteristic p by the by the hodge by the by the by the Cartier isomorphism so you mean you start from any ring any fd algebra in characteristic b and then the two patic the the two derived dirameclomologies are the same this is what you're saying using a simplification right this is in this all this blueprint of valga one on wait sorry i think there was some confusion uh i think akhil was i mean what akhil was saying only applies in the smooth case and then for general rings you just use the direct sum version that's what animation does for you so because it's a conjugation which is better so somehow right i mean you that's what the you don't think about the direct product version after uh proving this theorem in the smooth case okay so you prove in the smooth case what you said but it is not true in general probably probably it is not true you say yeah it is not true in general because one is going to be complete for the hodge filtration and the other is not in some singular setting yeah sorry that okay so you take the direct sum okay thank you okay yeah thanks bar go for car fine thank you i'm sorry for interrupting no no thanks uh yeah okay all right sorry so i guess i have maybe a little more than 15 minutes so so now what i want to do is explain the um let's talk about symptomic homology um so this yeah so this is sort of a general procedure of um defining logarithmic versions of these pietic homology theories and i want to sort of apply that to um to this this context of symptomic homology um and right so this is another one of these pietic homology theories and so let me start with the following definition which is due to uh frontend messing and kato so let uh r be a um formally smooth uh zp algebra um so uh right so first let me define it for i at least p minus one we're going to define zp of i of r to be the following uh to be the following object so this is going to live in the the p complete derived category of zp it's the fiber of phi minus uh p to the i which uh which goes from the sorry uh the i stage of the hodge filtration of so you take the dirham complex the p complete dirham complex of r and form the i stage of the hodge filtration and map that to uh uh the dirham complex of r so here we use we use a p complete dirham complex of r uh together with uh it's uh with the two pieces of structure given by the crystalline forbenius which is an endomorphism of the dirham complex so crystalline forbenius sorry uh and the hodge filtration uh so this this naturally produces an object uh living in the p complete derived category um of zp um and uh right so let's sorry let me call the zp of i uh fontan um i guess fontan messing in kato of of r so um right so this this definition is uh one can do a little bit better when um uh when i is small so if i is less than or equal to p minus two then one can define in addition to the forbenius one can define a divided forbenius phi divided by p to the i which goes from the i stage of the hodge filtration to the dirham complex itself so this is this is something that's going to be natural at least in the derived infinity category uh and then so let me define zp of i of r to be the fiber of um phi divided by p to the i minus the inclusion from omega r greater than or equal to i into omega r of dirham complex okay so this is i guess this is what one might call symptomic homology which are not no in the first one i is bigger than i can throw less than i is larger yeah yeah where is the hypothesis at number two on i used uh so i guess it's being used to define this phi divided by p to the i so well i guess yeah uh right so i mean i i guess you want to make this sort of natural so that okay yeah yeah i i think one can say it so for example one can you know one can define this sort of locally in the i mean one can define this using like a chris and then i think uh i think you will have an issue with the forbenius not being divisible enough that yeah thanks um okay okay so so right so this is some general construction and okay so we can define it let's i mean we can define this for all p complete algebras by animation and then we can also define it well so using the well for example using the procedure that i just explained or in this case you can just do it directly define on a log p attic algebras um right so using this infinite root stack so yeah so i guess one of the reasons these come up is that there are comparisons to there are comparisons from symptomic cohomology uh to uh to p attic nearby cycles so there are various comparisons i'm sorry you could do it in the law case you could either use the infinite root stack or suppose you just use log with differentials instead what's the how do they compare yeah i mean it would give you the same it would be yeah it would give you the same thing yeah because because because the logarithmic differentials are also the differentials on the infinite root stack um but i guess the infinite root stack would be helpful for when i want to do the integral version of these things um a little bit um yeah thanks okay so there are various comparisons in the subject uh so various comparisons between symptomic cohomology and uh p attic nearby cycles uh after truncation um so i think right so there there are many instances there are many uh examples of this in the literature so um for example uh so kato shows that if r is uh formally smooth over oc so c is algebraically closed non-arcomedian field you can i think you can weaken this whole bit so c only has to be perfectoid um then uh z p of i of r is the truncation of so you can consider this as a i mean z p of i i mean you can think of this as some sort of i mean as you let r vary it becomes some sort of sheaf on the pro etel site of r and uh well it's the truncation uh truncation tau less than or equal to i in the pro etel topology of the the p attic nearby cycle so of r j lower star of the usual z p of i where j going from spec of r join one over p into spec of r is the inclusion so if you if you take the generic fiber you have the the usual z p of i the take twists and the etel or pro etel topology you push that forward to z p of i and uh well i guess maybe you i should say spuff of r maybe i should say spuff of r because uh we're really restricting to the special fiber and uh and the statement is that it's the truncation in the pro etel topology of spuff of r of this of this p attic nearby cycles so this is for i less than or equal to p minus two you should say um so this is this is provided kato um for you thank you with dvr in kato what did the original of course it was less sophisticated than what you said but did they start from a dvr it's perfect the residue field and just considered this case i mean c oh yes sorry yes uh kato in and and if you but this technique you said is through using using found things i mean what you say is true that for any yes sorry what i'm saying is true but i think i think yeah so you're right i think kato is probably assuming that c is a completed algebraic closure um of like a p attic field but yeah thanks for that um and then uh there's work of kuri hara uh which uh which proves i guess again this the same statement and i won't keep repeating it but in the case where r is now is now formally smooth over dvr but now you give it the log structure that comes from the special fiber um so if you don't assume the law if you don't put in the log structure from the special fiber you'll get some contribution coming from logarithmic diram that forms from the special fiber which i think is the setup that kuri hara considers uh and then uh there's work of suji uh which considers uh are maybe formally smoother or semi-stable so let's say yeah r is semi-stable over oc and i think you're also allowed to have a divisor divisor at infinity here uh and in all ways so no uh bound on i uh but up to isogenic so uh right so somehow the zp of i's are for i uh greater than p minus two are not really the right thing they're only sort of the right thing up to isogenic uh but there's a comparison with piata herbicycles in this case um and uh uh and more recently worked by komez and nizio uh where r is semi-stable over dvr and again semi-stable allows some uh log structure um at infinity um and again rationally in all weights or rationally or up to isogenic in all weights um so it was only sort of more relatively recently that there were integral versions of these types of results uh because the because symptomic symptomic homology defined this way is not uh somehow it's not the right construction and integrally uh so only recently uh there are integral and the reason is that one has to replace these zp of i of fontan messing in kato with something a little bit more refined that comes from prismatic homology instead of diram homology um and so there's a definition which was made by batmorrow and schultz so they define object zp of i uh for all i of r so r is again this periodically complete ring and these arise as the associated graded's associated graded's of a filtration on uh on top on the I guess this object that arose in algebraic topology on topological cyclic homology of r with dp coefficients um so these can be also defined in terms of absolute prismatic homology these are defined as so zp of i of r is uh so this is discussed in the paper of batmorrow and schultz it's also a fiber of a divided forbenius minus the inclusion so minus the inclusion which goes from absolute prismatic homology of r except that now you've uh you've neigard completed completed with respect to the neigard filtration and you take the ith piece of the neigard filtration and you have a brachis and twist mapping to again neigard completed prismatic homology of r with this brachis and twist so this comes out sort of very naturally or this expression comes out very naturally from the expression in terms of topological cyclic homology and the sort of apparatus of uh of tc so so this is this is sort of an integral theory that works works in all weights um and um right so um so so for example in in work with antio uh moro and nikolas we showed that these zp of i is defined by batmorrow and schultz agree with the frontend messing and kato definition in low weights or sort of up to isogeny integrally um but these these zp of i is defined by batmorrow schultz work better integrally because uh so um sorry right so they they also show that uh they show that the zp of i's of r are truncated piatic nearby cycles uh in all weights uh if r is formally smooth over oc uh and this was extended to the semi-stable case uh r over oc by colmez uh dospenescu museo so r over oc semi-stable um right okay so yeah so what i wanted to explain is that right uh one can formulate a version of uh an integral version of these types of comparison results want to add one more item to those lists which occurs uh in uh in the case where you're over a dvr so uh and and where you do have to take into account the log structure coming from the special fiber as well as some sort of semi-stable log structure um so first is you can define the zp of i's in the log setting for example using the infinite root stack for example um and i yeah i guess i want to end with this theorem um with uh botan clausen which is going to state that basically the the the theorem of colmez and museo that you have this comparison between this rational comparison between uh truncated piatic nearby cycles and uh symptomic homology works integrally if you use these zp of i's so if r is semi-stable over oc and you can generalize this a little bit like you can allow these changes of semi-stable things as in colmez and museo um and you can allow a divisor infinity then the zp of i of r with log structure with the natural log structure are tau greater than or equal to i of piatic nearby cycles um so i just want to say a little bit uh i just want to say what goes into this i mean so somehow it would be really nice to have a purely sort of prismatic uh and the piatic proof of this um but what we actually do is uh is to is to use a connection between topological cyclic homology so these zp of i's are related to uh as in bms2 they're related to topological cyclic homology uh which turns out to be related to the piatic k theory of the generic fiber and then what we do is we uh we use we use the the block hadoe or bellinson licton bomb conjectures which give you a filtration on the k theory of the generic fiber and then you sort of try to match that up with the the filtration on on tc uh of log tc uh and and the piatic k theory of the generic fiber and the piatic k theory of the generic fiber is controlled by like these piatic vanishing cycles thanks to the block hadoe conjecture and so i think i believe this this type of idea and using k theory in this way is is due to museo um but uh in this context it would it would also be very nice to have a a purely prismatic approach so okay so i will stop here so thank you very much but there is no conjecture involved here oh sorry yeah i should i suppose you you don't need any conjecture sorry sorry sorry sorry i guess i'm yeah i should have said that so you're using like the block hadoe conjecture uh but yeah what i should say is this is a theorem of of uh weavadsky and roast yeah thanks okay uh but somehow this is like an elatic statement and it would be nice to sort of stay purely in the piatic world um but yeah so in the previous result so in this result you have the log structure uh uh used and in the the kind of semi-stable the number six or before no no before number six the the one where you had the uh uh okay you have the smooth case uh but then the you had also the uh the case of semi-stable which uh like the semi-stable case i will scroll by the spin this didn't use the log structure it uses the log structure sorry i maybe in a smooth case over oc you don't need the log structure because a log structure is sort of uniquely p-divisible so it you can put the log structure but it doesn't affect things and in the semi-stable case of an oc uh then i think you need the log structure yes um yeah i i have a question i don't know whether you can hear me hi yes i can hear you yeah okay so um i have a question so this um uh this uh way of of presenting things using infinite rule specs does it allow you to describe hiodokato theory uh i don't know i i would love to yeah i would be very interested in understanding that better but yeah i i don't know but thanks thank you for the question i was curious about that too so okay so it's a general question so in the arson's approach uh defining log structure on x is sending x to some some special stack which is locally like towing stack for example a 1 p.m and in the other approach instead of sending x to some stack you send a stack to to x stack yes so it's a sort of dual approach like uh i think in the terms of sanity in the Takeshi's expose you have two two things so even uh either you will send x to a curve or you send something to x yes so it's sort of a dual dual approach so in arson's approach uh it's a property of morphisms of uh or even uh objects log structure it could be read on the uh on the on the map to to this log log thing log stack and so i wonder where there are uh similarly you can uh so there are two questions first of all which type of stack would give you a log structure you said that uh somehow you can cut the log structure from something algebraic just a stack it's something like a root stack so what are the properties which would characterize the root stack and then and give you some some log structure and also uh what properties of log structure you could you read off on the this uh root stack like uh log smoothness not flatness et cetera um yeah thanks for those questions i think yeah i i believe that so in talpo and vestelius paper they explore sometimes so in particular i think they show that you can read off things like log flatness and log et al my kumar et alness in terms of what happens on the infinite root stack um i think i think they also discuss the relation um i think they also show that you can recover like a fine and saturated log scheme from the infinite root stack uh but it's not fully faithful but you can still recover it up to isomorphism and i think they also probably discuss which stacks are of this form i mean they for example they and they always have this particular form that like they they live over a scheme and the fibers are these like i i guess sort of profinyte tori um like classifying spaces is sort of internal characterization of these root stacks uh i'm not sure but it might be in their paper but i'm not sure off the top of my head but thanks so so the infinite root stack like if you have a if you just take a maybe i uh you said something about recovering the fine and saturated log scheme not not as an equivalence but only after isomorphism can we remember what it was or i believe i believe what they prove is that if you have if the if the infinite root stack if two infinite root stacks are isomorphic then the log schemes are isomorphic is my understanding um but on the other hand it i think they also show that it's not fully faithful that functor from log schemes to to stacks in this manner but for isomorphisms it is that is the isomorphisms correspond but not the maps is it the idea or that okay uh what i think the statement is just that if they're isomorphic as stacks then the log schemes are isomorphic i think it's not necessarily yeah not that the delta isomorphism on the set of isomorph okay okay yeah that's my that's my understanding yeah so here the in the last thing you said that you you interpret the zpi internal topological cyclic homology theory and paper of the general fiber so do use results relating to the k theory of the general fiber to some material to something involving a chemical what did you expect to see i mean which i i think that i think it's okay now but do you use such results yeah yeah i mean i guess yeah so we're using like the k theory the generic fiber is related to etiological homology and by the like blockado and valence and lectin bomb conjectures and so we're we're using that um and so yeah so basically the point is to match up like uh i mean you have you have something like tc of r i mean maybe with log structure or yeah so uh which is which is you know which is filtered by these zp of i's and then you have something like the k theory of r join one over p i mean let's just assume that r is like strictly Henselion so this is close to k of r to begin with uh and then then you're using the fact that this has a filtration by blockado valence and lectin bomb and this has a filtration by botmore and schultz and then the claim is that those can be made to match up and to do that one has to sort of take left arrived or animate the the right hand side and actually use these results of block and kato and followed by others of uh on controlling peatic vanishing cycles or peatic nearby cycles in terms of differential forms but you also need the k theory results which which compute the the k theory of our one over p yes we need the k theory results yeah okay you just explained to me that several years ago okay so uh is there any other question yeah i have a question so uh the tc of r would you would you wrote here is this just in a philosophical sense or did you really define it in the case where there is a log structure i mean so right so i i think you can define tc you can define log tc sorry i think maybe this also came from the chat so you can also define uh i mean you can define like log hoaxial tomology log tc and so forth using this this procedure because tc satisfies flat descent and behaves well with respect to animation and i think this will agree with you know i so i think has to hold nuts in uh for example consider version of log tc and i think it will be equivalent um so okay so you're using the if it looks that's the different tc etc of logarithmic things but does it also give you a definition of k theory of log log log schemes or log rings right oh yeah so one thing i should say is that it's not it's not the it's not the tc or the k theory of of sheaves on the log on the infinite root stack um but it's this thing defined by flat descent and so since k theory doesn't define doesn't satisfy flat descent i wouldn't uh see how to define yeah okay yeah thank you yeah but yeah i mean it would be nice if there was really a category here that one could just sort of apply it to the category and but here it's really being defined by flat descent uh well i think at least yeah i think at least you like for a log scheme you will you can do what yeah but i i don't know for yeah i mean so yeah i hope that for example some of the techniques uh that uh you know that the bar gov was explaining for working with these uh you know with absolute prismatic comology where um i mean so right so i mean this is you know this the statement about the zp of i is i mean these types of statements are i mean their statements equivalently about absolute prismatic comology um or some sort of filtered nigh and i guard eigenspace of prismatic comology and um so i mean these statements were proved for example by batman or schultz and comministus minuscunizio over oc uh using i mean using the fact that you have like nice expressions uh for like for a nth comology like these cutarum complexes and given by laida in particular and so forth um and i yeah i mean it would be i certainly hope it no will be something like that there will be some sort of something like an laida explanation for these types of statements so for okay and then maybe one could get hope for a purely pietic approach to these things um but um no okay so i don't know if i can just remember another technical point so when when you looked at the where it was going to like when you try to do the um omega i on some well maybe put it in this talk or in another maybe the lighting no you you mentioned the result the the the the wedge i of the cotangent comology is a shift yes but this was uh yes uh but this i think means that well of course this is in some derived sense but this means that you do uh just check uh cover or or hyper cover i think it was pulled for just check covers yes comology usually is defined in general as a limit of hyper covers which you cannot avoid i mean if you do unbalanced things so so so it uh so here this could be well i'm not working with this but this doesn't cause a problem here yeah so thanks for yeah right i mean so i think that is uh i mean as far as i know that's an open question of whether the cotangent complex and its wedge powers are hyper sheaves for the flat topology i think this was raised maybe in batmorrow and schultz uh uh it's paper second paper um but so i think in the setting in this particular setting like where you're taking at least when you're taking uh these infinite root stacks of sort of pretty nice things and then from what you're animating um there's no issue because everything for example is going to be like in the symptomic and then that's going to force everything to be uh um sort of bounded and formally bounded um and so so these issues of hyper completeness are are not going to not going to show up because you're not going to have uh stuff in comological degree going off to comological degree minus infinity um so nice thing yes so nice things like you start from a smooth with normal crossing divide okay then everything will be okay i understand then the convergence questions okay so the other and the other questions okay which case uh in the computer okay so let us uh speak it again for the illuminating but technical thank you thank you very much