 Well, thank you. Oh, it does work good So so thanks to the organizers for for putting this together, you know the workshop and the school last week Which I wasn't at in the thing last year like this is this is great and also I want to thank Craig and Mel for providing this This sort of fun space of mathematics for us all to to work in like it's it's really you know math as Craig said about Mel and that like I'm sure Baren to say about Craig like it would just math would be a lot Poor or feel to be a lot poorer without without their contributions and that's in some sense part of the the theme of my talk because You know Okay, so a conference honoring Mel and Craig of a conference on characteristic p-algebra. This is what the eighth ninth talk and there has been No talk about tight closure theory 20 years ago. That would be unthinkable, right? That would be impossible and I think that that is because of the You know the common wisdom so right so so Craig's periodization by the way, hi Craig Craig's periodization that he did yesterday. The reason I asked that question is he said 1985 to 2007 was smells, you know tight closure period, right and And and what is 2007 to me what that is is the year that the pre-print by Holger and and Paul Monsky went up on archive Showing that tight closure does not commute with arbitrary localizations. I Say arbitrary on purpose because they said carefully in their talk. I'm well in the in their paper That they don't that there's some that they that there are some cases They don't cover and including open localization like localizing at a single element And that's kind of my jumping off Point so I want to say I want to think of this talk as an invitation To kind of reinv you know to have a different perspective on the tight closure operation and similar operations That hopefully will make people want to kind of work in it again So okay, so Brenner Monsky's I haven't written anything yet, but whatever So Brenner Monsky's Paper I think Created the following common wisdom well we wanted tight closure to be a geometric operation right like in a real closure like for being as closure sort of is But it doesn't commute with localization in particular in Brenner Monsky's paper They showed that it doesn't work well with deformation theory and so therefore it couldn't be an Operation of geometric interest so let's so that the stuff that is actually of geometric interest is the The things that come from tight clue that arose in the study of tight closure theory namely test ideals and Certain F singularities such as F purity and strong left strong F regularity, right? But the thing is I think that there are some some geometry to do here That's that that they left that that everybody left on the table, which I'd like to present to you as a meal Now I'm not a geometry So I hope and and there are definitely people in this room And on this zoom that know a lot more geometry than than I do So so like I say an invitation, you know people who who know more geometry than I do like I'm gonna show you how to take the the the the tight closure of an ideal sheaf or you know a sheaf or a sheaf of modules will work will work the same way and and and sort of an invitation to sort of See if there is something geometric to do here Okay, so conventions Oh, and the other thing is Right, so what do I want to say? So our is just gonna be a commutative ring and When I talk about tight closure is gonna be a characteristic P ring P bigger than zero The theory in so so yeah, so I'm gonna basically I'm gonna show you for various closure or even pre closure operations on on Ideals in in in sort of a family of rings that's associated with the scheme I'll show you sort of when you can make a sheaf theory out of it And when that sheaf theory works fairly nicely, so there's sort of varying degrees of Niceness and that the coherence is gonna be kind of the big question at the end Like when one of these like you take the the the closure of a coherent sheaf when is it when is that a coherent sheaf? It'll be a sheaf, but I don't know when it's gonna be coherent assuming that so so properties of Preclosures I'm gonna call them new New in the sense. I mean nothing's new in mathematics, right? It's all there for us to discover But I mean that you know that I've identified that are important properties We're gonna have something called open persistence and something called glue ability Okay, and open persistence is gonna show you that there's a sheaf and glue ability means That You don't have to shrink it down. You'll see what I mean. You'll see what I mean and so Or I'm so I'm gonna show you this sheaf, and then we'll talk about closeness and then at the end and with a new Singularity type I Say new because of course conjecturally because this is gonna be between weak F regularity and F regularity So conjecture at so conjecturally it should just those should all be the same, right if you're because weak, you know We can play strong, you know Ian's gonna talk about some progress on that later in the week, I guess and But But anyway, it's it's new to the extent that if you don't know that that that week of regularity is F regularity then then you don't know then you don't know that this is either of them Okay, so So many words. All right, so definition Okay, so what's a closure? What's a pre-closure? So? Let s be a poset partially ordered set Okay, so a pre-closure is a function cl from s to s You know written You know x goes to x cl Such which is such that for all x and y in s x is bounded above by the closure of x and if x is Under y then the closure of x is under the closure of y Okay, and If cl is item potent i.e. the closure of things are closed We call it a closure So pre means might not be item potent. So this is this is all standard terminology from partially ordered set theory Okay, so I mean examples of that so why am I going to post sets is because You know if you're talking about ideals then you can talk about subsets and things but if you're talking about ideal sheaves then those aren't really sets and so you have to You have to but there's a poset structure Okay So So if If x is a scheme, I mean if you don't like that just think algebraic variety But no a scheme so like spec r then a a pre-closure CL on x For now is going to be on all the affine open You know rings that come from it right is a choice for each affine open you in X of a pre-closure also written CL on the ideals of OX of you So that's so anyway, so if you just have that that you can't really do much So that's why we need these these conditions of open persistence and globality So Definition yes, these are all definitions so far So and so of course I say pre because it might just be a closure, and then you took yeah, so pre-closure CL on a scheme X is openly persistent for any Containment of affine opens UV affine opens in X and G the corresponding ring homomorphism So here what u is spec? s v is spec r Right and G is like the the corresponding ring map going the other way For all ideals of r G of I closure is contained in Is closure so I call this openly persistent because it's persistent on these in the containment of opens And the reason I use persistence is because this is you know from from tight closure theory persistence is is You know you want to say that well, okay, so so for basically any ring map if your base ring is good enough Then you know the tight closure should persist along that ring map, but it turns out open persistence is a much easier thing to satisfy Basically because G of R not it's contained in s not But anyway so So fact if if I'm gonna say if CL is openly persents openly persistent on say spec R and What happens if you then I closure RF is contained in IRF closure for all F and R Right because that's just the basic affine the basic open sets and in spec R Okay, so Proposition I'll call this proposition a Tight closure is Openly persistent right so and here X on any netherian FP scheme And it's just because So basically the reason I won't get the proof I'll give a reason G So if R to s is open in in that sense right the corresponds to a containment of affine open sets Then G of R not is contained s not and In tight closure theory the only problem with persistence is that you know the things in our not might Like go to zero or go inside of the minimal primes of s and that just never happens. It's because of going down So it's it's flat and you know any Inclusion of opens is flat and so flat implies going down and that means that the minimal primes behave Well in such a way that G of R not is contained in s not um, okay So theorem 1 slash definition, so let CL be an openly persistent on a scheme X So we're gonna extend To a pre closure on The ideal sheaves I DSH the the post set of ideal sheaves on X OX I don't know the terminology As follows Okay so for For such an ideal sheaf. What do you do? and for s sorry for you open in X I need to so and s in OX you is in I of you If this is the definition of I of you I'm sorry. I closure of you if what if there if for any X and X there exists a affine open V with X in V and V is contained in you and s restricted to V is in I of V closure in OX of V which makes sense because OX of V is a ring and IV is an ideal in it and and we've defined the closure on all these Rings for all the affine open sets, right? so So anyway, so this at least gives you an assignment for each You of a set I closure of you, but in fact, this is this this actually makes yes No any ideal sheaf. Yeah later on coherence and cause a coherence will come up in a big way, but not yet Then I closure is in fact a sheaf of ideals and CL is a Here I'm going to say is a pre closure on the set of ideal on the post set of ideal. She is of X and if X is an aetherian and CL is item potent on each affine open then CL is item potent in other words you have an actual closure on the ideal sheaves and One more statement, which is part of this theorem moreover given an open you and S in OX you we have s in I closure of you if and only if there exists an open affine cover of you such that Alpha So what meaning that that you is the union of the you alphas and these are all open alphines inside of you for all alpha s restricted to you alpha is in I of you alpha closure in OX of you alpha and the proof is is you know straightforward But would take the next ten minutes if I wanted to actually do it because you know how proofs and sheaf theory are But it really is straightforward. It's just you know applying sheaf stuff to this Okay, but I said exists an open affine cover and I'm saying that you have to like for all these use you have to sort of find enough and for every X in in I guess in you Yeah Yeah, for every X in you you have to find sort of a different V and so on so but what you want is you want some Flexibility right you want to you want to be able to say oh well I just want once I go down to to like some affine open and you I want to say okay So I closure of that V should be I of V closure and that's what we need glue ability for So glue ability right so first of all corollary I star exists you take a Sheep of ideals and in a netherian FP scheme, and then you can take it's tight closure in this way Okay, so glue ability Definition let C L be an openly persistent closure operate pre closure operation on spec on spec R let's say C L by definition is glueable if For all collections F alpha in R not R not if you is the complement of the union of the minimal primes And for all G in the ideal generated by the F alphas and for all ideals I of R and For all X in R if X over one is in the Closure in IF alpha for all alpha Then X over one. I just need G in the radical of this Then X over one is in the closure of I G and these are taken in the corresponding localized rings and then For a scheme X C L is glueable over X if it's glueable on All affine opens My chalk broke, so let's see if there exists another one There's a good one Can I be here? I can be here, right? So proposition B So this is going to show that there's a connection with with commuting with localization at single elements If C L is openly persistent spec R IF closure equals I closure localized at F for all F and R Then it's global man. I spent so much time Preparing for you an example of a of a global one that does not commute with localization at at single elements But there isn't really time to present it. I'll just say what it is So I'll just say fact the converse Fails There exists a counter example arising I'll say arising from Zero local cosmology is it basically you have an idea like the closure is like if you have an ideal in a ring you you you delete all the Zero-dimensional primary components So that's that turns out to be a global closure operation that is not That that is not that doesn't commute with localization at single elements even in a two-dimensional Power series ring overfield Okay, okay So Proposition C Tight closure is global over any netherian FP scheme Yeah, and I wrote out the proof here, but But it is it's I mean it it comes if you've worked with tight closure theory You can sort of imagine how the the proof would go you look like you know G is in your radical of your ideal generator by your F alphas, but you might as well just take F1 through Fn For for each of the localized ones you take a ci that is kind of tests for the tight closure of your of your X in that thing but those that but because of the condition that all your f's avoided all the Minimal primes your ci's also have to then avoid the minimal primes of r when you delocalize and then you just take the product of the ci's and and you Present g as a lean or g to the n as a linear combination of you know some powers of of of f and then there you go And so now g to the n Times c x to the q is an i bracket q and then and then they're for all q and the errands can vary wildly, but that's that's okay Okay, so now I've talked through the proof And I guess that's if you have a you know big test element then there's an easier proof, but you don't actually need it for this so Theorem two and this is where the coherence is going to come in Let's see L be a glueable preclosure on a scheme X Let I be a quasi coherent. I'm just going to call it qc ideal sheaf s Be an o x of you where you is a open s is in I closure of you If and only if for all affine Opens V in you as restricted to V is in I of V closure Moreover For all affine Open V I Of the closure equals I closure of V Exactly what one wants right this means that That this is true of tight closure as well So what about quasi coherence of I closure So let R be a ring Let's see L be a glueable Preclosure on spec R Then the following are equivalent all affin R I f closure equals I closure localized at F B I Tilda closure is quasi coherent and C I Tilda closure is equal to I closure tilde So right so so somehow this this now comes to the to the four now the question of whether tight closure commutes with localization at Single elements for a particular ideal and for a particular ideal is exactly the question of whether when you take the corresponding Coherent sheaf of ideals you take its tight closure. Is that does that remain quasi coherent? Spent a lot of time on math sign it looking up stuff about Criteria for checking quasi coherence other than you know the definition or push forwards and Like I said, I don't know much So Okay, and Then Let's see Here's a new singularity type another you know, we just had one last talk now Let's have one this talk It's kind of a non-local f-singularity type. So let's say that R is Semi-affregular if for all ideal sheaves I I Equals I star and equivalently If X is spec R Is to say that RF is weakly F regular Brawl F and R and I know nobody's defined weakly F regular this This week, but it just means that every ideal is tightly closed So it's you know, so it's equivalent to say that every ideal in in our F is tightly closed for every F and So what do you have so you have? F regularity so just yeah two more results F regularity implies semi F regularity implies weak F regularity It's more or less obvious But Proposition if R is a Jacobson ring for instance finally finally generated over a field Then R Is semi of regularity? It's semi. I'm gonna say semi if and only if weak semi of regular if and only if weakly F regular and So then you might say oh, well, okay, so then maybe it's just the same as weak F regularity in general But if it is Then a big thing would be proved because here's another proposition Let R be weakly F regular suppose That for all prime ideals Such that R localizes P is weakly weakly F regular R localized at P is also semi F regular So for instance, if you knew that semi of regularity and weak F regularity were true for local rings even just local rings of R then R itself is F regular Not just semi of regular but F regular. So what does this mean? This means that for Jacobson rings they're equivalent But if you knew that they were equivalent in for arbitrary And in local rings, then you would know that then you would solve this major conjecture that we get for regularity localizes And I'm sorry for going over time but Thank you for your attention. We could have one question Oh, Craig has a question Do you have a question Craig? I think that was just a clap. That was a clap. Okay, I Guess I mean the only comment would be that last thing. It's it's like Murphy's result your equivalents Yeah, I guess it's the same kind of idea the proof is the same kind of idea well But well, no, it's it's just well, it's just because maximal ideals contract to maximal ideals Is is kind of where it comes from and and since we kept regularity is local on maximal ideals and That's kind of where it comes from. I don't think yeah, I didn't prove it like like Murphy's thing I didn't need I didn't need any accountability argument or anything Thank you. Yeah, I think the interest of time will Post on any other questions to the deep back. All right. Thank you