I have a dump of MySQL base sait from hosting. When I use this dump on my local server :
Warning: mysql_num_rows(): supplied argument is not a valid MySQL result resource in /home/u26773/omsk.arsec.ru/www/modules/protector/class/protector.php on line 45

Notice [PHP]: Undefined index: enable_badips in file class/xoopssecurity.php line 215
Notice [PHP]: Undefined index: language in file include/common.php line 152
Notice [PHP]: Undefined index: language in file include/common.php line 172
Notice [PHP]: Undefined index: use_ssl in file include/common.php line 183
Notice [PHP]: Undefined index: use_mysession in file include/common.php line 185
Warning [PHP]: session_start() [function.session-start]: Cannot send session cache limiter - headers already sent (output started at /home/u26773/omsk.arsec.ru/www/modules/protector/class/protector.php:45) in file include/common.php line 198
Notice [PHP]: Undefined index: banners in file include/common.php line 235
Notice [PHP]: Undefined index: debug_mode in file include/common.php line 235
Warning [PHP]: in_array() [function.in-array]: Wrong datatype for second argument in file include/common.php line 235
Notice [PHP]: Undefined index: theme_fromfile in file class/template.php line 66
Notice [PHP]: Undefined index: sitename in file class/theme.php line 417
Notice [PHP]: Undefined index: slogan in file class/theme.php line 418
Notice [PHP]: Undefined index: slogan in file class/theme.php line 440
Notice [PHP]: Undefined index: closesite in file include/common.php line 238
Notice [PHP]: Undefined index: theme_set in file class/theme.php line 253
Notice [PHP]: Undefined index: theme_set in file include/functions.php line 475
Notice [PHP]: Undefined index: language in file class/theme.php line 263
Notice [PHP]: Undefined index: theme_set in file include/functions.php line 475
Warning [Xoops]: Smarty error: unable to read resource: "db:system_redirect.html" in file class/smarty/Smarty.class.php line 1084
Notice [PHP]: Undefined index: debug_mode in file footer.php line 55
Warning [PHP]: in_array() [function.in-array]: Wrong datatype for second argument in file footer.php line 55
My server is 4.1.22-dump from 5.0 mysql

what did you use to import the database on to the local server?

also check include/common.php to see if you have any spaces before that shouldn't be there.