BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Jan Trlifaj (Charles University\, Prague)
DTSTART:20211012T160000Z
DTEND:20211012T170000Z
DTSTAMP:20260423T035826Z
UID:ONCAS/6
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ONCAS/6/">Cl
 osure properties of $\\displaystyle \\lim_\\longrightarrow \\mathcal C$</a
 >\nby Jan Trlifaj (Charles University\, Prague) as part of ONCAS Online No
 ncommutative Algebra Seminar\n\n\nAbstract\nLet $\\mathcal C$ be a class o
 f (right $R$-) modules closed under finite direct sums. If $\\mathcal C$ c
 onsists of finitely presented modules\, then the class $\\displaystyle \\l
 im_\\longrightarrow \\mathcal C$ of all direct limits of modules from $\\m
 athcal C$ is well-known to enjoy a number of closure properties. Moreover\
 , if $R \\in \\mathcal C$\, $\\mathcal C$ consists of FP$_2$-modules\, and
  $\\mathcal C$ is closed under extensions and direct summands\, then $\\di
 splaystyle \\lim_\\longrightarrow \\mathcal C$ can be described homologica
 lly: $\\displaystyle \\lim_\\longrightarrow \\mathcal C$ is the double per
 pendicular class of $\\mathcal C$ with respect to the Tor$_1^R$ bifunctor 
 [1]. \n\nThings change completely when $\\mathcal C$ is allowed to contain
  infinitely generated modules: $\\displaystyle \\lim_\\longrightarrow \\ma
 thcal C$ then need not even be closed under direct limits. After presentin
 g some positive general results (and their constraints)\, we will concentr
 ate on two particular cases: $\\mathcal C = add(M)$ and $\\mathcal C = Add
 (M)$\, for an arbitrary module $M$. We will prove that if $S = \\End M$ an
 d $\\mathcal F$ is the class of all flat right $S$-modules\, then $\\displ
 aystyle \\lim_\\longrightarrow add(M) = \\{ F \\otimes _S M \\mid F \\in \
 \mathcal F \\}$. For $\\displaystyle \\lim_\\longrightarrow Add(M)$\, we w
 ill have a similar formula\, involving the contratensor product $\\odot _S
 $ and direct limits of projective right $S$-contramodules (for $S$ endowed
  with the finite topology). We will also show that for various classes of 
 modules $\\mathcal D$\, if $M \\in \\mathcal D$ then $\\displaystyle \\lim
 _\\longrightarrow add(M) = \\displaystyle \\lim_\\longrightarrow Add(M)$. 
 However\, the equality remains open in general\, even for (infinitely gene
 rated) projective modules. \n\nThe talk is based on my recent joint work w
 ith Leonid Positselski [2].\n\n[1] L.Angeleri Hügel\, J. Trlifaj: Direct 
 limits of modules of finite projective dimension\, in Rings\, Modules\, Al
 gebras\, and Abelian Groups\, LNPAM 236\, M.Dekker\, New York 2004\, 27-44
 .\n\n[2] L.Positelski\, J.Trlifaj: Closure properties of $\\displaystyle \
 \lim_\\longrightarrow \\mathcal C$\, preprint.\n
LOCATION:https://researchseminars.org/talk/ONCAS/6/
END:VEVENT
END:VCALENDAR