Liverpool complete a precious victory. The tie is far from over because both sides created a load of chances in the game. Liverpool were grateful to Mignolet for some fine saves while Baumann made a few for Hoffenheim. But the outstanding moment came from Trent Alexander-Arnold, who crowned his first appearance in the Champions League with a magnificent goal to give Liverpool the lead.

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90+4 min: Encouraging composure from Liverpool as they take the ball down the other end and move it about well. Grujic tries to crown it with a delicate finish from the edge of the area but his sidefooter is tipped away by Baumann.

90 + 2 min: There are screen writers in Hollywood who would love to be able to ensure endings are as tense as Liverpool always seem to make them ...

90+1 min: Demiraby strokes a lovely freekick from the left into the danger zone.. and Hubner finds himself with a free header from nine yards! Surely he will punish Liverpool for their set-piece slackness .... no, he nods over the bar!

90 min: There will be at least four more minutes. Hoffenheim are raging forward in search of an equaliser...

90 min: Alexander-Arnold puts a dangerous cross behind for a corner. So that’s Liverpool facing a last-minute set-piece to hold on to a lead .... they do it well this time.

Liverpool substitution: Grujic on, Mané off. Any lead will do now for Klopp.

A long diagonal ball from the right sails over the Liverpool defence. Alexander-Arnold appeals in vain for offside but Lovren is keeping Uth on ... and the striker takes the ball down well and then drills a low shot from the edge of the box into the far corner! The tie is back in the balance...

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85 min: Liverpool continue to seek a third. Salah, whose sights have been just off today, overhits an attempted cross to Solanke, who had peeled off his marker to find space near the back post.

Liverpool substitution: Solanke on, Firmino off. I fancy the newcomer to score.

82 min: Hoffenheim raid down the left. A cross beyond Moreno is headed back into the path of Demirbay, who hits a half-volley into the ground and bouncing towards goals from 16 yards. Mignolet makes another fine save. That’s several so far.

80 min: Lovren, well positioned, clears a dangerous cross by Kaderabek despite being under pressure from Kramaric. He’s played well today.

78 min: There’s a third goal here for Liverpool, you know. They could claim their ticket for the group stages today. “I like your earlier quip about Alexander-Arnold looking for guidance in vain,” says Edouard. “As it stands, I think by the end of the 90 minutes the rest of Liverpool’s defensive line will be asking him how to proceed on the pitch. What a Champions League debut. Also, Phil Coutinho should be offended: it’s quite clear Barcelona haven’t been scouting Liverpool games. If they had, they’d be bidding for Mané instead.” That’ll come, no doubt.

Wijnaldum cleverly took a quick freekick in midfield and Firmino worked it wide on the left to Milner. He tried to pick out Salah at the back post but his cross took a nick off Nordtveit and looped over the keeper and into the net at the far post!

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73 min: Moreno curls over a freekick from the left. Can’s header never looks like hitting the target.

72 min: Hoffenheim are suddenly looking all punched out. Liverpool sense that and are going for a knockout blow....

71 min: Lovren meets another well delivered corner by Alexander-Arnold. It’s a powerful header from eight yards. But Baumann saves well.

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70 min: Hubner gives the ball away as he tries to tippy-tap his way forward from the back. Can slips an easy pass through to Salah, who romps into the box and bears down on goal before letting fly from 10 yards. Letting trickle, rather. It’s a comfortable save for Baumann and a real let-off for the hosts.

67 min: Alexander-Arnold takes a Liverpool corner - he strikes the ball beautifully, that much has been established - but Hoffenheim manage to clear this one.

66 min: Kramaric races past Alexander-Arnold again down the left. Gnabry meets his cross at the near post but can’t turn it on to the target.

65 min: Hoffenheim win three corners in a row. Demiraby’s deliveries are wicked. But Liverpool head them clear each time.

64 min: Kaderabek gets behind Liverpool down the left and floats a dangerouss cross towards the back post. Matip leaps high to head it behind for a corner. Mignolet rebukes Mané for failing to provide Moreno with any cover.

Liverpool substitution: Milner on, Henderson off.

61 min: Salah pings a nice pass through to Can, who had darted forward to provide an underlap. But Firmino demands a similar service and would be through on goal if he gets it ... but Can overhits the attempted pass, granting Hoffenheim a reprieve.

Here’s a young Alexander-Arnold helping to calm an old pro’s jitters before a match many years ago ...

57 min: “Liverpool should sell Moreno asap: he has been so bad bar an excellent interception for a cross,” blasts Ruth Purdue (and many others). You realise that’s not a persuasive sales pitch, right?

56 min: Alexander-Arnold booked for ... I’m not sure what. He seems bemused, too. I’d wonder if it might be a case of mistaken identity if not for the fact that everyone now knows who Trent Alexander Arnold is.

Hoffenheim substitutions: Nagelsmann has recognised Becakcic’s ordeal and decided to spare him further persecution from Mané. I doubt the former West Ham defender Nordtveit is going to fare any better. Also, Amiri has come on for Rupp.

51 min: Alexander-Arnold crosses from the right. It rools all the way over to Mané on the left. He dances past poor Becakic again and opens fire from a difficult angle. Baumann parries and Hoffenheim scramble clear. I’m no master strategist, but always giving the ball to Mané seems like a good ploy at this point. “Dear Paul, I was about to object to Phil Sawyer’s claim that roller coasters zigzag, but I realized he is right,” confides Dave Hannah. “When Lovren and Moreno are on the field, Liverpool regularly go off the rails defensively.”

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49 min: Zuber shows strength and speed to stop Salah from breaking through down the right. This is a very open game and there’s little sign of Liverpool sitting on their lead.

47 min: Mané skins Becackic down the left wing and continues on into the box. He tees up Firmino with a perfect low pass across the face of goal. Firmino meets it six yards out ... but fails to beat the keeper, who spread himself at his feet. A dink would have done it.

46 min: No changes during the break. Hoffenheim attack from the off, and Gnabry races past Alexander-Anrold and delivers a decent cross. Moreno, providing intelligent cover (that’s worth highlighting), clears wel.

Watching the replay of the penalty incident, it’s clear that the main reason that Lovren found himself in a one-on-one with Gnabry on the left-hand side of the box is that Moreno was up pressing the opposing keeper, who saw the gap and punted the ball long. Gnabry might have been offside, mind. The replay also confirms what an awful penalty it was by Kramaric, pretty much as bad it can be while being on target. Actually, that’s unfair on Pat Nevin.

“I was typing an email about Liverpool’s deficiencies and pre-season hopes already evaporating,” admits Phil Sawyer. “Then Alexander-Arnold scored. Then Moreno was left stranded and Hoffenheim nearly scored. After this half and the Watford match I think the only think I can safely say about this season is that I’m going to need to speak to my GP about possibly upping the blood pressure medication. We appear to be on a rollercoaster zigzagging wildly between Elation and Doomed.”

In other news, Everton have finally reached a deal with Swansea for Gylfii Sigurdsson. That and more developments can be found in this here transfer guide.

A very pleasing scoreline so far, thanks to a delightful goal by Alexander-Arnold following a penalty save by Mignolet. That’s unlikely to be the end of the scoring, however, as Hoffenheim are dangerous and Liverpool are sharp on the counter-attack.

45 +1 min: Lovren rises well to meet a Liverpool corner with a strong header. It bounces just past the far post!