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From beauty publicity to communications for the UN
June 13, 2018 — 12.23pm
More than ever in this fast-paced, digital-driven world, transferable skills are the key to being adaptable and employable. From developing social media marketing for lipstick and nailpolish in Sydney, Marley Tinnock was able to adapt her digital content and public relations experience into the fast-paced, international UN headquarters in Geneva.
Marley Tinnock works in PR for the United Nations.
Tinnock worked in public relations for Sydney agency Maxted Thomas for almost five years. She simultaneously maintained her freelance writing career, contributing fashion, style, design and sustainability features to fashion and lifestyle publications from 2011 to 2017.
In mid 2016, Tinnock left her account manager role, where she'd represented well-known brands. Far from the world of perfume, nail polish, lipstick and skin care, Tinnock packed her bags for Geneva, Switzerland to work in communications for the United Nations Development Programme. Founded to eradicate poverty, provide policy guidance and leadership skills to nearly 170 countries, the UNDP plays an essential role in creating sustainable business and innovation projects for communities where there is a need for external assistance. She is now a global communications officer and is currently assigned to the Central African Forest Initiative.
Tahira Matthews, of Stark Matthews, recognises how PR skills are transferable.
"The most important skill is an ability to clearly communicate, both verbally and through the written word, and also to listen – a big part of communication that is often overlooked," she says.
"Agility, determination, stamina and a whole lot of passion go quite far too. A lot of people with PR agency experience take the skills they've learnt from the dynamic, fast-paced nature of agency life into some incredible strategic communications in-house roles."
Since making the move to Europe, Tinnock hasn't looked back.
"Geneva – the European Headquarters for many UN agencies, is a hub for expats – so it's not hard to find an Australian. Between the UN expats and the huge number of financial firms based out of the city, it's often harder to find a local," Tinnock says.
"Although a French speaking city, the working language of the UN is largely English, so being a native speaker is an advantage – although I do envy my colleagues who are fluent in multiple languages."
Tinnock was able to apply her social media management skills to her initial junior communications officer role, where she handled the Twitter account. Her freelance journalism skills were exercised in interviewing experts, gathering analysis and project reports for the UNDP Europe website.
Marley Tinnock, second from left, at the inaugural Social Good Summit in Geneva.
"The UN recruitment systems can be strenuous," Tinnock recalls. "I did an exam and an interview for my first position, which lasted six months. I was lucky to have come across a supervisor who saw value in my years of experience in private sector, which is regarded as more competitive and innovative."
She says that after a recruit's first job, it is a matter of gaining confidence and experience before moving into bigger roles.
Jean-Paul Ong, Team Leader at Sarino Russo Job Access says transferable skills are essential to moving between private and public sectors.
"Every job has challenges and stress so having transferable, soft skills are important in every job. Resilience is vital. Relationship building skills and empathy are inarguably the key to gaining and maintaining work."


Sew Simple! A free beginner sewing pattern.
This is a free sewing pattern for sewing a very easy fabric wallet. This simple wallet project is also a great project for kids learning to sew, or for making quick gifts.


The next session of the Magdalen Iberian Medieval Studies Seminar is taking place on Friday, May 25th 2018. Professor Bienvenido Morros (Autonomous University of Barcelona) will give a lecture titled 'Las serranas del Arcipreste de Hita y las pastorelas francesas'. Dedicated to works of medieval Castilian poet Juan Ruiz — otherwise known as the Archpriest of Hita — the talk will explore the connection between the lyrical musings of Ruiz and French pastorals.
For further details about the event, please, follow this link. The lecture is open to the public and will be given in Spanish.


SINGAPORE - It is heartening to note that the Government is recognising the contributions of national servicemen through the National Service Recognition Award.
My children received this award in the form of cash deposited into their Post Secondary Education Accounts.
However, they cannot use it to repay the "loan" from my wife's Central Provident Fund (CPF) savings that was used to fund their polytechnic education prior to national service.
Since my children have also completed their university education overseas, they will not be able to use the money in their Post Secondary Education Accounts.
My sons are helping me to pay the bank loans for their university education, while also paying the "loan" from their mother's CPF account. This can be quite a burden for young men just embarking on their careers.
Can the CPF Board and Ministry of Education look into this matter?
I am certain many national servicemen are in the same situation of having to repay their parents' CPF "loans".


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A Felony Charge Against YBN Cordae From A Breonna Taylor Protest In Kentucky Has Been Dropped
by: Wongo Okon July 19, 2020
It has been four months since Breonna Taylor was tragically killed in her own home by Louisville, Kentucky police officers. In the four months since, officers have yet to be disciplined for her murder and as a result, protests have been nonstop in the months following her death. Many have taken their protest to the front door of Kentucky attorney general, Daniel Cameron, who has brushed off requests to have the police officers arrested from everyone from notable celebrities like Beyonce to hundreds more gathering daily in Kentucky.
Some of those protesters, 87 people including YBN Cordae, were arrested on a felony charge of intimidating a participant in a legal process. According to NBC News just a few days removed from the protest and the arrests, though, Jefferson County attorney Mike O'Connell has dropped the felony charges on all protestors.
Despite maintaining that the charges were reasonable, O'Connell said in a statement, "in the interest of justice and the promotion of the free exchange of ideas, we will dismiss that charge for each protester this past Tuesday."
Other notable names from the Kentucky protest included rapper Trae The Truth, Houston Texans wide receiver Kenny Stills, and The Real Housewives of Atlanta star Porsha Williams.
[via NBC News]
YBN Cordae is a Warner Music artist. Uproxx is an independent subsidiary of Warner Music Group.
Tags: Breonna Taylor, YBN Cordae
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redpajama_set_name: RedPajamaCommonCrawl


This year's Markex – Africa's premium trade show for marketing, promotion and events, was another smashing hit!
With annual corporate conferences and year-end functions nearly upon us, we were especially pleased to see exciting new offerings from some of our stalwart suppliers of custom-made trophies, exhibition stands and branded wall banners; as well as some really innovative corporate gift ideas, among a whole lot of other fine detail that usually goes into helping us connect brands to people in a memorable way.
The Britespark Communications team is super-stoked, and ready to bring your brand promise to life with everything from the latest in décor and set-up for events; to brand activation, out-of-home media; and endlessly creative printed communication ideas that will speak volumes to your audience.
Contact info@britespark.co.za for more info on our holistic brand marketing services.


Did you know? A portion from your payment every time you paint with us "Pays-It-Forward"!
We are passionate about giving back in our communities.
You can contact your local instructor about ministry parties in your community.
FIRST, God showed me that He wanted me to teach women who think they can't paint, HOW to paint, and get His Truth on it.
NEXT, I painted with one friend, Amy Spohn, and God clarified His vision while she was sitting at my dining room table. She encouraged me to follow His Voice and take the next step to plan a party.
THEN, that one party turned into a business only He could have begun, with the best team on the planet.
because we are passionate about showing Christ's love to people that are on our hearts.
Either way, we love what we do and YOU can join the fun!
Hope to paint with you soon! To God be the glory!
There are two ways to jump in: Host a Party or Join an Open Party.
Whichever you choose, when you paint with us, you're "paying it forward" in both your community and around the world.
THANK YOU for painting with Truth Be Told Art and helping us make a difference in your community!


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Let's go over these press release
6/02/2020 03:41:00 am Best Press Release Distribution Service, Free Press Release Sites, Press Release Sites, Press Release Submission Sites, Press Release Writing Service
You still need to be pitching to relevant journalists & bloggers interested in your topic/industry.
You still need to remember about trying to establish a human connection with whoever you're writing to, and not mass mailing everybody.
Let's go over these press release Best Press Release Distribution basics in regard to writing about a new product.
In this guide, I'll show you how you can how to write a press release for a new product to grab the attention of journalists and media influencers and Press Release Services ultimately get your product seen by your target audience.
Towards the bottom, you'll find a few useful new product launch press release examples that'll get you inspired.
If you're short on time, check out our new product announcement press release template. It includes everything you need to have the correct format. However, for the best results, I recommend reading this whole post.
Product launch press release format
Include the following in your product press release:
Headline – an attention-grabbing headline about your product. Make it count!
Lead – a brief overview of the story, including answers to the questions: Who? What? When? Why? With what effect?
Dateline – to confirm that you're providing the most up-to-date information.
Body – an explanation of the information provided earlier in the lead, starting from the most newsworthy info about your product and finishing with secondary details.
Company info – a short paragraph about your company Free Press Release Submission Sites and its products to help journalists understand the nature of the business and make writing about it easier.
Media contact information – the author or company's/agency's contact information.
Find journalists and media influencers interested in your specific product
A key factor in getting your Press Release Submission Sites picked up by the media is pitching to relevant journalists and bloggers—people that are interested in writing about your type of product.
An easy way to find relevant media contacts interested in your product is by using a media contacts database.
You can refine your search by filtering where a journalist is located, or what topics/industries he or she is interested in. This is a much better approach than the "spray and pray" method, hoping that someone will be interested in your story.
Media lists often have an extensive list of topics that journalists are Free Press Release Sites focused on, ranging from beer or cosmetics to cloud computing.
FIND MEDIA CONTACTS INTERESTED IN YOUR PRODUCT
If you have a bit of time, it's worth doing some more in-depth research.
I recommend spending a bit more time on it and making sure it's attention-grabbing and mentions your product's USP (unique selling proposition).
Add exciting features and benefits that'll make people stop what they're doing and read what your product has to offer. Try including numbers in your headline/e-mail subject – they're proven to improve open rates.
Remember – your headline needs something that'll get your audience intrigued. Stay away from boring titles such as [Company] Launches "New Product".
If readers are asking themselves "so what?", then you're doing it wrong.
If you're out of ideas – Grammarly has a guide on creating a catchy headline.
Bubble T Cosmetics created a catchy headline by mentioning their collaboration with influencer Ling.KT
Focus on the unique features and benefits of your product
It's doesn't make sense to write about a product that seems just like every other, does it?
Focus on making your product stand out from the rest by emphasizing on unique features and benefits. Depending on your product and target audience, you'll want to focus more on one or the other.
If you're targeting a very particular and specific audience, especially in the B2B sector it can be a better idea to focus on the technical features of your product. Stick to the point and mention the objectively superior qualities of your product.
Otherwise, it's best to bring attention to your product by showcasing how it'll benefit your customers and improve their day-to-day lives. Think about how using your product can be easier, simpler or more enjoyable than what the competition has to offer.
I've added a list of questions to make it easier:
Why is it worth writing about this particular product?
What are its unique features?
How will it contribute to the consumers' lives?
What makes it different from other products?
Can you somehow compare it to the products offered by your competitors?
Does your product come in different varieties and sizes?
Is there a formal product launch ceremony?
Be visual—include high-quality photos and videos displaying your new product
Unless you have a boring, non-visual product, adding photos & videos to your product press release is an absolute must.
If you want your product to get noticed, you should support it with relevant pictures, photos, videos, logos, and even animated gifs.
If you're unfamiliar with visual press releases, check out these multimedia press release templates and examples.
However, remember toavoid attachments as much as possible.
That means you shouldn't send out emails with any Press Release Writing Service attached images or videos (especially large files), resulting in a gigantic and possibly confusing email that will clog up journalists' inboxes.
But what if you want to send out a press release that contains lots of high-quality and high-res images or videos (and because it's a way to make your product more attractive for the press—why wouldn't you)?
Instead of adding attachments to your product press release email, try this:
Create a visual press release (using a press release creator like Prowly or using a CMS like WordPress)
Add your product photos, videos & other rich media (such as Facebook posts or Twitter conversations) directly to your press release
(Optional) post it online to your brand newsroom, a place where you keep all your company press releases
Send your press release to journalists by using a shareable link (plus, you'll be able to track open and click rates and see who's reading your emails)
New product press release examples
Below you'll find 3 different types of new product press release examples to inspire you to create your own.
Each press release was created with Prowly's Video Press Release Creator, which lets you easily create visual press releases (great for showing off your product!) with drag & drop elements and add photos, videos, call-to-action buttons and social media conversations to your press release.
For example, if you're writing a press release about a new line of natural cosmetics, check which journalists that you Best Press Release Distribution Service found are actually interested in natural and organic products, and not just cosmetics in general. It'll improve the chances of your product press release getting picked up and published.
Write a press release headline that gets your product noticed
The headline of your press release (and also the subject of your media pitch email) is crucial to getting your press release read or even email opened.
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Let's go over these press release Reviewed by Press Release Power on 6/02/2020 03:41:00 am Rating: 5
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Keybass
Vampire Weekend Releases New Album, Father of The Bride
Vampire Weekend's long-awaited fourth album Father of the Bride is released today
Bass Magazine
Ending months of fevered anticipation punctuated by three brilliant double-A-side singles and a slew of over-the-top positive early reviews, Vampire Weekend's long-awaited fourth album Father of the Bride is released today, May 3rd, on Columbia Records. The album is available to purchase or stream here.
Vampire Weekend will celebrate FOTB's release with an all-day show Sunday, May 5th at New York City's recently reopened Webster Hall. Over the course of the morning and afternoon, the band will play three sets including a performance of the new album in its entirety. Bagels for breakfast and pizza for lunch will be included. See http://vampireweekend.shop.ticketstoday.com for more information.
Simultaneous to the May 5th all-day affair will be a very special episode of Time Crisis with Ezra Koenig. This installment of the internet radio show created and hosted by Ezra will feature an in-depth dissection of FOTB with the album's co-producer Ariel Rechtshaid and de facto Time Crisis co-host Jake Longstreth. Listen live here and on demand here.
And on Tuesday, May 7th, Vampire Weekend will make its first television appearance in five years, with a debut performance on The Tonight Show Starring Jimmy Fallon.
The May 5th all-day experience concludes Vampire Weekend's New York State Tour of intimate venues, which brought an "unforgettable… truly transcendent experience"(Buffalo News) to Buffalo and Kingston. The Father of the Bride North American Tour begins the weekend of May 16-19 with a headlining set at the Hangout Festival in Gulf Shores, Alabama. The band's U.S. dates currently extend through an October 8-9 two-night stand at Red Rocks in Colorado, and include Vampire Weekend's most ambitious hometown date ever: a September 6th in-the-round performance for 18,000 fans at New York's Madison Square Garden.
North American Father of the Bride Tour 2019
05/05/19 - New York, NY – Webster Hall
05/17-19/19 – Gulf Shores, AL – Hangout Festival
06/05/19 — Toronto, ON — RBC Echo Beach
06/07/19 — Indianapolis, IN — Farm Bureau Insurance Lawn at White River State Park
06/08/19 — Milwaukee, WI — BMO Harris Pavilion
06/09/19 — Minneapolis, MN — The Armory
06/11/19 — Madison, WI — The Sylvee
06/12/19 — St. Louis, MO — Fabulous Fox Theatre
06/14/19 — Cleveland, OH — Jacobs Pavilion at Nautica
06/15/19 — Cincinnati, OH — PNC Pavilion
06/16/19 — Chicago, IL — Huntington Bank Pavilion at Northerly Island
06/18/19 — Nashville, TN — Ascend Amphitheater
06/21/19 — Raleigh, NC — Red Hat Amphitheater
06/22/19 — Charlotte, NC — Charlotte Metro Credit Union Amphitheatre
06/23/19 – Dover, DE – Firefly Music Festival
07/27-28/19 – Detroit, MI – Mo Pop Festival
08/16/19 — Oklahoma City, OK — The Criterion
08/17/19 — Houston, TX — White Oak Music Hall
08/18/19 — Dallas, TX —The Pavilion at Toyota Music Factory
08/20/19 — Austin, TX — ACL Live
08/24/19 — Miami, FL — James L. Knight Center
08/25/19 — St. Augustine, FL — St. Augustine Amphitheatre
08/27/19 — Atlanta, GA — Fox Theatre
08/30/19 — Norfolk, VA — Ted Constant Convocation Center
09/03/19 — Boston, MA — Agganis Arena
09/04/19 — Philadelphia, PA — Mann Center for the Performing Arts
09/06/19 — New York, NY — Madison Square Garden
09/08/19 — Montreal, QC — MTELUS
09/20-22/19 – Las Vegas, NV – Life Is Beautiful Festival
09/25/19 — Vancouver, BC — Festival Lawn at Deer Lake Park
09/27/19 — Seattle, WA — WaMu Theater
09/28/19 — Portland, OR — McMenamins Edgefield
10/01/19 — San Francisco, CA — Bill Graham Civic Auditorium
10/02/19 — Los Angeles, CA — Hollywood Bowl
10/03/19 — San Diego, CA — CalCoast Credit Union Open Air Theatre at SDSU
10/06/19 — Salt Lake City, UT— The Complex
10/08/19 — Denver, CO— Red Rocks Amphitheatre
10/09/19 – Denver, CO – Red Rocks Amphitheatre
Fender Expands Pedals Line With Six New Pedals
Source Audio Unveils the C4 Synth Pedal
Vox Introduces the Mini Superbeetle Bass Amp
Aguilar Announces the DCB-M4 and DCB-M5 Pickups
Paz Lenchantin and Pixies Release Kaleidoscopic Music Video "On Graveyard Hill"
Dunlop Releases New ILOVEDUST Jacquard Straps
Blackstar Amplification Launches the amPlug 2 FLY
Hannah Moorhead and Hard Rockers The Giraffes to Release New Album, Flower Of The Cosmos
Slash Announces Live Album "Living The Dream Tour" With Todd Kerns on Bass
© 2019 Bass Magazine


Approximately 10 miles south-east of Porbander is the village of Odedar. This is the village of the Odedra Mahers. Originally known as the Sumra-Rajputs from the region of Sindh. Acts of bravery and the fight for righteousness still exist in these small but rich in culture villages across the planes of the Kathiyavad region in Saurashtra. This is one of the many true stories that enrich the history of the region and its people. This is the story of Maher – Kardva Odedra.
In 1740 (British calendar), Kardva, the son of Ala-bhai Odedra from Odedar was born. Kardva had five brothers and four sisters. The youngest sister – Moonthi, who had two sons Arjun and Vinja, was married to Nebho in the village of Kadchche. Kardva was married to Janji whose parents lived in the village of Bakharla near Porbandar.
During this period, Mangrol was in the rein of Sheikh Darbar and Junagadh was governed by Nawab Hamid Khan Babi, who then reported to the central Mogul Empire. In those times the local Rajput clans would be given lands and villages as a token of appreciation for giving backing to Maharajas at times of war. The Rajputs of those given villages and land would then be the 'Gurasia'. Arjun, the nephew of Kardva, was the 'Gurasia' of the village Foolarama, which is near the town of Kutiyana.
Kardva was well respected and admired individual in the area for his fairness and righteousness. In the village of Odedar he was the leader and gave most of his time for the well-being of the village and its people. Kardva's reputation grew to the region of Junagadh. The Nawab very soon made close ties with Kardva.
One day, Amarji who was the Nawab's right hand man was on his way to Mangrol and happened to see the rich and vast crop yield that grew in lands around the village of Foolarama. He returned to Junagadh and informed the Nawab. Arjun Kadchcha and his father Nebho were summoned to the Nawab's court in Junagadh. They were both told that the village of Foolarama would have to pay yield-tax ('vero') for the crops that they grew. Arjun and Nebho returned to their village very concerned and decided to inform Kardva. They both went to Odedar and told Kardva what had happened. Kardva assured them that all would be fine and that he would pay a visit to the Nawab.
Kardva went to Junagadh and in the courts informed the Nawab that the village of Foolarama was that of the Gurasias who was Arjun, his nephew and Junagadh does not have any right for any tax on that land. The Nawab listened to all this and agreed that Kardva was right and this should not have happened. Amarji, who was also present, was very angry but could not question the Nawab's decision.
Time went on and one day Amarji was heading back to Junagadh from Mangrol. When he was between Meethi and Bagsora, near Dharm Vav he met Kardva's nephew – Arjun. When Amarji saw Arjun, his blood began to boil. Amarji shouted to Arjun: "Why do you hide behind your uncle if you call yourself a man!" Arjun hand went to his sword. Amarji's men took their swords out and ran towards Arjun. Arjun began to unleash his sword. Swiping it from side to side and cutting Amarji's men. Arjun had no chance. He was outnumbered and soon he was overpowered. Amarji's men constantly pushing their swords into the body of Arjun, he fell to the ground and died.
Kardva was furious with the Nawab. He sent his sister and his family to the village of Bakharla and got ready to face the might of the Nawab. With a Burdeye-Pagdi on his head, a Rajput sword on his side and a rhino-skin shield on his back, Kardva headed towards Junagadh.
On the way to Junagadh he visited Jaimal Lakha in the village of Padharia. Jaimal Lakha was a close friend of Kardva and had 1200 Rajput warriors under his hand. Kardva informed Jaimal of what had gone on and that he intended to take revenge for the killing of his nephew. Jaimal assured Kardva that he and his men would back him when ever he commanded them. Kardva informed Jaimal that he first intended to find out who gave the orders. He would go in the gates of Junagadh disguised as a 'Sadhoo'(Holy man).
Guard: " …this is no place to get any ganja. This is the Prince's place of learning. Nobody is permitted to go inside".
Kardva: " Write what I say on the cooper plate!".
Nawab: " Say what you will".
Kardva: " Nawab…. The door that I have arrived into Jungadh will be named Gate of Kardva. My nephew was killed unjustly and for this I order you to pay a fine of 900 'cori'. (Ordering a fine to a Nawab was considered a major insult to him) And the Chhokh from where I take this fine will be known as Kardva Chhokh!. Also after I get out from here you will not come after me until the ninth day.
ld start to look for him after the nine days. From there Kardva and Vajdev got on a horse and rode out.
Kardva decided to go to the Kingdom of Gonal, where Bha-Koombhaji reined. Bha-Koombhaji did not have the courage to confront the might of Jungadh and refused to assist Kardva. Kardva then rode to Bhavnager where Vakhatsinji reined. There too he was refused. He then headed west to the Okha region to a town called Positra. Positra was a fort and had the strength of gunpowder and cannons. The Vaghers of Okha vowed to give Kardva assistance.
The nine days were soon up. The Nawab of Junagadh ordered the head of Kardva to be brought to him. Like a swarm of wasps from a hive, the Junagadh armies surged out of the gates of Jungadh. Searching for days, finally the armies were informed that Kardva and Vajdev were at Positra. The army arrived at Positra and found that the fort was inaccessible. The army could not get over the walls nor through the walls so they decided to dig tunnels under the walls of the fort.
The armies of Junagadh began to crawl their way through and the battle began. Kardva swinging his sword from side to side. His sword in one hand and shield in the other, cutting and slicing his enemies into two he led the attack. The battle went on for a month. But death was getting closer to the courageous Kardva. He and his men were becoming outnumbered by the shear number of men from the Junagadh armies.
On the day of Thursday, the second day of the month of 'Fagan' in the Vikram-swanth calendar year of 1831 (1775 British calendar), Kardva and Vajdev lost their lives fighting and became Martyrs of truth and righteousness.
To date the dynasty of Arjun still live in the village of Kadchhe near Kutiyana in Saurashtra and are known as the Nebhanie family. And the dynasty of Kardva are still living in the village of Bhakharla and are known as the Arjunka-Odedras.


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Home Science & Hi-Tech Activists see Google's $1 billion Bay Area housing pledge as 'big win'...
Activists see Google's $1 billion Bay Area housing pledge as 'big win' amid affordability crisis
Can Google, which has helped fuel the Bay Area's housing crisis, become a better neighbor?
Activists who have held rallies, marches and town halls are praising Google's $1 billion pledge for up to 20,000 housing units in the uber-expensive Bay Area as a win after years of grassroots pressure — but they also believe the company, and Silicon Valley more broadly, must do more to mitigate the region's skyrocketing cost of living.
The dramatic expansion of tech giants like Google, Facebook, and Apple in California, encompassing San Francisco, San Jose, as well as cities like Mountain View, Menlo Park and Cupertino, combined with not enough new housing construction, has severely increased costs and displaced many working-class and middle-class residents.
In San Francisco, where Facebook, Uber and Twitter have offices, a recent report from Zumper put the median monthly rent for a one-bedroom apartment at $3,700. In San Jose, where Google is planning to build a massive new campus for 20,000 employees that will feature housing, retail and offices, the average apartment studio rents for $2,000 as the city struggles with an escalating housing shortage and about 4,000 homeless residents.
Google this week announced a massive investment in housing following years of activist pressure.
(Jeff Barrera/Silicon Valley Rising/Getty)
The commitment announced in a blog post by CEO Sundar Pichai includes $750 million worth of Google's existing office space that will be converted into about 15,000 housing units, $250 million for incentives for other developers to construct 5,000 units of affordable housing and $50 million for nonprofits that help homeless people find shelter.
It's worth noting, as well, that Google was already offering to build some 5,700 housing units in a major plan for the North Bayshore area of Mountain View, the home of its headquarters. The North Bayshore plan has not been approved yet.
Advocates who have pressured Google and the tech industry, in general, offered cautious praise, while local politicians said the announcement was a strong first step.
"This announcement demonstrates that Google is listening to the thousands of people in San Jose and around the Bay Area who have demanded the tech giant take responsibility for its role in the housing crisis. This shows that when communities speak up, they can better shape a better approach to tech growth," Maria Noel Fernandez, campaign director for Silicon Valley Rising, a coalition of activist and labor groups pushing for a more inclusive tech sector, told Fox News via email.
California State Senator Scott Wiener, who is chair of the senate's housing committee, called the pledge a "major investment."
"To solve our housing emergency, all of us — government, business, housing advocates, neighbors – must work together. Today's announcement is a strong step in the right direction," Wiener said on Twitter.
Fernandez also added a note of caution to her praise, saying that her organization wants to "make sure Google partners with diverse communities so tech development creates real opportunity for working families instead of more displacement."
California Gov. Gavin Newsom said he hopes Google's announcement inspires other tech giants to follow suit.
"Google recognizes that it has an important role to play in addressing California's cost crisis," Newsom said.
San Jose Mayor Sam Liccardo, who made affordable housing a central pillar of his campaign, also said Google's plan was a good step.
"We look forward to working with Google to ensure today's announcement manifests into housing that will benefit thousands of San Jose residents struggling under the burden of high rents," he said in a statement posted to Twitter.
A June report from Working Partnerships USA, a grassroots group fighting for "just economy," concluded that Google and San Jose can prevent further rent hikes, overcrowding, massive commutes and displacement by "subsidizing the development of 5,284 affordable homes and helping produce 12,450 market-rate units by 2030."
Although there are many details to be worked out and activists will continue to pressure Google, the overall response was positive.
Activism from Google employees has prompted discussions about the search giant's previous efforts to expand in China, sexual harassment and alleged retaliation as well as its treatment of contractors, vendors and temporary employees.
By Christopher Carbone
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Rates are on hold - but for how much longer?
At the monthly board meeting on Tuesday, the RBA kept the cash rate at 2.5 per cent for the 12th consecutive meeting, marking the longest period of steady rates since 2006.
The Reserve Bank of Australia is on track to break its record for the longest spell of cash rate inertia, with most major economists forecasting the central bank will sit on its hands well into 2015.
RBA governor Glenn Stevens said in the statement accompanying Tuesday's rate decision that "the most prudent course is likely to be a period of stability in interest rates," despite the Australian dollar being stuck in the tight trading range between US92¢ and US93¢. The statement had sparked market speculation about when the "period of stability" will end.
Economists agree that the next move by the central bank will be a tightening of monetary policy. But they have produced wildly varying forecasts for the timing of that move from as early as February 2015 (Commonwealth Bank of Australia) to as late as the first quarter of 2016 (Bank of America Merrill Lynch).
At the monthly board meeting on Tuesday, the central bank kept the cash rate on hold at 2.5 per cent for the 12th consecutive meeting, marking the longest period of steady rates since 2006.
The longest the Reserve Bank has kept interest rates on hold was more than 18 years ago, when the central bank kept the rates at 7.5 per cent for 17 consecutive meetings between February 1995 and July 1996. If the central bank were to match that record, that would imply no change to current settings until April next year, matching most analyst forecasts.
A number of financial houses have cast their votes on an interest rate rise in early 2015, with CBA and Barclays leading the way.
On CBA's analysis, the interest rate rise could be as early as February 2015.
"The RBA has been too pessimistic on growth and too optimistic on inflation," CBA economist Diana Mousina said.
Ms Mousina said there were a "large stockpile" of potential projects that could receive approval, including a number of L&G projects in Queensland.
She also said the RBA overestimated how much the removal of the carbon tax will take out from the downgraded inflation forecast.
But the housing bubble is not frothy enough to be a key consideration for an interest rate rise, Ms Mousina said.
On CBA's forecast, the Australian dollar is forecast to reach above the market forecast at US94¢ by the 2014 end.
ANZ, UBS and HSBC predict the hike will be in the second quarter of 2015, while JP Morgan and Westpac forecast August next year.
On JP Morgan's analysis, the rate hike will not kick in until August 2015 despite the high exchange rate.
"The domestic economy is underperforming...but we doubt it is soft enough for the RBA to contemplate a rate cut in the near term," Stephen Walters, JP Morgan's chief economist, said in a note.
"Nor does the elevated Australian dollar justify easing at this stage – the RBA has other tools in its kit to deal with currency strength.
"Similarly, we suspect officials are wary that a rate cut now would add to growing signs of frothiness in the housing market."
On UBS's forecast, the central bank is expected to raise the interest rate to 2.75 per cent in May 2015 and to 3 per cent in June 2015 as the labour market recovers. But until then, the interest rate is likely to remain static.
"It is the most boring central bank in the world for the market," reckons UBS interest rate strategist Andrew Lilley.
Bank of America predicts the Reserve Bank will raise the cash rate in the 2016 first quarter, which would see the interest rate being on hold for two and a half years. NAB and RBC Markets are betting on a rate hike in late 2015.
"Historically the Reserve Bank has never initiated a cycle of raising rates until after unemployment has clearly peaked," Bank of America Merrill Lynch chief economist Saul Eslake said.
The monetary policy decision on Tuesday indicated the central bank's concern for a slack labour market, Mr Eslake said.
"They (the RBA) conceded for the first time there was certain degree of slack in the labour market.
"[The] unemployment rate is a key indicator of the amount of slack spare capacity in the economy."
Mr Eslake predicted the unemployment rate to peak in late 2015 in the 6.5 and 6.75 per cent range.
On BoA numbers, the local currency is expected to plunge to US80¢ by the 2015 year-end on the back of falling commodity prices, the completion of construction work in resources and LNG projects and the US Federal Reserve lifting interest rates.
RBC Capital cast its vote on a rate rise in the fourth quarter of 2015.
"As long as there are signs of policy traction, particularly in housing, rate cuts are unlikely," RBC Capital head of strategy Su-Lin Ong said in a note.
"But, the modest rotation of growth, degree of labour market slack, and well-behaved inflation outlook suggest rate hikes are also not on the agenda."
Ms Ong said any rate change was likely to be on the sidelines for the foreseeable future, given accommodative settings for growth and limitations of monetary policy.
Economists' snap shot: When will the "period of stability" end?
CBA: First hike will be in February.
TD Securities: Expect a rise in March.
Barclays: RBA will start "to raise rates in Q1 2015 as the economy improves and with the cash rate at 3.5 per cent by the end of that year".
HSBC: hike in June quarter of 2015, "though much depends on the [Aussie dollar] outlook".
ANZ: Tightening cycle begins in May 2015.
JP Morgan: "First step along the road to policy normalisation in August next year".
Westpac: "Our official forecast is for a 25 basispoint increase in the cash rate at the board meeting in August next year."
RBC Capital Markets: "Given a number of cyclical as well as structural challenges, we remain comfortable with our view for an extended period of steady cash at 2.5 per cent until [the fourth quarter] 2015".
NAB: The bank "still expects the next move in the cash rate will be up, but not until late 2015".
BoA: "Our forecast remains for policy to be tightened in [the first quarter of] 2016"


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Results for all evidence | Drugs and Technologies | British National Formulary - BNF
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What might be a classroom of the future?
How can we design existing space to be more dynamic?
How might a space support different learning styles?
How can we be more intentional about this "third space"?
If a classroom were no longer filled with immovable desks aligned in rows, could it become a space that invited–even required–student collaboration, creativity, and critical thinking? What if the front of the room disappeared and students owned the walls–writing, wondering, and questioning? What if the physical space could be transformed in seconds, flexing to support the moment's work? What if lesson plans gave way to student learning experiences where the student, not just the teacher, designed the curriculum and the learning?
In conjunction with Cannon Design/The Third Teacher+, a school design consulting group in Chicago, and led by Laura Deisley and upper school dean of academic affairs, Marsha Little, the nine fellows have already engaged in significant professional development, as they focus on their role as designers–of curriculum, space, and student experiences. They're teaching in renovated rooms with mobile desks, tables, and chairs; copious writable surfaces; and versatile technology.
As the traditional boundaries of room ownership and discipline-specific hallways fade, we're already seeing innovative collaboration between teachers and a breaking down of boundaries that seemed etched in stone. Ultimately, our goal is simple and unchanged: offer the best possible learning experience to students as they become excellent collaborators, creators, critical thinkers, and communicators. We are excited to explore how that work might be more fully realized in the Model Classrooms as the year unfolds.
My fellow Norcross High School librarian Jennifer Lund and I were interested in the Lovett model since their work aligned with our vision for not only redesigning our physical space and the kinds of learning experiences we want our library to facilitate, but to also help us recast our roles here as instructional partners more deeply embedded in the day to day work of teaching and learning with our faculty and students as we aspire to craft a more participatory culture of learning, a vision that also aligns with our school's literacy plan. Inspired by the work and vision of Brian Mathews at Virginia Tech and the model of Connected Learning, we envision the NHS library as a vibrant innovation hub of sorts to incubate, ignite and inspire larger change throughout our entire building, ultimately transforming learner experiences in a more seamless and cohesive way. As part of our strategic plan for the revamping of our physical space, we have proposed housing a prototype space for teaching and learning where we work with faculty and students to pilot, assess, model, and assess learning experiences. We want to enable our teachers and students to take risks as learners, to reflect and learn with them, and to help then share and distribute those insights across multiple content areas for cross-pollination of innovative strategies.
Because our vision paralleled the design drivers and physical design elements embodied in the six prototype classrooms at Lovett, we reached out to my friend and colleague Laura Deisley, the Director of Strategic Innovation at Lovett, to see if we could come for a visit and to have a conversation about their work. I know from my previous collaboration with Laura through Reimagine Ed that she is someone passionate about learning, libraries, and inquiry. Laura and Upper School Dean of Academic Affairs, Marsha Little, graciously accepted our request. Our visit yesterday, accompanied by friend and colleague Holly Frilot, Media Instructional Coach for our school district, was nothing short of energizing.
As I watched students actively participating in learning experiences and discovering how the shift in physical space shifted teachers' pedagogical stances, I literally quivered with excitement and joy—seeing how such simple yet profound changes were yielding impact and transformative learning experiences for both teachers and students was such a tremendous joy. In my previous experiences and now here at Norcross, I have sought to break down the artificial barriers that seem to separate libraries from their communities and create siloed learning experiences. The Lovett model reminds me that this scale of change is possible, scale that is needed to help our school grow into a learning community that embraces shared ownership of learning and connects contexts of academic driven, interest powered, and peer/mentor supported learning. Because the teachers float through the day from one model classroom to another, they do not have a teacher desk or other teacher-oriented paraphernalia plastered all over the room. Instead, the student handprint is embedded in each room with student created content; I find Lovett's concept of learning spaces co-curated by students to be quite powerful and participatory as it invites collaboratively created content and honors fluidity in the roles of experts and novices by positioning teachers and students as co-teachers and co-learners. The purposeful and contextualized application of technology, rooted in learner needs and sound instructional design, was also refreshing. I heartily applaud the cohort's willingness to engage in this kind of messy, organic work and to grow from the friction/cognitive dissonance that comes from being outside of one's comfort zone.
Variations in furniture styles and functions as well as heights for flexible seating and multiple types of collaborative work. Soft seating as well as hard seating were incorporated into the rooms.
Writable surfaces: tinted glass affixed to the wall, mobile whiteboards on wheels (a favorite of the students and staff), IdeaPaint walls, writable tabletops. (note: Laura advises that if you are in an older building that has had multiple layers of paint or less than ideal drywall, you might want to consider the glass surfaces or mobile whiteboards (6 feet in length is optimal) since ghosting and the durability of the paint could potentially be problematic.
"Zones" of rooms for smaller groups or paired conferencing.
No teacher desk taking up the space.
Students and teachers capture the student work each period with their cameras; Evernote is also a great app for capturing and sharing the content on the walls.
Strategic lecture designed to facilitate, rather than dominate, the conversation, conversation that is fueled by students being contributors and active participants. Some teachers also time themselves to limit themselves to short chunks of lecture at a time.
Rooms feel clean yet cozy; they are quickly and easily transformed since the space is open and supports transitions and reconfigurations of seating/workspaces/students.
Time for individual reflection and work as well as active, collaborative time with small and whole group discussions/conversations for learning.
Experiencing this learning ecology firsthand affirms our belief that our library can be the Story Studio prototype learning space for Norcross High; we see it as a springboard to future work in developing and working with a cohorts here to scale the pilot and more explicitly situate our work in the context of our classrooms. We are deeply appreciative that our principal, Mr. Will Bishop, supports this vision and the kind of work we want to do; we also appreciate the encouragement and feedback from assistant principal John Decarvalho and our LSTC (Local School Technology Coordinator) Dr. Victoria Dodd. By being a catalyst for designing learning experiences with our teachers and students that will meet our learners at their point of need, we can expand the possibilities for our library as both a learning space and a change agent that creates more diverse entry points for all learners, helps young people leverage the knowledge across multiple learning boundaries, and honors multiple ways of learning. We also see this an opportunity to more deliberately examine and expand the ways our library functions as a sponsor of literacy here at Norcross High. Ultimately, we hope to crowdsource with our students and faculty our own "learning playbook" that we can share with others as we engage in conversations for rewriting and composing new narratives of learning for everyone who is part of our learning community.
A heartfelt thank you Laura, Marsha, the faculty, and students of Lovett for so generously sharing their wisdom and for giving us the opportunity to experience their learning ecology firsthand!
The work that Buffy and Jen are doing at NHS will be nothing less than transformational. I am truly excited about their ideas of using space to affect pedagogy in positive and dynamic ways and feel privileged to have the opportunity to learn from them firsthand.
As we discussed yesterday, I believe deeply in the opportunity for libraries to be catalysts, activators of contemporary teaching and learning, in their school communities. You have blazed the trail both in public and school libraries, and you and the Norcross/Gwinnettt County team have a wonderful vision and will lead us all yet again along a vibrant pathway. I eagerly await what will unfold!
Thank you for your generous reflection, your very kind words, and the opportunity to have you as a colleague and friend.
Poetry and Possibilities: Do They Live at Your Library?


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Nancy Gerdt, Co-Chair; Jim Mosher, Co-Chair; Michele Mosher, Treasurer; Diane McCormick, Events Team Co-Leader; Linda Skeff, Restoration Team Co-Leader; Mary Hammer, Anita Lande, Judi Sherman, April Zilber, Kristen Praly, Katelyn Praly, Nina Moore, Karen Holl.
Felton Library Friends hosted the Library Festival on Saturday, May 20, on the new library site. The purpose of the Festival was to celebrate the future Library and park, display site plans for the two-plus acre site, and share ways for community members to become involved. The event kicked off the on-line auction for the figures at feltonlibraryfriends.org. 91+ volunteers flourished in the heat and hosted several craft activities and served food/drink.
The events committee had a great time getting to know each other better and benefitting from our combined strengths. The diversity of talent and resources were great. A very successful day!
The online bidding is going well, and is up to around $3,000. Bidding ends at 5 pm June 16.
We have started to rebuild our account after contributions to the CEQA application, the Measure S campaign, and funding the Nature Explore workshops. New Leaf Community Day, which netted $1524, and the event and raffle, which netted $2,421, have brought our account with FSCPL to $13,704. As our fiscal agent, FSCPL charges administrative fees of 5% on all deposits.
The site plan for the library/park project will be submitted for review the week of June 12. This summer, a library estimator will come up with a formal estimate for the library building, parking lot, and furnishings. County Parks will work on an estimate for the park side that is not covered by Measure S funds. Both estimates will help determine our goals for fundraising. In anticipation of applying for grants, we will be working on securing "letters of intent" from community partners. We hope to use these for the grant applications and to determine how different community groups envision using the library/park.
The development team has been working with Supervisor McPherson's office and the County Parks Department to name the park area across from Bull Creek the Karen Christensen Nature Discovery Park, honoring Ms. Christensen, a Felton resident who passed away last year. She is well known as an environmental advocate and champion whose work is well exemplified in the plans for the new park. The naming may also promote important fundraising opportunities in the environmental community. FLF filed a letter in support of the naming proposal to the County Parks Department (available on request) which will be considered first by the County Parks Commission and then the Board of Supervisors. Passage is accepted.
Development team co-chairs Jim Mosher and Ralph Miljanich conducted with a prospective fundraising consultant, Lana Weeks, who has extensive experience conducting major donor campaigns for environmental causes and was a colleague and friend of Karen Christensen. Lana will be coming to Felton Sunday and Monday, June 11-12 and will meet with members of the fundraising team to discuss a possible major donor campaign that could include an endowment fund. There was discussion regarding how such a campaign would be conducted, including possible fiscal agents and campaign committee members.
The team is also working with the County Parks Department on a grant proposal to the California Department of Parks and Wildlife. This looks to be a very good possibility for funding the park as the grant criteria fit well with the plans we are developing. The proposal will be due September 1; FLF will be expected to provide drafts of much of the grant narrative. Funding will go directly to the county parks department; the department expects to ask for between $200,000 and $270,000. Supervisor Mark Stone has been contacted and was enthusiastic about this funding idea. A possible problem is that we must obtain CEQA approval by October 31 to qualify for the grant. FLF is working with Supervisor McPherson's office and county staff to insure CEQA approval is obtained in a timely manner.
They have hired Vivian Rogers as E.D. The board is in the process of reorganizing and Jim is serving as our chapter representative.
The library is planning a system-wide celebration of its 150th Anniversary next year, and would like to have a volunteer from each chapter on the planning committee. Please let Michele know if you are interested.
An explanation of the new governing structure included information on the Joint Powers Board and the Library Advisory Committee. Also mentioned was the great work being conducted by library staff in the areas of homelessness and mental health. Agendas and Minutes are available at the Santa Cruz Public Libraries website.
Next meeting will be Wednesday, August 30, 7:00 pm, at the Felton Library.


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HudsonValleyPress
The Heart of the Community
Pillars of the Community 2019
Cornerstone Family Healthcare's 20th Annual Pillars of the Community Gala held at Anthony's Pier 9 on Saturday, November 2, 2019.
Kenneth J Steier, DO, Executive Dean of TouroCOM accepts the Pillar Award from Linda S. Muller, President & CEO Cornerstone Family Healthcare during the 20th Annual Pillars of the Community Gala held at Anthony's Pier 9 on Saturday, November 2, 2019. Hudson Valley Press/CHUCK STEWART, JR.
Michael Gilfeather, President & CEO of Orange Bank & Trust accepts the Pillar Award from Linda S. Muller, President & CEO Cornerstone Family Healthcare during the 20th Annual Pillars of the Community Gala held at Anthony's Pier 9 on Saturday, November 2, 2019. Hudson Valley Press/CHUCK STEWART, JR.
Joseph (left) and Mary Ellen Sayegh (right) received the Lifetime Achievement Award from Linda S. Muller, President & CEO Cornerstone Family Healthcare during the 20th Annual Pillars of the Community Gala held at Anthony's Pier 9 on Saturday, November 2, 2019. Hudson Valley Press/CHUCK STEWART, JR.
Linda S. Muller, President & CEO Cornerstone Family Healthcare offers remarks during the 20th Annual Pillars of the Community Gala held at Anthony's Pier 9 on Saturday, November 2, 2019. Hudson Valley Press/CHUCK STEWART, JR.
Cornerstone Family Healthcare's 20th Annual Pillars of the Community Gala held at Anthony's Pier 9 on Saturday, November 2, 2019. Hudson Valley Press/CHUCK STEWART, JR.
Lisa and Josh Sommers were the Masters of Cermonies for Cornerstone Family Healthcare's 20th Annual Pillars of the Community Gala held at Anthony's Pier 9 on Saturday, November 2, 2019. Hudson Valley Press/CHUCK STEWART, JR.
Board Chairman Wayne Day offers remarks during Cornerstone Family Healthcare's 20th Annual Pillars of the Community Gala held at Anthony's Pier 9 on Saturday, November 2, 2019. Hudson Valley Press/CHUCK STEWART, JR.
John and Ruth Smith enjoy a dance during Cornerstone Family Healthcare's 20th Annual Pillars of the Community Gala held at Anthony's Pier 9 on Saturday, November 2, 2019. Hudson Valley Press/CHUCK STEWART, JR.
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text:
\section{Introduction}

Let $R$ be a commutative Noetherian ring of Krull dimension $d$. A classical result of Serre \cite{Serre} says that
every finitely generated projective $R$-module $P$ of rank $>d$ splits off a free summand. This is the  best possible
result in general as it is evidenced from  the well-known example of ``the tangent bundle over real algebraic sphere of dimension two".
Therefore the question  ``splitting off a free summand" becomes subtle when $rank(P)=d$.
If $R$ is a reduced affine algebra over an algebraically
closed field, then for a rank $d$ projective $R$-module $P$, M.P. Murthy \cite{Mu94} defined an obstruction class $c_d(P)$ in
the group $F^dK_0(A)$. Further assuming $F^dK_0(A)$ has no $(d-1)!$ torsion, he proved that $c_d(P)=0$
if and only if $P$ splits off a free summand of rank one.

For a commutative Noetherian ring $R$ of dimension $d$, Bhatwadekar--Raja Sridharan
(\cite{BR98},\cite{BR00})
defined an obstruction group called Euler class group, denoted by $\E^d(R)$. Assume $\mbox{$\mathbb Q$}}	\newcommand{\BR}{\mbox{$\mathbb R$} \subset R$, then
given a projective $R$-module of rank $d$, Bhatwadekar--Raja Sridharan defined an obstruction class $e_d(P)$
and  proved $e_d(P)=0$ in $\E^d(R)$ if and only if $P$ splits off a free summand of rank one.
Later, for a smooth scheme $X$ of dimension $n$, Barge--Morel \cite{BM}
defined the Chow--Witt group $\wt{CH}^j(X)$ ($j\geq 0$) and  associated to each vector bundle $E$ of rank $n$ with trivial determinant an Euler class
$\wt{c_n}(E)$ in $\wt{CH}^n(X)$. Let $A$ be a smooth affine domain of dimension $n$ and $P$ a finitely generated projective $A$-module of rank $n$.
Then it was proved that $\wt{c_n}(P)=0$ if and only if $P\by \sim Q\op A$ for $n=2$ in \cite{BM} (see also  \cite{F08}), $n=3$ in
\cite{FS09} and $n\geq 4$ in \cite{M012}.

A recent result of Marco Schlichting \cite{Marco16} proved a similar kind of result for a commutative Noetherian ring $R$
of dimension $d$ all of whose residue fields are infinite. Precisely, given a rank $d$ oriented projective $R$-module $P$,
he defined a class $e(P)$ in $H^d_{Zar}(R, \mathcal{K}_d^{MW})$ such that $e(P)=0$ if and only if
 $P$ splits off a free summand of rank one.

 One of the aims  of this  article is to provide a class of examples of commutative Noetherian rings $R$ of dimension $d$
 such that every rank $d$ projective $R$-module splits off a free summand of rank one. We prove the following:

 \begin{theorem}\label{unim}
   Let $R$ be a commutative Noetherian domain of dimension $d-1$ ($d\geq 1$) and $I$ an ideal of $R$.
   Define $A:=R[It] $ or $R[It, t^{-1}]$ (note that $dim(A)\leq d$).
  Let $P$ be a projective $A$-module  of rank  $d$. Then  $P\by \sim Q\op A$ for some projective $A$-module $Q$.
 \end{theorem}

 In particular, if $\mbox{$\mathbb Q$}}	\newcommand{\BR}{\mbox{$\mathbb R$} \subset A$, then the obstruction class $e_d(P)$ defined by Bhatwadekar--Raja Sridharan \cite{BR00}
 is zero in $\E^d(A)$. Also, in the view of Schlichting's result \cite{Marco16},
 if we assume that all residue fields of $A$ are infinite,
 then $e(P)$ defined by Schlichting is zero in $H^d_{Zar}(A, \mathcal{K}_d^{MW})$. For $A=R[t]$ and for birational
 overrings of $R[t]$, a similar type of result is proved by Plumstead \cite{P} and Rao (\cite{Rao82}, \cite{RaoTh}) respectively.

 In this direction, a parallel problem is ``the cancellation problem". Let $P$ be a projective module over a commutative
 Noetherian ring $R$ of dimension $d$ such that $rank(P)>d$. Then Bass \cite{Bass64} proved that $P$ is cancellative
 i.e. $P\op Q \by \sim P'\op Q \Rightarrow P\by \sim P'$. Again this is the best possible result in general as it is
 evidenced by the same well-known example ``tangent bundle over the real algebraic sphere of dimension two". However Suslin
 (\cite{Su77b}) proved that if $R$ is an affine algebra of dimension $d$ over an algebraically closed field,
 then every projective $R$-module of rank $d$ is cancellative. We enlarge the class of rings by proving the
 following result.

 \begin{theorem}\label{cancel}
  Let $R$ be a commutative Noetherian domain of dimension $d-1$ ($d\geq 1$) and $I$ an ideal of $R$.
   Define $A:=R[It] $ or $R[It, t^{-1}]$.
  Then every finitely generated projective $A$-module  of rank  $d$ is cancellative.
 \end{theorem}

  For $A=R[t]$ and for birational
 overrings of $R[t]$, a  similar type of result is proved by Plumstead \cite{P} and Rao (\cite{Rao82}, \cite{RaoTh}) respectively.

 The following result follows from our  result  Theorem \ref{cancel} and a result of Wiemers \cite[Theorem]{Wi1}.

 \begin{corollary}
  Let $R$ be a commutative Noetherian domain of dimension $d-1$ ($d\geq 1$) such that $1/d! \in R$ and $I$ an ideal of $R$.
   Define $A:=R[It] $ or $R[It, t^{-1}]$.
  Then every finitely generated projective $A[X_1,\ldots,X_n]$-module  of rank  $d$ is cancellative.
 \end{corollary}

As an application of our result, we prove the following result.

\begin{theorem}
 Let $R$ be a commutative Noetherian domain of dimension $d$ and $I$ an ideal of $R$. Let $M$ be  a finitely
 generated module over   $ A:=R[It]$ or $R[It, t^{-1}]$.
 Then $M$ is generated by $e(M):= Sup_{\p}\{\mu_{\p}(M)+dim(A/\p)\}$
 elements.
\end{theorem}

Let $A$ be a domain of dimension $n$, $R=A[X_1,\ldots,X_m]$ and $I$ the ideal of $R$ generated by $(X_1,\ldots,X_m)$.
Then $R[It]=A[X_1,\ldots,X_m,X_1t,\ldots,X_mt]$. Note that in this case $R[It]$ becomes a monoid algebra $A[M]$,
where $M$ is the monoid generated by $(X_1,\ldots,X_m,X_1t,\ldots,X_mt)$. In this case Gubeladze \cite{G1} conjectured
that every projective $A[M]$-module of rank $>n$ splits off a free summand of rank one. In  Theorem \ref{unim}, we have verified it affirmatively  but our
rank-dimension condition is not optimal. We note that the second author and  Keshari \cite{KeS} studied the problem of existence of
unimodular elements over monoid algebras. But their results do not cover the above monoid algebra.

{\bf Acknowledgement:} We would like to thank the referee for carefully reading the paper and  some useful comments.

 \section{Notations, Rees algebras and some properties}
   {\it Throughout the paper, we assume that all the modules are finitely generated.}

   Let $A$ be a commutative ring and $Q$ a  $A$-module. We say $p\in Q$ is unimodular
   if the order ideal $O_Q(p)=\{\phi(p) \,|\,\, \phi \in Q^*=\mbox{\rm Hom\,}}	\newcommand{\Sing}{\mbox{\rm Sing}(Q,A)\}$ equals
   $A$. Let $\p \in \mbox{\rm Spec\,}}	\newcommand{\hh}{\mbox{\rm ht\,}(A)$. An element $q\in Q$ is said to be a basic element of $Q$ at
   $\p$ if $q \notin \p Q_{\p}$. We say $q$ is a basic element of $Q$ if it is a basic element of $Q$
   at every prime ideal of $A$.  Let $\mu_{\p}(Q)$ denote the minimum number of generators of $Q_{\p}$
   over $A_{\p}$.
   Let $P$ be a projective $A$-module. We say $P$ is cancellative if
   $P\op Q\cong P' \op Q\Rightarrow P\cong P'$  for all  projective $A$-modules $P'$ and $Q$.

   The set of all unimodular elements in $Q$ is denoted by
   $\mbox{\rm Um}}		\newcommand{\SL}{\mbox{\rm SL}(Q)$. We write $\E_n(A)$ for the group generated by the set of all
   $n\times n$ elementary matrices over $A$ and $\mbox{\rm Um}}		\newcommand{\SL}{\mbox{\rm SL}_n(A)$ for
   $\mbox{\rm Um}}		\newcommand{\SL}{\mbox{\rm SL}(A^n)$. We denote by $\mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}_A(Q )$, the group of all $A$-automorphisms
   of $Q$.

   For an ideal $J$ of $A$, we denote by $\E(A \oplus Q, J)$, the subgroup
   of $\mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}_A (A\oplus Q )$ generated by  the automorphisms
   $\Delta_{a\phi}= \bigl( \begin{smallmatrix} 1 & a\phi \\ 0&id_Q
   \end{smallmatrix} \bigr)$ and $\Gamma_{q}= \bigl( \begin{smallmatrix}
   1 & 0 \\ q&id_Q
   \end{smallmatrix} \bigr)$ with $a\in J$, $\phi \in Q^*$ and $q\in JQ$.
   Further, we shall write $\E(A\oplus Q)$ for $\E(A\oplus Q, A)$.
   We denote by $\mbox{\rm Um}}		\newcommand{\SL}{\mbox{\rm SL}(A\oplus Q, J)$, the set of all $(a,q) \in \mbox{\rm Um}}		\newcommand{\SL}{\mbox{\rm SL}(A\oplus
   Q )$ with $a \in 1 + J$ and $q \in JQ$.

   {\bf Generalized dimension:} Let $R$ be a commutative ring and $\mbox{$\mathcal S$}}	\newcommand{\CT}{\mbox{$\mathcal T$} \subset \mbox{\rm Spec\,}}	\newcommand{\hh}{\mbox{\rm ht\,}(R)$.
   Let $\gd: \mbox{$\mathcal S$}}	\newcommand{\CT}{\mbox{$\mathcal T$}\rightarrow}		\newcommand{\lra}{\longrightarrow \BN\cup \{0\}$ be a function. Define a partial order on $\mbox{$\mathcal S$}}	\newcommand{\CT}{\mbox{$\mathcal T$}$
   as $\p<<\mbox{$\mathfrak q$}}	\newcommand{\mr}{\mbox{$\mathfrak r$}$ if $\p\subset \mbox{$\mathfrak q$}}	\newcommand{\mr}{\mbox{$\mathfrak r$}$ and $\gd(\p)>\gd(\mbox{$\mathfrak q$}}	\newcommand{\mr}{\mbox{$\mathfrak r$})$. We say that $\gd$ is
   a generalized dimension function on $\mbox{$\mathcal S$}}	\newcommand{\CT}{\mbox{$\mathcal T$}$ if for any ideal $I$ of $R$,
   $V(I)\cap \mbox{$\mathcal S$}}	\newcommand{\CT}{\mbox{$\mathcal T$}$ has only a finite number of minimal elements with respect to
   $<<$. We say that $R$ has the generalized dimension $d$ if $d=min_{\gd}(max_{\p\in \mbox{\rm Spec\,}}	\newcommand{\hh}{\mbox{\rm ht\,}(R)} \gd(\p))$. The notion
   of the generalized dimension was introduced by Plumstead in \cite{P}.

   For example, the standard dimension function $\gd(\p)=coheight(\p):=\dim(R/\p)$ is a generalized dimension function.
   Thus, the generalized dimension of $R$ is $\leq$ the Krull dimension of $R$.
   Observe that if $s\in R$ is such that $R/(s)$ and $R_s$ have the generalized dimension $\leq d$, then the generalized
   dimension of $R\leq d$. Indeed, if $\gd_1$ and $\gd_2$ are generalized dimension functions on $R/sR$ and $ R_s$ respectively
   with $\gd_i\leq d$. Then we define $\gd:\mbox{\rm Spec\,}}	\newcommand{\hh}{\mbox{\rm ht\,}(R)\rightarrow}		\newcommand{\lra}{\longrightarrow \BN\cup \{0\}$ as follows $\gd(\p)=\gd_1(\p)$ if $s\in \p$ and
   $\gd(\p)=\gd_2(\p)$ if $s\notin \p$. Now clearly $\gd$ is a generalized dimension function on $R$ with $\gd \leq d$.

   We note down an example of
   Plumstead \cite[Example 4]{P} where he has given a ring having generalized dimension $<$ Krull
   dimension. Take $A:=R[X]$, where $R$ is a ring having an element $s\in rad(R)$ with $\dim(R/sR)< \dim(R)$.
   Then Plumstead \cite{P} proved that the generalized dimension of $A<\dim(A)$.

   {\bf The Rees algebras and the extended Rees algebras:} Let $R$ be a commutative Noetherian ring of dimension $d$ and $I$
   an ideal of $R$. Then the algebra
   $$R[It]:=\{\sum_{i=0}^{n}a_it^i: n\in \BN, a_i\in I^i\}=\op_{n\geq0}I^nt^n$$
   is called the Rees algebra of $R$ with respect to $I$. Sometimes it is also called the blow-up algebra. These
   algebras arise naturally in the process of blowing-up a variety along a subvariety. In the case of an affine variety
   $V(I)\subset \mbox{\rm Spec\,}}	\newcommand{\hh}{\mbox{\rm ht\,}(R)$,  the blowing-up is the natural map from $Proj(R[It])\rightarrow}		\newcommand{\lra}{\longrightarrow \mbox{\rm Spec\,}}	\newcommand{\hh}{\mbox{\rm ht\,}(R)$. It is a fact that
   dimension of $R[It]\leq d+1$ (see \cite[Theorem 1.3]{Va}). Further if $I$ is not contained in any minimal primes of $R$, then the dimension
   of $R[It]=d+1$ (see \cite[Theorem 1.3]{Va}). For further properties of $R[It]$, we refer the reader to \cite{Va}.

   One defines the extended Rees algebra $R[It, t^{-1}]$ with respect to an ideal $I$ of $R$ as  a subring of $R[t,t^{-1}]$
    as follows
   $$R[It, t^{-1}]=\{\sum_{i=-n}^{n}a_it^i: n\in \BN, a_i\in I^i\}=\op_{n\in \BZ}I^nt^n,$$
   where $I^n=R$ for $n\leq0$.
   It is easy to observe that $R[It, t^{-1}]$ is birational
   to $R[It]$, hence the dimension of $R[It, t^{-1}]\leq d+1$.

\section{Splitting off and cancellation results for  Rees algebras}

 \begin{lemma}\label{mcset}
 	Let $R$ be a commutative ring, $S\subset R$ be a multiplicative subset and $I\subset R$ an
 	ideal.  Then $S^{-1}(R[It])=S^{-1}R[(S^{-1}I)t]$.
 \end{lemma}
 \begin{proof}
 	By definition $S^{-1}(R[It])=S^{-1}R \oplus S^{-1}(IR)t\oplus S^{-1}(IR)^2t^2\oplus \cdots$ and
 	$$S^{-1}R[(S^{-1}I)t]=S^{-1}R \oplus S^{-1}I(S^{-1}R)t\oplus (S^{-1}I(S^{-1}R))^2t^2\oplus \cdots .$$

 	 Since localization commutes with direct sums,  we have
 	$S^{-1}(R[It])=S^{-1}R[(S^{-1}I)t]$.
 	$\hfill \blacksquare}	\newcommand{\gl}{\lambda$
 	\end{proof}

  \begin{lemma}\label{lem:gen-dim}
 	Let $A$ be a commutative Noetherian ring of dimension $d\geq 1$ and let
 	$s$ be a non-zero-divisor of $A$.
 	Then the generalized dimension of $A_{1+sA}$ is $\leq d-1$.
 \end{lemma}
 \begin{proof}
 	Let $\mathcal{P}_1$ be the set of all primes of  $A_{1+sA}$ which contains $s$ and
 	$\mathcal{P}_2:=\{\mathfrak{p} \in {\rm Spec}(A_{1+sA}): ht(\mathfrak{p)}<d\}$. We claim that
 	$${\rm Spec}(A_{1+sA})=\mathcal{P}_1\cup \mathcal{P}_2.$$
 	Let $\mathfrak{p}\in {\rm Spec}(A_{1+sA})$ be a prime ideal of height $d$. Hence $\mathfrak{p}$ is
 	a maximal ideal of $A_{1+sA}$. We claim that $s\in \p$. Suppose not, then $\p+(s)=(1)$.  This implies
 	that there exists $p\in \p$ such that $ap= 1+sb$. But $1+sb$ is a unit in $A_{1+sA}$ which is a
 	contradiction to the fact $1+sb=pa\in \p$.

 	This establishes the claim.
 	Now following \cite[Example 2]{P},
 	we get that the generalized dimension of $A_{1+sA}$ is $\leq d-1$.
 	$\hfill \blacksquare}	\newcommand{\gl}{\lambda$
 \end{proof}
 \begin{lemma}(Plumstead \cite{P})\label{lem:plum}
 	Let A be a commutative Noetherian ring of generalized dimension $d$ and $P$ a projective
 	$A$-module of rank $\geq d+1$. Then

 	$(1)$
 	$P$ has a unimodular element.
 	More generally, if $M$ is  a finitely generated $A$-module such that $\mu_{\p}(M)\geq d$
 	for all $\p \in {\rm Spec}(A)$, then $M$ has a basic element.

 	$(2)$ $P$ is cancellative. In fact $\E(A\oplus P )$ acts transitively on $\mbox{\rm Um}}		\newcommand{\SL}{\mbox{\rm SL}(A\oplus P)$.
 \end{lemma}
 \begin{proof}
 	This is an observation made by Plumstead in \cite[page 1421, paragraph 4]{P} except part $(2)$
 	second statement. For this, let $(a,p)\in \mbox{\rm Um}}		\newcommand{\SL}{\mbox{\rm SL}(A\oplus P)$.
 	By Eisenbud--Evans Theorem (see the version in \cite[\S 1]{P}),
 	there exists an element $q\in P$ such that $p+aq$ is a unimodular element. Hence there exists $\psi \in P^*$ such that $\psi(p+aq)=1$.
 	Now we have  $\Gamma_{-aq-p}\Delta_{\psi}\Delta_{-a\psi}\Gamma_{q}(a,p)^t=(1,0)^t$ ($v^t $=transpose of the vector $v$). This finishes the proof.
 	$\hfill \blacksquare}	\newcommand{\gl}{\lambda$
 \end{proof}

 \begin{lemma}\label{lem:gen-dim-res}
 	Let $R$ be a commutative Noetherian ring of dimension $d$ and let $A:=R[It]$ or $R[It,t^{-1}]$.
 	Let  $s$ be a non-zero-divisor of $R$ and $P$ a projective $A_{1+sA}$-module of rank $\geq d+1$. Then

 	$(1)$ $P$ has a unimodular element. More generally, if $M$ is  a finitely generated $A$-module such that $\mu_{\p}(M)\geq d$
 	for all $\p \in {\rm Spec}(A)$, then $M$ has a basic element.

 	$(2)$ $P$ is cancellative. More generally, $\E(A\oplus P )$ acts transitively on $\mbox{\rm Um}}		\newcommand{\SL}{\mbox{\rm SL}(A\oplus P)$.

 \end{lemma}
 \begin{proof}
 	This  is a direct consequence of  Lemma \ref{lem:gen-dim} and  Lemma \ref{lem:plum}.
 	$\hfill \blacksquare}	\newcommand{\gl}{\lambda$
 \end{proof}

\begin{theorem}\label{thm:unimodular}
 Let $R$ be a commutative Noetherian domain of dimension $d$ and $I$ an ideal of $R$.

 $(1)$ Let $P$ be a projective module over the Rees algebra $R[It]$ such that the rank of $P$ is $>d$. Then  $P$
 has a unimodular element.

 $(2)$ Let $Q$ be a projective module over the extended Rees algebra $R[It, t^{-1}]$ such that the  rank
  of  $Q$ is $>d$. Then $Q$ has a unimodular element.
\end{theorem}
\begin{proof}
$(1)$
Let $A=R[It]$. Since $R$ is a domain, so is $R[t]$. Since $A$ is a subring of $R[t]$, we conclude
that $A$ is a domain.
If $I=(0)$, then $A=R$. In this case, the result follows from a classical result of Serre \cite{Serre}.
If $I=(1)$, then $A=R[t]$. In this case, the result follows from \cite[Corollary 4]{P}.
So, we assume that $(0)\neq I\neq (1)$. We already noted that $dim(A)\leq d+1$.
In fact here $dim(A)=d+1$ since $I\neq 0$. Hence in the view of a classical
result of Serre \cite{Serre}, we only have to consider the case when the rank of $P$ is $d+1$.

 Let $S$ be the set of all non-zero-divisors of $R$. Then by Lemma \ref{mcset}, we have
  $S^{-1}R[It]\cong R'[t]$, where $R'$ is
 a field. Therefore $S^{-1}P$ is a free $S^{-1}A$-module. Since $P$ is finitely generated, there exists $s\in S$ such that $P_s$ is a free $A_s$-module.

Let $a$ be  a non-zero non-unit element of $I$. Let us consider the following commutative diagram of rings

\[
   	\xymatrix{
   		R[It] \ar@{->}[r]^{p_1}
   		\ar@{->}[d]^{p_2}
   		&R[It]_{as}\cong R_{as}[t]
   		\ar@{->}[d]^{j_1}
   		\\
   		R[It]_{1+asR[It]} \ar@{->}[r]^{j_2}
   		&(R_{as}[t])_{1+asR[It]}.
   	}
   	\]
   	Since $a\in I$, we have $IR_a=R_a$. Hence we get  the isomorphism $R[It]_{as}\cong R_{as}[t]$
   	using Lemma \ref{mcset}.
   	It is easy to see that ${\rm Spec}(R[It])={\rm Spec}(R[It]_{as}) \cup {\rm Spec}(R[It]_{1+asR[It]})$.
   	Hence the above commutative diagram of rings is a localization Cartesian square of rings.
Since $P_{as}\cong R[It]_{as}^{d+1}$, $P_{as}$ has a unimodular element $p_1$.
Since $\dim(R[It])\leq d+1$, $\dim(R[It]/asR[It])\leq d$ (recall that $as$ is  a non-zero-divisor in $R[It]$  since $R[It]$ is a domain). By Lemma \ref{lem:gen-dim-res},
$P_{1+asR[It]}$ has a unimodular element $p_2$. Hence we can write
$P_{1+asR[It]}\cong R[It]_{1+asR[It]}\op Q$,
where $Q$  is a  projective module over $R[It]_{1+asR[It]}$.

Observe that $B:=(R_{as}[t])_{1+asR[It]}=T^{-1}R'[t]$,
where $R'=R_{as(1+asR)}$ is of dimension $d-1$ which follows from the arguments of Lemma \ref{lem:gen-dim} and $T\subset R$ is a multiplicatively closed set. Hence  we have $P_{as(1+asR[It])}\cong B^{d+1}$. Let the images of $p_1$ and $p_2$ be the unimodular
 vectors $\overline u$
and $\overline v$ respectively.
By a result of Rao \cite[Theorem 5.1(I)]{Rao82} (see also \cite[Theorem 6.2]{BRoy}), there exists $\gs\in \E_{d+1}(B)$ such that
 $\gs(\overline{u})=(\overline{v})$. Now by \cite[Corollary 3.2]{Roy82} (see for more explanations \cite[Proposition 3.2]{BLR}), we can write
$\gs=(\gs_1)_{as}(\gs_2)_{1+asR[It]}$, where $\gs_1\in \E_{d+1}(R[It]_{as})$, $\gs_2\in \E(R[It]_{1+asR[It]}\op Q)$.
Hence suitably changing $p_1$, $p_2$, we can assume that $\overline{u}=\overline{v}$.
 Therefore by a sheaf patching on a two cover, we get
a unimodular element $p$ in $P$. Now we explain this  more algebraically. The above Cartesian
square of rings induces the following Cartesian square of projective modules

\[
\xymatrix{
	P \ar@{->}[r]^{p_1}
	\ar@{->}[d]^{p_2}
	&P_{as}
	\ar@{->}[d]^{j_1}
	\\
	P_{1+asR[It]} \ar@{->}[r]^{j_2}
	&P_{as(1+asR[It])}.
}
\]
Having a unimodular element in $P$ is equivalent to an existence of a surjective
homomorphism $\phi: P \rightarrow R[It]$. The above arguments show that we have a surjection locally, hence by the Cartesian square diagram, we get
a surjection globally.
 This finishes the proof of $(1)$.

$(2)$ Proof of this part is  a verbatim copy of the proof of Part $(1)$.
$\hfill \blacksquare}	\newcommand{\gl}{\lambda$
\end{proof}

We prove the following criterion for cancellation.

\begin{lemma}\label{lem:criteria}
 {\bf (Criterion for cancellation)} Let $R$ be a commutative Noetherian ring of dimension $d$ and $P$ a projective
 $R$-module of rank $n$. Let $s,t\in R$ such that $sR+tR=R$.
 Let $(b,p)\in \mbox{\rm Um}}		\newcommand{\SL}{\mbox{\rm SL}(R\op P)$.
 Assume that there exist
 $\gs_1\in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R_s\op P_s)$, $\gs_2\in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R_t\op P_t)$ such that
 $((b,p)_{s})\gs_1=(1,0)$,  $((b,p)_t)\gs_2=(1,0)$ respectively.

 $(1)$  If we  define $\eta=(\gs_1)_t^{-1}(\gs_2)_{s}$, then
 $\eta= \bigl( \begin{smallmatrix} 1 & 0 \\ * & \theta}	\newcommand{\gu}{\upsilon
\end{smallmatrix} \bigr)$, where $\theta}	\newcommand{\gu}{\upsilon \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}{(P_{st})}$.

$(2)$ Further if
  $\eta=(\eta_2)_t(\eta_1)_s$, where $\eta_2 = \bigl( \begin{smallmatrix} 1 & 0 \\ *_2 & \theta}	\newcommand{\gu}{\upsilon_s
 \end{smallmatrix} \bigr) \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R_{s}\op P_{s})$ and  $\eta_1=  \bigl( \begin{smallmatrix} 1 & 0 \\ *_1 & \theta}	\newcommand{\gu}{\upsilon_t
 \end{smallmatrix} \bigr) \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R_{t}\op P_{t})$, then there exists $\phi \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R\op P)$ such that $(b,p)\phi=(1,0)$.
\end{lemma}

\begin{proof}
	For $(1)$, we observe that
 $(1,0)\eta=(1,0)$. Hence it is easy to see that

  $\eta= \bigl( \begin{smallmatrix} 1 & 0 \\ * & \theta}	\newcommand{\gu}{\upsilon
\end{smallmatrix} \bigr)$, where $\theta}	\newcommand{\gu}{\upsilon \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(P_{st})$.
 Consider the following Cartesian square
 \[
   	\xymatrix{
   		R \ar@{->}[r]^{p_1}
   		\ar@{->}[d]^{p_2}
   		&R_s
   		\ar@{->}[d]^{j_1}
   		\\
   		R_t \ar@{->}[r]^{j_2}
   		& R_{st}.
   	}
   	\]

 We set  $\gs_s=\gs_1\eta_2$ and $\gs_t=\gs_2\eta_1^{-1}$. Then we observe that $((b,p)_{s})\gs_s=(1,0)$,  $((b,p)_t)\gs_t=(1,0)$
and $(\gs_s)_t=(\gs_t)_s$ (recall that $\eta=(\gs_1)_t^{-1}(\gs_2)_{s}=(\eta_2)_t(\eta_1)_s$).
Now by
a standard  patching argument, we will get an automorphism $\phi\in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R\op P)$ such that $(b,p)\phi=(1,0)$. Now we explain how we get the automorphisn $\phi$. The following is a commutative
diagram of modules where the front square and the back square are  Cartesian.

 \[
\xymatrix{
	R\oplus P \ar@{->}[rr]\ar@{->}[rd]^{\phi}\ar@{->}[dd] &&R_s\oplus P_s \ar@{->}[dd]\ar@{->}[rd]^{\phi_s}
	\\
	&R\oplus P \ar@{->}[rr]	\ar@{->}[dd] &&R_s\oplus P_s \ar@{->}[dd]
	\\
	R_t\oplus P_t \ar@{->}[rr]\ar@{->}[rd]^{\phi_t}& & R_{st}\oplus P_{st}\ar@{->}[rd]^{\phi_{st}}
	\\
 &R_t\oplus P_t \ar@{->}[rr] && R_{st}\oplus P_{st}
}
\]
We get the homomorphism $\phi$ by the universal property of the Cartesian square. We observe that $\phi$
is  locally an isomorphism and it sends $(b,p)$ locally to $(1,0)$. Hence we conclude that $\phi$ is an isomorphism
and it sends $(b,p)$ to $(1,0)$.
$\hfill \blacksquare}	\newcommand{\gl}{\lambda$
\end{proof}

\begin{lemma}\label{lem:splitting}
	Continuing with the notations as in Lemma \ref{lem:criteria},  we further assume that $P_s$ is free.
	Then we can write  $\eta=(\eta_2)_t(\eta_1)_s$, where $\eta_2 = \bigl( \begin{smallmatrix} 1 & 0 \\ *_2 & \theta}	\newcommand{\gu}{\upsilon_s
	\end{smallmatrix} \bigr) \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R_{s}\op P_{s})$ and  $\eta_1=  \bigl( \begin{smallmatrix} 1 & 0 \\ *_1 & \theta}	\newcommand{\gu}{\upsilon_t
	\end{smallmatrix} \bigr) \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R_{t}\op P_{t})$ i.e. $(1,0)\eta_i=(1,0)$ for $i=1,2$.

	\end{lemma}
\begin{proof}
	By \cite[Corollary 3.2]{Roy82}, we can write
	$\eta=(\eta_1)_t(\eta_2)_{s}$ such that $\eta_1\in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R_s\op P_{s})$ and $\eta_2\in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R_t\op P_t)$.  Since $(1,0)\eta=(1,0)$,  the explicit computations of $\eta_1$, $\eta_2$ in the proof of \cite[Proposition 3.1]{Roy82} can be modified suitably to get $\eta=(\eta_1)_t(\eta_2)_{s}$
  such that	$(1,0)\eta_1=(1,0)$ and $(1,0)\eta_2=(1,0)$.
	Hence we have  $\eta_1=\bigl( \begin{smallmatrix} 1 & 0 \\ *_1 & \theta}	\newcommand{\gu}{\upsilon_1
	\end{smallmatrix} \bigr) \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R_{s}\op P_{s})$ and $\eta_2=\bigl( \begin{smallmatrix} 1 &  0\\ *_2 & \theta}	\newcommand{\gu}{\upsilon_2
	\end{smallmatrix} \bigr) \in\mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R_t\op P_t) $.

	Alternatively, one can use Quillen splitting lemma to get the required splitting as follows.
	Consider the elementary group $\E(R_{st}[Z]\oplus P_{st}[Z])$, where $Z$ is  a variable.
	It is easy to see that we can find $\alpha(Z) \in \E(R_{st}[Z]\oplus P_{st}[Z])$ with the properties
	$(1,0) \alpha(Z) = (1,0)$,  $\alpha(1) = \eta$, and $\alpha(0) = Id$. By
	Quillen's splitting lemma \cite[Theorem 1, paragraph 2]{Qu}, for $g = (s)^N$ with large $N$, we have
	\begin{eqnarray*}
		\alpha(Z) &=& (\alpha(Z)\alpha(gZ)^{-1})_t\alpha(gZ)_{s},
	\end{eqnarray*}
	with $\alpha(Z)\alpha(gZ)^{-1} \in Aut(R_{s}[Z]\oplus P_{s}[Z])$, $\alpha(gZ) \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R_t[Z]\oplus P_t[Z])$.
	Note that $(1,0)\alpha(gZ) = (1,0)$; hence $(1,0)\{(\alpha(Z)\alpha(gZ)^{-1})\}= (1,0)$.
	Specializing $Z = 1$ gives the required splitting of $\eta = (\eta_1)_t(\eta_2)_{s}$,
	with $\eta_1 \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R_t\oplus P_t)$, $\eta_2 \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R_{s}\oplus P_{s})$ such that
	$(1,0)\eta_i  = (1,0)$  for $i = 1, 2$.
	$\hfill \blacksquare}	\newcommand{\gl}{\lambda$
	\end{proof}

\begin{theorem}\label{thm:canc}
 Let $R$ be a commutative Noetherian domain of dimension $d$ and $I$ an ideal of $R$.

 $(1)$ Let $P$ be a projective module over the Rees algebra $R[It]$ such that $rank(P)>d$.
 Let $(b,p)\in \mbox{\rm Um}}		\newcommand{\SL}{\mbox{\rm SL}(R[It]\op P)$. Then  there exists $\gs \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R[It]\op P)$ such that $(b,p)\gs=(1,0)$.

 $(2)$  Let $P$ be a projective module over the extended  Rees algebra $R[It,t^{-1}]$ such that $rank(P)>d$.
 Let $(b,p)\in \mbox{\rm Um}}		\newcommand{\SL}{\mbox{\rm SL}(R[It,t^{-1}]\oplus P)$. Then  there exists $\gs \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R[It, t^{-1}]\op P)$ such that $(b,p)\gs=(1,0)$.
 \end{theorem}

\begin{proof}

	Let $A=R[It]$. Since $R$ is a domain, so is $R[t]$. Since $A$ is a subring of $R[t]$, we conclude
	that $A$ is a domain.
	If $I=(0)$, then $A=R$. In this case, the result follows from a  result of Bass \cite{Bass64}.
	If $I=(1)$, then $A=R[t]$. In this case, the result follows from \cite[Corollary 2]{P}. So, we assume that $(0)\neq I\neq (1)$.
	 Note that in this case $\dim(A)= d+1$. Hence in the view of a classical
	result of Bass \cite{Bass64}, we only have to consider the case when the rank of $P$ is $d+1$.

	Let $S$ be the set of all non-zero-divisors of $R$. Then using Lemma \ref{mcset},  we have $S^{-1}R[It]\cong R'[t]$ where $R'$ is
	a field. Therefore $S^{-1}P$ is a free $S^{-1}A$-module. Since $P$ is finitely generated, there exists $s\in S$ such that $P_s$ is a free $A_s$-module.

 Let $(b,p)\in \mbox{\rm Um}}		\newcommand{\SL}{\mbox{\rm SL}(R[It]\op P)$ and $a$ a non-zero non-unit element of $I$. We have already observed the following Cartesian square of rings in the proof of  Theorem \ref{thm:unimodular}
 \[
   	\xymatrix{
   		R[It] \ar@{->}[r]^{p_1}
   		\ar@{->}[d]^{p_2}
   		&R[It]_{as}\cong R_{as}[t]
   		\ar@{->}[d]^{j_1}
   		\\
   		R[It]_{1+asR[It]} \ar@{->}[r]^{j_2}
   		&(R_{as}[t])_{1+asR[It]}.
   	}
   	\]
  Let $T:=1+asR[It]$.  Since $P_{as}$ is free,  using a result of Suslin \cite[Theorem 2.6]{Su77},  there exists $\gs_1\in \E(R[It]_{as}\op P_{as})$ such that
  $((b,p)_{as})\gs_1=(1,0)$ for $d\geq 1$. For $d=0$, $R$ becomes a field. Also in this case it is easy to see that
   there exists $\gs_1\in \E(R[It]_{as}\op P_{as})$ such that
  $((b,p)_{as})\gs_1=(1,0)$.

  By   Lemma \ref{lem:gen-dim-res}, there exists
  $\gs_2\in \E(R[It]_T\op P_T)$ such that  $((b,p)_T)\gs_2=(1,0)$.  Let $\eta=(\gs_1)_T^{-1}(\gs_2)_{as}$.
  Since $(1,0)\eta=(1,0)$,

  $\eta= \bigl( \begin{smallmatrix} 1 & 0 \\ * & \theta}	\newcommand{\gu}{\upsilon
\end{smallmatrix} \bigr)$, where $\theta}	\newcommand{\gu}{\upsilon \in \mbox{\rm GL}}		\newcommand{\tm}{\mbox{\small \mm}_{d+1}(B)$, where $B=(R_{as}[t])_{1+asR[It]}$.
In fact $\theta}	\newcommand{\gu}{\upsilon \in \SL_{d+1}(B)$, since $\eta$ is elementary.

By Lemma \ref{lem:splitting}, we have $\eta = (\eta_1)_T(\eta_2)_{as}$,
with $\eta_1 \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R[It]_T\oplus P_T)$, $\eta_2 \in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R_{as}[t]\oplus P_{as})$ such that
$(1,0)\eta_i  = (1,0)$  for $i = 1, 2$.
Now by Lemma \ref{lem:criteria}, we will get an automorphism $\phi\in \mbox{\rm Aut\,}}	\newcommand{\End}{\mbox{\rm End\,}(R[It]\op P)$ such that $(b,p)\phi=(1,0)$.
This completes the proof of $(1)$.

$(2)$ The proof of the second part is a verbatim copy of the first part. But one needs to use the cancellation result
for the Laurent extensions (see \cite[Theorem 7.2]{Su77}).
$\hfill \blacksquare}	\newcommand{\gl}{\lambda $
\end{proof}

\section{Applications}

The following is the Eisenbud--Evans estimate on the number of generators of a module over the (extended) Rees algebras.

\begin{theorem}
 Let $R$ be a commutative Noetherian domain of dimension $d$ and $I$ an ideal of $R$. Let $M$ be  a non-zero finitely
 generated module over  $A:=R[It]$ or $R[It,t^{-1}]$. Then $M$ is generated by $e(M):= Sup_{\p}\{\mu_{\p}(M)+\dim(A/\p)\}$
 elements.
\end{theorem}
\begin{proof}
 Suppose $M$ is generated by $n$ elements such that $n>e(M)>d$. Then it is enough to prove that $M$
 is generated by $(n-1)$ elements. Since $M$ is generated by $n$ elements, we have the following
 surjective map $A^n\rightarrow}		\newcommand{\lra}{\longrightarrow M\rightarrow}		\newcommand{\lra}{\longrightarrow 0$. Let $K$ be the kernel of this map. Since $A$ is Noetherian, $K$ is finitely generated. Also, $K$ is a torsion-free module since $K$ is a submodule of $A^n$.
  Let $S$ be the  set of all non-zero divisors of $R$. Then $S^{-1}A
  \cong k[t]$, where $k$ is a field.
Since $S^{-1}K$ is a torsion-free module over the PID $k[t]$, by the structure theorem of finite generated modules over a PID, we get
that $S^{-1}K$ is $S^{-1}A$-free. Since $K$ is finitely generated, there exists $s\in S$ such
that $K_s$ is $A_s$-free.

Let $x_1$ be a basic element of $K_s$. We set $T=1+sA$. By  Lemma \ref{lem:plum}, we get
that $K_T$ contains a basic element $x_2$  which is a unimodular element in $A^n_T$. Now patching $x_1$ and $x_2$, we get a basic element $x$ in $K$ which is a unimodular element $x\in A^n$.
 Hence we can write $A^n\cong xA\op P$. Now we observe that locally $P$
surjects onto $M$. Hence $P$ surjects onto $M$ globally. Note that rank of $P\geq n-1>d$. Hence by the cancellation result Theorem \ref{thm:canc}, we have
$P\cong A^{n-1}$. Therefore $M$ is generated by $n-1$ elements. This finishes the proof.
$\hfill \blacksquare}	\newcommand{\gl}{\lambda$

\end{proof}

The following is a ${\rm K}_1$-analog of the above results.

\begin{theorem}
 Let $R$ be a commutative domain of dimension $d$ and $I\subset R$ an ideal of $R$. Then for $n\geq max\{3,d+2\}$, the
 natural map $\phi: \mbox{\rm GL}}		\newcommand{\tm}{\mbox{\small \mm}_n(R[It])/\E_n(R[It])\rightarrow}		\newcommand{\lra}{\longrightarrow {\rm K}_1(R[It])$ is an isomorphism.
\end{theorem}
\begin{proof}
 Surjectivity follows from Theorem \ref{thm:canc}. We have to prove injectivity i.e. stably elementary
 is elementary. We grade $A:=R[It]$ as $A=R\op It\op I^2t^2\op \cdots$, where $A_0=R$ and $A_+=It\op I^2t^2\op \cdots$.
 Let $\alpha}	\newcommand{\gb}{\beta\in \mbox{\rm GL}}		\newcommand{\tm}{\mbox{\small \mm}_n(A)$ which is stably elementary. Multiplying by elementary matrices, we can assume that
 $\alpha}	\newcommand{\gb}{\beta\in \mbox{\rm GL}}		\newcommand{\tm}{\mbox{\small \mm}_n(A,A_+)$. Let $\p \in \mbox{\rm Spec\,}}	\newcommand{\hh}{\mbox{\rm ht\,}(R)$. Note that the generalized dimension of $R_{\p}[I_{\p}t]\leq d$.

  By usual estimation, we get $\alpha}	\newcommand{\gb}{\beta_{\p}\in \E_n(R_{\p}[I_{\p}t])$. Then by following Gubeladze \cite[Proposition 7.3]{G2},
we get  $\alpha}	\newcommand{\gb}{\beta\in \E_n(R[It])$. This completes the proof.
 $\hfill \blacksquare}	\newcommand{\gl}{\lambda$
\end{proof}

{\small

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